Coverage for local/lib/python2.7/site-packages/sage/tests/book_stein_modform.py : 100%
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""" This file contains a bunch of tests extracted from the published book 'Modular Forms: a Computational Approach' by William Stein, AMS 2007.
sage: G = SL(2,ZZ); G Special Linear Group of degree 2 over Integer Ring sage: S, T = G.gens() sage: S [ 0 1] [-1 0] sage: T [1 1] [0 1] sage: delta_qexp(6) q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6) sage: bernoulli(12) -691/2730 sage: bernoulli(50) 495057205241079648212477525/66 sage: len(str(bernoulli(10000))) 27706 sage: E4 = eisenstein_series_qexp(4, 3) sage: E6 = eisenstein_series_qexp(6, 3) sage: E4^6 1/191102976000000 + 1/132710400000*q + 203/44236800000*q^2 + O(q^3) sage: E4^3*E6^2 1/3511517184000 - 1/12192768000*q - 377/4064256000*q^2 + O(q^3) sage: E6^4 1/64524128256 - 1/32006016*q + 241/10668672*q^2 + O(q^3) sage: victor_miller_basis(28,5) [ 1 + 15590400*q^3 + 36957286800*q^4 + O(q^5), q + 151740*q^3 + 61032448*q^4 + O(q^5), q^2 + 192*q^3 - 8280*q^4 + O(q^5) ] sage: R.<q> = QQ[['q']] sage: F4 = 240 * eisenstein_series_qexp(4,3) sage: F6 = -504 * eisenstein_series_qexp(6,3) sage: F4^3 1 + 720*q + 179280*q^2 + O(q^3) sage: Delta = (F4^3 - F6^2)/1728; Delta q - 24*q^2 + O(q^3) sage: F4^3 - 720*Delta 1 + 196560*q^2 + O(q^3) sage: M = ModularForms(1,36, prec=6).echelon_form() sage: M.basis() [ 1 + 6218175600*q^4 + 15281788354560*q^5 + O(q^6), q + 57093088*q^4 + 37927345230*q^5 + O(q^6), q^2 + 194184*q^4 + 7442432*q^5 + O(q^6), q^3 - 72*q^4 + 2484*q^5 + O(q^6) ] sage: T2 = M.hecke_matrix(2); T2 [ 34359738369 0 6218175600 9026867482214400] [ 0 0 34416831456 5681332472832] [ 0 1 194184 -197264484] [ 0 0 -72 -54528] sage: T2.charpoly().factor() (x - 34359738369) * (x^3 - 139656*x^2 - 59208339456*x - 1467625047588864) sage: bernoulli_mod_p(23) [1, 4, 13, 17, 13, 6, 10, 5, 10, 9, 15] sage: set_modsym_print_mode ('modular') sage: M = ModularSymbols(11, 2) sage: M.basis() ({Infinity, 0}, {-1/8, 0}, {-1/9, 0}) sage: S = M.cuspidal_submodule() sage: S.integral_basis() # basis over ZZ. ({-1/8, 0}, {-1/9, 0}) sage: set_modsym_print_mode ('manin') # set it back sage: continued_fraction(4/7).convergents() [0, 1, 1/2, 4/7] sage: M = ModularSymbols(2,2) sage: M Modular Symbols space of dimension 1 for Gamma_0(2) of weight 2 with sign 0 over Rational Field sage: M.manin_generators() [(0,1), (1,0), (1,1)]
sage: M = ModularSymbols(3,2) sage: M.manin_generators() [(0,1), (1,0), (1,1), (1,2)]
