/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([-1, 2, 7, -2, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([31, 31, w^5 - 6*w^3 - w^2 + 5*w]) primes_array = [ [7, 7, w^4 - w^3 - 4*w^2 + w + 1],\ [8, 2, w^5 - w^4 - 5*w^3 + 2*w^2 + 5*w],\ [17, 17, -w^2 + w + 2],\ [23, 23, -w^4 + 2*w^3 + 3*w^2 - 4*w - 1],\ [25, 5, -w^3 + w^2 + 4*w],\ [31, 31, -w^3 + 4*w + 1],\ [31, 31, w^5 - 6*w^3 - w^2 + 5*w],\ [41, 41, -w^5 + w^4 + 5*w^3 - 2*w^2 - 6*w - 1],\ [47, 47, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 2],\ [49, 7, -w^5 - w^4 + 7*w^3 + 6*w^2 - 8*w - 3],\ [71, 71, -w^4 + 5*w^2 + w - 3],\ [71, 71, w^4 - 5*w^2 - 2*w + 4],\ [73, 73, -2*w^5 + w^4 + 10*w^3 - 9*w - 1],\ [73, 73, -w^5 + 6*w^3 + 2*w^2 - 5*w - 1],\ [79, 79, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 3],\ [89, 89, w^5 - 7*w^3 - w^2 + 9*w],\ [97, 97, 2*w^5 - 2*w^4 - 10*w^3 + 5*w^2 + 10*w - 1],\ [103, 103, w^5 - w^4 - 4*w^3 + w^2 + 3*w + 2],\ [103, 103, -2*w^5 + w^4 + 11*w^3 - w^2 - 11*w - 1],\ [103, 103, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],\ [103, 103, w^5 + w^4 - 7*w^3 - 5*w^2 + 9*w + 1],\ [113, 113, 2*w^5 - w^4 - 11*w^3 + w^2 + 10*w],\ [121, 11, w^4 - 5*w^2 - 3*w + 3],\ [121, 11, 2*w^5 - w^4 - 10*w^3 + w^2 + 7*w - 1],\ [121, 11, -2*w^5 + 2*w^4 + 9*w^3 - 5*w^2 - 6*w + 3],\ [127, 127, w^5 - 5*w^3 - 3*w^2 + 2*w + 3],\ [127, 127, -w^4 + w^3 + 3*w^2 - w + 2],\ [137, 137, -2*w^5 + w^4 + 9*w^3 + w^2 - 6*w - 2],\ [137, 137, -w^5 + w^4 + 6*w^3 - 3*w^2 - 8*w],\ [137, 137, -2*w^5 + 12*w^3 + 3*w^2 - 14*w - 3],\ [137, 137, -w^5 + w^4 + 4*w^3 - w^2 - 2*w - 3],\ [151, 151, -w^5 + 2*w^4 + 4*w^3 - 8*w^2 - 3*w + 7],\ [167, 167, -w^5 + w^4 + 5*w^3 - 2*w^2 - 4*w - 2],\ [169, 13, -2*w^5 + w^4 + 10*w^3 + w^2 - 8*w - 2],\ [193, 193, 2*w^5 - w^4 - 10*w^3 + 10*w],\ [199, 199, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 1],\ [199, 199, -w^5 - w^4 + 7*w^3 + 7*w^2 - 10*w - 6],\ [199, 199, w^5 - w^4 - 5*w^3 + 4*w^2 + 4*w - 5],\ [223, 223, -2*w^5 + w^4 + 11*w^3 - 2*w^2 - 12*w + 1],\ [223, 223, w^5 - w^4 - 4*w^3 + 4*w + 4],\ [233, 233, w^5 - 5*w^3 - w^2 + 2*w + 1],\ [233, 233, -w^4 + w^3 + 6*w^2 - 2*w - 6],\ [239, 239, -2*w^3 + 2*w^2 + 7*w],\ [239, 239, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 4],\ [241, 241, -w^5 + 2*w^4 + 4*w^3 - 7*w^2 - 4*w + 3],\ [241, 241, w^5 + w^4 - 8*w^3 - 