/* 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([2, 0, -4, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([31,31,w^3 - w^2 - 4*w + 1]) primes_array = [ [2, 2, w],\ [17, 17, -w^2 - w + 3],\ [17, 17, -w^3 - w^2 + 3*w + 1],\ [17, 17, w^3 - w^2 - 3*w + 1],\ [17, 17, w^2 - w - 3],\ [31, 31, w^3 + w^2 - 2*w - 3],\ [31, 31, -w^3 + w^2 + 4*w - 1],\ [31, 31, w^3 + w^2 - 4*w - 1],\ [31, 31, -w^3 + w^2 + 2*w - 3],\ [47, 47, -2*w^3 + w^2 + 5*w - 1],\ [47, 47, 2*w^3 + w^2 - 6*w - 1],\ [47, 47, -2*w^3 + w^2 + 6*w - 1],\ [47, 47, 2*w^3 + w^2 - 5*w - 1],\ [49, 7, w^2 + 1],\ [49, 7, -2*w^2 + 3],\ [79, 79, -w^3 - w^2 + 4*w - 1],\ [79, 79, -w^3 + w^2 + 2*w - 5],\ [79, 79, w^3 + w^2 - 2*w - 5],\ [79, 79, w^3 - w^2 - 4*w - 1],\ [81, 3, -3],\ [97, 97, -2*w^3 + 8*w + 1],\ [97, 97, -2*w^3 + 4*w - 1],\ [97, 97, -2*w^3 + 4*w + 1],\ [97, 97, 2*w^3 - 8*w + 1],\ [113, 113, w^3 - w^2 - w + 3],\ [113, 113, 2*w^3 - w^2 - 7*w + 1],\ [113, 113, 2*w^3 + w^2 - 7*w - 1],\ [113, 113, w^3 + w^2 - w - 3],\ [127, 127, 2*w^3 + 2*w^2 - 6*w - 3],\ [127, 127, 2*w^2 - 2*w - 5],\ [127, 127, 2*w^2 + 2*w - 5],\ [127, 127, -2*w^3 + 2*w^2 + 6*w - 3],\ [191, 191, 2*w^3 - 9*w - 3],\ [191, 191, -w^3 + 3*w^2 + 3*w - 5],\ [191, 191, -w^3 - 3*w^2 + 3*w + 5],\ [191, 191, -2*w^3 + 9*w - 3],\ [193, 193, -4*w^3 - w^2 + 12*w + 1],\ [193, 193, 3*w^3 - w^2 - 11*w + 1],\ [193, 193, -3*w^3 - w^2 + 11*w + 1],\ [193, 193, -w^3 + 2*w^2 + 5*w - 7],\ [223, 223, 3*w^3 - w^2 - 10*w + 1],\ [223, 223, w^3 - 5*w - 5],\ [223, 223, -w^3 + 5*w - 5],\ [223, 223, -3*w^3 - w^2 + 10*w + 1],\ [239, 239, -2*w^2 + w - 1],\ [239, 239, -w^3 - 2*w^2 + 3*w + 9],\ [239, 239, w^3 - 2*w^2 - 3*w + 9],\ [239, 239, -2*w^2 - w - 1],\ [241, 241, -w^3 + w^2 + 2*w - 7],\ [241, 241, -w^3 - 2*w^2 + w + 5],\ [241, 241, w^3 - 2*w^2 - w + 5],\ [241, 241, w^3 + w^2 - 2*w - 7],\ [257, 257, -w^3 - 3*w^2 + w + 7],\ [257, 257, -2*w^2 + 3*w + 5],\ [257, 257, -2*w^2 - 3*w + 5],\ [257, 257, w^3 - 3*w^2 - w + 7],\ [271, 271, 3*w^3 - 10*w + 3],\ [271, 271, w^3 + 3],\ [271, 271, -w^3 + 3],\ [271, 271, -3*w^3 + 10*w + 3],\ [337, 337, -w^3 + w^2 + w - 5],\ [337, 337, -2*w^3 + w^2 + 7*w + 1],\ [337, 337, 2*w^3 + w^2 - 7*w + 1],\ [337, 337, w^3 + w^2 - w - 5],\ [353, 353, -w^3 + 6*w - 1],\ [353, 353, 3*w^3 - 8*w + 1],\ [353, 353, 3*w^3 - 8*w - 1],\ [353, 353, w^3 - 6*w - 1],\ [367, 367, -2*w^3 + w^2 + 8*w + 1],\ [367, 367, -2*w^3 + w^2 + 4*w - 5],\ [367, 367, 2*w^3 + w^2 - 4*w - 5],\ [367, 367, 2*w^3 + w^2 - 8*w + 1],\ [383, 383, -w^3 - w^2 + 3*w - 3],\ [383, 383, w^2 - w - 7],\ [383, 383, w^2 + w - 7],\ [383, 383, w^3 - w^2 - 3*w - 3],\ [401, 401, 3*w^3 - w^2 - 9*w - 1],\ [401, 401, w^3 - w^2 - 4*w + 7],\ [401, 401, w^3 + w^2 - 4*w - 7],\ [401, 401, -3*w^3 - w^2 + 9*w - 1],\ [431, 431, 2*w^2 + 2*w - 7],\ [431, 431, 2*w^3 - 2*w^2 - 6*w + 1],\ [431, 431, -2*w^3 - 2*w^2 + 6*w + 1],\ [431, 431, 2*w^2 - 2*w - 7],\ [433, 433, w^3 - 2*w - 5],\ [433, 433, w^3 - 4*w - 5],\ [433, 433, -w^3 + 4*w - 5],\ [433, 433, -w^3 + 2*w - 5],\ [449, 449, 3*w^3 + 3*w^2 - 9*w - 5],\ [449, 449, w^2 + 4*w - 1],\ [449, 449, -w^2 + 4*w + 1],\ [449, 449, -3*w^3 + 3*w^2 + 9*w - 5],\ [463, 463, -w^3 + w^2 + 3*w - 7],\ [463, 463, -w^2 - w - 3],\ [463, 463, -w^2 + w - 3],\ [463, 463, w^3 + w^2 - 3*w - 7],\ [479, 479, 2*w^3 - w^2 - 9*w + 3],\ [479, 479, 3*w^3 - w^2 - 7*w + 1],\ [479, 479, -3*w^3 - w^2 + 7*w + 1],\ [479, 479, -2*w^3 - w^2 + 9*w + 3],\ [529, 23, -w^2 - 3],\ [529, 23, w^2 - 7],\ [577, 577, -3*w^3 + 11*w - 5],\ [577, 577, 3*w^3 + w^2 - 12*w - 3],\ [577, 577, -3*w^3 + w^2 + 12*w - 3],\ [577, 577, 3*w^3 - 11*w - 5],\ [593, 593, -2*w^3 - 3*w^2 + 5*w + 1],\ [593, 593, 2*w^3 + 4*w^2 - 6*w - 9],\ [593, 593, -2*w^3 + 4*w^2 + 6*w - 9],\ [593, 593, 2*w^3 - 3*w^2 - 5*w + 1],\ [607, 607, -w^3 + 2*w^2 + 7*w - 3],\ [607, 607, 4*w^3 + 2*w^2 - 11*w - 5],\ [607, 607, -4*w^3 + 2*w^2 + 11*w - 5],\ [607, 607, -3*w^2 - 2*w + 9],\ [625, 5, -5],\ [641, 641, w^3 - 4*w^2 - 4*w + 7],\ [641, 641, 3*w^3 - w^2 - 9*w - 3],\ [641, 641, -3*w^3 - w^2 + 9*w - 3],\ [641, 641, -w^3 - 4*w^2 + 4*w + 7],\ [673, 673, 2*w^3 - 2*w^2 - 7*w + 1],\ [673, 673, -2*w^3 + w^2 + w - 3],\ [673, 673, -2*w^3 - w^2 + w + 3],\ [673, 673, 2*w^3 + 2*w^2 - 7*w - 1],\ [719, 719, -w^3 - 4*w^2 + 4*w + 3],\ [719, 719, w^3 - w^2 + 5],\ [719, 719, -w^3 - w^2 + 5],\ [719, 719, -w^3 + 4*w^2 + 4*w - 3],\ [751, 751, 3*w^3 - 3*w^2 - 8*w + 3],\ [751, 751, w^3 + 3*w^2 - 6*w - 9],\ [751, 751, -w^3 + 3*w^2 + 6*w - 9],\ [751, 751, -3*w^3 - 3*w^2 + 8*w + 3],\ [769, 769, w^3 - 4*w^2 - 4*w + 9],\ [769, 769, w^3 + 4*w^2 - 2*w - 7],\ [769, 769, -w^3 + 4*w^2 + 2*w - 7],\ [769, 769, w^3 + 4*w^2 - 4*w - 9],\ [863, 863, -w^3 + 4*w^2 + 3*w - 9],\ [863, 863, 4*w^2 - w - 7],\ [863, 863, 4*w^2 + w - 7],\ [863, 863, w^3 + 4*w^2 - 3*w - 9],\ [881, 881, 2*w^2 - 3*w - 7],\ [881, 881, 3*w^3 + 2*w^2 - 9*w - 1],\ [881, 881, 3*w^3 - 2*w^2 - 9*w + 1],\ [881, 881, 2*w^2 + 3*w - 7],\ [911, 911, -w^3 - w^2 + 7*w + 1],\ [911, 911, 4*w^3 - w^2 - 11*w + 3],\ [911, 911, 4*w^3 + w^2 - 11*w - 3],\ [911, 911, w^3 - w^2 - 7*w + 1],\ [929, 929, 2*w^3 - 10*w + 3],\ [929, 929, 4*w^3 - 10*w - 3],\ [929, 929, -4*w^3 + 10*w - 3],\ [929, 929, 2*w^3 - 10*w - 3],\ [977, 977, -4*w^3 + 3*w^2 + 11*w - 11],\ [977, 977, 4*w^3 + 3*w^2 - 16*w - 9],\ [977, 977, -2*w^3 + 5*w^2 + 5*w - 15],\ [977, 977, 4*w^3 + 3*w^2 - 11*w - 11],\ [991, 991, -4*w^3 + 11*w + 1],\ [991, 991, w^3 - 7*w - 1],\ [991, 991, -w^3 + 7*w - 1],\ [991, 991, 4*w^3 - 11*w + 1]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, -2, 4, -2, -2, 4, -1, 10, 4, -4, 8, 2, 8, 10, -8, 0, 0, -12, 6, -4, 10, -2, -2, -14, 6, 0, -6, -18, 2, -4, 8, -16, 10, -8, 4, -20, -16, 14, -10, -22, 6, 6, -24, 12, 24, 0, -24, 12, 30, 0, 6, -6, -10, 14, -10, 26, 28, -8, 10, -14, 18, 30, 6, -18, -6, -18, -30, -18, -4, -28, -28, -4, 8, -16, 20, 8, -6, 6, 6, -30, 20, 32, 20, -16, 26, 20, 14, -28, -18, -18, -6, 0, -22, -16, 20, 32, 20, 14, -10, 8, -26, -26, -24, 24, -42, 42, 16, 4, -38, -26, -14, -32, 40, -32, -2, 0, -24, 30, 18, -22, 38, -34, -22, -24, -24, -48, 42, 26, -40, -16, 32, 28, -26, 10, -14, 0, 48, -24, 54, 50, -28, -58, 50, -50, 4, 28, 10, -6, -18, -42, -6, -50, -8, 46, 34, 8, -4, -4, -16] 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^3 - w^2 - 4*w + 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]