Dirichlet characters#

A DirichletCharacter is the extension of a homomorphism

\[(\ZZ/N\ZZ)^* \to R^*,\]

for some ring \(R\), to the map \(\ZZ/N\ZZ \to R\) obtained by sending those \(x\in\ZZ/N\ZZ\) with \(\gcd(N,x)>1\) to \(0\).

EXAMPLES:

Sage

sage: G = DirichletGroup(35)
sage: x = G.gens()
sage: e = x[0]*x[1]^2; e
Dirichlet character modulo 35 of conductor 35
 mapping 22 |--> zeta12^3, 31 |--> zeta12^2 - 1
sage: e.order()
12

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(35))
>>> x = G.gens()
>>> e = x[Integer(0)]*x[Integer(1)]**Integer(2); e
Dirichlet character modulo 35 of conductor 35
 mapping 22 |--> zeta12^3, 31 |--> zeta12^2 - 1
>>> e.order()
12

This illustrates a canonical coercion:

Sage

sage: e = DirichletGroup(5, QQ).0
sage: f = DirichletGroup(5, CyclotomicField(4)).0
sage: e*f
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(5), QQ).gen(0)
>>> f = DirichletGroup(Integer(5), CyclotomicField(Integer(4))).gen(0)
>>> e*f
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4

AUTHORS:

William Stein (2005-09-02): Fixed bug in comparison of Dirichlet characters. It was checking that their values were the same, but not checking that they had the same level!
William Stein (2006-01-07): added more examples
William Stein (2006-05-21): added examples of everything; fix a lot of tiny bugs and design problem that became clear when creating examples.
Craig Citro (2008-02-16): speed up __call__ method for Dirichlet characters, miscellaneous fixes
Julian Rueth (2014-03-06): use UniqueFactory to cache DirichletGroups

class sage.modular.dirichlet.DirichletCharacter(parent, x, check=True)[source]#

Bases: MultiplicativeGroupElement

A Dirichlet character.

bar()[source]#

Return the complex conjugate of this Dirichlet character.

EXAMPLES:

Sage

sage: e = DirichletGroup(5).0
sage: e
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4
sage: e.bar()
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(5)).gen(0)
>>> e
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4
>>> e.bar()
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4

base_ring()[source]#

Return the base ring of this Dirichlet character.

EXAMPLES:

Sage

sage: G = DirichletGroup(11)
sage: G.gen(0).base_ring()
Cyclotomic Field of order 10 and degree 4
sage: G = DirichletGroup(11, RationalField())
sage: G.gen(0).base_ring()
Rational Field

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(11))
>>> G.gen(Integer(0)).base_ring()
Cyclotomic Field of order 10 and degree 4
>>> G = DirichletGroup(Integer(11), RationalField())
>>> G.gen(Integer(0)).base_ring()
Rational Field

bernoulli(k, algorithm='recurrence', cache=True, **opts)[source]#

Return the generalized Bernoulli number \(B_{k,eps}\).

INPUT:

k – a non-negative integer
algorithm – either 'recurrence' (default) or 'definition'
cache – if True, cache answers
**opts – optional arguments; not used directly, but passed to the bernoulli() function if this is called

OUTPUT:

Let \(\varepsilon\) be a (not necessarily primitive) character of modulus \(N\). This function returns the generalized Bernoulli number \(B_{k,\varepsilon}\), as defined by the following identity of power series (see for example [DI1995], Section 2.2):

\[\sum_{a=1}^N \frac{\varepsilon(a) t e^{at}}{e^{Nt}-1} = \sum_{k=0}^{\infty} \frac{B_{k,\varepsilon}}{k!} t^k.\]

ALGORITHM:

The 'recurrence' algorithm computes generalized Bernoulli numbers via classical Bernoulli numbers using the formula in [Coh2007], Proposition 9.4.5; this is usually optimal. The definition algorithm uses the definition directly.

Warning

In the case of the trivial Dirichlet character modulo 1, this function returns \(B_{1,\varepsilon} = 1/2\), in accordance with the above definition, but in contrast to the value \(B_1 = -1/2\) for the classical Bernoulli number. Some authors use an alternative definition giving \(B_{1,\varepsilon} = -1/2\); see the discussion in [Coh2007], Section 9.4.1.

EXAMPLES:

Sage

sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.bernoulli(5)
7430/13*zeta12^3 - 34750/13*zeta12^2 - 11380/13*zeta12 + 9110/13
sage: eps = DirichletGroup(9).0
sage: eps.bernoulli(3)
10*zeta6 + 4
sage: eps.bernoulli(3, algorithm="definition")
10*zeta6 + 4

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(13))
>>> e = G.gen(0)
>>> e.bernoulli(Integer(5))
7430/13*zeta12^3 - 34750/13*zeta12^2 - 11380/13*zeta12 + 9110/13
>>> eps = DirichletGroup(Integer(9)).gen(0)
>>> eps.bernoulli(Integer(3))
10*zeta6 + 4
>>> eps.bernoulli(Integer(3), algorithm="definition")
10*zeta6 + 4

change_ring(R)[source]#

Return the base extension of self to R.

INPUT:

R – either a ring admitting a conversion map from the base ring of self, or a ring homomorphism with the base ring of self as its domain

EXAMPLES:

Sage

sage: e = DirichletGroup(7, QQ).0
sage: f = e.change_ring(QuadraticField(3, 'a'))
sage: f.parent()
Group of Dirichlet characters modulo 7 with values in Number Field in a
 with defining polynomial x^2 - 3 with a = 1.732050807568878?

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(7), QQ).gen(0)
>>> f = e.change_ring(QuadraticField(Integer(3), 'a'))
>>> f.parent()
Group of Dirichlet characters modulo 7 with values in Number Field in a
 with defining polynomial x^2 - 3 with a = 1.732050807568878?

Sage

sage: e = DirichletGroup(13).0
sage: e.change_ring(QQ)
Traceback (most recent call last):
...
TypeError: Unable to coerce zeta12 to a rational

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(13)).gen(0)
>>> e.change_ring(QQ)
Traceback (most recent call last):
...
TypeError: Unable to coerce zeta12 to a rational

We test the case where \(R\) is a map (Issue #18072):

Sage

sage: K.<i> = QuadraticField(-1)
sage: chi = DirichletGroup(5, K)[1]
sage: chi(2)
i
sage: f = K.complex_embeddings()[0]
sage: psi = chi.change_ring(f)
sage: psi(2)
-1.83697019872103e-16 - 1.00000000000000*I

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> chi = DirichletGroup(Integer(5), K)[Integer(1)]
>>> chi(Integer(2))
i
>>> f = K.complex_embeddings()[Integer(0)]
>>> psi = chi.change_ring(f)
>>> psi(Integer(2))
-1.83697019872103e-16 - 1.00000000000000*I

conductor()[source]#

Compute and return the conductor of this character.

EXAMPLES:

Sage

sage: G.<a,b> = DirichletGroup(20)
sage: a.conductor()
4
sage: b.conductor()
5
sage: (a*b).conductor()
20

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20), names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> a.conductor()
4
>>> b.conductor()
5
>>> (a*b).conductor()
20

conrey_number()[source]#

Return the Conrey number for this character.

This is a positive integer coprime to \(q\) that identifies a Dirichlet character of modulus \(q\).

See https://www.lmfdb.org/knowledge/show/character.dirichlet.conrey

EXAMPLES:

Sage

sage: # needs sage.rings.number_field
sage: chi4 = DirichletGroup(4).gen()
sage: chi4.conrey_number()
3
sage: chi = DirichletGroup(24)([1,-1,-1]); chi
Dirichlet character modulo 24 of conductor 24
mapping 7 |--> 1, 13 |--> -1, 17 |--> -1
sage: chi.conrey_number()
5

sage: chi = DirichletGroup(60)([1,-1,I])
sage: chi.conrey_number()
17

sage: chi = DirichletGroup(420)([1,-1,-I,1])
sage: chi.conrey_number()
113

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> chi4 = DirichletGroup(Integer(4)).gen()
>>> chi4.conrey_number()
3
>>> chi = DirichletGroup(Integer(24))([Integer(1),-Integer(1),-Integer(1)]); chi
Dirichlet character modulo 24 of conductor 24
mapping 7 |--> 1, 13 |--> -1, 17 |--> -1
>>> chi.conrey_number()
5

>>> chi = DirichletGroup(Integer(60))([Integer(1),-Integer(1),I])
>>> chi.conrey_number()
17

>>> chi = DirichletGroup(Integer(420))([Integer(1),-Integer(1),-I,Integer(1)])
>>> chi.conrey_number()
113

decomposition()[source]#

Return the decomposition of self as a product of Dirichlet characters of prime power modulus, where the prime powers exactly divide the modulus of this character.

EXAMPLES:

Sage

sage: G.<a,b> = DirichletGroup(20)
sage: c = a*b
sage: d = c.decomposition(); d
[Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4]
sage: d[0].parent()
Group of Dirichlet characters modulo 4 with values
 in Cyclotomic Field of order 4 and degree 2
sage: d[1].parent()
Group of Dirichlet characters modulo 5 with values
 in Cyclotomic Field of order 4 and degree 2

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20), names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> c = a*b
>>> d = c.decomposition(); d
[Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4]
>>> d[Integer(0)].parent()
Group of Dirichlet characters modulo 4 with values
 in Cyclotomic Field of order 4 and degree 2
>>> d[Integer(1)].parent()
Group of Dirichlet characters modulo 5 with values
 in Cyclotomic Field of order 4 and degree 2

We cannot multiply directly, since coercion of one element into the other parent fails in both cases:

Sage

sage: d[0]*d[1] == c
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: 'Group of Dirichlet
characters modulo 4 with values in Cyclotomic Field of order 4 and
degree 2' and 'Group of Dirichlet characters modulo 5 with values
in Cyclotomic Field of order 4 and degree 2'

Python

>>> from sage.all import *
>>> d[Integer(0)]*d[Integer(1)] == c
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: 'Group of Dirichlet
characters modulo 4 with values in Cyclotomic Field of order 4 and
degree 2' and 'Group of Dirichlet characters modulo 5 with values
in Cyclotomic Field of order 4 and degree 2'

We can multiply if we are explicit about where we want the multiplication to take place.

