Congruence subgroup \(\Gamma_H(N)\)#
AUTHORS:
Jordi Quer
David Loeffler
- class sage.modular.arithgroup.congroup_gammaH.GammaH_class(level, H, Hlist=None)#
Bases:
CongruenceSubgroup
The congruence subgroup \(\Gamma_H(N)\) for some subgroup \(H \trianglelefteq (\ZZ / N\ZZ)^\times\), which is the subgroup of \(\SL_2(\ZZ)\) consisting of matrices of the form \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) with \(N \mid c\) and \(a, b \in H\).
- atkin_lehner_matrix(Q)#
Return the matrix of the Atkin–Lehner–Li operator \(W_Q\) associated to an exact divisor \(Q\) of \(N\), where \(N\) is the level of this group; that is, \(gcd(Q, N/Q) = 1\).
Note
We follow the conventions of [AL1978] here, so \(W_Q\) is given by the action of any matrix of the form \(\begin{pmatrix} Qx & y \\ Nz & Qw \end{pmatrix}\) where \(x,y,z,w\) are integers such that \(y = 1 \bmod Q\), \(x = 1 \bmod N/Q\), and \(det(W_Q) = Q\). For convenience, we actually always choose \(x = y = 1\).
INPUT:
Q
(integer): an integer dividing \(N\), where \(N\) is the level of this group. If this divisor does not satisfy \(gcd(Q, N/Q) = 1\), it will be replaced by the unique integer with this property having the same prime factors as \(Q\).
EXAMPLES:
sage: Gamma1(994).atkin_lehner_matrix(71) [ 71 1] [4970 71] sage: Gamma1(996).atkin_lehner_matrix(2) [ 4 1] [-996 -248] sage: Gamma1(15).atkin_lehner_matrix(7) Traceback (most recent call last): ... ValueError: Q must divide the level
- characters_mod_H(sign=None, galois_orbits=False)#
Return the characters of \((\ZZ / N\ZZ)^*\), of the specified sign, which are trivial on H.
INPUT:
sign
(default: None): if not None, return only characters of the given signgalois_orbits
(default: False): if True, return only one character from each Galois orbit.
EXAMPLES:
sage: GammaH(5, [-1]).characters_mod_H() [Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1, Dirichlet character modulo 5 of conductor 1 mapping 2 |--> 1] sage: Gamma1(31).characters_mod_H(galois_orbits=True,sign=-1) [Dirichlet character modulo 31 of conductor 31 mapping 3 |--> zeta30, Dirichlet character modulo 31 of conductor 31 mapping 3 |--> zeta30^3, Dirichlet character modulo 31 of conductor 31 mapping 3 |--> zeta30^5, Dirichlet character modulo 31 of conductor 31 mapping 3 |--> -1] sage: GammaH(31, [-1]).characters_mod_H(sign=-1) []
- coset_reps()#
Return a set of coset representatives for self \ SL2Z.
EXAMPLES:
sage: list(Gamma1(3).coset_reps()) [ [1 0] [-1 0] [ 0 -1] [ 0 1] [1 0] [-1 0] [ 0 -1] [ 0 1] [0 1], [ 0 -1], [ 1 0], [-1 0], [1 1], [-1 -1], [ 1 2], [-1 -2] ] sage: len(list(Gamma1(31).coset_reps())) == 31**2 - 1 True
- dimension_cusp_forms(k=2)#
Return the dimension of the space of weight k cusp forms for this group. For \(k \ge 2\), this is given by a standard formula in terms of k and various invariants of the group; see Diamond + Shurman, “A First Course in Modular Forms”, section 3.5 and 3.6. If k is not given, default to k = 2.
For dimensions of spaces of cusp forms with character for Gamma1, use the dimension_cusp_forms method of the Gamma1 class, or the standalone function dimension_cusp_forms().
For weight 1 cusp forms, there is no simple formula for the dimensions, so we first try to rule out nonzero cusp forms existing via Riemann-Roch, and if this fails, we trigger computation of the cusp form space using Schaeffer’s algorithm; this can be quite expensive in large levels.
