Congruence subgroup \(\Gamma_0(N)\)

class sage.modular.arithgroup.congroup_gamma0.Gamma0_class(level)[source]

Bases: GammaH_class

The congruence subgroup \(\Gamma_0(N)\).

coset_reps()[source]

Return representatives for the right cosets of this congruence subgroup in \(\SL_2(\ZZ)\) as a generator object.

Use list(self.coset_reps()) to obtain coset reps as a list.

EXAMPLES:

sage: list(Gamma0(5).coset_reps())
[
[1 0]  [ 0 -1]  [1 0]  [ 0 -1]  [ 0 -1]  [ 0 -1]
[0 1], [ 1  0], [1 1], [ 1  2], [ 1  3], [ 1  4]
]
sage: list(Gamma0(4).coset_reps())
[
[1 0]  [ 0 -1]  [1 0]  [ 0 -1]  [ 0 -1]  [1 0]
[0 1], [ 1  0], [1 1], [ 1  2], [ 1  3], [2 1]
]
sage: list(Gamma0(1).coset_reps())
[
[1 0]
[0 1]
]

>>> from sage.all import *
>>> list(Gamma0(Integer(5)).coset_reps())
[
[1 0]  [ 0 -1]  [1 0]  [ 0 -1]  [ 0 -1]  [ 0 -1]
[0 1], [ 1  0], [1 1], [ 1  2], [ 1  3], [ 1  4]
]
>>> list(Gamma0(Integer(4)).coset_reps())
[
[1 0]  [ 0 -1]  [1 0]  [ 0 -1]  [ 0 -1]  [1 0]
[0 1], [ 1  0], [1 1], [ 1  2], [ 1  3], [2 1]
]
>>> list(Gamma0(Integer(1)).coset_reps())
[
[1 0]
[0 1]
]
dimension_new_cusp_forms(k=2, p=0)[source]

Return the dimension of the space of new (or \(p\)-new) weight \(k\) cusp forms for this congruence subgroup.

INPUT:

  • k – integer (default: 2); the weight. Not fully implemented for \(k = 1\).

  • p – integer (default: 0); if nonzero, compute the \(p\)-new subspace

OUTPUT: integer

ALGORITHM:

This comes from the formula given in Theorem 1 of http://www.math.ubc.ca/~gerg/papers/downloads/DSCFN.pdf

EXAMPLES:

sage: Gamma0(11000).dimension_new_cusp_forms()
240
sage: Gamma0(11000).dimension_new_cusp_forms(k=1)
0
sage: Gamma0(22).dimension_new_cusp_forms(k=4)
3
sage: Gamma0(389).dimension_new_cusp_forms(k=2,p=17)
32

>>> from sage.all import *
>>> Gamma0(Integer(11000)).dimension_new_cusp_forms()
240
>>> Gamma0(Integer(11000)).dimension_new_cusp_forms(k=Integer(1))
0
>>> Gamma0(Integer(22)).dimension_new_cusp_forms(k=Integer(4))
3
>>> Gamma0(Integer(389)).dimension_new_cusp_forms(k=Integer(2),p=Integer(17))
32
divisor_subgroups()[source]

Return the subgroups of SL2Z of the form Gamma0(M) that contain this subgroup, i.e. those for M a divisor of N.

EXAMPLES:

sage: Gamma0(24).divisor_subgroups()
[Modular Group SL(2,Z),
 Congruence Subgroup Gamma0(2),
 Congruence Subgroup Gamma0(3),
 Congruence Subgroup Gamma0(4),
 Congruence Subgroup Gamma0(6),
 Congruence Subgroup Gamma0(8),
 Congruence Subgroup Gamma0(12),
 Congruence Subgroup Gamma0(24)]

>>> from sage.all import *
>>> Gamma0(Integer(24)).divisor_subgroups()
[Modular Group SL(2,Z),
 Congruence Subgroup Gamma0(2),
 Congruence Subgroup Gamma0(3),
 Congruence Subgroup Gamma0(4),
 Congruence Subgroup Gamma0(6),
 Congruence Subgroup Gamma0(8),
 Congruence Subgroup Gamma0(12),
 Congruence Subgroup Gamma0(24)]
gamma_h_subgroups()[source]

Return the subgroups of the form \(\Gamma_H(N)\) contained in self, where \(N\) is the level of self.