Sage

sage: G(d[0])*G(d[1]) == c
True

Python

>>> from sage.all import *
>>> G(d[Integer(0)])*G(d[Integer(1)]) == c
True

Conductors that are divisible by various powers of 2 present some problems as the multiplicative group modulo \(2^k\) is trivial for \(k = 1\) and non-cyclic for \(k \ge 3\):

Sage

sage: (DirichletGroup(18).0).decomposition()
[Dirichlet character modulo 2 of conductor 1,
 Dirichlet character modulo 9 of conductor 9 mapping 2 |--> zeta6]
sage: (DirichletGroup(36).0).decomposition()
[Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1,
 Dirichlet character modulo 9 of conductor 1 mapping 2 |--> 1]
sage: (DirichletGroup(72).0).decomposition()
[Dirichlet character modulo 8 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
 Dirichlet character modulo 9 of conductor 1 mapping 2 |--> 1]

Python

>>> from sage.all import *
>>> (DirichletGroup(Integer(18)).gen(0)).decomposition()
[Dirichlet character modulo 2 of conductor 1,
 Dirichlet character modulo 9 of conductor 9 mapping 2 |--> zeta6]
>>> (DirichletGroup(Integer(36)).gen(0)).decomposition()
[Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1,
 Dirichlet character modulo 9 of conductor 1 mapping 2 |--> 1]
>>> (DirichletGroup(Integer(72)).gen(0)).decomposition()
[Dirichlet character modulo 8 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
 Dirichlet character modulo 9 of conductor 1 mapping 2 |--> 1]

element()[source]#

Return the underlying \(\ZZ/n\ZZ\)-module vector of exponents.

EXAMPLES:

Sage

sage: G.<a,b> = DirichletGroup(20)
sage: a.element()
(2, 0)
sage: b.element()
(0, 1)

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20), names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> a.element()
(2, 0)
>>> b.element()
(0, 1)

Note

The constructor of DirichletCharacter sets the cache of element() or of values_on_gens(). The cache of one of these methods needs to be set for the other method to work properly, these caches have to be stored when pickling an instance of DirichletCharacter.

extend(M)[source]#

Return the extension of this character to a Dirichlet character modulo the multiple M of the modulus.

EXAMPLES:

Sage

sage: G.<a,b> = DirichletGroup(20)
sage: H.<c> = DirichletGroup(4)
sage: c.extend(20)
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: a
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: c.extend(20) == a
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20), names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> H = DirichletGroup(Integer(4), names=('c',)); (c,) = H._first_ngens(1)
>>> c.extend(Integer(20))
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
>>> a
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
>>> c.extend(Integer(20)) == a
True

fixed_field()[source]#

Given a Dirichlet character, this will return the abelian extension fixed by the kernel of the corresponding Galois character.

OUTPUT:

a number field

EXAMPLES:

Sage

sage: G = DirichletGroup(37)
sage: chi = G.0
sage: psi = chi^18
sage: psi.fixed_field()
Number Field in a with defining polynomial x^2 + x - 9


sage: G = DirichletGroup(7)
sage: chi = G.0^2
sage: chi
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1
sage: chi.fixed_field()
Number Field in a with defining polynomial x^3 + x^2 - 2*x - 1

sage: G = DirichletGroup(31)
sage: chi = G.0
sage: chi^6
Dirichlet character modulo 31 of conductor 31 mapping 3 |--> zeta30^6
sage: psi = chi^6
sage: psi.fixed_field()
Number Field in a with defining polynomial x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(37))
>>> chi = G.gen(0)
>>> psi = chi**Integer(18)
>>> psi.fixed_field()
Number Field in a with defining polynomial x^2 + x - 9


>>> G = DirichletGroup(Integer(7))
>>> chi = G.gen(0)**Integer(2)
>>> chi
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1
>>> chi.fixed_field()
Number Field in a with defining polynomial x^3 + x^2 - 2*x - 1

>>> G = DirichletGroup(Integer(31))
>>> chi = G.gen(0)
>>> chi**Integer(6)
Dirichlet character modulo 31 of conductor 31 mapping 3 |--> zeta30^6
>>> psi = chi**Integer(6)
>>> psi.fixed_field()
Number Field in a with defining polynomial x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

fixed_field_polynomial(algorithm='pari')[source]#

Given a Dirichlet character, this will return a polynomial generating the abelian extension fixed by the kernel of the corresponding Galois character.

ALGORITHM: (Sage)

A formula by Gauss for the products of periods; see Disquisitiones §343. See the source code for more.

OUTPUT:

a polynomial with integer coefficients

EXAMPLES:

Sage

sage: G = DirichletGroup(37)
sage: chi = G.0
sage: psi = chi^18
sage: psi.fixed_field_polynomial()
x^2 + x - 9

sage: G = DirichletGroup(7)
sage: chi = G.0^2
sage: chi
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1
sage: chi.fixed_field_polynomial()
x^3 + x^2 - 2*x - 1

sage: G = DirichletGroup(31)
sage: chi = G.0
sage: chi^6
Dirichlet character modulo 31 of conductor 31 mapping 3 |--> zeta30^6
sage: psi = chi^6
sage: psi.fixed_field_polynomial()
x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

sage: G = DirichletGroup(7)
sage: chi = G.0
sage: chi.fixed_field_polynomial()
x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

sage: G = DirichletGroup(1001)
sage: chi = G.0
sage: psi = chi^3
sage: psi.order()
2
sage: psi.fixed_field_polynomial(algorithm="pari")
x^2 + x + 2

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(37))
>>> chi = G.gen(0)
>>> psi = chi**Integer(18)
>>> psi.fixed_field_polynomial()
x^2 + x - 9

>>> G = DirichletGroup(Integer(7))
>>> chi = G.gen(0)**Integer(2)
>>> chi
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1
>>> chi.fixed_field_polynomial()
x^3 + x^2 - 2*x - 1

>>> G = DirichletGroup(Integer(31))
>>> chi = G.gen(0)
>>> chi**Integer(6)
Dirichlet character modulo 31 of conductor 31 mapping 3 |--> zeta30^6
>>> psi = chi**Integer(6)
>>> psi.fixed_field_polynomial()
x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

>>> G = DirichletGroup(Integer(7))
>>> chi = G.gen(0)
>>> chi.fixed_field_polynomial()
x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

>>> G = DirichletGroup(Integer(1001))
>>> chi = G.gen(0)
>>> psi = chi**Integer(3)
>>> psi.order()
2
>>> psi.fixed_field_polynomial(algorithm="pari")
x^2 + x + 2

With the Sage implementation:

Sage

sage: G = DirichletGroup(37)
sage: chi = G.0
sage: psi = chi^18
sage: psi.fixed_field_polynomial(algorithm="sage")
x^2 + x - 9

sage: G = DirichletGroup(7)
sage: chi = G.0^2
sage: chi
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1
sage: chi.fixed_field_polynomial(algorithm="sage")
x^3 + x^2 - 2*x - 1

sage: G = DirichletGroup(31)
sage: chi = G.0
sage: chi^6
Dirichlet character modulo 31 of conductor 31 mapping 3 |--> zeta30^6
sage: psi = chi^6
sage: psi.fixed_field_polynomial(algorithm="sage")
x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

sage: G = DirichletGroup(7)
sage: chi = G.0
sage: chi.fixed_field_polynomial(algorithm="sage")
x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

sage: G = DirichletGroup(1001)
sage: chi = G.0
sage: psi = chi^3
sage: psi.order()
2
sage: psi.fixed_field_polynomial(algorithm="sage")
x^2 + x + 2

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(37))
>>> chi = G.gen(0)
>>> psi = chi**Integer(18)
>>> psi.fixed_field_polynomial(algorithm="sage")
x^2 + x - 9

>>> G = DirichletGroup(Integer(7))
>>> chi = G.gen(0)**Integer(2)
>>> chi
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6 - 1
>>> chi.fixed_field_polynomial(algorithm="sage")
x^3 + x^2 - 2*x - 1

>>> G = DirichletGroup(Integer(31))
>>> chi = G.gen(0)
>>> chi**Integer(6)
Dirichlet character modulo 31 of conductor 31 mapping 3 |--> zeta30^6
>>> psi = chi**Integer(6)
>>> psi.fixed_field_polynomial(algorithm="sage")
x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

>>> G = DirichletGroup(Integer(7))
>>> chi = G.gen(0)
>>> chi.fixed_field_polynomial(algorithm="sage")
x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

>>> G = DirichletGroup(Integer(1001))
>>> chi = G.gen(0)
>>> psi = chi**Integer(3)
>>> psi.order()
2
>>> psi.fixed_field_polynomial(algorithm="sage")
x^2 + x + 2

The algorithm must be one of \(sage\) or \(pari\):

Sage

sage: G = DirichletGroup(1001)
sage: chi = G.0
sage: psi = chi^3
sage: psi.order()
2
sage: psi.fixed_field_polynomial(algorithm="banana")
Traceback (most recent call last):
...
NotImplementedError: algorithm must be one of 'pari' or 'sage'

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(1001))
>>> chi = G.gen(0)
>>> psi = chi**Integer(3)
>>> psi.order()
2
>>> psi.fixed_field_polynomial(algorithm="banana")
Traceback (most recent call last):
...
NotImplementedError: algorithm must be one of 'pari' or 'sage'

galois_orbit(sort=True)[source]#

Return the orbit of this character under the action of the absolute Galois group of the prime subfield of the base ring.

EXAMPLES:

Sage

sage: G = DirichletGroup(30); e = G.1
sage: e.galois_orbit()
[Dirichlet character modulo 30 of conductor 5 mapping 11 |--> 1, 7 |--> -zeta4,
 Dirichlet character modulo 30 of conductor 5 mapping 11 |--> 1, 7 |--> zeta4]

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(30)); e = G.gen(1)
>>> e.galois_orbit()
[Dirichlet character modulo 30 of conductor 5 mapping 11 |--> 1, 7 |--> -zeta4,
 Dirichlet character modulo 30 of conductor 5 mapping 11 |--> 1, 7 |--> zeta4]

Another example:

Sage

sage: G = DirichletGroup(13)
sage: G.galois_orbits()
[
[Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1],
...,
[Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -1]
]
sage: e = G.0
sage: e
Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12
sage: e.galois_orbit()
[Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta12^3 + zeta12,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12^3 - zeta12,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta12]
sage: e = G.0^2; e
Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12^2
sage: e.galois_orbit()
[Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12^2,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta12^2 + 1]

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(13))
>>> G.galois_orbits()
[
[Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1],
...,
[Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -1]
]
>>> e = G.gen(0)
>>> e
Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12
>>> e.galois_orbit()
[Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta12^3 + zeta12,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12^3 - zeta12,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta12]
>>> e = G.gen(0)**Integer(2); e
Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12^2
>>> e.galois_orbit()
[Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12^2,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta12^2 + 1]

A non-example:

Sage

sage: chi = DirichletGroup(7, Integers(9), zeta = Integers(9)(2)).0
sage: chi.galois_orbit()
Traceback (most recent call last):
...
TypeError: Galois orbits only defined if base ring is an integral domain

Python

>>> from sage.all import *
>>> chi = DirichletGroup(Integer(7), Integers(Integer(9)), zeta = Integers(Integer(9))(Integer(2))).gen(0)
>>> chi.galois_orbit()
Traceback (most recent call last):
...
TypeError: Galois orbits only defined if base ring is an integral domain

gauss_sum(a=1)[source]#

Return a Gauss sum associated to this Dirichlet character.

