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# -*- coding: utf-8 -*- """ The Eisenstein Subspace """ from __future__ import absolute_import from six.moves import range
from sage.structure.all import Sequence from sage.misc.all import verbose import sage.rings.all as rings from sage.categories.all import Objects from sage.matrix.all import Matrix from sage.rings.all import CyclotomicField from sage.arith.all import lcm, euler_phi
from . import eis_series from . import element from . import submodule
class EisensteinSubmodule(submodule.ModularFormsSubmodule): """ The Eisenstein submodule of an ambient space of modular forms. """ def __init__(self, ambient_space): """ Return the Eisenstein submodule of the given space.
EXAMPLES::
sage: E = ModularForms(23,4).eisenstein_subspace() ## indirect doctest sage: E Eisenstein subspace of dimension 2 of Modular Forms space of dimension 7 for Congruence Subgroup Gamma0(23) of weight 4 over Rational Field sage: E == loads(dumps(E)) True """ already_echelonized=True)
def _repr_(self): """ Return the string representation of self.
EXAMPLES::
sage: E = ModularForms(23,4).eisenstein_subspace() ## indirect doctest sage: E._repr_() 'Eisenstein subspace of dimension 2 of Modular Forms space of dimension 7 for Congruence Subgroup Gamma0(23) of weight 4 over Rational Field' """
def eisenstein_submodule(self): """ Return the Eisenstein submodule of self. (Yes, this is just self.)
EXAMPLES::
sage: E = ModularForms(23,4).eisenstein_subspace() sage: E == E.eisenstein_submodule() True """
def modular_symbols(self, sign=0): r""" Return the corresponding space of modular symbols with given sign. This will fail in weight 1.
.. warning::
If sign != 0, then the space of modular symbols will, in general, only correspond to a *subspace* of this space of modular forms. This can be the case for both sign +1 or -1.
EXAMPLES::
sage: E = ModularForms(11,2).eisenstein_submodule() sage: M = E.modular_symbols(); M Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field sage: M.sign() 0
sage: M = E.modular_symbols(sign=-1); M Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field
sage: E = ModularForms(1,12).eisenstein_submodule() sage: E.modular_symbols() Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
sage: eps = DirichletGroup(13).0 sage: E = EisensteinForms(eps^2, 2) sage: E.modular_symbols() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 and level 13, weight 2, character [zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2
sage: E = EisensteinForms(eps, 1); E Eisenstein subspace of dimension 1 of Modular Forms space of character [zeta12] and weight 1 over Cyclotomic Field of order 12 and degree 4 sage: E.modular_symbols() Traceback (most recent call last): ... ValueError: the weight must be at least 2 """
class EisensteinSubmodule_params(EisensteinSubmodule): def parameters(self): r""" Return a list of parameters for each Eisenstein series spanning self. That is, for each such series, return a triple of the form (`\psi`, `\chi`, level), where `\psi` and `\chi` are the characters defining the Eisenstein series, and level is the smallest level at which this series occurs.
EXAMPLES::
sage: ModularForms(24,2).eisenstein_submodule().parameters() [(Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, 2), ... Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, 24)] sage: EisensteinForms(12,6).parameters()[-1] (Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1, Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1, 12)
sage: pars = ModularForms(DirichletGroup(24).0,3).eisenstein_submodule().parameters() sage: [(x[0].values_on_gens(),x[1].values_on_gens(),x[2]) for x in pars] [((1, 1, 1), (-1, 1, 1), 1), ((1, 1, 1), (-1, 1, 1), 2), ((1, 1, 1), (-1, 1, 1), 3), ((1, 1, 1), (-1, 1, 1), 6), ((-1, 1, 1), (1, 1, 1), 1), ((-1, 1, 1), (1, 1, 1), 2), ((-1, 1, 1), (1, 1, 1), 3), ((-1, 1, 1), (1, 1, 1), 6)] sage: EisensteinForms(DirichletGroup(24).0,1).parameters() [(Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 1), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 2), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 3), (Dirichlet character modulo 24 of conductor 1 mapping 7 |--> 1, 13 |--> 1, 17 |--> 1, Dirichlet character modulo 24 of conductor 4 mapping 7 |--> -1, 13 |--> 1, 17 |--> 1, 6)] """ P = eis_series.compute_eisenstein_params(self.level(), self.weight()) else:
def new_submodule(self, p=None): r""" Return the new submodule of self.