sage: M = ModularSymbols(6,2) sage: M.manin_generators() [(0,1), (1,0), (1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,3), (2,5), (3,1), (3,2)] sage: M = ModularSymbols(2,2) sage: [x.lift_to_sl2z(2) for x in M.manin_generators()] [[1, 0, 0, 1], [0, -1, 1, 0], [1, 0, 1, 1]] sage: M = ModularSymbols(6,2) sage: x = M.manin_generators()[9] sage: x (2,5) sage: x.lift_to_sl2z(6) [1, 2, 2, 5] sage: M = ModularSymbols(2,2) sage: M.manin_basis() [1] sage: [M.manin_generators()[i] for i in M.manin_basis()] [(1,0)] sage: M = ModularSymbols(6,2) sage: M.manin_basis() [1, 10, 11] sage: [M.manin_generators()[i] for i in M.manin_basis()] [(1,0), (3,1), (3,2)] sage: M.basis() ((1,0), (3,1), (3,2)) sage: [x.modular_symbol_rep() for x in M.basis()] [{Infinity, 0}, {0, 1/3}, {-1/2, -1/3}] sage: M = ModularSymbols(2,2) sage: M.manin_gens_to_basis() [-1] [ 1] [ 0] sage: M = ModularSymbols(2,2) sage: x = (1,0); M(x) (1,0) sage: M( (3,1) ) # entries are reduced modulo 2 first 0 sage: M( (10,19) ) -(1,0) sage: M = ModularSymbols(6,2) sage: M.manin_gens_to_basis() [-1 0 0] [ 1 0 0] [ 0 0 0] [ 0 -1 1] [ 0 -1 0] [ 0 -1 1] [ 0 0 0] [ 0 1 -1] [ 0 0 -1] [ 0 1 -1] [ 0 1 0] [ 0 0 1] sage: M = ModularSymbols(6,2) sage: M((0,1)) -(1,0) sage: M((1,2)) -(3,1) + (3,2) sage: HeilbronnCremona(2).to_list() [[1, 0, 0, 2], [2, 0, 0, 1], [2, 1, 0, 1], [1, 0, 1, 2]] sage: HeilbronnCremona(3).to_list() [[1, 0, 0, 3], [3, 1, 0, 1], [1, 0, 1, 3], [3, 0, 0, 1], [3, -1, 0, 1], [-1, 0, 1, -3]] sage: HeilbronnCremona(5).to_list() [[1, 0, 0, 5], [5, 2, 0, 1], [2, 1, 1, 3], [1, 0, 3, 5], [5, 1, 0, 1], [1, 0, 1, 5], [5, 0, 0, 1], [5, -1, 0, 1], [-1, 0, 1, -5], [5, -2, 0, 1], [-2, 1, 1, -3], [1, 0, -3, 5]] sage: len(HeilbronnCremona(37)) 128 sage: len(HeilbronnCremona(389)) 1892 sage: len(HeilbronnCremona(2003)) 11662 sage: M = ModularSymbols(2,2) sage: M.T(2).matrix() [1] sage: M = ModularSymbols(6, 2) sage: M.T(2).matrix() [ 2 1 -1] [-1 0 1] [-1 -1 2] sage: M.T(3).matrix() [3 2 0] [0 1 0] [2 2 1] sage: M.T(3).fcp() # factored characteristic polynomial (x - 3) * (x - 1)^2 sage: M = ModularSymbols(39, 2) sage: T2 = M.T(2) sage: T2.matrix() [ 3 0 -1 0 0 1 1 -1 0] [ 0 0 2 0 -1 1 0 1 -1] [ 0 1 0 -1 1 1 0 1 -1] [ 0 0 1 0 0 1 0 1 -1] [ 0 -1 2 0 0 1 0 1 -1] [ 0 0 1 1 0 1 1 -1 0] [ 0 0 0 -1 0 1 1 2 0] [ 0 0 0 1 0 0 2 0 1] [ 0 0 -1 0 0 0 1 0 2] sage: T2.fcp() # factored characteristic polynomial (x - 1)^2 * (x - 3)^3 * (x^2 + 2*x - 1)^2 sage: T2 = M.T(2).matrix() sage: T5 = M.T(5).matrix() sage: T2*T5 - T5*T2 == 0 True sage: T5.charpoly().factor() (x - 2)^2 * (x - 6)^3 * (x^2 - 8)^2 sage: M = ModularSymbols(39, 2) sage: M.T(2).decomposition() [ Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(39) of weight 2 with sign 0 over Rational Field, Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 9 for Gamma_0(39) of weight 2 with sign 0 over Rational Field, Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 9 for Gamma_0(39) of weight 2 with sign 0 over Rational Field ] sage: M = ModularSymbols(2, 2) sage: M.boundary_map() Hecke module morphism boundary map defined by the matrix [ 1 -1] Domain: Modular Symbols space of dimension 1 for Gamma_0(2) of weight ... Codomain: Space of Boundary Modular Symbols for Congruence Subgroup Gamma0(2) ... sage: M.cuspidal_submodule() Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 1 for Gamma_0(2) of weight 2 with sign 0 over Rational Field sage: M = ModularSymbols(11, 2) sage: M.boundary_map().matrix() [ 1 -1] [ 0 0] [ 0 0] sage: M.cuspidal_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field sage: S = M.cuspidal_submodule(); S Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field sage: S.basis() ((1,8), (1,9)) sage: S.T(2).matrix() [-2 0] [ 0 -2] sage: S.T(3).matrix() [-1 0] [ 0 -1] sage: S.T(5).matrix() [1 0] [0 1] sage: E = EllipticCurve([0,-1,1,-10,-20]) sage: 2 + 1 - E.Np(2) -2 sage: 3 + 1 - E.Np(3) -1 sage: 5 + 1 - E.Np(5) 1 sage: 7 + 1 - E.Np(7) -2 sage: [S.T(p).matrix()[0,0] for p in [2,3,5,7]] [-2, -1, 1, -2] sage: M = ModularSymbols(11); M.basis() ((1,0), (1,8), (1,9)) sage: S = M.cuspidal_submodule(); S Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field sage: S.T(2).matrix() [-2 0] [ 0 -2] sage: S.T(3).matrix() [-1 0] [ 0 -1] sage: M = ModularSymbols(33) sage: S = M.cuspidal_submodule(); S Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field sage: R.<q> = PowerSeriesRing(QQ) sage: v = [S.T(n).matrix()[0,0] for n in range(1,6)] sage: f00 = sum(v[n-1]*q^n for n in range(1,6)) + O(q^6) sage: f00 q - q^2 - q^3 + q^4 + O(q^6) sage: v = [S.T(n).matrix()[0,1] for n in range(1,6)] sage: f01 = sum(v[n-1]*q^n for n in range(1,6)) + O(q^6) sage: f01 -2*q^3 + O(q^6) sage: v = [S.T(n).matrix()[1,0] for n in range(1,6)] sage: f10 = sum(v[n-1]*q^n for n in range(1,6)) + O(q^6) sage: f10 q^3 + O(q^6) sage: v = [S.T(n).matrix()[1,1] for n in range(1,6)] sage: f11 = sum(v[n-1]*q^n for n in range(1,6)) + O(q^6) sage: f11 q - 2*q^2 + 2*q^4 + q^5 + O(q^6) sage: M = ModularSymbols(23) sage: S = M.cuspidal_submodule() sage: S Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Rational Field sage: f = S.q_expansion_cuspforms(6) sage: f(0,0) q - 2/3*q^2 + 1/3*q^3 - 1/3*q^4 - 4/3*q^5 + O(q^6) sage: f(0,1) O(q^6) sage: f(1,0) -1/3*q^2 + 2/3*q^3 + 1/3*q^4 - 2/3*q^5 + O(q^6) sage: S.q_expansion_basis(6) [ q - q^3 - q^4 + O(q^6), q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6) ] sage: R = Integers(49) sage: R Ring of integers modulo 49 sage: R.unit_gens() (3,) sage: Integers(25).unit_gens() (2,) sage: Integers(100).unit_gens() (51, 77) sage: Integers(200).unit_gens() (151, 101, 177) sage: Integers(2005).unit_gens() (402, 1206) sage: Integers(200000000).unit_gens() (174218751, 51562501, 187109377) sage: list(DirichletGroup(5)) [Dirichlet character modulo 5 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4, Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1, Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4] sage: list(DirichletGroup(5, QQ)) [Dirichlet character modulo 5 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1] sage: G = DirichletGroup(200) sage: G Group of Dirichlet characters modulo 200 with values in Cyclotomic Field