The Gauss sum associated to \(\chi\) is

\[g_a(\chi) = \sum_{r \in \ZZ/m\ZZ} \chi(r)\,\zeta^{ar},\]

where \(m\) is the modulus of \(\chi\) and \(\zeta\) is a primitive \(m^{th}\) root of unity.

FACTS: If the modulus is a prime \(p\) and the character is nontrivial, then the Gauss sum has absolute value \(\sqrt{p}\).

CACHING: Computed Gauss sums are not cached with this character.

EXAMPLES:

Sage

sage: G = DirichletGroup(3)
sage: e = G([-1])
sage: e.gauss_sum(1)
2*zeta6 - 1
sage: e.gauss_sum(2)
-2*zeta6 + 1
sage: norm(e.gauss_sum())
3

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(3))
>>> e = G([-Integer(1)])
>>> e.gauss_sum(Integer(1))
2*zeta6 - 1
>>> e.gauss_sum(Integer(2))
-2*zeta6 + 1
>>> norm(e.gauss_sum())
3

Sage

sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.gauss_sum()
-zeta156^46 + zeta156^45 + zeta156^42 + zeta156^41 + 2*zeta156^40
 + zeta156^37 - zeta156^36 - zeta156^34 - zeta156^33 - zeta156^31
 + 2*zeta156^30 + zeta156^28 - zeta156^24 - zeta156^22 + zeta156^21
 + zeta156^20 - zeta156^19 + zeta156^18 - zeta156^16 - zeta156^15
 - 2*zeta156^14 - zeta156^10 + zeta156^8 + zeta156^7 + zeta156^6
 + zeta156^5 - zeta156^4 - zeta156^2 - 1
sage: factor(norm(e.gauss_sum()))
13^24

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(13))
>>> e = G.gen(0)
>>> e.gauss_sum()
-zeta156^46 + zeta156^45 + zeta156^42 + zeta156^41 + 2*zeta156^40
 + zeta156^37 - zeta156^36 - zeta156^34 - zeta156^33 - zeta156^31
 + 2*zeta156^30 + zeta156^28 - zeta156^24 - zeta156^22 + zeta156^21
 + zeta156^20 - zeta156^19 + zeta156^18 - zeta156^16 - zeta156^15
 - 2*zeta156^14 - zeta156^10 + zeta156^8 + zeta156^7 + zeta156^6
 + zeta156^5 - zeta156^4 - zeta156^2 - 1
>>> factor(norm(e.gauss_sum()))
13^24

See also

sage.arith.misc.gauss_sum() for general finite fields
sage.rings.padics.misc.gauss_sum() for a \(p\)-adic version

gauss_sum_numerical(prec=53, a=1)[source]#

Return a Gauss sum associated to this Dirichlet character as an approximate complex number with prec bits of precision.

INPUT:

prec – integer (default: 53), bits of precision
a – integer, as for gauss_sum().

The Gauss sum associated to \(\chi\) is

\[g_a(\chi) = \sum_{r \in \ZZ/m\ZZ} \chi(r)\,\zeta^{ar},\]

where \(m\) is the modulus of \(\chi\) and \(\zeta\) is a primitive \(m^{th}\) root of unity.

EXAMPLES:

Sage

sage: G = DirichletGroup(3)
sage: e = G.0
sage: abs(e.gauss_sum_numerical())
1.7320508075...
sage: sqrt(3.0)
1.73205080756888
sage: e.gauss_sum_numerical(a=2)
-...e-15 - 1.7320508075...*I
sage: e.gauss_sum_numerical(a=2, prec=100)
4.7331654313260708324703713917e-30 - 1.7320508075688772935274463415*I
sage: G = DirichletGroup(13)
sage: H = DirichletGroup(13, CC)
sage: e = G.0
sage: f = H.0
sage: e.gauss_sum_numerical()
-3.07497205... + 1.8826966926...*I
sage: f.gauss_sum_numerical()
-3.07497205... + 1.8826966926...*I
sage: abs(e.gauss_sum_numerical())
3.60555127546...
sage: abs(f.gauss_sum_numerical())
3.60555127546...
sage: sqrt(13.0)
3.60555127546399

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(3))
>>> e = G.gen(0)
>>> abs(e.gauss_sum_numerical())
1.7320508075...
>>> sqrt(RealNumber('3.0'))
1.73205080756888
>>> e.gauss_sum_numerical(a=Integer(2))
-...e-15 - 1.7320508075...*I
>>> e.gauss_sum_numerical(a=Integer(2), prec=Integer(100))
4.7331654313260708324703713917e-30 - 1.7320508075688772935274463415*I
>>> G = DirichletGroup(Integer(13))
>>> H = DirichletGroup(Integer(13), CC)
>>> e = G.gen(0)
>>> f = H.gen(0)
>>> e.gauss_sum_numerical()
-3.07497205... + 1.8826966926...*I
>>> f.gauss_sum_numerical()
-3.07497205... + 1.8826966926...*I
>>> abs(e.gauss_sum_numerical())
3.60555127546...
>>> abs(f.gauss_sum_numerical())
3.60555127546...
>>> sqrt(RealNumber('13.0'))
3.60555127546399

is_even()[source]#

Return True if and only if \(\varepsilon(-1) = 1\).

EXAMPLES:

Sage

sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.is_even()
False
sage: e(-1)
-1
sage: [e.is_even() for e in G]
[True, False, True, False, True, False, True, False, True, False, True, False]

sage: G = DirichletGroup(13, CC)
sage: e = G.0
sage: e.is_even()
False
sage: e(-1)
-1.000000...
sage: [e.is_even() for e in G]
[True, False, True, False, True, False, True, False, True, False, True, False]

sage: G = DirichletGroup(100000, CC)
sage: G.1.is_even()
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(13))
>>> e = G.gen(0)
>>> e.is_even()
False
>>> e(-Integer(1))
-1
>>> [e.is_even() for e in G]
[True, False, True, False, True, False, True, False, True, False, True, False]

>>> G = DirichletGroup(Integer(13), CC)
>>> e = G.gen(0)
>>> e.is_even()
False
>>> e(-Integer(1))
-1.000000...
>>> [e.is_even() for e in G]
[True, False, True, False, True, False, True, False, True, False, True, False]

>>> G = DirichletGroup(Integer(100000), CC)
>>> G.gen(1).is_even()
True

Note that is_even need not be the negation of is_odd, e.g., in characteristic 2:

Sage

sage: G.<e> = DirichletGroup(13, GF(4,'a'))
sage: e.is_even()
True
sage: e.is_odd()
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(13), GF(Integer(4),'a'), names=('e',)); (e,) = G._first_ngens(1)
>>> e.is_even()
True
>>> e.is_odd()
True

is_odd()[source]#

Return True if and only if \(\varepsilon(-1) = -1\).

EXAMPLES:

Sage

sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.is_odd()
True
sage: [e.is_odd() for e in G]
[False, True, False, True, False, True, False, True, False, True, False, True]

sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.is_odd()
True
sage: [e.is_odd() for e in G]
[False, True, False, True, False, True, False, True, False, True, False, True]

sage: G = DirichletGroup(100000, CC)
sage: G.0.is_odd()
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(13))
>>> e = G.gen(0)
>>> e.is_odd()
True
>>> [e.is_odd() for e in G]
[False, True, False, True, False, True, False, True, False, True, False, True]

>>> G = DirichletGroup(Integer(13))
>>> e = G.gen(0)
>>> e.is_odd()
True
>>> [e.is_odd() for e in G]
[False, True, False, True, False, True, False, True, False, True, False, True]

>>> G = DirichletGroup(Integer(100000), CC)
>>> G.gen(0).is_odd()
True

Note that is_even need not be the negation of is_odd, e.g., in characteristic 2:

Sage

sage: G.<e> = DirichletGroup(13, GF(4,'a'))
sage: e.is_even()
True
sage: e.is_odd()
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(13), GF(Integer(4),'a'), names=('e',)); (e,) = G._first_ngens(1)
>>> e.is_even()
True
>>> e.is_odd()
True

is_primitive()[source]#

Return True if and only if this character is primitive, i.e., its conductor equals its modulus.

EXAMPLES:

Sage

sage: G.<a,b> = DirichletGroup(20)
sage: a.is_primitive()
False
sage: b.is_primitive()
False
sage: (a*b).is_primitive()
True
sage: G.<a,b> = DirichletGroup(20, CC)
sage: a.is_primitive()
False
sage: b.is_primitive()
False
sage: (a*b).is_primitive()
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20), names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> a.is_primitive()
False
>>> b.is_primitive()
False
>>> (a*b).is_primitive()
True
>>> G = DirichletGroup(Integer(20), CC, names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> a.is_primitive()
False
>>> b.is_primitive()
False
>>> (a*b).is_primitive()
True

is_trivial()[source]#

Return True if this is the trivial character, i.e., has order 1.

EXAMPLES:

Sage

sage: G.<a,b> = DirichletGroup(20)
sage: a.is_trivial()
False
sage: (a^2).is_trivial()
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20), names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> a.is_trivial()
False
>>> (a**Integer(2)).is_trivial()
True

jacobi_sum(char, check=True)[source]#

Return the Jacobi sum associated to these Dirichlet characters (i.e., J(self,char)).

This is defined as

\[J(\chi, \psi) = \sum_{a \in \ZZ / N\ZZ} \chi(a) \psi(1-a)\]

where \(\chi\) and \(\psi\) are both characters modulo \(N\).

