Congruence subgroup \(\Gamma_1(N)\)

class sage.modular.arithgroup.congroup_gamma1.Gamma1_class(level)[source]

Bases: GammaH_class

The congruence subgroup \(\Gamma_1(N)\).

dimension_cusp_forms(k=2, eps=None, algorithm='CohenOesterle')[source]

Return the dimension of the space of cusp forms for self, or the dimension of the subspace corresponding to the given character if one is supplied.

INPUT:

  • k – integer (default: 2); the weight

  • eps – either None or a Dirichlet character modulo N, where N is the level of this group. If this is None, then the dimension of the whole space is returned; otherwise, the dimension of the subspace of forms of character eps.

  • algorithm – either 'CohenOesterle' (the default) or 'Quer'. This specifies the method to use in the case of nontrivial character: either the Cohen–Oesterle formula as described in Stein’s book, or by Möbius inversion using the subgroups GammaH (a method due to Jordi Quer). Ignored for weight 1.

EXAMPLES:

We compute the same dimension in two different ways

sage: # needs sage.rings.number_field
sage: K = CyclotomicField(3)
sage: eps = DirichletGroup(7*43,K).0^2
sage: G = Gamma1(7*43)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(3))
>>> eps = DirichletGroup(Integer(7)*Integer(43),K).gen(0)**Integer(2)
>>> G = Gamma1(Integer(7)*Integer(43))

Via Cohen–Oesterle:

sage: Gamma1(7*43).dimension_cusp_forms(2, eps)                             # needs sage.rings.number_field
28

>>> from sage.all import *
>>> Gamma1(Integer(7)*Integer(43)).dimension_cusp_forms(Integer(2), eps)                             # needs sage.rings.number_field
28

Via Quer’s method:

sage: Gamma1(7*43).dimension_cusp_forms(2, eps, algorithm='Quer')           # needs sage.rings.number_field
28

>>> from sage.all import *
>>> Gamma1(Integer(7)*Integer(43)).dimension_cusp_forms(Integer(2), eps, algorithm='Quer')           # needs sage.rings.number_field
28

Some more examples:

sage: G.<eps> = DirichletGroup(9)
sage: [Gamma1(9).dimension_cusp_forms(k, eps) for k in [1..10]]
[0, 0, 1, 0, 3, 0, 5, 0, 7, 0]
sage: [Gamma1(9).dimension_cusp_forms(k, eps^2) for k in [1..10]]
[0, 0, 0, 2, 0, 4, 0, 6, 0, 8]

>>> from sage.all import *
>>> G = DirichletGroup(Integer(9), names=('eps',)); (eps,) = G._first_ngens(1)
>>> [Gamma1(Integer(9)).dimension_cusp_forms(k, eps) for k in (ellipsis_range(Integer(1),Ellipsis,Integer(10)))]
[0, 0, 1, 0, 3, 0, 5, 0, 7, 0]
>>> [Gamma1(Integer(9)).dimension_cusp_forms(k, eps**Integer(2)) for k in (ellipsis_range(Integer(1),Ellipsis,Integer(10)))]
[0, 0, 0, 2, 0, 4, 0, 6, 0, 8]

In weight 1, we can sometimes rule out cusp forms existing via Riemann-Roch, but if this does not work, we trigger computation of the cusp forms space via Schaeffer’s algorithm:

sage: chi = [u for u in DirichletGroup(40) if u(-1) == -1 and u(21) == 1][0]
sage: Gamma1(40).dimension_cusp_forms(1, chi)
0
sage: G = DirichletGroup(57); chi = (G.0) * (G.1)^6
sage: Gamma1(57).dimension_cusp_forms(1, chi)
1

>>> from sage.all import *
>>> chi = [u for u in DirichletGroup(Integer(40)) if u(-Integer(1)) == -Integer(1) and u(Integer(21)) == Integer(1)][Integer(0)]
>>> Gamma1(Integer(40)).dimension_cusp_forms(Integer(1), chi)
0
>>> G = DirichletGroup(Integer(57)); chi = (G.gen(0)) * (G.gen(1))**Integer(6)
>>> Gamma1(Integer(57)).dimension_cusp_forms(Integer(1), chi)
1
dimension_eis(k=2, eps=None, algorithm='CohenOesterle')[source]

Return the dimension of the space of Eisenstein series forms for self, or the dimension of the subspace corresponding to the given character if one is supplied.