4*w^2 + 10*w],\ [241, 241, -w^5 + 6*w^3 + 2*w^2 - 8*w - 2],\ [241, 241, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 4*w + 5],\ [257, 257, -w^4 + 6*w^2 + w - 4],\ [257, 257, -2*w^5 + 3*w^4 + 8*w^3 - 8*w^2 - 6*w + 1],\ [257, 257, 3*w^5 - w^4 - 15*w^3 - 2*w^2 + 10*w + 2],\ [263, 263, w^5 - w^4 - 6*w^3 + 5*w^2 + 7*w - 5],\ [263, 263, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 8*w + 1],\ [263, 263, w^5 - w^4 - 4*w^3 + 2*w^2 + 2],\ [271, 271, w^5 - w^4 - 5*w^3 + 4*w^2 + 4*w - 4],\ [271, 271, -2*w^4 + w^3 + 10*w^2 - 6],\ [271, 271, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 3*w - 5],\ [271, 271, w^5 + w^4 - 7*w^3 - 5*w^2 + 8*w + 2],\ [281, 281, 3*w^5 - 2*w^4 - 16*w^3 + 3*w^2 + 15*w],\ [289, 17, 2*w^5 - 11*w^3 - 4*w^2 + 8*w + 3],\ [311, 311, 2*w^4 - w^3 - 9*w^2 - 2*w + 4],\ [313, 313, -w^4 - w^3 + 7*w^2 + 5*w - 6],\ [313, 313, w^2 + 2*w - 2],\ [313, 313, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 8*w],\ [337, 337, 2*w^5 - w^4 - 11*w^3 + 11*w - 1],\ [337, 337, 2*w^5 - 2*w^4 - 10*w^3 + 5*w^2 + 9*w + 1],\ [337, 337, -w^5 - w^4 + 7*w^3 + 5*w^2 - 8*w - 1],\ [343, 7, w^5 + w^4 - 7*w^3 - 5*w^2 + 9*w],\ [353, 353, -2*w^5 + w^4 + 9*w^3 + 2*w^2 - 5*w - 4],\ [353, 353, 2*w^5 - w^4 - 9*w^3 - w^2 + 3*w + 3],\ [353, 353, w^5 - w^4 - 5*w^3 + 3*w^2 + 3*w - 3],\ [353, 353, 3*w^5 - 2*w^4 - 14*w^3 + w^2 + 10*w + 2],\ [359, 359, w^5 + w^4 - 6*w^3 - 7*w^2 + 4*w + 2],\ [367, 367, -2*w^5 + 2*w^4 + 11*w^3 - 5*w^2 - 14*w + 1],\ [383, 383, 2*w^5 + w^4 - 13*w^3 - 7*w^2 + 14*w + 2],\ [401, 401, w^3 - 4*w + 2],\ [409, 409, 3*w^5 - 3*w^4 - 15*w^3 + 7*w^2 + 14*w - 3],\ [409, 409, w^5 + w^4 - 8*w^3 - 5*w^2 + 10*w + 2],\ [431, 431, -w^3 - w^2 + 5*w + 4],\ [431, 431, -w^5 - w^4 + 7*w^3 + 6*w^2 - 10*w - 3],\ [431, 431, -w^5 + 3*w^4 + 3*w^3 - 11*w^2 - 2*w + 4],\ [433, 433, -2*w^3 + 2*w^2 + 5*w - 3],\ [433, 433, -w^5 - w^4 + 6*w^3 + 6*w^2 - 3*w - 4],\ [439, 439, 3*w^5 - 3*w^4 - 13*w^3 + 5*w^2 + 8*w - 1],\ [439, 439, -2*w^5 + 2*w^4 + 11*w^3 - 5*w^2 - 13*w + 2],\ [449, 449, -w^5 - w^4 + 7*w^3 + 7*w^2 - 9*w - 4],\ [457, 457, 2*w^5 - 12*w^3 - 3*w^2 + 11*w],\ [463, 463, -3*w^5 + w^4 + 16*w^3 + w^2 - 14*w - 1],\ [463, 463, -w^5 + 2*w^4 + 4*w^3 - 7*w^2 - 5*w + 4],\ [479, 479, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 7*w + 1],\ [479, 479, -w^5 + w^4 + 6*w^3 - 4*w^2 - 9*w + 1],\ [479, 479, w^4 - 2*w^3 - 3*w^2 + 4*w - 2],\ [487, 487, -w^5 + 6*w^3 + w^2 - 4*w + 2],\ [503, 503, -w^4 + w^3 + 4*w^2 + w - 2],\ [521, 521, -w^3 + w^2 + 5*w - 3],\ [521, 521, -w^4 + 3*w^3 + 2*w^2 - 7*w - 1],\ [529, 23, -w^5 - w^4 + 6*w^3 + 7*w^2 - 6*w - 2],\ [569, 569, 2*w^5 - 3*w^4 - 8*w^3 + 7*w^2 + 5*w + 1],\ [569, 569, -w^5 + 6*w^3 + 2*w^2 - 6*w - 5],\ [569, 569, -3*w^5 + 2*w^4 + 15*w^3 - 2*w^2 - 13*w + 1],\ [569, 569, -w^4 + 7*w^2 + w - 7],\ [577, 577, -w^4 + w^3 + 4*w^2 - 4*w - 2],\ [577, 577, -w^5 - w^4 + 8*w^3 + 4*w^2 - 12*w],\ [593, 593, 2*w^5 - w^4 - 11*w^3 + w^2 + 12*w + 2],\ [593, 593, 3*w^5 - w^4 - 16*w^3 - 2*w^2 + 13*w + 2],\ [593, 593, w^5 - w^4 - 5*w^3 + 3*w^2 + 2*w - 2],\ [593, 593, -w^5 + 7*w^3 - 10*w - 1],\ [599, 599, 2*w^5 - 2*w^4 - 9*w^3 + 4*w^2 + 8*w],\ [601, 601, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 5*w + 6],\ [607, 607, w^5 - w^4 - 4*w^3 + 3*w + 5],\ [607, 607, -2*w^5 + 3*w^4 + 9*w^3 - 9*w^2 - 9*w + 2],\ [617, 617, -2*w^5 + w^4 + 10*w^3 + w^2 - 7*w - 1],\ [617, 617, w^4 - w^3 - 6*w^2 + 3*w + 7],\ [617, 617, 2*w^5 + w^4 - 13*w^3 - 8*w^2 + 14*w + 5],\ [617, 617, 2*w^4 - 3*w^3 - 6*w^2 + 6*w],\ [625, 5, w^4 + w^3 - 6*w^2 - 4*w + 4],\ [631, 631, -w^5 + 2*w^4 + 3*w^3 - 7*w^2 + w + 6],\ [641, 641, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],\ [647, 647, 3*w^5 - 3*w^4 - 15*w^3 + 7*w^2 + 14*w - 2],\ [647, 647, -w^5 + 2*w^4 + 4*w^3 - 8*w^2 - 4*w + 5],\ [673, 673, 2*w^4 - 2*w^3 - 7*w^2 + 3*w + 1],\ [673, 673, -2*w^5 + 2*w^4 + 9*w^3 - 5*w^2 - 7*w + 4],\ [673, 673, 2*w^4 - 2*w^3 - 7*w^2 + 3*w + 2],\ [673, 673, -w^5 + 5*w^3 + 2*w^2 - 2*w - 3],\ [719, 719, -w^4 + w^3 + 4*w^2 - w - 5],\ [719, 719, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 10*w - 1],\ [727, 727, -w^5 - w^4 + 6*w^3 + 7*w^2 - 5*w - 2],\ [729, 3, -3],\ [743, 743, 2*w^4 - w^3 - 8*w^2 + 4],\ [743, 743, -2*w^5 + w^4 + 11*w^3 + w^2 - 12*w - 2],\ [751, 751, -3*w^5 + w^4 + 15*w^3 + 3*w^2 - 11*w - 2],\ [751, 751, -w^4 + w^3 + 5*w^2 - w - 7],\ [761, 761, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + w + 3],\ [761, 761, w^4 - 2*w^3 - 3*w^2 + 6*w + 3],\ [761, 761, -2*w^5 + 3*w^4 + 9*w^3 - 8*w^2 - 10*w + 1],\ [769, 769, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w],\ [809, 809, w^5 - w^4 - 4*w^3 + 3*w^2 - w - 4],\ [823, 823, -3*w^5 + 3*w^4 + 15*w^3 - 7*w^2 - 