EXAMPLES:

Sage

sage: D = DirichletGroup(13)
sage: e = D.0
sage: f = D[-2]
sage: e.jacobi_sum(f)
3*zeta12^2 + 2*zeta12 - 3
sage: f.jacobi_sum(e)
3*zeta12^2 + 2*zeta12 - 3
sage: p = 7
sage: DP = DirichletGroup(p)
sage: f = DP.0
sage: e.jacobi_sum(f)
Traceback (most recent call last):
...
NotImplementedError: Characters must be from the same Dirichlet Group.

sage: all_jacobi_sums = [(DP[i].values_on_gens(),
....:                     DP[j].values_on_gens(),
....:                     DP[i].jacobi_sum(DP[j]))
....:                    for i in range(p - 1) for j in range(i, p - 1)]
sage: for s in all_jacobi_sums:
....:     print(s)
((1,), (1,), 5)
((1,), (zeta6,), -1)
((1,), (zeta6 - 1,), -1)
((1,), (-1,), -1)
((1,), (-zeta6,), -1)
((1,), (-zeta6 + 1,), -1)
((zeta6,), (zeta6,), -zeta6 + 3)
((zeta6,), (zeta6 - 1,), 2*zeta6 + 1)
((zeta6,), (-1,), -2*zeta6 - 1)
((zeta6,), (-zeta6,), zeta6 - 3)
((zeta6,), (-zeta6 + 1,), 1)
((zeta6 - 1,), (zeta6 - 1,), -3*zeta6 + 2)
((zeta6 - 1,), (-1,), 2*zeta6 + 1)
((zeta6 - 1,), (-zeta6,), -1)
((zeta6 - 1,), (-zeta6 + 1,), -zeta6 - 2)
((-1,), (-1,), 1)
((-1,), (-zeta6,), -2*zeta6 + 3)
((-1,), (-zeta6 + 1,), 2*zeta6 - 3)
((-zeta6,), (-zeta6,), 3*zeta6 - 1)
((-zeta6,), (-zeta6 + 1,), -2*zeta6 + 3)
((-zeta6 + 1,), (-zeta6 + 1,), zeta6 + 2)

Python

>>> from sage.all import *
>>> D = DirichletGroup(Integer(13))
>>> e = D.gen(0)
>>> f = D[-Integer(2)]
>>> e.jacobi_sum(f)
3*zeta12^2 + 2*zeta12 - 3
>>> f.jacobi_sum(e)
3*zeta12^2 + 2*zeta12 - 3
>>> p = Integer(7)
>>> DP = DirichletGroup(p)
>>> f = DP.gen(0)
>>> e.jacobi_sum(f)
Traceback (most recent call last):
...
NotImplementedError: Characters must be from the same Dirichlet Group.

>>> all_jacobi_sums = [(DP[i].values_on_gens(),
...                     DP[j].values_on_gens(),
...                     DP[i].jacobi_sum(DP[j]))
...                    for i in range(p - Integer(1)) for j in range(i, p - Integer(1))]
>>> for s in all_jacobi_sums:
...     print(s)
((1,), (1,), 5)
((1,), (zeta6,), -1)
((1,), (zeta6 - 1,), -1)
((1,), (-1,), -1)
((1,), (-zeta6,), -1)
((1,), (-zeta6 + 1,), -1)
((zeta6,), (zeta6,), -zeta6 + 3)
((zeta6,), (zeta6 - 1,), 2*zeta6 + 1)
((zeta6,), (-1,), -2*zeta6 - 1)
((zeta6,), (-zeta6,), zeta6 - 3)
((zeta6,), (-zeta6 + 1,), 1)
((zeta6 - 1,), (zeta6 - 1,), -3*zeta6 + 2)
((zeta6 - 1,), (-1,), 2*zeta6 + 1)
((zeta6 - 1,), (-zeta6,), -1)
((zeta6 - 1,), (-zeta6 + 1,), -zeta6 - 2)
((-1,), (-1,), 1)
((-1,), (-zeta6,), -2*zeta6 + 3)
((-1,), (-zeta6 + 1,), 2*zeta6 - 3)
((-zeta6,), (-zeta6,), 3*zeta6 - 1)
((-zeta6,), (-zeta6 + 1,), -2*zeta6 + 3)
((-zeta6 + 1,), (-zeta6 + 1,), zeta6 + 2)

Let’s check that trivial sums are being calculated correctly:

Sage

sage: N = 13
sage: D = DirichletGroup(N)
sage: g = D(1)
sage: g.jacobi_sum(g)
11
sage: sum([g(x)*g(1-x) for x in IntegerModRing(N)])
11

Python

>>> from sage.all import *
>>> N = Integer(13)
>>> D = DirichletGroup(N)
>>> g = D(Integer(1))
>>> g.jacobi_sum(g)
11
>>> sum([g(x)*g(Integer(1)-x) for x in IntegerModRing(N)])
11

And sums where exactly one character is nontrivial (see Issue #6393):

Sage

sage: G = DirichletGroup(5); X = G.list(); Y=X[0]; Z=X[1]
sage: Y.jacobi_sum(Z)
-1
sage: Z.jacobi_sum(Y)
-1

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(5)); X = G.list(); Y=X[Integer(0)]; Z=X[Integer(1)]
>>> Y.jacobi_sum(Z)
-1
>>> Z.jacobi_sum(Y)
-1

Now let’s take a look at a non-prime modulus:

Sage

sage: N = 9
sage: D = DirichletGroup(N)
sage: g = D(1)
sage: g.jacobi_sum(g)
3

Python

>>> from sage.all import *
>>> N = Integer(9)
>>> D = DirichletGroup(N)
>>> g = D(Integer(1))
>>> g.jacobi_sum(g)
3

We consider a sum with values in a finite field:

Sage

sage: g = DirichletGroup(17, GF(9,'a')).0
sage: g.jacobi_sum(g**2)
2*a

Python

>>> from sage.all import *
>>> g = DirichletGroup(Integer(17), GF(Integer(9),'a')).gen(0)
>>> g.jacobi_sum(g**Integer(2))
2*a

kernel()[source]#

Return the kernel of this character.

OUTPUT: Currently the kernel is returned as a list. This may change.

EXAMPLES:

Sage

sage: G.<a,b> = DirichletGroup(20)
sage: a.kernel()
[1, 9, 13, 17]
sage: b.kernel()
[1, 11]

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20), names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> a.kernel()
[1, 9, 13, 17]
>>> b.kernel()
[1, 11]

kloosterman_sum(a=1, b=0)[source]#

Return the “twisted” Kloosterman sum associated to this Dirichlet character.

This includes Gauss sums, classical Kloosterman sums, Salié sums, etc.

The Kloosterman sum associated to \(\chi\) and the integers a,b is

\[K(a,b,\chi) = \sum_{r \in (\ZZ/m\ZZ)^\times} \chi(r)\,\zeta^{ar+br^{-1}},\]

where \(m\) is the modulus of \(\chi\) and \(\zeta\) is a primitive \(m\) th root of unity. This reduces to the Gauss sum if \(b=0\).

This method performs an exact calculation and returns an element of a suitable cyclotomic field; see also kloosterman_sum_numerical(), which gives an inexact answer (but is generally much quicker).

CACHING: Computed Kloosterman sums are not cached with this character.

EXAMPLES:

Sage

sage: G = DirichletGroup(3)
sage: e = G([-1])
sage: e.kloosterman_sum(3,5)
-2*zeta6 + 1
sage: G = DirichletGroup(20)
sage: e = G([1 for u in G.unit_gens()])
sage: e.kloosterman_sum(7,17)
-2*zeta20^6 + 2*zeta20^4 + 4

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(3))
>>> e = G([-Integer(1)])
>>> e.kloosterman_sum(Integer(3),Integer(5))
-2*zeta6 + 1
>>> G = DirichletGroup(Integer(20))
>>> e = G([Integer(1) for u in G.unit_gens()])
>>> e.kloosterman_sum(Integer(7),Integer(17))
-2*zeta20^6 + 2*zeta20^4 + 4

kloosterman_sum_numerical(prec=53, a=1, b=0)[source]#

Return the Kloosterman sum associated to this Dirichlet character as an approximate complex number with prec bits of precision.

See also kloosterman_sum(), which calculates the sum exactly (which is generally slower).

INPUT:

prec – integer (default: 53), bits of precision
a – integer, as for kloosterman_sum()
b – integer, as for kloosterman_sum().

EXAMPLES:

Sage

sage: G = DirichletGroup(3)
sage: e = G.0

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(3))
>>> e = G.gen(0)

The real component of the numerical value of e is near zero:

Sage

sage: v = e.kloosterman_sum_numerical()
sage: v.real() < 1.0e15
True
sage: v.imag()
1.73205080756888
sage: G = DirichletGroup(20)
sage: e = G.1
sage: e.kloosterman_sum_numerical(53,3,11)
3.80422606518061 - 3.80422606518061*I

Python

>>> from sage.all import *
>>> v = e.kloosterman_sum_numerical()
>>> v.real() < RealNumber('1.0e15')
True
>>> v.imag()
1.73205080756888
>>> G = DirichletGroup(Integer(20))
>>> e = G.gen(1)
>>> e.kloosterman_sum_numerical(Integer(53),Integer(3),Integer(11))
3.80422606518061 - 3.80422606518061*I

level()[source]#

Synonym for modulus.

EXAMPLES:

Sage

sage: e = DirichletGroup(100, QQ).0
sage: e.level()
100

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(100), QQ).gen(0)
>>> e.level()
100

lfunction(prec=53, algorithm='pari')[source]#

Return the L-function of self.

The result is a wrapper around a PARI L-function or around the lcalc program.

INPUT:

prec – precision (default 53)
algorithm – ‘pari’ (default) or ‘lcalc’

EXAMPLES:

Sage

sage: G.<a,b> = DirichletGroup(20)
sage: L = a.lfunction(); L
PARI L-function associated to Dirichlet character modulo 20
of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: L(4)
0.988944551741105

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20), names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> L = a.lfunction(); L
PARI L-function associated to Dirichlet character modulo 20
of conductor 4 mapping 11 |--> -1, 17 |--> 1
>>> L(Integer(4))
0.988944551741105

With the algorithm “lcalc”:

Sage

sage: a = a.primitive_character()
sage: L = a.lfunction(algorithm='lcalc'); L
L-function with complex Dirichlet coefficients
sage: L.value(4)  # abs tol 1e-8
0.988944551741105 + 0.0*I

Python

>>> from sage.all import *
>>> a = a.primitive_character()
>>> L = a.lfunction(algorithm='lcalc'); L
L-function with complex Dirichlet coefficients
>>> L.value(Integer(4))  # abs tol 1e-8
0.988944551741105 + 0.0*I

lmfdb_page()[source]#

Open the LMFDB web page of the character in a browser.