EXAMPLES:
sage: GammaH(31, [23]).dimension_cusp_forms(10) 69 sage: GammaH(31, [7]).dimension_cusp_forms(1) 1
- dimension_new_cusp_forms(k=2, p=0)#
Return the dimension of the space of new (or \(p\)-new) weight \(k\) cusp forms for this congruence subgroup.
INPUT:
k
- an integer (default: 2), the weight. Not fully implemented for k = 1.p
- integer (default: 0); if nonzero, compute the \(p\)-new subspace.
OUTPUT: Integer
EXAMPLES:
sage: GammaH(33,[2]).dimension_new_cusp_forms() 3 sage: Gamma1(4*25).dimension_new_cusp_forms(2, p=5) 225 sage: Gamma1(33).dimension_new_cusp_forms(2) 19 sage: Gamma1(33).dimension_new_cusp_forms(2,p=11) 21
- divisor_subgroups()#
Given this congruence subgroup \(\Gamma_H(N)\), return all subgroups \(\Gamma_G(M)\) for \(M\) a divisor of \(N\) and such that \(G\) is equal to the image of \(H\) modulo \(M\).
EXAMPLES:
sage: G = GammaH(33,[2]); G Congruence Subgroup Gamma_H(33) with H generated by [2] sage: G._list_of_elements_in_H() [1, 2, 4, 8, 16, 17, 25, 29, 31, 32] sage: G.divisor_subgroups() [Modular Group SL(2,Z), Congruence Subgroup Gamma0(3), Congruence Subgroup Gamma0(11), Congruence Subgroup Gamma_H(33) with H generated by [2]]
- extend(M)#
Return the subgroup of \(\Gamma_0(M)\), for \(M\) a multiple of \(N\), obtained by taking the preimage of this group under the reduction map; in other words, the intersection of this group with \(\Gamma_0(M)\).
EXAMPLES:
sage: G = GammaH(33, [2]) sage: G.extend(99) Congruence Subgroup Gamma_H(99) with H generated by [2, 17, 68] sage: G.extend(11) Traceback (most recent call last): ... ValueError: M (=11) must be a multiple of the level (33) of self
- gamma0_coset_reps()#
Return a set of coset representatives for self \ Gamma0(N), where N is the level of self.
EXAMPLES:
sage: GammaH(108, [1,-1]).gamma0_coset_reps() [ [1 0] [-43 -2] [ 31 2] [-49 -5] [ 25 3] [-19 -3] [0 1], [108 5], [108 7], [108 11], [108 13], [108 17], [-17 -3] [ 47 10] [ 13 3] [ 41 11] [ 7 2] [-37 -12] [108 19], [108 23], [108 25], [108 29], [108 31], [108 35], [-35 -12] [ 29 11] [ -5 -2] [ 23 10] [-11 -5] [ 53 26] [108 37], [108 41], [108 43], [108 47], [108 49], [108 53] ]
- generators(algorithm='farey')#
Return generators for this congruence subgroup. The result is cached.
INPUT:
algorithm
(string): eitherfarey
(default) ortodd-coxeter
.
If
algorithm
is set to"farey"
, then the generators will be calculated using Farey symbols, which will always return a minimal generating set. Seefarey_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: GammaH(7, [2]).generators() [ [1 1] [ 2 -1] [ 4 -3] [0 1], [ 7 -3], [ 7 -5] ] sage: GammaH(7, [2]).generators(algorithm="todd-coxeter") [ [1 1] [-13 4] [ 15 4] [-3 -1] [ 1 -1] [1 0] [1 1] [-3 -1] [0 1], [ 42 -13], [-49 -13], [ 7 2], [ 0 1], [7 1], [0 1], [ 7 2], [-13 4] [-5 -1] [-5 -2] [-10 3] [ 1 0] [ 2 -1] [1 0] [ 42 -13], [21 4], [28 11], [ 63 -19], [-7 1], [ 7 -3], [7 1], [-3 -1] [ 15 -4] [ 2 -1] [-5 1] [ 8 -3] [11 5] [-13 -4] [ 7 2], [ 49 -13], [ 7 -3], [14 -3], [-21 8], [35 16], [-42 -13] ]
- image_mod_n()#
Return the image of this group in \(SL(2, \ZZ / N\ZZ)\).