INPUT:

  • k – integer (default: 2); the weight

  • eps – either None or a Dirichlet character modulo N, where N is the level of this group. If this is None, then the dimension of the whole space is returned; otherwise, the dimension of the subspace of Eisenstein series of character eps.

  • algorithm – either 'CohenOesterle' (the default) or 'Quer'. This specifies the method to use in the case of nontrivial character: either the Cohen–Oesterle formula as described in Stein’s book, or by Möbius inversion using the subgroups GammaH (a method due to Jordi Quer).

AUTHORS:

  • William Stein - Cohen–Oesterle algorithm

  • Jordi Quer - algorithm based on GammaH subgroups

  • David Loeffler (2009) - code refactoring

EXAMPLES:

The following two computations use different algorithms:

sage: [Gamma1(36).dimension_eis(1,eps) for eps in DirichletGroup(36)]
[0, 4, 3, 0, 0, 2, 6, 0, 0, 2, 3, 0]
sage: [Gamma1(36).dimension_eis(1,eps,algorithm='Quer') for eps in DirichletGroup(36)]
[0, 4, 3, 0, 0, 2, 6, 0, 0, 2, 3, 0]

>>> from sage.all import *
>>> [Gamma1(Integer(36)).dimension_eis(Integer(1),eps) for eps in DirichletGroup(Integer(36))]
[0, 4, 3, 0, 0, 2, 6, 0, 0, 2, 3, 0]
>>> [Gamma1(Integer(36)).dimension_eis(Integer(1),eps,algorithm='Quer') for eps in DirichletGroup(Integer(36))]
[0, 4, 3, 0, 0, 2, 6, 0, 0, 2, 3, 0]

So do these:

sage: [Gamma1(48).dimension_eis(3,eps) for eps in DirichletGroup(48)]
[0, 12, 0, 4, 0, 8, 0, 4, 12, 0, 4, 0, 8, 0, 4, 0]
sage: [Gamma1(48).dimension_eis(3,eps,algorithm='Quer') for eps in DirichletGroup(48)]
[0, 12, 0, 4, 0, 8, 0, 4, 12, 0, 4, 0, 8, 0, 4, 0]

>>> from sage.all import *
>>> [Gamma1(Integer(48)).dimension_eis(Integer(3),eps) for eps in DirichletGroup(Integer(48))]
[0, 12, 0, 4, 0, 8, 0, 4, 12, 0, 4, 0, 8, 0, 4, 0]
>>> [Gamma1(Integer(48)).dimension_eis(Integer(3),eps,algorithm='Quer') for eps in DirichletGroup(Integer(48))]
[0, 12, 0, 4, 0, 8, 0, 4, 12, 0, 4, 0, 8, 0, 4, 0]
dimension_modular_forms(k=2, eps=None, algorithm='CohenOesterle')[source]

Return the dimension of the space of modular forms for self, or the dimension of the subspace corresponding to the given character if one is supplied.

INPUT:

  • k – integer (default: 2); the weight

  • eps – either None or a Dirichlet character modulo N, where N is the level of this group. If this is None, then the dimension of the whole space is returned; otherwise, the dimension of the subspace of forms of character eps.

  • algorithm – either 'CohenOesterle' (the default) or 'Quer'. This specifies the method to use in the case of nontrivial character: either the Cohen–Oesterle formula as described in Stein’s book, or by Möbius inversion using the subgroups GammaH (a method due to Jordi Quer).

EXAMPLES:

sage: # needs sage.rings.number_field
sage: K = CyclotomicField(3)
sage: eps = DirichletGroup(7*43,K).0^2
sage: G = Gamma1(7*43)
sage: G.dimension_modular_forms(2, eps)
32
sage: G.dimension_modular_forms(2, eps, algorithm='Quer')
32

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(3))
>>> eps = DirichletGroup(Integer(7)*Integer(43),K).gen(0)**Integer(2)
>>> G = Gamma1(Integer(7)*Integer(43))
>>> G.dimension_modular_forms(Integer(2), eps)
32
>>> G.dimension_modular_forms(Integer(2), eps, algorithm='Quer')
32
dimension_new_cusp_forms(k=2, eps=None, p=0, algorithm='CohenOesterle')[source]

Dimension of the new subspace (or \(p\)-new subspace) of cusp forms of weight \(k\) and character \(\varepsilon\).