See https://www.lmfdb.org

EXAMPLES:

Sage

sage: # needs sage.rings.number_field
sage: E = DirichletGroup(4).gen()
sage: E.lmfdb_page()  # optional -- webbrowser

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> E = DirichletGroup(Integer(4)).gen()
>>> E.lmfdb_page()  # optional -- webbrowser

maximize_base_ring()[source]#

Let

\[\varepsilon : (\ZZ/N\ZZ)^* \to \QQ(\zeta_n)\]

be a Dirichlet character. This function returns an equal Dirichlet character

\[\chi : (\ZZ/N\ZZ)^* \to \QQ(\zeta_m)\]

where \(m\) is the least common multiple of \(n\) and the exponent of \((\ZZ/N\ZZ)^*\).

EXAMPLES:

Sage

sage: G.<a,b> = DirichletGroup(20,QQ)
sage: b.maximize_base_ring()
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> -1
sage: b.maximize_base_ring().base_ring()
Cyclotomic Field of order 4 and degree 2
sage: DirichletGroup(20).base_ring()
Cyclotomic Field of order 4 and degree 2

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20),QQ, names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> b.maximize_base_ring()
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> -1
>>> b.maximize_base_ring().base_ring()
Cyclotomic Field of order 4 and degree 2
>>> DirichletGroup(Integer(20)).base_ring()
Cyclotomic Field of order 4 and degree 2

minimize_base_ring()[source]#

Return a Dirichlet character that equals this one, but over as small a subfield (or subring) of the base ring as possible.

Note

This function is currently only implemented when the base ring is a number field. It is the identity function in characteristic p.

EXAMPLES:

Sage

sage: G = DirichletGroup(13)
sage: e = DirichletGroup(13).0
sage: e.base_ring()
Cyclotomic Field of order 12 and degree 4
sage: e.minimize_base_ring().base_ring()
Cyclotomic Field of order 12 and degree 4
sage: (e^2).minimize_base_ring().base_ring()
Cyclotomic Field of order 6 and degree 2
sage: (e^3).minimize_base_ring().base_ring()
Cyclotomic Field of order 4 and degree 2
sage: (e^12).minimize_base_ring().base_ring()
Rational Field

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(13))
>>> e = DirichletGroup(Integer(13)).gen(0)
>>> e.base_ring()
Cyclotomic Field of order 12 and degree 4
>>> e.minimize_base_ring().base_ring()
Cyclotomic Field of order 12 and degree 4
>>> (e**Integer(2)).minimize_base_ring().base_ring()
Cyclotomic Field of order 6 and degree 2
>>> (e**Integer(3)).minimize_base_ring().base_ring()
Cyclotomic Field of order 4 and degree 2
>>> (e**Integer(12)).minimize_base_ring().base_ring()
Rational Field

modulus()[source]#

Return the modulus of this character.

EXAMPLES:

Sage

sage: e = DirichletGroup(100, QQ).0
sage: e.modulus()
100
sage: e.conductor()
4

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(100), QQ).gen(0)
>>> e.modulus()
100
>>> e.conductor()
4

multiplicative_order()[source]#

Return the order of this character.

EXAMPLES:

Sage

sage: e = DirichletGroup(100).1
sage: e.order()    # same as multiplicative_order, since group is multiplicative
20
sage: e.multiplicative_order()
20
sage: e = DirichletGroup(100).0
sage: e.multiplicative_order()
2

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(100)).gen(1)
>>> e.order()    # same as multiplicative_order, since group is multiplicative
20
>>> e.multiplicative_order()
20
>>> e = DirichletGroup(Integer(100)).gen(0)
>>> e.multiplicative_order()
2

primitive_character()[source]#

Return the primitive character associated to self.

EXAMPLES:

Sage

sage: e = DirichletGroup(100).0; e
Dirichlet character modulo 100 of conductor 4 mapping 51 |--> -1, 77 |--> 1
sage: e.conductor()
4
sage: f = e.primitive_character(); f
Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1
sage: f.modulus()
4

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(100)).gen(0); e
Dirichlet character modulo 100 of conductor 4 mapping 51 |--> -1, 77 |--> 1
>>> e.conductor()
4
>>> f = e.primitive_character(); f
Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1
>>> f.modulus()
4

restrict(M)[source]#

Return the restriction of this character to a Dirichlet character modulo the divisor M of the modulus, which must also be a multiple of the conductor of this character.

EXAMPLES:

Sage

sage: e = DirichletGroup(100).0
sage: e.modulus()
100
sage: e.conductor()
4
sage: e.restrict(20)
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: e.restrict(4)
Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1
sage: e.restrict(50)
Traceback (most recent call last):
...
ValueError: conductor(=4) must divide M(=50)

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(100)).gen(0)
>>> e.modulus()
100
>>> e.conductor()
4
>>> e.restrict(Integer(20))
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
>>> e.restrict(Integer(4))
Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1
>>> e.restrict(Integer(50))
Traceback (most recent call last):
...
ValueError: conductor(=4) must divide M(=50)

values()[source]#

Return a list of the values of this character on each integer between 0 and the modulus.

EXAMPLES:

Sage

sage: e = DirichletGroup(20)(1)
sage: e.values()
[0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1]
sage: e = DirichletGroup(20).gen(0)
sage: e.values()
[0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1]
sage: e = DirichletGroup(20).gen(1)
sage: e.values()
[0, 1, 0, -zeta4, 0, 0, 0, zeta4, 0, -1, 0, 1, 0, -zeta4, 0, 0, 0, zeta4, 0, -1]
sage: e = DirichletGroup(21).gen(0) ; e.values()
[0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1]
sage: e = DirichletGroup(21, base_ring=GF(37)).gen(0) ; e.values()
[0, 1, 36, 0, 1, 36, 0, 0, 36, 0, 1, 36, 0, 1, 0, 0, 1, 36, 0, 1, 36]
sage: e = DirichletGroup(21, base_ring=GF(3)).gen(0) ; e.values()
[0, 1, 2, 0, 1, 2, 0, 0, 2, 0, 1, 2, 0, 1, 0, 0, 1, 2, 0, 1, 2]

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(20))(Integer(1))
>>> e.values()
[0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1]
>>> e = DirichletGroup(Integer(20)).gen(Integer(0))
>>> e.values()
[0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1]
>>> e = DirichletGroup(Integer(20)).gen(Integer(1))
>>> e.values()
[0, 1, 0, -zeta4, 0, 0, 0, zeta4, 0, -1, 0, 1, 0, -zeta4, 0, 0, 0, zeta4, 0, -1]
>>> e = DirichletGroup(Integer(21)).gen(Integer(0)) ; e.values()
[0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1]
>>> e = DirichletGroup(Integer(21), base_ring=GF(Integer(37))).gen(Integer(0)) ; e.values()
[0, 1, 36, 0, 1, 36, 0, 0, 36, 0, 1, 36, 0, 1, 0, 0, 1, 36, 0, 1, 36]
>>> e = DirichletGroup(Integer(21), base_ring=GF(Integer(3))).gen(Integer(0)) ; e.values()
[0, 1, 2, 0, 1, 2, 0, 0, 2, 0, 1, 2, 0, 1, 0, 0, 1, 2, 0, 1, 2]

Sage

sage: chi = DirichletGroup(100151, CyclotomicField(10)).0
sage: ls = chi.values() ; ls[0:10]
[0,
1,
-zeta10^3,
-zeta10,
-zeta10,
1,
zeta10^3 - zeta10^2 + zeta10 - 1,
zeta10,
zeta10^3 - zeta10^2 + zeta10 - 1,
zeta10^2]

Python

>>> from sage.all import *
>>> chi = DirichletGroup(Integer(100151), CyclotomicField(Integer(10))).gen(0)
>>> ls = chi.values() ; ls[Integer(0):Integer(10)]
[0,
1,
-zeta10^3,
-zeta10,
-zeta10,
1,
zeta10^3 - zeta10^2 + zeta10 - 1,
zeta10,
zeta10^3 - zeta10^2 + zeta10 - 1,
zeta10^2]

values_on_gens()[source]#

Return a tuple of the values of self on the standard generators of \((\ZZ/N\ZZ)^*\), where \(N\) is the modulus.

EXAMPLES:

Sage

sage: # needs sage.rings.number_field
sage: e = DirichletGroup(16)([-1, 1])
sage: e.values_on_gens ()
(-1, 1)

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> e = DirichletGroup(Integer(16))([-Integer(1), Integer(1)])
>>> e.values_on_gens ()
(-1, 1)

Note

class sage.modular.dirichlet.DirichletGroupFactory[source]#

Bases: UniqueFactory

Construct a group of Dirichlet characters modulo \(N\).

INPUT:

N – positive integer
base_ring – commutative ring; the value ring for the characters in this group (default: the cyclotomic field \(\QQ(\zeta_n)\), where \(n\) is the exponent of \((\ZZ/N\ZZ)^*\))
zeta – (optional) root of unity in base_ring
zeta_order – (optional) positive integer; this must be the order of zeta if both are specified
names – ignored (needed so G.<...> = DirichletGroup(...) notation works)
integral – boolean (default: False); whether to replace the default cyclotomic field by its rings of integers as the base ring. This is ignored if base_ring is not None.

OUTPUT:

The group of Dirichlet characters modulo \(N\) with values in a subgroup \(V\) of the multiplicative group \(R^*\) of base_ring. This is the group of homomorphisms \((\ZZ/N\ZZ)^* \to V\) with pointwise multiplication. The group \(V\) is determined as follows:

If both zeta and zeta_order are omitted, then \(V\) is taken to be \(R^*\), or equivalently its \(n\)-torsion subgroup, where \(n\) is the exponent of \((\ZZ/N\ZZ)^*\). Many operations, such as finding a set of generators for the group, are only implemented if \(V\) is cyclic and a generator for \(V\) can be found.
If zeta is specified, then \(V\) is taken to be the cyclic subgroup of \(R^*\) generated by zeta. If zeta_order is also given, it must be the multiplicative order of zeta; this is useful if the base ring is not exact or if the order of zeta is very large.
If zeta is not specified but zeta_order is, then \(V\) is taken to be the group of roots of unity of order dividing zeta_order in \(R\). In this case, \(R\) must be a domain (so \(V\) is cyclic), and \(V\) must have order zeta_order. Furthermore, a generator zeta of \(V\) is computed, and an error is raised if such zeta cannot be found.