15*w],\ [823, 823, -2*w^5 + 10*w^3 + 5*w^2 - 7*w - 1],\ [823, 823, 3*w^5 - w^4 - 17*w^3 + 16*w - 2],\ [823, 823, w^5 - 6*w^3 - 3*w^2 + 9*w + 4],\ [839, 839, -w^4 + 5*w^2 + 2*w - 6],\ [857, 857, -w^5 - w^4 + 8*w^3 + 6*w^2 - 11*w - 4],\ [857, 857, -3*w^5 + w^4 + 15*w^3 + 3*w^2 - 12*w - 3],\ [857, 857, 2*w^5 - 12*w^3 - 3*w^2 + 13*w + 4],\ [857, 857, w^5 - w^4 - 6*w^3 + 5*w^2 + 7*w - 4],\ [863, 863, 3*w^5 - 17*w^3 - 5*w^2 + 16*w + 3],\ [863, 863, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 11*w + 1],\ [863, 863, 3*w^5 - w^4 - 15*w^3 - 3*w^2 + 12*w + 4],\ [881, 881, 3*w^5 - 3*w^4 - 14*w^3 + 7*w^2 + 12*w - 2],\ [887, 887, -3*w^5 + 2*w^4 + 14*w^3 - w^2 - 11*w - 2],\ [887, 887, -w^5 - w^4 + 7*w^3 + 4*w^2 - 8*w],\ [887, 887, 2*w^5 - 2*w^4 - 8*w^3 + 2*w^2 + 3*w + 3],\ [887, 887, w^5 - w^4 - 6*w^3 + 2*w^2 + 9*w],\ [911, 911, -3*w^5 + w^4 + 16*w^3 + w^2 - 14*w - 3],\ [919, 919, -2*w^5 + 2*w^4 + 9*w^3 - 3*w^2 - 7*w - 4],\ [919, 919, -3*w^5 + 2*w^4 + 15*w^3 - 4*w^2 - 12*w + 2],\ [929, 929, -3*w^5 + w^4 + 16*w^3 + 3*w^2 - 14*w - 4],\ [929, 929, -w^5 - w^4 + 6*w^3 + 8*w^2 - 5*w - 4],\ [929, 929, -2*w^5 + 2*w^4 + 10*w^3 - 3*w^2 - 11*w - 3],\ [937, 937, 2*w^4 - 3*w^3 - 7*w^2 + 5*w + 2],\ [937, 937, w^2 + w - 4],\ [953, 953, 4*w^5 - 3*w^4 - 19*w^3 + 4*w^2 + 14*w],\ [961, 31, w^5 - 6*w^3 - 2*w^2 + 5*w + 5],\ [961, 31, 2*w^5 - 3*w^4 - 8*w^3 + 8*w^2 + 5*w - 5],\ [991, 991, -3*w^5 + 3*w^4 + 15*w^3 - 8*w^2 - 14*w + 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-2, 0, 0, -2, -2, 6, 1, -10, 6, 2, 8, 12, -16, 14, -4, -6, 6, -8, -16, -8, -18, -6, -8, -14, -14, -8, 20, 22, -2, 8, -10, -6, -4, -4, -2, 28, -20, -8, 24, -20, -6, -8, -16, -8, -18, 28, 18, -18, 8, 6, -26, 16, 0, 16, -2, -2, -12, 22, -6, -20, -4, 8, 34, 22, 32, -20, 2, 12, -26, 6, -6, -8, 0, -12, -6, -34, -26, -14, -22, 24, -8, -14, -34, 0, -16, -2, 22, -28, -2, 0, -42, -32, -24, -40, -18, -34, -34, 26, -46, 10, 2, -18, 42, -44, -2, -14, -20, -8, -8, 8, -8, 38, -42, -38, -34, -34, 8, -4, 34, 48, -28, 32, 46, 18, 24, -36, -48, 46, -54, 16, 46, 30, 32, -36, -6, -16, 52, -28, 22, 4, -24, 0, -38, -14, 6, -10, 6, -10, -36, -2, 48, -40, 56, -28, 20, -44, 40, 30, 24, 36, 22, 2, -14, 10, 46, 56] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([31, 31, w^5 - 6*w^3 - w^2 + 5*w])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]