EXAMPLES:

sage: G = Gamma0(11)
sage: G.gamma_h_subgroups()  # optional - gap_package_polycyclic
[Congruence Subgroup Gamma0(11),
 Congruence Subgroup Gamma_H(11) with H generated by [3],
 Congruence Subgroup Gamma_H(11) with H generated by [10],
 Congruence Subgroup Gamma1(11)]
sage: G = Gamma0(12)
sage: G.gamma_h_subgroups()  # optional - gap_package_polycyclic
[Congruence Subgroup Gamma0(12),
 Congruence Subgroup Gamma_H(12) with H generated by [7],
 Congruence Subgroup Gamma_H(12) with H generated by [11],
 Congruence Subgroup Gamma_H(12) with H generated by [5],
 Congruence Subgroup Gamma1(12)]

>>> from sage.all import *
>>> G = Gamma0(Integer(11))
>>> G.gamma_h_subgroups()  # optional - gap_package_polycyclic
[Congruence Subgroup Gamma0(11),
 Congruence Subgroup Gamma_H(11) with H generated by [3],
 Congruence Subgroup Gamma_H(11) with H generated by [10],
 Congruence Subgroup Gamma1(11)]
>>> G = Gamma0(Integer(12))
>>> G.gamma_h_subgroups()  # optional - gap_package_polycyclic
[Congruence Subgroup Gamma0(12),
 Congruence Subgroup Gamma_H(12) with H generated by [7],
 Congruence Subgroup Gamma_H(12) with H generated by [11],
 Congruence Subgroup Gamma_H(12) with H generated by [5],
 Congruence Subgroup Gamma1(12)]
generators(algorithm='farey')[source]

Return generators for this congruence subgroup.

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: Gamma0(3).generators()
[
[1 1]  [-1  1]
[0 1], [-3  2]
]
sage: Gamma0(3).generators(algorithm='todd-coxeter')
[
[1 1]  [-1  0]  [ 1 -1]  [1 0]  [1 1]  [-1  0]  [ 1  0]
[0 1], [ 0 -1], [ 0  1], [3 1], [0 1], [ 3 -1], [-3  1]
]
sage: SL2Z.gens()
(
[ 0 -1]  [1 1]
[ 1  0], [0 1]
)

>>> from sage.all import *
>>> Gamma0(Integer(3)).generators()
[
[1 1]  [-1  1]
[0 1], [-3  2]
]
>>> Gamma0(Integer(3)).generators(algorithm='todd-coxeter')
[
[1 1]  [-1  0]  [ 1 -1]  [1 0]  [1 1]  [-1  0]  [ 1  0]
[0 1], [ 0 -1], [ 0  1], [3 1], [0 1], [ 3 -1], [-3  1]
]
>>> SL2Z.gens()
(
[ 0 -1]  [1 1]
[ 1  0], [0 1]
)
index()[source]

Return the index of self in the full modular group.

This is given by

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

EXAMPLES:

sage: [Gamma0(n).index() for n in [1..19]]
[1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20]
sage: Gamma0(32041).index()
32220

>>> from sage.all import *
>>> [Gamma0(n).index() for n in (ellipsis_range(Integer(1),Ellipsis,Integer(19)))]
[1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20]
>>> Gamma0(Integer(32041)).index()
32220
is_even()[source]

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

Since \(\Gamma0(N)\) always contains the matrix -1, this always returns True.

EXAMPLES:

sage: Gamma0(12).is_even()
True
sage: SL2Z.is_even()
True

>>> from sage.all import *
>>> Gamma0(Integer(12)).is_even()
True
>>> SL2Z.is_even()
True
is_subgroup(right)[source]

Return True if self is a subgroup of right.

EXAMPLES:

sage: G = Gamma0(20)
sage: G.is_subgroup(SL2Z)
True
sage: G.is_subgroup(Gamma0(4))
True
sage: G.is_subgroup(Gamma0(20))
True
sage: G.is_subgroup(Gamma0(7))
False
sage: G.is_subgroup(Gamma1(20))
False
sage: G.is_subgroup(GammaH(40, []))
False
sage: Gamma0(80).is_subgroup(GammaH(40, [31, 21, 17]))
True
sage: Gamma0(2).is_subgroup(Gamma1(2))
True

>>> from sage.all import *
>>> G = Gamma0(Integer(20))
>>> G.is_subgroup(SL2Z)
True
>>> G.is_subgroup(Gamma0(Integer(4)))
True
>>> G.is_subgroup(Gamma0(Integer(20)))
True
>>> G.is_subgroup(Gamma0(Integer(7)))
False
>>> G.is_subgroup(Gamma1(Integer(20)))
False
>>> G.is_subgroup(GammaH(Integer(40), []))
False
>>> Gamma0(Integer(80)).is_subgroup(GammaH(Integer(40), [Integer(31), Integer(21), Integer(17)]))
True
>>> Gamma0(Integer(2)).is_subgroup(Gamma1(Integer(2)))
True
ncusps()[source]

Return the number of cusps of this subgroup \(\Gamma_0(N)\).