EXAMPLES:

The default base ring is a cyclotomic field of order the exponent of \((\ZZ/N\ZZ)^*\):

Sage

sage: DirichletGroup(20)
Group of Dirichlet characters modulo 20 with values
 in Cyclotomic Field of order 4 and degree 2

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(20))
Group of Dirichlet characters modulo 20 with values
 in Cyclotomic Field of order 4 and degree 2

We create the group of Dirichlet character mod 20 with values in the rational numbers:

Sage

sage: G = DirichletGroup(20, QQ); G
Group of Dirichlet characters modulo 20 with values in Rational Field
sage: G.order()
4
sage: G.base_ring()
Rational Field

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20), QQ); G
Group of Dirichlet characters modulo 20 with values in Rational Field
>>> G.order()
4
>>> G.base_ring()
Rational Field

The elements of G print as lists giving the values of the character on the generators of \((Z/NZ)^*\):

Sage

sage: list(G)
[Dirichlet character modulo 20 of conductor 1 mapping 11 |--> 1, 17 |--> 1,
 Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1,
 Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> -1,
 Dirichlet character modulo 20 of conductor 20 mapping 11 |--> -1, 17 |--> -1]

Python

>>> from sage.all import *
>>> list(G)
[Dirichlet character modulo 20 of conductor 1 mapping 11 |--> 1, 17 |--> 1,
 Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1,
 Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> -1,
 Dirichlet character modulo 20 of conductor 20 mapping 11 |--> -1, 17 |--> -1]

Next we construct the group of Dirichlet character mod 20, but with values in \(\QQ(\zeta_n)\):

Sage

sage: G = DirichletGroup(20)
sage: G.1
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20))
>>> G.gen(1)
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4

We next compute several invariants of G:

Sage

sage: G.gens()
(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1,
 Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4)
sage: G.unit_gens()
(11, 17)
sage: G.zeta()
zeta4
sage: G.zeta_order()
4

Python

>>> from sage.all import *
>>> G.gens()
(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1,
 Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4)
>>> G.unit_gens()
(11, 17)
>>> G.zeta()
zeta4
>>> G.zeta_order()
4

In this example we create a Dirichlet group with values in a number field:

Sage

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^4 + 1)
sage: DirichletGroup(5, K)
Group of Dirichlet characters modulo 5 with values
 in Number Field in a with defining polynomial x^4 + 1

Python

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(4) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> DirichletGroup(Integer(5), K)
Group of Dirichlet characters modulo 5 with values
 in Number Field in a with defining polynomial x^4 + 1

An example where we give zeta, but not its order:

Sage

sage: G = DirichletGroup(5, K, a); G
Group of Dirichlet characters modulo 5 with values in the group of order 8
 generated by a in Number Field in a with defining polynomial x^4 + 1
sage: G.list()
[Dirichlet character modulo 5 of conductor 1 mapping 2 |--> 1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> a^2,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -a^2]

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(5), K, a); G
Group of Dirichlet characters modulo 5 with values in the group of order 8
 generated by a in Number Field in a with defining polynomial x^4 + 1
>>> G.list()
[Dirichlet character modulo 5 of conductor 1 mapping 2 |--> 1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> a^2,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -a^2]

We can also restrict the order of the characters, either with or without specifying a root of unity:

Sage

sage: DirichletGroup(5, K, zeta=-1, zeta_order=2)
Group of Dirichlet characters modulo 5 with values in the group of order 2
 generated by -1 in Number Field in a with defining polynomial x^4 + 1
sage: DirichletGroup(5, K, zeta_order=2)
Group of Dirichlet characters modulo 5 with values in the group of order 2
 generated by -1 in Number Field in a with defining polynomial x^4 + 1

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(5), K, zeta=-Integer(1), zeta_order=Integer(2))
Group of Dirichlet characters modulo 5 with values in the group of order 2
 generated by -1 in Number Field in a with defining polynomial x^4 + 1
>>> DirichletGroup(Integer(5), K, zeta_order=Integer(2))
Group of Dirichlet characters modulo 5 with values in the group of order 2
 generated by -1 in Number Field in a with defining polynomial x^4 + 1

Sage

sage: G.<e> = DirichletGroup(13)
sage: loads(G.dumps()) == G
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(13), names=('e',)); (e,) = G._first_ngens(1)
>>> loads(G.dumps()) == G
True

Sage

sage: G = DirichletGroup(19, GF(5))
sage: loads(G.dumps()) == G
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(19), GF(Integer(5)))
>>> loads(G.dumps()) == G
True

We compute a Dirichlet group over a large prime field:

Sage

sage: p = next_prime(10^40)
sage: g = DirichletGroup(19, GF(p)); g
Group of Dirichlet characters modulo 19 with values
 in Finite Field of size 10000000000000000000000000000000000000121

Python

>>> from sage.all import *
>>> p = next_prime(Integer(10)**Integer(40))
>>> g = DirichletGroup(Integer(19), GF(p)); g
Group of Dirichlet characters modulo 19 with values
 in Finite Field of size 10000000000000000000000000000000000000121

Note that the root of unity has small order, i.e., it is not the largest order root of unity in the field:

Sage

sage: g.zeta_order()
2

Python

>>> from sage.all import *
>>> g.zeta_order()
2

Sage

sage: r4 = CyclotomicField(4).ring_of_integers()
sage: G = DirichletGroup(60, r4)
sage: G.gens()
(Dirichlet character modulo 60 of conductor 4
   mapping 31 |--> -1, 41 |--> 1, 37 |--> 1,
 Dirichlet character modulo 60 of conductor 3
   mapping 31 |--> 1, 41 |--> -1, 37 |--> 1,
 Dirichlet character modulo 60 of conductor 5
   mapping 31 |--> 1, 41 |--> 1, 37 |--> zeta4)
sage: val = G.gens()[2].values_on_gens()[2] ; val
zeta4
sage: parent(val)
Gaussian Integers generated by zeta4 in Cyclotomic Field of order 4 and degree 2
sage: r4_29_0 = r4.residue_field(r4.ideal(29).factor()[0][0]); r4_29_0(val)
doctest:warning ... DeprecationWarning: ...
17
sage: r4_29_0(val) * GF(29)(3)
22
sage: r4_29_0(G.gens()[2].values_on_gens()[2]) * 3
22
sage: parent(r4_29_0(G.gens()[2].values_on_gens()[2]) * 3)
Residue field of Fractional ideal (-2*zeta4 + 5)

Python

>>> from sage.all import *
>>> r4 = CyclotomicField(Integer(4)).ring_of_integers()
>>> G = DirichletGroup(Integer(60), r4)
>>> G.gens()
(Dirichlet character modulo 60 of conductor 4
   mapping 31 |--> -1, 41 |--> 1, 37 |--> 1,
 Dirichlet character modulo 60 of conductor 3
   mapping 31 |--> 1, 41 |--> -1, 37 |--> 1,
 Dirichlet character modulo 60 of conductor 5
   mapping 31 |--> 1, 41 |--> 1, 37 |--> zeta4)
>>> val = G.gens()[Integer(2)].values_on_gens()[Integer(2)] ; val
zeta4
>>> parent(val)
Gaussian Integers generated by zeta4 in Cyclotomic Field of order 4 and degree 2
>>> r4_29_0 = r4.residue_field(r4.ideal(Integer(29)).factor()[Integer(0)][Integer(0)]); r4_29_0(val)
doctest:warning ... DeprecationWarning: ...
17
>>> r4_29_0(val) * GF(Integer(29))(Integer(3))
22
>>> r4_29_0(G.gens()[Integer(2)].values_on_gens()[Integer(2)]) * Integer(3)
22
>>> parent(r4_29_0(G.gens()[Integer(2)].values_on_gens()[Integer(2)]) * Integer(3))
Residue field of Fractional ideal (-2*zeta4 + 5)

Sage

sage: DirichletGroup(60, integral=True)
Group of Dirichlet characters modulo 60 with values in
 Gaussian Integers generated by zeta4 in Cyclotomic Field of order 4 and degree 2
sage: parent(DirichletGroup(60, integral=True).gens()[2].values_on_gens()[2])
Gaussian Integers generated by zeta4 in Cyclotomic Field of order 4 and degree 2

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(60), integral=True)
Group of Dirichlet characters modulo 60 with values in
 Gaussian Integers generated by zeta4 in Cyclotomic Field of order 4 and degree 2
>>> parent(DirichletGroup(Integer(60), integral=True).gens()[Integer(2)].values_on_gens()[Integer(2)])
Gaussian Integers generated by zeta4 in Cyclotomic Field of order 4 and degree 2

If the order of zeta cannot be determined automatically, we can specify it using zeta_order:

Sage

sage: DirichletGroup(7, CC, zeta=exp(2*pi*I/6))                                 # needs sage.symbolic
Traceback (most recent call last):
...
NotImplementedError: order of element not known

sage: DirichletGroup(7, CC, zeta=exp(2*pi*I/6), zeta_order=6)                   # needs sage.symbolic
Group of Dirichlet characters modulo 7 with values in the group of order 6
 generated by 0.500000000000000 + 0.866025403784439*I
 in Complex Field with 53 bits of precision

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(7), CC, zeta=exp(Integer(2)*pi*I/Integer(6)))                                 # needs sage.symbolic
Traceback (most recent call last):
...
NotImplementedError: order of element not known

>>> DirichletGroup(Integer(7), CC, zeta=exp(Integer(2)*pi*I/Integer(6)), zeta_order=Integer(6))                   # needs sage.symbolic
Group of Dirichlet characters modulo 7 with values in the group of order 6
 generated by 0.500000000000000 + 0.866025403784439*I
 in Complex Field with 53 bits of precision

If the base ring is not a domain (in which case the group of roots of unity is not necessarily cyclic), some operations still work, such as creation of elements:

Sage

sage: G = DirichletGroup(5, Zmod(15)); G
Group of Dirichlet characters modulo 5 with values in Ring of integers modulo 15
sage: chi = G([13]); chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 13
sage: chi^2
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 4
sage: chi.multiplicative_order()
4

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(5), Zmod(Integer(15))); G
Group of Dirichlet characters modulo 5 with values in Ring of integers modulo 15
>>> chi = G([Integer(13)]); chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 13
>>> chi**Integer(2)
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 4
>>> chi.multiplicative_order()
4

Other operations only work if zeta is specified:

Sage

sage: G.gens()
Traceback (most recent call last):
...
NotImplementedError: factorization of polynomials over rings
with composite characteristic is not implemented
sage: G = DirichletGroup(5, Zmod(15), zeta=2); G
Group of Dirichlet characters modulo 5 with values in the group of order 4
 generated by 2 in Ring of integers modulo 15
sage: G.gens()
(Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 2,)

Python

>>> from sage.all import *
>>> G.gens()
Traceback (most recent call last):
...
NotImplementedError: factorization of polynomials over rings
with composite characteristic is not implemented
>>> G = DirichletGroup(Integer(5), Zmod(Integer(15)), zeta=Integer(2)); G
Group of Dirichlet characters modulo 5 with values in the group of order 4
 generated by 2 in Ring of integers modulo 15
>>> G.gens()
(Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 2,)

create_key(N, base_ring=None, zeta=None, zeta_order=None, names=None, integral=False)[source]#: Create a key that uniquely determines a Dirichlet group.

create_object(version, key, **extra_args)[source]#

Create the object from the key (extra arguments are ignored).

