Congruence Subgroup $$\Gamma_1(N)$$¶

class sage.modular.arithgroup.congroup_gamma1.Gamma1_class(level)

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

TESTS:

sage: [Gamma1(n).genus() for n in prime_range(2,100)]
[0, 0, 0, 0, 1, 2, 5, 7, 12, 22, 26, 40, 51, 57, 70, 92, 117, 126, 155, 176, 187, 222, 247, 287, 345]
sage: [Gamma1(n).index() for n in [1..10]]
[1, 3, 8, 12, 24, 24, 48, 48, 72, 72]

sage: [Gamma1(n).dimension_cusp_forms() for n in [1..20]]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 5, 2, 7, 3]
sage: [Gamma1(n).dimension_cusp_forms(1) for n in [1..20]]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
sage: [Gamma1(4).dimension_cusp_forms(k) for k in [1..20]]
[0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8]
sage: Gamma1(23).dimension_cusp_forms(1)
Traceback (most recent call last):
...
NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general

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

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 - an 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 Moebius inversion using the subgroups GammaH (a method due to Jordi Quer).

EXAMPLES:

We compute the same dimension in two different ways

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


Via Cohen–Oesterle:

sage: Gamma1(7*43).dimension_cusp_forms(2, eps)
28


Via Quer’s method:

sage: Gamma1(7*43).dimension_cusp_forms(2, eps, algorithm="Quer")
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]

dimension_eis(k=2, eps=None, algorithm='CohenOesterle')

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 - an 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 Moebius 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]


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]

dimension_modular_forms(k=2, eps=None, algorithm='CohenOesterle')

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 - an 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 Moebius inversion using the subgroups GammaH (a method due to Jordi Quer).

EXAMPLES:

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

dimension_new_cusp_forms(k=2, eps=None, p=0, algorithm='CohenOesterle')

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

INPUT:

• k - an 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 Moebius 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]


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]


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]

generators(algorithm='farey')

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.

EXAMPLE:

sage: Gamma1(3).generators()
[
[1 1]  [ 1 -1]
[0 1], [ 3 -2]
]
sage: Gamma1(3).generators(algorithm="todd-coxeter")
[
[1 1]  [-20   9]  [ 4  1]  [-5 -2]  [ 1 -1]  [1 0]  [1 1]  [-5  2]
[0 1], [ 51 -23], [-9 -2], [ 3  1], [ 0  1], [3 1], [0 1], [12 -5],

[ 1  0]  [ 4 -1]  [ -5   3]  [ 1 -1]  [ 7 -3]  [ 4 -1]  [ -5   3]
[-3  1], [ 9 -2], [-12   7], [ 3 -2], [12 -5], [ 9 -2], [-12   7]
]

index()

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

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

EXAMPLE:

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]

is_even()

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

is_subgroup(right)

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

ncusps()

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]

nu2()

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.

EXAMPLE:

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]

nu3()

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.

EXAMPLE:

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]

sage.modular.arithgroup.congroup_gamma1.Gamma1_constructor(N)

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()
True

sage.modular.arithgroup.congroup_gamma1.is_Gamma1(x)

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)
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


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