of order 20 and degree 8 sage: G.exponent() 20 sage: G.gens() (Dirichlet character modulo 200 of conductor 4 mapping 151 |--> -1, 101 |--> 1, 177 |--> 1, Dirichlet character modulo 200 of conductor 8 mapping 151 |--> 1, 101 |--> -1, 177 |--> 1, Dirichlet character modulo 200 of conductor 25 mapping 151 |--> 1, 101 |--> 1, 177 |--> zeta20) sage: K = G.base_ring() sage: zeta = K.0 sage: eps = G([1,-1,zeta^5]) sage: eps Dirichlet character modulo 200 of conductor 40 mapping 151 |--> 1, 101 |--> -1, 177 |--> zeta20^5 sage: eps(3) zeta20^5 sage: -zeta^15 zeta20^5 sage: kronecker(151,200) 1 sage: kronecker(101,200) -1 sage: kronecker(177,200) 1 sage: G = DirichletGroup(20) sage: G.galois_orbits() [ [Dirichlet character modulo 20 of conductor 20 mapping 11 |--> -1, 17 |--> -1], [Dirichlet character modulo 20 of conductor 20 mapping 11 |--> -1, 17 |--> -zeta4, Dirichlet character modulo 20 of conductor 20 mapping 11 |--> -1, 17 |--> zeta4], [Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1], [Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> -1], [Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> -zeta4, Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4], [Dirichlet character modulo 20 of conductor 1 mapping 11 |--> 1, 17 |--> 1] ] sage: G = DirichletGroup(11, QQ); G Group of Dirichlet characters modulo 11 with values in Rational Field sage: list(G) [Dirichlet character modulo 11 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 11 of conductor 11 mapping 2 |--> -1] sage: eps = G.0 # 0th generator for Dirichlet group sage: eps Dirichlet character modulo 11 of conductor 11 mapping 2 |--> -1 sage: G.unit_gens() (2,) sage: eps(2) -1 sage: eps(3) 1 sage: [eps(11*n) for n in range(10)] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] sage: R = CyclotomicField(4) sage: CyclotomicField(4) Cyclotomic Field of order 4 and degree 2 sage: G = DirichletGroup(15, R) sage: G Group of Dirichlet characters modulo 15 with values in Cyclotomic Field of order 4 and degree 2 sage: list(G) [Dirichlet character modulo 15 of conductor 1 mapping 11 |--> 1, 7 |--> 1, Dirichlet character modulo 15 of conductor 3 mapping 11 |--> -1, 7 |--> 1, Dirichlet character modulo 15 of conductor 5 mapping 11 |--> 1, 7 |--> zeta4, Dirichlet character modulo 15 of conductor 15 mapping 11 |--> -1, 7 |--> zeta4, Dirichlet character modulo 15 of conductor 5 mapping 11 |--> 1, 7 |--> -1, Dirichlet character modulo 15 of conductor 15 mapping 11 |--> -1, 7 |--> -1, Dirichlet character modulo 15 of conductor 5 mapping 11 |--> 1, 7 |--> -zeta4, Dirichlet character modulo 15 of conductor 15 mapping 11 |--> -1, 7 |--> -zeta4] sage: e = G.1 sage: e(4) -1 sage: e(-1) -1 sage: e(5) 0 sage: [e(n) for n in range(15)] [0, 1, zeta4, 0, -1, 0, 0, zeta4, -zeta4, 0, 0, 1, 0, -zeta4, -1] sage: G = DirichletGroup(15, GF(5)); G Group of Dirichlet characters modulo 15 with values in