INPUT:

  • k – integer (default: 2)

  • eps – a Dirichlet character

  • p – a prime (default: 0); just the \(p\)-new subspace if given

  • algorithm – either 'CohenOesterle' (the default) or 'Quer'. This specifies the method to use in the case of nontrivial character: either the Cohen–Oesterle formula as described in Stein’s book, or by Möbius inversion using the subgroups GammaH (a method due to Jordi Quer).

EXAMPLES:

sage: G = DirichletGroup(9)
sage: eps = G.0^3
sage: eps.conductor()
3
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [Gamma1(9).dimension_cusp_forms(k, eps) for k in [2..10]]
[0, 0, 0, 2, 0, 4, 0, 6, 0]
sage: [Gamma1(9).dimension_new_cusp_forms(k, eps, 3) for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]

>>> from sage.all import *
>>> G = DirichletGroup(Integer(9))
>>> eps = G.gen(0)**Integer(3)
>>> eps.conductor()
3
>>> [Gamma1(Integer(9)).dimension_new_cusp_forms(k, eps) for k in (ellipsis_range(Integer(2),Ellipsis,Integer(10)))]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
>>> [Gamma1(Integer(9)).dimension_cusp_forms(k, eps) for k in (ellipsis_range(Integer(2),Ellipsis,Integer(10)))]
[0, 0, 0, 2, 0, 4, 0, 6, 0]
>>> [Gamma1(Integer(9)).dimension_new_cusp_forms(k, eps, Integer(3)) for k in (ellipsis_range(Integer(2),Ellipsis,Integer(10)))]
[0, 0, 0, 2, 0, 2, 0, 2, 0]

Double check using modular symbols (independent calculation):

sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace().dimension()  for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
sage: [ModularSymbols(eps,k,sign=1).cuspidal_subspace().new_subspace(3).dimension()  for k in [2..10]]
[0, 0, 0, 2, 0, 2, 0, 2, 0]

>>> from sage.all import *
>>> [ModularSymbols(eps,k,sign=Integer(1)).cuspidal_subspace().new_subspace().dimension()  for k in (ellipsis_range(Integer(2),Ellipsis,Integer(10)))]
[0, 0, 0, 2, 0, 2, 0, 2, 0]
>>> [ModularSymbols(eps,k,sign=Integer(1)).cuspidal_subspace().new_subspace(Integer(3)).dimension()  for k in (ellipsis_range(Integer(2),Ellipsis,Integer(10)))]
[0, 0, 0, 2, 0, 2, 0, 2, 0]

Another example at level 33:

sage: G = DirichletGroup(33)
sage: eps = G.1
sage: eps.conductor()
11
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1) for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1, algorithm='Quer') for k in [2..4]]
[0, 4, 0]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2) for k in [2..4]]
[2, 0, 6]
sage: [Gamma1(33).dimension_new_cusp_forms(k, G.1^2, p=3) for k in [2..4]]
[2, 0, 6]

>>> from sage.all import *
>>> G = DirichletGroup(Integer(33))
>>> eps = G.gen(1)
>>> eps.conductor()
11
>>> [Gamma1(Integer(33)).dimension_new_cusp_forms(k, G.gen(1)) for k in (ellipsis_range(Integer(2),Ellipsis,Integer(4)))]
[0, 4, 0]
>>> [Gamma1(Integer(33)).dimension_new_cusp_forms(k, G.gen(1), algorithm='Quer') for k in (ellipsis_range(Integer(2),Ellipsis,Integer(4)))]
[0, 4, 0]
>>> [Gamma1(Integer(33)).dimension_new_cusp_forms(k, G.gen(1)**Integer(2)) for k in (ellipsis_range(Integer(2),Ellipsis,Integer(4)))]
[2, 0, 6]
>>> [Gamma1(Integer(33)).dimension_new_cusp_forms(k, G.gen(1)**Integer(2), p=Integer(3)) for k in (ellipsis_range(Integer(2),Ellipsis,Integer(4)))]
[2, 0, 6]
generators(algorithm='farey')[source]

Return generators for this congruence subgroup. The result is cached.