EXAMPLES::
sage: e = EisensteinForms(Gamma0(225), 2).new_submodule(); e Modular Forms subspace of dimension 3 of Modular Forms space of dimension 42 for Congruence Subgroup Gamma0(225) of weight 2 over Rational Field sage: e.basis() [ q + O(q^6), q^2 + O(q^6), q^4 + O(q^6) ] """
def _parameters_character(self): """ Return the character defining self.
EXAMPLES::
sage: EisensteinForms(DirichletGroup(33).1,5)._parameters_character() Dirichlet character modulo 33 of conductor 11 mapping 23 |--> 1, 13 |--> zeta10 """
def change_ring(self, base_ring): """ Return self as a module over base_ring.
EXAMPLES::
sage: E = EisensteinForms(12,2) ; E Eisenstein subspace of dimension 5 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(12) of weight 2 over Rational Field sage: E.basis() [ 1 + O(q^6), q + 6*q^5 + O(q^6), q^2 + O(q^6), q^3 + O(q^6), q^4 + O(q^6) ] sage: E.change_ring(GF(5)) Eisenstein subspace of dimension 5 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(12) of weight 2 over Finite Field of size 5 sage: E.change_ring(GF(5)).basis() [ 1 + O(q^6), q + q^5 + O(q^6), q^2 + O(q^6), q^3 + O(q^6), q^4 + O(q^6) ] """
def eisenstein_series(self): """ Return the Eisenstein series that span this space (over the algebraic closure).
EXAMPLES::
sage: EisensteinForms(11,2).eisenstein_series() [ 5/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6) ] sage: EisensteinForms(1,4).eisenstein_series() [ 1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6) ] sage: EisensteinForms(1,24).eisenstein_series() [ 236364091/131040 + q + 8388609*q^2 + 94143178828*q^3 + 70368752566273*q^4 + 11920928955078126*q^5 + O(q^6) ] sage: EisensteinForms(5,4).eisenstein_series() [ 1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6), 1/240 + q^5 + O(q^6) ] sage: EisensteinForms(13,2).eisenstein_series() [ 1/2 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6) ]
sage: E = EisensteinForms(Gamma1(7),2) sage: E.set_precision(4) sage: E.eisenstein_series() [ 1/4 + q + 3*q^2 + 4*q^3 + O(q^4), 1/7*zeta6 - 3/7 + q + (-2*zeta6 + 1)*q^2 + (3*zeta6 - 2)*q^3 + O(q^4), q + (-zeta6 + 2)*q^2 + (zeta6 + 2)*q^3 + O(q^4), -1/7*zeta6 - 2/7 + q + (2*zeta6 - 1)*q^2 + (-3*zeta6 + 1)*q^3 + O(q^4), q + (zeta6 + 1)*q^2 + (-zeta6 + 3)*q^3 + O(q^4) ]
sage: eps = DirichletGroup(13).0^2 sage: ModularForms(eps,2).eisenstein_series() [ -7/13*zeta6 - 11/13 + q + (2*zeta6 + 1)*q^2 + (-3*zeta6 + 1)*q^3 + (6*zeta6 - 3)*q^4 - 4*q^5 + O(q^6), q + (zeta6 + 2)*q^2 + (-zeta6 + 3)*q^3 + (3*zeta6 + 3)*q^4 + 4*q^5 + O(q^6) ]
sage: M = ModularForms(19,3).eisenstein_subspace() sage: M.eisenstein_series() [ ]
sage: M = ModularForms(DirichletGroup(13).0, 1) sage: M.eisenstein_series() [ -1/13*zeta12^3 + 6/13*zeta12^2 + 4/13*zeta12 + 2/13 + q + (zeta12 + 1)*q^2 + zeta12^2*q^3 + (zeta12^2 + zeta12 + 1)*q^4 + (-zeta12^3 + 1)*q^5 + O(q^6) ]
sage: M = ModularForms(GammaH(15, [4]), 4) sage: M.eisenstein_series() [ 1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6), 1/240 + q^3 + O(q^6), 1/240 + q^5 + O(q^6), 1/240 + O(q^6), 1 + q - 7*q^2 - 26*q^3 + 57*q^4 + q^5 + O(q^6), 1 + q^3 + O(q^6), q + 7*q^2 + 26*q^3 + 57*q^4 + 125*q^5 + O(q^6), q^3 + O(q^6) ] """ None, t, chi, psi) for \ chi, psi, t in P], immutable=True, cr = True, universe=Objects())
def new_eisenstein_series(self): r""" Return a list of the Eisenstein series in this space that are new.