EXAMPLES:

sage: [Gamma0(n).ncusps() for n in [1..19]]
[1, 2, 2, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 4, 6, 2, 8, 2]
sage: [Gamma0(n).ncusps() for n in prime_range(2,100)]
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

>>> from sage.all import *
>>> [Gamma0(n).ncusps() for n in (ellipsis_range(Integer(1),Ellipsis,Integer(19)))]
[1, 2, 2, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 4, 6, 2, 8, 2]
>>> [Gamma0(n).ncusps() for n in prime_range(Integer(2),Integer(100))]
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
nu2()[source]

Return the number of elliptic points of order 2 for this congruence subgroup \(\Gamma_0(N)\).

The number of these is given by a standard formula: 0 if \(N\) is divisible by 4 or any prime congruent to -1 mod 4, and otherwise \(2^d\) where d is the number of odd primes dividing \(N\).

EXAMPLES:

sage: Gamma0(2).nu2()
1
sage: Gamma0(4).nu2()
0
sage: Gamma0(21).nu2()
0
sage: Gamma0(1105).nu2()
8
sage: [Gamma0(n).nu2() for n in [1..19]]
[1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0]

>>> from sage.all import *
>>> Gamma0(Integer(2)).nu2()
1
>>> Gamma0(Integer(4)).nu2()
0
>>> Gamma0(Integer(21)).nu2()
0
>>> Gamma0(Integer(1105)).nu2()
8
>>> [Gamma0(n).nu2() for n in (ellipsis_range(Integer(1),Ellipsis,Integer(19)))]
[1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0]
nu3()[source]

Return the number of elliptic points of order 3 for this congruence subgroup \(\Gamma_0(N)\). The number of these is given by a standard formula: 0 if \(N\) is divisible by 9 or any prime congruent to -1 mod 3, and otherwise \(2^d\) where d is the number of primes other than 3 dividing \(N\).

EXAMPLES:

sage: Gamma0(2).nu3()
0
sage: Gamma0(3).nu3()
1
sage: Gamma0(9).nu3()
0
sage: Gamma0(7).nu3()
2
sage: Gamma0(21).nu3()
2
sage: Gamma0(1729).nu3()
8

>>> from sage.all import *
>>> Gamma0(Integer(2)).nu3()
0
>>> Gamma0(Integer(3)).nu3()
1
>>> Gamma0(Integer(9)).nu3()
0
>>> Gamma0(Integer(7)).nu3()
2
>>> Gamma0(Integer(21)).nu3()
2
>>> Gamma0(Integer(1729)).nu3()
8
sage.modular.arithgroup.congroup_gamma0.Gamma0_constructor(N)[source]

Return the congruence subgroup Gamma0(N).

EXAMPLES:

sage: G = Gamma0(51) ; G # indirect doctest
Congruence Subgroup Gamma0(51)
sage: G == Gamma0(51)
True
sage: G is Gamma0(51)
True

>>> from sage.all import *
>>> G = Gamma0(Integer(51)) ; G # indirect doctest
Congruence Subgroup Gamma0(51)
>>> G == Gamma0(Integer(51))
True
>>> G is Gamma0(Integer(51))
True
sage.modular.arithgroup.congroup_gamma0.is_Gamma0(x)[source]

Return True if x is a congruence subgroup of type Gamma0.

EXAMPLES:

sage: from sage.modular.arithgroup.all import is_Gamma0
sage: is_Gamma0(SL2Z)
doctest:warning...
DeprecationWarning: The function is_Gamma0 is deprecated; use 'isinstance(..., Gamma0_class)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
True
sage: is_Gamma0(Gamma0(13))
True
sage: is_Gamma0(Gamma1(6))
False

>>> from sage.all import *
>>> from sage.modular.arithgroup.all import is_Gamma0
>>> is_Gamma0(SL2Z)
doctest:warning...
DeprecationWarning: The function is_Gamma0 is deprecated; use 'isinstance(..., Gamma0_class)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
True
>>> is_Gamma0(Gamma0(Integer(13)))
True
>>> is_Gamma0(Gamma1(Integer(6)))
False