INPUT:

  • algorithm – string; either 'farey' (default) or 'todd-coxeter'

If algorithm is set to 'farey', then the generators will be calculated using Farey symbols, which will always return a minimal generating set. See farey_symbol for more information.

If algorithm is set to 'todd-coxeter', a simpler algorithm based on Todd-Coxeter enumeration will be used. This tends to return far larger sets of generators.

EXAMPLES:

sage: Gamma1(3).generators()
[
[1 1]  [ 1 -1]
[0 1], [ 3 -2]
]
sage: Gamma1(3).generators(algorithm='todd-coxeter')
[
[1 1]  [-2  1]  [1 1]  [ 1 -1]  [1 0]  [1 1]  [-5  2]  [ 1  0]
[0 1], [-3  1], [0 1], [ 0  1], [3 1], [0 1], [12 -5], [-3  1],

[ 1 -1]  [ 1 -1]  [ 4 -1]  [ -5   3]
[ 3 -2], [ 3 -2], [ 9 -2], [-12   7]
]

>>> from sage.all import *
>>> Gamma1(Integer(3)).generators()
[
[1 1]  [ 1 -1]
[0 1], [ 3 -2]
]
>>> Gamma1(Integer(3)).generators(algorithm='todd-coxeter')
[
[1 1]  [-2  1]  [1 1]  [ 1 -1]  [1 0]  [1 1]  [-5  2]  [ 1  0]
[0 1], [-3  1], [0 1], [ 0  1], [3 1], [0 1], [12 -5], [-3  1],
<BLANKLINE>
[ 1 -1]  [ 1 -1]  [ 4 -1]  [ -5   3]
[ 3 -2], [ 3 -2], [ 9 -2], [-12   7]
]
index()[source]

Return the index of self in the full modular group. This is given by the formula

\[\begin{split}N^2 \prod_{\substack{p \mid N \\ \text{$p$ prime}}} \left( 1 - \frac{1}{p^2}\right).\end{split}\]

EXAMPLES:

sage: Gamma1(180).index()
20736
sage: [Gamma1(n).projective_index() for n in [1..16]]
[1, 3, 4, 6, 12, 12, 24, 24, 36, 36, 60, 48, 84, 72, 96, 96]

>>> from sage.all import *
>>> Gamma1(Integer(180)).index()
20736
>>> [Gamma1(n).projective_index() for n in (ellipsis_range(Integer(1),Ellipsis,Integer(16)))]
[1, 3, 4, 6, 12, 12, 24, 24, 36, 36, 60, 48, 84, 72, 96, 96]
is_even()[source]

Return True precisely if this subgroup contains the matrix -1.

EXAMPLES:

sage: Gamma1(1).is_even()
True
sage: Gamma1(2).is_even()
True
sage: Gamma1(15).is_even()
False

>>> from sage.all import *
>>> Gamma1(Integer(1)).is_even()
True
>>> Gamma1(Integer(2)).is_even()
True
>>> Gamma1(Integer(15)).is_even()
False
is_subgroup(right)[source]

Return True if self is a subgroup of right.

EXAMPLES:

sage: Gamma1(3).is_subgroup(SL2Z)
True
sage: Gamma1(3).is_subgroup(Gamma1(5))
False
sage: Gamma1(3).is_subgroup(Gamma1(6))
False
sage: Gamma1(6).is_subgroup(Gamma1(3))
True
sage: Gamma1(6).is_subgroup(Gamma0(2))
True
sage: Gamma1(80).is_subgroup(GammaH(40, []))
True
sage: Gamma1(80).is_subgroup(GammaH(40, [21]))
True

>>> from sage.all import *
>>> Gamma1(Integer(3)).is_subgroup(SL2Z)
True
>>> Gamma1(Integer(3)).is_subgroup(Gamma1(Integer(5)))
False
>>> Gamma1(Integer(3)).is_subgroup(Gamma1(Integer(6)))
False
>>> Gamma1(Integer(6)).is_subgroup(Gamma1(Integer(3)))
True
>>> Gamma1(Integer(6)).is_subgroup(Gamma0(Integer(2)))
True
>>> Gamma1(Integer(80)).is_subgroup(GammaH(Integer(40), []))
True
>>> Gamma1(Integer(80)).is_subgroup(GammaH(Integer(40), [Integer(21)]))
True
ncusps()[source]

Return the number of cusps of this subgroup \(\Gamma_1(N)\).