Finite Field of size 5 sage: list(G) [Dirichlet character modulo 15 of conductor 1 mapping 11 |--> 1, 7 |--> 1, Dirichlet character modulo 15 of conductor 3 mapping 11 |--> 4, 7 |--> 1, Dirichlet character modulo 15 of conductor 5 mapping 11 |--> 1, 7 |--> 2, Dirichlet character modulo 15 of conductor 15 mapping 11 |--> 4, 7 |--> 2, Dirichlet character modulo 15 of conductor 5 mapping 11 |--> 1, 7 |--> 4, Dirichlet character modulo 15 of conductor 15 mapping 11 |--> 4, 7 |--> 4, Dirichlet character modulo 15 of conductor 5 mapping 11 |--> 1, 7 |--> 3, Dirichlet character modulo 15 of conductor 15 mapping 11 |--> 4, 7 |--> 3] sage: e = G.1 sage: e(-1) 4 sage: e(2) 2 sage: e(5) 0 sage: [e(n) for n in range(15)] [0, 1, 2, 0, 4, 0, 0, 2, 3, 0, 0, 1, 0, 3, 4] sage: G = DirichletGroup(5) sage: e = G.0 sage: e(2) zeta4 sage: e.bernoulli(1) -1/5*zeta4 - 3/5 sage: e.bernoulli(9) -108846/5*zeta4 - 176868/5 sage: E = EisensteinForms(Gamma1(13),2) sage: E.eisenstein_series() [ 1/2 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6), -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), -zeta6 + q + (2*zeta6 - 1)*q^2 + (3*zeta6 - 2)*q^3 + (-2*zeta6 - 1)*q^4 + 6*q^5 + O(q^6), q + (zeta6 + 1)*q^2 + (zeta6 + 2)*q^3 + (zeta6 + 2)*q^4 + 6*q^5 + O(q^6), -1 + q - q^2 + 4*q^3 + 3*q^4 - 4*q^5 + O(q^6), q + q^2 + 4*q^3 + 3*q^4 + 4*q^5 + O(q^6), zeta6 - 1 + q + (-2*zeta6 + 1)*q^2 + (-3*zeta6 + 1)*q^3 + (2*zeta6 - 3)*q^4 + 6*q^5 + O(q^6), q + (-zeta6 + 2)*q^2 + (-zeta6 + 3)*q^3 + (-zeta6 + 3)*q^4 + 6*q^5 + O(q^6), 7/13*zeta6 - 18/13 + q + (-2*zeta6 + 3)*q^2 + (3*zeta6 - 2)*q^3 + (-6*zeta6 + 3)*q^4 - 4*q^5 + O(q^6), q + (-zeta6 + 3)*q^2 + (zeta6 + 2)*q^3 + (-3*zeta6 + 6)*q^4 + 4*q^5 + O(q^6) ] sage: e = E.eisenstein_series() sage: for e in E.eisenstein_series(): ....: print(e.parameters()) (Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, 13) (Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta6, 1) (Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta6, Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, 1) (Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta6 - 1, 1) (Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta6 - 1, Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, 1) (Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -1, 1) (Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -1, Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, 1) (Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta6, 1) (Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta6, Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, 1) (Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta6 + 1, 1) (Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta6 + 1, Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1, 1) sage: dimension_cusp_forms(Gamma0(2007),2) 221 sage: dimension_eis(Gamma0(2007),2) 7 sage: dimension_modular_forms(Gamma0(2007),2) 