EXAMPLES:
sage: Gamma0(3).image_mod_n() Matrix group over Ring of integers modulo 3 with 2 generators ( [2 0] [1 1] [0 2], [0 1] )
- index()#
Return the index of
self
in SL2Z.EXAMPLES:
sage: [G.index() for G in Gamma0(40).gamma_h_subgroups()] [72, 144, 144, 144, 144, 288, 288, 288, 288, 144, 288, 288, 576, 576, 144, 288, 288, 576, 576, 144, 288, 288, 576, 576, 288, 576, 1152]
- is_even()#
Return True precisely if this subgroup contains the matrix -1.
EXAMPLES:
sage: GammaH(10, [3]).is_even() True sage: GammaH(14, [1]).is_even() False
- is_subgroup(other)#
Return True if self is a subgroup of right, and False otherwise.
EXAMPLES:
sage: GammaH(24,[7]).is_subgroup(SL2Z) True sage: GammaH(24,[7]).is_subgroup(Gamma0(8)) True sage: GammaH(24, []).is_subgroup(GammaH(24, [7])) True sage: GammaH(24, []).is_subgroup(Gamma1(24)) True sage: GammaH(24, [17]).is_subgroup(GammaH(24, [7])) False sage: GammaH(1371, [169]).is_subgroup(GammaH(457, [169])) True
- ncusps()#
Return the number of orbits of cusps (regular or otherwise) for this subgroup.
EXAMPLES:
sage: GammaH(33,[2]).ncusps() 8 sage: GammaH(32079, [21676]).ncusps() 28800
AUTHORS:
Jordi Quer
- nirregcusps()#
Return the number of irregular cusps for this subgroup.
EXAMPLES:
sage: GammaH(3212, [2045, 2773]).nirregcusps() 720
- nregcusps()#
Return the number of orbits of regular cusps for this subgroup. A cusp is regular if we may find a parabolic element generating the stabiliser of that cusp whose eigenvalues are both +1 rather than -1. If G contains -1, all cusps are regular.
EXAMPLES:
sage: GammaH(20, [17]).nregcusps() 4 sage: GammaH(20, [17]).nirregcusps() 2 sage: GammaH(3212, [2045, 2773]).nregcusps() 1440 sage: GammaH(3212, [2045, 2773]).nirregcusps() 720
AUTHOR:
Jordi Quer
- nu2()#
Return the number of orbits of elliptic points of order 2 for this group.
EXAMPLES:
sage: [H.nu2() for n in [1..10] for H in Gamma0(n).gamma_h_subgroups()] [1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0] sage: GammaH(33,[2]).nu2() 0 sage: GammaH(5,[2]).nu2() 2
AUTHORS:
Jordi Quer
- nu3()#
Return the number of orbits of elliptic points of order 3 for this group.
EXAMPLES:
sage: [H.nu3() for n in [1..10] for H in Gamma0(n).gamma_h_subgroups()] [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] sage: GammaH(33,[2]).nu3() 0 sage: GammaH(7,[2]).nu3() 2
AUTHORS:
Jordi Quer
- reduce_cusp(c)#
Compute a minimal representative for the given cusp c. Returns a cusp c’ which is equivalent to the given cusp, and is in lowest terms with minimal positive denominator, and minimal positive numerator for that denominator.
Two cusps \(u_1/v_1\) and \(u_2/v_2\) are equivalent modulo \(\Gamma_H(N)\) if and only if
\[v_1 = h v_2 \bmod N\quad \text{and}\quad u_1 = h^{-1} u_2 \bmod {\rm gcd}(v_1,N)\]or
\[v_1 = -h v_2 \bmod N\quad \text{and}\quad u_1 = -h^{-1} u_2 \bmod {\rm gcd}(v_1,N)\]for some \(h \in H\).