This is only called if the object was not found in the cache.

class sage.modular.dirichlet.DirichletGroup_class(base_ring, modulus, zeta, zeta_order)[source]#

Bases: WithEqualityById, Parent

Group of Dirichlet characters modulo \(N\) with values in a ring \(R\).

Element[source]#: alias of DirichletCharacter

base_extend(R)[source]#

Return the base extension of self to R.

INPUT:

R – either a ring admitting a coercion map from the base ring of self, or a ring homomorphism with the base ring of self as its domain

EXAMPLES:

Sage

sage: G = DirichletGroup(7,QQ); G
Group of Dirichlet characters modulo 7 with values in Rational Field
sage: H = G.base_extend(CyclotomicField(6)); H
Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and degree 2

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(7),QQ); G
Group of Dirichlet characters modulo 7 with values in Rational Field
>>> H = G.base_extend(CyclotomicField(Integer(6))); H
Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and degree 2

Note that the root of unity can change:

Sage

sage: H.zeta()
zeta6

Python

>>> from sage.all import *
>>> H.zeta()
zeta6

This method (in contrast to change_ring()) requires a coercion map to exist:

Sage

sage: G.base_extend(ZZ)
Traceback (most recent call last):
...
TypeError: no coercion map from Rational Field to Integer Ring is defined

Python

>>> from sage.all import *
>>> G.base_extend(ZZ)
Traceback (most recent call last):
...
TypeError: no coercion map from Rational Field to Integer Ring is defined

Base-extended Dirichlet groups do not silently get roots of unity with smaller order than expected (Issue #6018):

Sage

sage: G = DirichletGroup(10, QQ).base_extend(CyclotomicField(4))
sage: H = DirichletGroup(10, CyclotomicField(4))
sage: G is H
True

sage: G3 = DirichletGroup(31, CyclotomicField(3))
sage: G5 = DirichletGroup(31, CyclotomicField(5))
sage: K30 = CyclotomicField(30)
sage: G3.gen(0).base_extend(K30) * G5.gen(0).base_extend(K30)
Dirichlet character modulo 31 of conductor 31 mapping 3 |--> -zeta30^7 + zeta30^5 + zeta30^4 + zeta30^3 - zeta30 - 1

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(10), QQ).base_extend(CyclotomicField(Integer(4)))
>>> H = DirichletGroup(Integer(10), CyclotomicField(Integer(4)))
>>> G is H
True

>>> G3 = DirichletGroup(Integer(31), CyclotomicField(Integer(3)))
>>> G5 = DirichletGroup(Integer(31), CyclotomicField(Integer(5)))
>>> K30 = CyclotomicField(Integer(30))
>>> G3.gen(Integer(0)).base_extend(K30) * G5.gen(Integer(0)).base_extend(K30)
Dirichlet character modulo 31 of conductor 31 mapping 3 |--> -zeta30^7 + zeta30^5 + zeta30^4 + zeta30^3 - zeta30 - 1

When a root of unity is specified, base extension still works if the new base ring is not an integral domain:

Sage

sage: f = DirichletGroup(17, ZZ, zeta=-1).0
sage: g = f.base_extend(Integers(15))
sage: g(3)
14
sage: g.parent().zeta()
14

Python

>>> from sage.all import *
>>> f = DirichletGroup(Integer(17), ZZ, zeta=-Integer(1)).gen(0)
>>> g = f.base_extend(Integers(Integer(15)))
>>> g(Integer(3))
14
>>> g.parent().zeta()
14

change_ring(R, zeta=None, zeta_order=None)[source]#

Return the base extension of self to R.

INPUT:

R – either a ring admitting a conversion map from the base ring of self, or a ring homomorphism with the base ring of self as its domain
zeta – (optional) root of unity in R
zeta_order – (optional) order of zeta

EXAMPLES:

Sage

sage: G = DirichletGroup(7,QQ); G
Group of Dirichlet characters modulo 7 with values in Rational Field
sage: G.change_ring(CyclotomicField(6))                                     # needs sage.rings.number_field
Group of Dirichlet characters modulo 7 with values in
 Cyclotomic Field of order 6 and degree 2

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(7),QQ); G
Group of Dirichlet characters modulo 7 with values in Rational Field
>>> G.change_ring(CyclotomicField(Integer(6)))                                     # needs sage.rings.number_field
Group of Dirichlet characters modulo 7 with values in
 Cyclotomic Field of order 6 and degree 2

decomposition()[source]#

Return the Dirichlet groups of prime power modulus corresponding to primes dividing modulus.

(Note that if the modulus is 2 mod 4, there will be a “factor” of \((\ZZ/2\ZZ)^*\), which is the trivial group.)

EXAMPLES:

Sage

sage: DirichletGroup(20).decomposition()
[
Group of Dirichlet characters modulo 4 with values in Cyclotomic Field of order 4 and degree 2,
Group of Dirichlet characters modulo 5 with values in Cyclotomic Field of order 4 and degree 2
]
sage: DirichletGroup(20,GF(5)).decomposition()
[
Group of Dirichlet characters modulo 4 with values in Finite Field of size 5,
Group of Dirichlet characters modulo 5 with values in Finite Field of size 5
]

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(20)).decomposition()
[
Group of Dirichlet characters modulo 4 with values in Cyclotomic Field of order 4 and degree 2,
Group of Dirichlet characters modulo 5 with values in Cyclotomic Field of order 4 and degree 2
]
>>> DirichletGroup(Integer(20),GF(Integer(5))).decomposition()
[
Group of Dirichlet characters modulo 4 with values in Finite Field of size 5,
Group of Dirichlet characters modulo 5 with values in Finite Field of size 5
]

exponent()[source]#

Return the exponent of this group.

EXAMPLES:

Sage

sage: DirichletGroup(20).exponent()
4
sage: DirichletGroup(20,GF(3)).exponent()
2
sage: DirichletGroup(20,GF(2)).exponent()
1
sage: DirichletGroup(37).exponent()
36

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(20)).exponent()
4
>>> DirichletGroup(Integer(20),GF(Integer(3))).exponent()
2
>>> DirichletGroup(Integer(20),GF(Integer(2))).exponent()
1
>>> DirichletGroup(Integer(37)).exponent()
36

galois_orbits(v=None, reps_only=False, sort=True, check=True)[source]#

Return a list of the Galois orbits of Dirichlet characters in self, or in v if v is not None.

INPUT:

v – (optional) list of elements of self
reps_only – (optional: default False) if True only returns representatives for the orbits.
sort – (optional: default True) whether to sort the list of orbits and the orbits themselves (slightly faster if False).
check – (default: True) whether or not to explicitly coerce each element of v into self.

The Galois group is the absolute Galois group of the prime subfield of Frac(R). If R is not a domain, an error will be raised.

EXAMPLES:

Sage

sage: DirichletGroup(20).galois_orbits()
[
[Dirichlet character modulo 20 of conductor 20 mapping 11 |--> -1, 17 |--> -1],
...,
[Dirichlet character modulo 20 of conductor 1 mapping 11 |--> 1, 17 |--> 1]
]
sage: DirichletGroup(17, Integers(6), zeta=Integers(6)(5)).galois_orbits()
Traceback (most recent call last):
...
TypeError: Galois orbits only defined if base ring is an integral domain
sage: DirichletGroup(17, Integers(9), zeta=Integers(9)(2)).galois_orbits()
Traceback (most recent call last):
...
TypeError: Galois orbits only defined if base ring is an integral domain

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(20)).galois_orbits()
[
[Dirichlet character modulo 20 of conductor 20 mapping 11 |--> -1, 17 |--> -1],
...,
[Dirichlet character modulo 20 of conductor 1 mapping 11 |--> 1, 17 |--> 1]
]
>>> DirichletGroup(Integer(17), Integers(Integer(6)), zeta=Integers(Integer(6))(Integer(5))).galois_orbits()
Traceback (most recent call last):
...
TypeError: Galois orbits only defined if base ring is an integral domain
>>> DirichletGroup(Integer(17), Integers(Integer(9)), zeta=Integers(Integer(9))(Integer(2))).galois_orbits()
Traceback (most recent call last):
...
TypeError: Galois orbits only defined if base ring is an integral domain

gen(n=0)[source]#

Return the n-th generator of self.

EXAMPLES:

Sage

sage: G = DirichletGroup(20)
sage: G.gen(0)
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: G.gen(1)
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4
sage: G.gen(2)
Traceback (most recent call last):
...
IndexError: n(=2) must be between 0 and 1

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20))
>>> G.gen(Integer(0))
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
>>> G.gen(Integer(1))
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4
>>> G.gen(Integer(2))
Traceback (most recent call last):
...
IndexError: n(=2) must be between 0 and 1

Sage

sage: G.gen(-1)
Traceback (most recent call last):
...
IndexError: n(=-1) must be between 0 and 1

Python

>>> from sage.all import *
>>> G.gen(-Integer(1))
Traceback (most recent call last):
...
IndexError: n(=-1) must be between 0 and 1

gens()[source]#

Return generators of self.

EXAMPLES:

Sage

sage: G = DirichletGroup(20)
sage: G.gens()
(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1, Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4)

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20))
>>> G.gens()
(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1, Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4)

integers_mod()[source]#

Return the group of integers \(\ZZ/N\ZZ\) where \(N\) is the modulus of self.

EXAMPLES:

Sage

sage: G = DirichletGroup(20)
sage: G.integers_mod()
Ring of integers modulo 20

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20))
>>> G.integers_mod()
Ring of integers modulo 20

list()[source]#

Return a list of the Dirichlet characters in this group.

EXAMPLES:

Sage

sage: DirichletGroup(5).list()
[Dirichlet character modulo 5 of conductor 1 mapping 2 |--> 1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4]

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(5)).list()
[Dirichlet character modulo 5 of conductor 1 mapping 2 |--> 1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4]

modulus()[source]#

Return the modulus of self.

EXAMPLES:

Sage

sage: G = DirichletGroup(20)
sage: G.modulus()
20

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20))
>>> G.modulus()
20

ngens()[source]#

Return the number of generators of self.

