Elements of Hecke modular forms spaces¶
AUTHORS:
- Jonas Jermann (2013): initial version
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class
sage.modular.modform_hecketriangle.element.FormsElement(parent, rat)¶ Bases:
sage.modular.modform_hecketriangle.graded_ring_element.FormsRingElement(Hecke) modular forms.
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ambient_coordinate_vector()¶ Return the coordinate vector of
selfwith respect toself.parent().ambient_space().gens().The returned coordinate vector is an element of
self.parent().module().Note
This uses the corresponding function of the parent. If the parent has not defined a coordinate vector function or an ambient module for coordinate vectors then an exception is raised by the parent (default implementation).
EXAMPLES:
sage: from sage.modular.modform_hecketriangle.space import ModularForms sage: MF = ModularForms(n=4, k=24, ep=-1) sage: MF.gen(0).ambient_coordinate_vector().parent() Vector space of dimension 3 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring sage: MF.gen(0).ambient_coordinate_vector() (1, 0, 0) sage: subspace = MF.subspace([MF.gen(0), MF.gen(2)]) sage: subspace.gen(0).ambient_coordinate_vector().parent() Vector space of degree 3 and dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring Basis matrix: [1 0 0] [0 0 1] sage: subspace.gen(0).ambient_coordinate_vector() (1, 0, 0) sage: subspace.gen(0).ambient_coordinate_vector() == subspace.ambient_coordinate_vector(subspace.gen(0)) True
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coordinate_vector()¶ Return the coordinate vector of
selfwith respect toself.parent().gens().Note
This uses the corresponding function of the parent. If the parent has not defined a coordinate vector function or a module for coordinate vectors then an exception is raised by the parent (default implementation).
EXAMPLES:
sage: from sage.modular.modform_hecketriangle.space import ModularForms sage: MF = ModularForms(n=4, k=24, ep=-1) sage: MF.gen(0).coordinate_vector().parent() Vector space of dimension 3 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring sage: MF.gen(0).coordinate_vector() (1, 0, 0) sage: subspace = MF.subspace([MF.gen(0), MF.gen(2)]) sage: subspace.gen(0).coordinate_vector().parent() Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring sage: subspace.gen(0).coordinate_vector() (1, 0) sage: subspace.gen(0).coordinate_vector() == subspace.coordinate_vector(subspace.gen(0)) True
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lseries(num_prec=None, max_imaginary_part=0, max_asymp_coeffs=40)¶ Return the L-series of
selfifselfis modular and holomorphic.This relies on the (pari) based function
Dokchitser.INPUT:
num_prec– An integer denoting the to-be-used numerical precision.- If integer
num_prec=None(default) the default numerical precision of the parent ofselfis used.
max_imaginary_part– A real number (default: 0), indicating up to which- imaginary part the L-series is going to be studied.
max_asymp_coeffs– An integer (default: 40).
OUTPUT:
An interface to Tim Dokchitser’s program for computing L-series, namely the series given by the Fourier coefficients of
self.EXAMPLES:
sage: from sage.modular.modform.eis_series import eisenstein_series_lseries sage: from sage.modular.modform_hecketriangle.space import ModularForms sage: f = ModularForms(n=3, k=4).E4()/240 sage: L = f.lseries() sage: L L-series associated to the modular form 1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + O(q^5) sage: L.conductor 1 sage: L(1).prec() 53 sage: L.check_functional_equation() < 2^(-50) True sage: L(1) -0.0304484570583... sage: abs(L(1) - eisenstein_series_lseries(4)(1)) < 2^(-53) True sage: L.derivative(1, 1) -0.0504570844798... sage: L.derivative(1, 2)/2 -0.0350657360354... sage: L.taylor_series(1, 3) -0.0304484570583... - 0.0504570844798...*z - 0.0350657360354...*z^2 + O(z^3) sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True) sage: sum([coeffs[k] * ZZ(k)^(-10) for k in range(1,len(coeffs))]).n(53) 1.00935215408... sage: L(10) 1.00935215649... sage: f = ModularForms(n=6, k=4).E4() sage: L = f.lseries(num_prec=200) sage: L.conductor 3 sage: L.check_functional_equation() < 2^(-180) True sage: L(1) -2.92305187760575399490414692523085855811204642031749788... sage: L(1).prec() 200 sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True) sage: sum([coeffs[k] * ZZ(k)^(-10) for k in range(1,len(coeffs))]).n(53) 24.2281438789... sage: L(10).n(53) 24.2281439447... sage: f = ModularForms(n=8, k=6, ep=-1).E6() sage: L = f.lseries() sage: L.check_functional_equation() < 2^(-45) True sage: L.taylor_series(3, 3) 0.000000000000... + 0.867197036668...*z + 0.261129628199...*z^2 + O(z^3) sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True) sage: sum([coeffs[k]*k^(-10) for k in range(1,len(coeffs))]).n(53) -13.0290002560... sage: L(10).n(53) -13.0290184579... sage: f = (ModularForms(n=17, k=24).Delta()^2) # long time sage: L = f.lseries() # long time sage: L.check_functional_equation() < 2^(-50) # long time True sage: L.taylor_series(12, 3) # long time 0.000683924755280... - 0.000875942285963...*z + 0.000647618966023...*z^2 + O(z^3) sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True) # long time sage: sum([coeffs[k]*k^(-30) for k in range(1,len(coeffs))]).n(53) # long time 9.31562890589...e-10 sage: L(30).n(53) # long time 9.31562890589...e-10 sage: f = ModularForms(n=infinity, k=2, ep=-1).f_i() sage: L = f.lseries() sage: L.check_functional_equation() < 2^(-50) True sage: L.taylor_series(1, 3) 0.000000000000... + 5.76543616701...*z + 9.92776715593...*z^2 + O(z^3) sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True) sage: sum([coeffs[k] * ZZ(k)^(-10) for k in range(1,len(coeffs))]).n(53) -23.9781792831... sage: L(10).n(53) -23.9781792831...
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