EXAMPLES::
sage: E = EisensteinForms(25, 4) sage: E.new_eisenstein_series() [q + 7*zeta4*q^2 - 26*zeta4*q^3 - 57*q^4 + O(q^6), q - 9*q^2 - 28*q^3 + 73*q^4 + O(q^6), q - 7*zeta4*q^2 + 26*zeta4*q^3 - 57*q^4 + O(q^6)] """
def _compute_q_expansion_basis(self, prec=None, new=False): """ Compute a q-expansion basis for self to precision prec.
EXAMPLES::
sage: EisensteinForms(22,4)._compute_q_expansion_basis(6) [1 + O(q^6), q + 28*q^3 - 8*q^4 + 126*q^5 + O(q^6), q^2 + 9*q^4 + O(q^6), O(q^6)] sage: EisensteinForms(22,4)._compute_q_expansion_basis(15) [1 + O(q^15), q + 28*q^3 - 8*q^4 + 126*q^5 + 344*q^7 - 72*q^8 + 757*q^9 - 224*q^12 + 2198*q^13 + O(q^15), q^2 + 9*q^4 + 28*q^6 + 73*q^8 + 126*q^10 + 252*q^12 + 344*q^14 + O(q^15), q^11 + O(q^15)] """ prec = self.prec() else:
else: else: # restrict scalars from L to K
else:
def _q_expansion(self, element, prec): """ Compute a q-expansion for a given element of self, expressed as a vector of coefficients for the basis vectors of self, viewing self as a subspace of the corresponding space of modular forms.
EXAMPLES::
sage: E = EisensteinForms(17,4) sage: (11*E.0 + 3*E.1).q_expansion(20) 11 + 3*q + 27*q^2 + 84*q^3 + 219*q^4 + 378*q^5 + 756*q^6 + 1032*q^7 + 1755*q^8 + 2271*q^9 + 3402*q^10 + 3996*q^11 + 6132*q^12 + 6594*q^13 + 9288*q^14 + 10584*q^15 + 14043*q^16 + 17379*q^17 + 20439*q^18 + 20580*q^19 + O(q^20) sage: E._q_expansion([0,0,0,0,11,3],20) 11 + 3*q + 27*q^2 + 84*q^3 + 219*q^4 + 378*q^5 + 756*q^6 + 1032*q^7 + 1755*q^8 + 2271*q^9 + 3402*q^10 + 3996*q^11 + 6132*q^12 + 6594*q^13 + 9288*q^14 + 10584*q^15 + 14043*q^16 + 17379*q^17 + 20439*q^18 + 20580*q^19 + O(q^20) """
class EisensteinSubmodule_g0_Q(EisensteinSubmodule_params): r""" Space of Eisenstein forms for `\Gamma_0(N)`. """
class EisensteinSubmodule_gH_Q(EisensteinSubmodule_params): r""" Space of Eisenstein forms for `\Gamma_H(N)`. """ def _parameters_character(self): """ Return the character defining self. Since self is a space of Eisenstein forms on GammaH(N) rather than a space with fixed character, we return the group GammaH(N) itself.