EXAMPLES:
sage: GammaH(6,[5]).reduce_cusp(5/3) 1/3 sage: GammaH(12,[5]).reduce_cusp(Cusp(8,9)) 1/3 sage: GammaH(12,[5]).reduce_cusp(5/12) Infinity sage: GammaH(12,[]).reduce_cusp(Cusp(5,12)) 5/12 sage: GammaH(21,[5]).reduce_cusp(Cusp(-9/14)) 1/7 sage: Gamma1(5).reduce_cusp(oo) Infinity sage: Gamma1(5).reduce_cusp(0) 0
- restrict(M)#
Return the subgroup of \(\Gamma_0(M)\), for \(M\) a divisor of \(N\), obtained by taking the image of this group under reduction modulo \(N\).
EXAMPLES:
sage: G = GammaH(33,[2]) sage: G.restrict(11) Congruence Subgroup Gamma0(11) sage: G.restrict(1) Modular Group SL(2,Z) sage: G.restrict(15) Traceback (most recent call last): ... ValueError: M (=15) must be a divisor of the level (33) of self
- to_even_subgroup()#
Return the smallest even subgroup of \(SL(2, \ZZ)\) containing self.
EXAMPLES:
sage: GammaH(11, [4]).to_even_subgroup() Congruence Subgroup Gamma0(11) sage: Gamma1(11).to_even_subgroup() Congruence Subgroup Gamma_H(11) with H generated by [10]
- sage.modular.arithgroup.congroup_gammaH.GammaH_constructor(level, H)#
Return the congruence subgroup \(\Gamma_H(N)\), which is the subgroup of \(SL_2(\ZZ)\) consisting of matrices of the form \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) with \(N | c\) and \(a, b \in H\), for \(H\) a specified subgroup of \((\ZZ/N\ZZ)^\times\).
INPUT:
level – an integer
- H – either 0, 1, or a list
If H is a list, return \(\Gamma_H(N)\), where \(H\) is the subgroup of \((\ZZ/N\ZZ)^*\) generated by the elements of the list.
If H = 0, returns \(\Gamma_0(N)\).
If H = 1, returns \(\Gamma_1(N)\).
EXAMPLES:
sage: GammaH(11,0) # indirect doctest Congruence Subgroup Gamma0(11) sage: GammaH(11,1) Congruence Subgroup Gamma1(11) sage: GammaH(11,[10]) Congruence Subgroup Gamma_H(11) with H generated by [10] sage: GammaH(11,[10,1]) Congruence Subgroup Gamma_H(11) with H generated by [10] sage: GammaH(14,[10]) Traceback (most recent call last): ... ArithmeticError: The generators [10] must be units modulo 14
- sage.modular.arithgroup.congroup_gammaH.is_GammaH(x)#
Return True if x is a congruence subgroup of type GammaH.
EXAMPLES:
sage: from sage.modular.arithgroup.all import is_GammaH sage: is_GammaH(GammaH(13, [2])) True sage: is_GammaH(Gamma0(6)) True sage: is_GammaH(Gamma1(6)) True sage: is_GammaH(sage.modular.arithgroup.congroup_generic.CongruenceSubgroup(5)) False
- sage.modular.arithgroup.congroup_gammaH.mumu(N)#
Return 0 if any cube divides \(N\). Otherwise return \((-2)^v\) where \(v\) is the number of primes that exactly divide \(N\).
This is similar to the Möbius function.
INPUT:
N
– an integer at least 1
OUTPUT: Integer
EXAMPLES:
sage: from sage.modular.arithgroup.congroup_gammaH import mumu sage: mumu(27) 0 sage: mumu(6*25) 4 sage: mumu(7*9*25) -2 sage: mumu(9*25) 1