228 sage: dimension_new_cusp_forms(Gamma0(11),12) 8 sage: dimension_cusp_forms(Gamma0(11),12) 10 sage: dimension_new_cusp_forms(Gamma0(2007),12) 1017 sage: dimension_cusp_forms(Gamma0(2007),12) 2460 sage: dimension_cusp_forms(Gamma1(2007),2) 147409 sage: dimension_eis(Gamma1(2007),2) 3551 sage: dimension_modular_forms(Gamma1(2007),2) 150960 sage: G = DirichletGroup(2007) sage: e = prod(G.gens(), G(1)) sage: dimension_cusp_forms(e,2) 222 sage: dimension_cusp_forms(e,3) 0 sage: dimension_cusp_forms(e,4) 670 sage: dimension_cusp_forms(e,24) 5150 sage: dimension_eis(e,2) 4 sage: dimension_eis(e,3) 0 sage: dimension_eis(e,4) 4 sage: dimension_eis(e,24) 4 sage: G = DirichletGroup(2007, QQ) sage: e = prod(G.gens(), G(1)) sage: dimension_new_cusp_forms(e,2) 76 sage: p = 389 sage: k = GF(p) sage: a = k(7/13); a 210 sage: a.rational_reconstruction() 7/13 sage: M = MatrixSpace(QQ,4,8) sage: A = M([[-9,6,7,3,1,0,0,0],[-10,3,8,2,0,1,0,0],[3,-6,2,8,0,0,1,0],[-8,-6,-8,6,0,0,0,1]]) sage: A41 = MatrixSpace(GF(41),4,8)(A) sage: E41 = A41.echelon_form() sage: B = A.matrix_from_columns([0,1,2,4]) sage: E = B^(-1)*A sage: B = A.matrix_from_columns([0,1,2,3]) sage: M = ModularSymbols(Gamma0(6)); M Modular Symbols space of dimension 3 for Gamma_0(6) of weight 2 with sign 0 over Rational Field sage: M.new_subspace() Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(6) of weight 2 with sign 0 over Rational Field sage: M.old_subspace() Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 3 for Gamma_0(6) of weight 2 with sign 0 over Rational Field sage: G = DirichletGroup(13) sage: G Group of Dirichlet characters modulo 13 with values in Cyclotomic Field of order 12 and degree 4 sage: dimension_modular_forms(Gamma1(13),2) 13 sage: [dimension_modular_forms(e,2) for e in G] [1, 0, 3, 0, 2, 0, 2, 0, 2, 0, 3, 0] sage: G = DirichletGroup(100) sage: G Group of Dirichlet characters modulo 100 with values in Cyclotomic Field of order 20 and degree 8 sage: dimension_modular_forms(Gamma1(100),2) 370 sage: v = [dimension_modular_forms(e,2) for e in G]; v [24, 0, 0, 17, 18, 0, 0, 17, 18, 0, 0, 21, 18, 0, 0, 17, 18, 0, 0, 17, 24, 0, 0, 17, 18, 0, 0, 17, 18, 0, 0, 21, 18, 0, 0, 17, 18, 0, 0, 17] sage: sum(v) 370 sage: S = CuspForms(Gamma0(45), 2, prec=14); S Cuspidal subspace of dimension 3 of Modular Forms space of dimension 10 for Congruence Subgroup Gamma0(45) of weight 2 over Rational Field sage: S.basis() [ q - q^4 - q^10 - 2*q^13 + O(q^14), q^2 - q^5 - 3*q^8 + 4*q^11 + O(q^14), q^3 - q^6 - q^9 - q^12 + O(q^14) ] sage: S.new_subspace().basis() [ q + q^2 - q^4 - q^5 - 3*q^8 - q^10 + 4*q^11 - 2*q^13 + O(q^14) ] sage: CuspForms(Gamma0(9),2) Cuspidal subspace of dimension 0 of Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(9) of weight 2 over Rational Field sage: CuspForms(Gamma0(15),2, prec=10).basis() [ q - q^2 - q^3 - q^4 + q^5 + q^6 + 3*q^8 + q^9 + O(q^10) ] """ |