EXAMPLES::
sage: EisensteinForms(GammaH(9, [4]),4)._parameters_character() Congruence Subgroup Gamma_H(9) with H generated by [4] """
def _convert_matrix_from_modsyms_eis(self, A): r""" Given a matrix acting on the space of modular symbols corresponding to this space, calculate the matrix of the operator it induces on this space itself. Used for Hecke and diamond operators.
This is a minor modification of the code used for cusp forms, which is required because modular symbols "don't see the constant term": the modular symbol method calculates the matrix of the operator with respect to the unique basis of the modular forms space for which the *non-constant* coefficients are in echelon form, and we need to modify this to get a matrix with respect to the basis we're actually using.
EXAMPLES::
sage: EisensteinForms(Gamma1(6), 3).hecke_matrix(3) # indirect doctest [ 1 0 72 0] [ 0 0 36 -9] [ 0 0 9 0] [ 0 1 -4 10] """
def _compute_hecke_matrix(self, n, bound=None): r""" Calculate the matrix of the Hecke operator `T_n` acting on this space, via modular symbols.
INPUT:
- n: a positive integer
- bound: an integer such that any element of this space with coefficients a_1, ..., a_b all zero must be the zero element. If this turns out not to be true, the code will increase the bound and try again. Setting bound = None is equivalent to setting bound = self.dimension().
OUTPUT:
- a matrix (over `\QQ`)
ALGORITHM:
This uses the usual pairing between modular symbols and modular forms, but in a slightly non-standard way. As for cusp forms, we can find a basis for this space made up of forms with q-expansions `c_m(f) = a_{i,j}(T_m)`, where `T_m` denotes the matrix of the Hecke operator on the corresponding modular symbols space. Then `c_m(T_n f) = a_{i,j}(T_n* T_m)`. But we can't find the constant terms by this method, so an extra step is required.
EXAMPLES::
sage: EisensteinForms(Gamma1(6), 3).hecke_matrix(3) # indirect doctest [ 1 0 72 0] [ 0 0 36 -9] [ 0 0 9 0] [ 0 1 -4 10] """
def _compute_diamond_matrix(self, d): r""" Calculate the matrix of the diamond bracket operator <d> on this space, using modular symbols.
EXAMPLES::
sage: E = EisensteinForms(Gamma1(7), 3) sage: E._compute_diamond_matrix(3) [ 27 126 294 770 2142 3528] [56/3 85 200 530 1445 2408] [11/3 14 22 66 233 392] [ -1 -3 -3 -11 -51 -87] [ -1 -4 -7 -20 -67 -112] [-1/3 -2 -6 -15 -34 -56] """
class EisensteinSubmodule_g1_Q(EisensteinSubmodule_gH_Q): r""" Space of Eisenstein forms for `\Gamma_1(N)`. """ def _parameters_character(self): """ Return the character defining self. Since self is a space of Eisenstein forms on `\Gamma_1(N)`, all characters modulo the level are possible, so we return the level.
EXAMPLES::
sage: EisensteinForms(Gamma1(7),4)._parameters_character() 7 """
class EisensteinSubmodule_eps(EisensteinSubmodule_params): """ Space of Eisenstein forms with given Dirichlet character.