EXAMPLES:

Sage

sage: G = DirichletGroup(20)
sage: G.ngens()
2

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(20))
>>> G.ngens()
2

order()[source]#

Return the number of elements of self.

This is the same as len(self).

EXAMPLES:

Sage

sage: DirichletGroup(20).order()
8
sage: DirichletGroup(37).order()
36

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(20)).order()
8
>>> DirichletGroup(Integer(37)).order()
36

random_element()[source]#

Return a random element of self.

The element is computed by multiplying a random power of each generator together, where the power is between 0 and the order of the generator minus 1, inclusive.

EXAMPLES:

Sage

sage: D = DirichletGroup(37)
sage: g = D.random_element()
sage: g.parent() is D
True
sage: g**36
Dirichlet character modulo 37 of conductor 1 mapping 2 |--> 1
sage: S = set(D.random_element().conductor() for _ in range(100))
sage: while S != {1, 37}:
....:     S.add(D.random_element().conductor())

sage: D = DirichletGroup(20)
sage: g = D.random_element()
sage: g.parent() is D
True
sage: g**4
Dirichlet character modulo 20 of conductor 1 mapping 11 |--> 1, 17 |--> 1
sage: S = set(D.random_element().conductor() for _ in range(100))
sage: while S != {1, 4, 5, 20}:
....:     S.add(D.random_element().conductor())

sage: D = DirichletGroup(60)
sage: g = D.random_element()
sage: g.parent() is D
True
sage: g**4
Dirichlet character modulo 60 of conductor 1 mapping 31 |--> 1, 41 |--> 1, 37 |--> 1
sage: S = set(D.random_element().conductor() for _ in range(100))
sage: while S != {1, 3, 4, 5, 12, 15, 20, 60}:
....:     S.add(D.random_element().conductor())

Python

>>> from sage.all import *
>>> D = DirichletGroup(Integer(37))
>>> g = D.random_element()
>>> g.parent() is D
True
>>> g**Integer(36)
Dirichlet character modulo 37 of conductor 1 mapping 2 |--> 1
>>> S = set(D.random_element().conductor() for _ in range(Integer(100)))
>>> while S != {Integer(1), Integer(37)}:
...     S.add(D.random_element().conductor())

>>> D = DirichletGroup(Integer(20))
>>> g = D.random_element()
>>> g.parent() is D
True
>>> g**Integer(4)
Dirichlet character modulo 20 of conductor 1 mapping 11 |--> 1, 17 |--> 1
>>> S = set(D.random_element().conductor() for _ in range(Integer(100)))
>>> while S != {Integer(1), Integer(4), Integer(5), Integer(20)}:
...     S.add(D.random_element().conductor())

>>> D = DirichletGroup(Integer(60))
>>> g = D.random_element()
>>> g.parent() is D
True
>>> g**Integer(4)
Dirichlet character modulo 60 of conductor 1 mapping 31 |--> 1, 41 |--> 1, 37 |--> 1
>>> S = set(D.random_element().conductor() for _ in range(Integer(100)))
>>> while S != {Integer(1), Integer(3), Integer(4), Integer(5), Integer(12), Integer(15), Integer(20), Integer(60)}:
...     S.add(D.random_element().conductor())

unit_gens()[source]#

Return the minimal generators for the units of \((\ZZ/N\ZZ)^*\), where \(N\) is the modulus of self.

EXAMPLES:

Sage

sage: DirichletGroup(37).unit_gens()
(2,)
sage: DirichletGroup(20).unit_gens()
(11, 17)
sage: DirichletGroup(60).unit_gens()
(31, 41, 37)
sage: DirichletGroup(20,QQ).unit_gens()
(11, 17)

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(37)).unit_gens()
(2,)
>>> DirichletGroup(Integer(20)).unit_gens()
(11, 17)
>>> DirichletGroup(Integer(60)).unit_gens()
(31, 41, 37)
>>> DirichletGroup(Integer(20),QQ).unit_gens()
(11, 17)

zeta()[source]#

Return the chosen root of unity in the base ring.

EXAMPLES:

Sage

sage: DirichletGroup(37).zeta()
zeta36
sage: DirichletGroup(20).zeta()
zeta4
sage: DirichletGroup(60).zeta()
zeta4
sage: DirichletGroup(60,QQ).zeta()
-1
sage: DirichletGroup(60, GF(25,'a')).zeta()
2

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(37)).zeta()
zeta36
>>> DirichletGroup(Integer(20)).zeta()
zeta4
>>> DirichletGroup(Integer(60)).zeta()
zeta4
>>> DirichletGroup(Integer(60),QQ).zeta()
-1
>>> DirichletGroup(Integer(60), GF(Integer(25),'a')).zeta()
2

zeta_order()[source]#

Return the order of the chosen root of unity in the base ring.

EXAMPLES:

Sage

sage: DirichletGroup(20).zeta_order()
4
sage: DirichletGroup(60).zeta_order()
4
sage: DirichletGroup(60, GF(25,'a')).zeta_order()
4
sage: DirichletGroup(19).zeta_order()
18

Python

>>> from sage.all import *
>>> DirichletGroup(Integer(20)).zeta_order()
4
>>> DirichletGroup(Integer(60)).zeta_order()
4
>>> DirichletGroup(Integer(60), GF(Integer(25),'a')).zeta_order()
4
>>> DirichletGroup(Integer(19)).zeta_order()
18

sage.modular.dirichlet.TrivialCharacter(N, base_ring=Rational Field)[source]#

Return the trivial character of the given modulus, with values in the given base ring.

EXAMPLES:

Sage

sage: t = trivial_character(7)
sage: [t(x) for x in [0..20]]
[0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]
sage: t(1).parent()
Rational Field
sage: trivial_character(7, Integers(3))(1).parent()
Ring of integers modulo 3

Python

>>> from sage.all import *
>>> t = trivial_character(Integer(7))
>>> [t(x) for x in (ellipsis_range(Integer(0),Ellipsis,Integer(20)))]
[0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]
>>> t(Integer(1)).parent()
Rational Field
>>> trivial_character(Integer(7), Integers(Integer(3)))(Integer(1)).parent()
Ring of integers modulo 3

sage.modular.dirichlet.is_DirichletCharacter(x)[source]#

Return True if x is of type DirichletCharacter.

EXAMPLES:

Sage

sage: from sage.modular.dirichlet import is_DirichletCharacter
sage: is_DirichletCharacter(trivial_character(3))
doctest:warning...
DeprecationWarning: The function is_DirichletCharacter is deprecated;
use 'isinstance(..., DirichletCharacter)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True
sage: is_DirichletCharacter([1])
False

Python

>>> from sage.all import *
>>> from sage.modular.dirichlet import is_DirichletCharacter
>>> is_DirichletCharacter(trivial_character(Integer(3)))
doctest:warning...
DeprecationWarning: The function is_DirichletCharacter is deprecated;
use 'isinstance(..., DirichletCharacter)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True
>>> is_DirichletCharacter([Integer(1)])
False

sage.modular.dirichlet.is_DirichletGroup(x)[source]#

Return True if x is a Dirichlet group.

EXAMPLES:

Sage

sage: from sage.modular.dirichlet import is_DirichletGroup
sage: is_DirichletGroup(DirichletGroup(11))
doctest:warning...
DeprecationWarning: The function is_DirichletGroup is deprecated; use 'isinstance(..., DirichletGroup_class)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
True
sage: is_DirichletGroup(11)
False
sage: is_DirichletGroup(DirichletGroup(11).0)
False

Python

>>> from sage.all import *
>>> from sage.modular.dirichlet import is_DirichletGroup
>>> is_DirichletGroup(DirichletGroup(Integer(11)))
doctest:warning...
DeprecationWarning: The function is_DirichletGroup is deprecated; use 'isinstance(..., DirichletGroup_class)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
True
>>> is_DirichletGroup(Integer(11))
False
>>> is_DirichletGroup(DirichletGroup(Integer(11)).gen(0))
False

sage.modular.dirichlet.kronecker_character(d)[source]#

Return the quadratic Dirichlet character (d/.) of minimal conductor.

EXAMPLES:

Sage

sage: kronecker_character(97*389*997^2)
Dirichlet character modulo 37733 of conductor 37733
 mapping 1557 |--> -1, 37346 |--> -1

Python

>>> from sage.all import *
>>> kronecker_character(Integer(97)*Integer(389)*Integer(997)**Integer(2))
Dirichlet character modulo 37733 of conductor 37733
 mapping 1557 |--> -1, 37346 |--> -1

Sage

sage: a = kronecker_character(1)
sage: b = DirichletGroup(2401,QQ)(a)    # NOTE -- over QQ!
sage: b.modulus()
2401

Python

>>> from sage.all import *
>>> a = kronecker_character(Integer(1))
>>> b = DirichletGroup(Integer(2401),QQ)(a)    # NOTE -- over QQ!
>>> b.modulus()
2401

AUTHORS:

Jon Hanke (2006-08-06)

sage.modular.dirichlet.kronecker_character_upside_down(d)[source]#

Return the quadratic Dirichlet character (./d) of conductor d, for d > 0.

EXAMPLES:

Sage

sage: kronecker_character_upside_down(97*389*997^2)
Dirichlet character modulo 37506941597 of conductor 37733
 mapping 13533432536 |--> -1, 22369178537 |--> -1, 14266017175 |--> 1

Python

>>> from sage.all import *
>>> kronecker_character_upside_down(Integer(97)*Integer(389)*Integer(997)**Integer(2))
Dirichlet character modulo 37506941597 of conductor 37733
 mapping 13533432536 |--> -1, 22369178537 |--> -1, 14266017175 |--> 1

AUTHORS:

Jon Hanke (2006-08-06)

sage.modular.dirichlet.trivial_character(N, base_ring=Rational Field)[source]#

Return the trivial character of the given modulus, with values in the given base ring.

EXAMPLES:

Sage

sage: t = trivial_character(7)
sage: [t(x) for x in [0..20]]
[0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]
sage: t(1).parent()
Rational Field
sage: trivial_character(7, Integers(3))(1).parent()
Ring of integers modulo 3

Python

>>> from sage.all import *
>>> t = trivial_character(Integer(7))
>>> [t(x) for x in (ellipsis_range(Integer(0),Ellipsis,Integer(20)))]
[0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]
>>> t(Integer(1)).parent()
Rational Field
>>> trivial_character(Integer(7), Integers(Integer(3)))(Integer(1)).parent()
Ring of integers modulo 3