EXAMPLES:

sage: [Gamma1(n).ncusps() for n in [1..15]]
[1, 2, 2, 3, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 16]
sage: [Gamma1(n).ncusps() for n in prime_range(2, 100)]
[2, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96]

>>> from sage.all import *
>>> [Gamma1(n).ncusps() for n in (ellipsis_range(Integer(1),Ellipsis,Integer(15)))]
[1, 2, 2, 3, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 16]
>>> [Gamma1(n).ncusps() for n in prime_range(Integer(2), Integer(100))]
[2, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96]
nu2()[source]

Calculate the number of orbits of elliptic points of order 2 for this subgroup \(\Gamma_1(N)\). This is known to be 0 if N > 2.

EXAMPLES:

sage: Gamma1(2).nu2()
1
sage: Gamma1(457).nu2()
0
sage: [Gamma1(n).nu2() for n in [1..16]]
[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

>>> from sage.all import *
>>> Gamma1(Integer(2)).nu2()
1
>>> Gamma1(Integer(457)).nu2()
0
>>> [Gamma1(n).nu2() for n in (ellipsis_range(Integer(1),Ellipsis,Integer(16)))]
[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
nu3()[source]

Calculate the number of orbits of elliptic points of order 3 for this subgroup \(\Gamma_1(N)\). This is known to be 0 if N > 3.

EXAMPLES:

sage: Gamma1(2).nu3()
0
sage: Gamma1(3).nu3()
1
sage: Gamma1(457).nu3()
0
sage: [Gamma1(n).nu3() for n in [1..10]]
[1, 0, 1, 0, 0, 0, 0, 0, 0, 0]

>>> from sage.all import *
>>> Gamma1(Integer(2)).nu3()
0
>>> Gamma1(Integer(3)).nu3()
1
>>> Gamma1(Integer(457)).nu3()
0
>>> [Gamma1(n).nu3() for n in (ellipsis_range(Integer(1),Ellipsis,Integer(10)))]
[1, 0, 1, 0, 0, 0, 0, 0, 0, 0]
sage.modular.arithgroup.congroup_gamma1.Gamma1_constructor(N)[source]

Return the congruence subgroup \(\Gamma_1(N)\).

EXAMPLES:

sage: Gamma1(5) # indirect doctest
Congruence Subgroup Gamma1(5)
sage: G = Gamma1(23)
sage: G is Gamma1(23)
True
sage: G is GammaH(23, [1])
True
sage: TestSuite(G).run()
sage: G is loads(dumps(G))
True

>>> from sage.all import *
>>> Gamma1(Integer(5)) # indirect doctest
Congruence Subgroup Gamma1(5)
>>> G = Gamma1(Integer(23))
>>> G is Gamma1(Integer(23))
True
>>> G is GammaH(Integer(23), [Integer(1)])
True
>>> TestSuite(G).run()
>>> G is loads(dumps(G))
True
sage.modular.arithgroup.congroup_gamma1.is_Gamma1(x)[source]

Return True if x is a congruence subgroup of type Gamma1.

EXAMPLES:

sage: from sage.modular.arithgroup.all import is_Gamma1
sage: is_Gamma1(SL2Z)
doctest:warning...
DeprecationWarning: The function is_Gamma1 is deprecated; use 'isinstance(..., Gamma1_class)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
False
sage: is_Gamma1(Gamma1(13))
True
sage: is_Gamma1(Gamma0(6))
False
sage: is_Gamma1(GammaH(12, [])) # trick question!
True
sage: is_Gamma1(GammaH(12, [5]))
False

>>> from sage.all import *
>>> from sage.modular.arithgroup.all import is_Gamma1
>>> is_Gamma1(SL2Z)
doctest:warning...
DeprecationWarning: The function is_Gamma1 is deprecated; use 'isinstance(..., Gamma1_class)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
False
>>> is_Gamma1(Gamma1(Integer(13)))
True
>>> is_Gamma1(Gamma0(Integer(6)))
False
>>> is_Gamma1(GammaH(Integer(12), [])) # trick question!
True
>>> is_Gamma1(GammaH(Integer(12), [Integer(5)]))
False