EXAMPLES::
sage: e = DirichletGroup(27,CyclotomicField(3)).0**2 sage: M = ModularForms(e,2,prec=10).eisenstein_subspace() sage: M.dimension() 6
sage: M.eisenstein_series() [ -1/3*zeta6 - 1/3 + q + (2*zeta6 - 1)*q^2 + q^3 + (-2*zeta6 - 1)*q^4 + (-5*zeta6 + 1)*q^5 + O(q^6), -1/3*zeta6 - 1/3 + q^3 + O(q^6), q + (-2*zeta6 + 1)*q^2 + (-2*zeta6 - 1)*q^4 + (5*zeta6 - 1)*q^5 + O(q^6), q + (zeta6 + 1)*q^2 + 3*q^3 + (zeta6 + 2)*q^4 + (-zeta6 + 5)*q^5 + O(q^6), q^3 + O(q^6), q + (-zeta6 - 1)*q^2 + (zeta6 + 2)*q^4 + (zeta6 - 5)*q^5 + O(q^6) ] sage: M.eisenstein_subspace().T(2).matrix().fcp() (x + 2*zeta3 + 1) * (x + zeta3 + 2) * (x - zeta3 - 2)^2 * (x - 2*zeta3 - 1)^2 sage: ModularSymbols(e,2).eisenstein_subspace().T(2).matrix().fcp() (x + 2*zeta3 + 1) * (x + zeta3 + 2) * (x - zeta3 - 2)^2 * (x - 2*zeta3 - 1)^2
sage: M.basis() [ 1 - 3*zeta3*q^6 + (-2*zeta3 + 2)*q^9 + O(q^10), q + (5*zeta3 + 5)*q^7 + O(q^10), q^2 - 2*zeta3*q^8 + O(q^10), q^3 + (zeta3 + 2)*q^6 + 3*q^9 + O(q^10), q^4 - 2*zeta3*q^7 + O(q^10), q^5 + (zeta3 + 1)*q^8 + O(q^10) ]
""" # TODO #def _compute_q_expansion_basis(self, prec): #B = EisensteinSubmodule_params._compute_q_expansion_basis(self, prec) #raise NotImplementedError, "must restrict scalars down correctly."
def cyclotomic_restriction(L,K): r""" Given two cyclotomic fields L and K, compute the compositum M of K and L, and return a function and the index [M:K]. The function is a map that acts as follows (here `M = Q(\zeta_m)`):
INPUT:
element alpha in L
OUTPUT:
a polynomial `f(x)` in `K[x]` such that `f(\zeta_m) = \alpha`, where we view alpha as living in `M`. (Note that `\zeta_m` generates `M`, not `L`.)
EXAMPLES::
sage: L = CyclotomicField(12) ; N = CyclotomicField(33) ; M = CyclotomicField(132) sage: z, n = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction(L,N) sage: n 2
sage: z(L.0) -zeta33^19*x sage: z(L.0)(M.0) zeta132^11
sage: z(L.0^3-L.0+1) (zeta33^19 + zeta33^8)*x + 1 sage: z(L.0^3-L.0+1)(M.0) zeta132^33 - zeta132^11 + 1 sage: z(L.0^3-L.0+1)(M.0) - M(L.0^3-L.0+1) 0 """ """ Function returned by cyclotomic restriction.
INPUT:
element alpha in L
OUTPUT:
a polynomial `f(x)` in `K[x]` such that `f(\zeta_m) = \alpha`, where we view alpha as living in `M`. (Note that `\zeta_m` generates `M`, not `L`.)
EXAMPLES::
sage: L = CyclotomicField(12) sage: N = CyclotomicField(33) sage: g, n = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction(L,N) sage: g(L.0) -zeta33^19*x """ else: euler_phi(L.zeta_order())//euler_phi(K.zeta_order())
def cyclotomic_restriction_tower(L,K): """ Suppose L/K is an extension of cyclotomic fields and L=Q(zeta_m). This function computes a map with the following property:
INPUT:
an element alpha in L
OUTPUT:
a polynomial `f(x)` in `K[x]` such that `f(zeta_m) = alpha`.
EXAMPLES::
sage: L = CyclotomicField(12) ; K = CyclotomicField(6) sage: z = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction_tower(L,K) sage: z(L.0) x sage: z(L.0^2+L.0) x + zeta6 """ raise ValueError("K must be contained in L") raise ValueError("K (= Q(\zeta_%s)) is not contained in L (= Q(\zeta_%s))"%(K._n(), L._n())) """ Function returned by cyclotomic_restriction_tower.
INPUT:
an element alpha in L
OUTPUT:
a polynomial `f(x)` in `K[x]` such that `f(zeta_m) = alpha`.
EXAMPLES::
sage: L = CyclotomicField(121) ; K = CyclotomicField(11) sage: z = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction_tower(L,K) sage: z(L.0) x sage: z(L.0^11) zeta11 """
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