/* 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, 5, 2, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([14, 14, -w^2 + 2]) primes_array = [ [2, 2, w],\ [7, 7, w^4 - w^3 - 6*w^2 + w + 5],\ [13, 13, -w^4 + w^3 + 6*w^2 - w - 3],\ [16, 2, w^4 - w^3 - 6*w^2 + 2*w + 5],\ [19, 19, -w^4 + 7*w^2 + 2*w - 3],\ [23, 23, w^3 - w^2 - 4*w + 1],\ [23, 23, -w^4 + 7*w^2 + 4*w - 7],\ [25, 5, -w^2 + w + 3],\ [31, 31, w^2 - w - 1],\ [31, 31, w^4 - 6*w^2 - 2*w + 3],\ [31, 31, w^3 - 2*w^2 - 4*w + 3],\ [67, 67, 2*w^3 - 3*w^2 - 8*w + 3],\ [79, 79, -w^4 + 6*w^2 + 3*w - 1],\ [83, 83, 2*w^4 - 14*w^2 - 7*w + 11],\ [83, 83, w^4 - 2*w^3 - 4*w^2 + 6*w + 1],\ [83, 83, -w^4 + 8*w^2 + 3*w - 7],\ [97, 97, w^4 - 2*w^3 - 4*w^2 + 4*w + 1],\ [101, 101, 2*w^4 - 14*w^2 - 6*w + 11],\ [101, 101, -w^4 + 8*w^2 + 2*w - 9],\ [107, 107, -w^3 + 2*w^2 + 4*w - 1],\ [107, 107, -3*w^4 + 2*w^3 + 17*w^2 + 3*w - 9],\ [107, 107, -2*w^3 + 3*w^2 + 8*w - 5],\ [125, 5, -3*w^4 + w^3 + 19*w^2 + 5*w - 11],\ [137, 137, -w^4 + w^3 + 5*w^2 - w - 5],\ [137, 137, 2*w^4 - w^3 - 12*w^2 - 4*w + 7],\ [139, 139, 4*w^4 - 3*w^3 - 23*w^2 + 13],\ [139, 139, w^4 - 8*w^2 - 3*w + 9],\ [149, 149, 3*w^4 - 2*w^3 - 18*w^2 - 2*w + 11],\ [149, 149, w^4 - w^3 - 4*w^2 - w - 1],\ [157, 157, -w^3 + 2*w^2 + 3*w - 5],\ [163, 163, 2*w^4 - 2*w^3 - 10*w^2 + 2*w + 3],\ [167, 167, 5*w^4 - 2*w^3 - 32*w^2 - 7*w + 23],\ [173, 173, 2*w^4 - w^3 - 10*w^2 - 5*w + 3],\ [181, 181, w^4 - 2*w^3 - 4*w^2 + 4*w + 3],\ [191, 191, w^4 - 2*w^3 - 5*w^2 + 6*w + 3],\ [193, 193, 2*w^4 - 14*w^2 - 6*w + 7],\ [197, 197, -2*w^4 + 3*w^3 + 8*w^2 - 5*w + 1],\ [197, 197, w^4 - w^3 - 5*w^2 - 2*w + 3],\ [197, 197, -2*w^4 + 13*w^2 + 8*w - 9],\ [199, 199, -w^3 + w^2 + 6*w - 1],\ [211, 211, w^4 - 8*w^2 - 2*w + 11],\ [211, 211, -2*w^4 + 2*w^3 + 10*w^2 - w - 5],\ [211, 211, 2*w^4 - w^3 - 12*w^2 - 4*w + 9],\ [223, 223, -6*w^4 + 3*w^3 + 38*w^2 + 6*w - 31],\ [223, 223, -2*w^4 + w^3 + 11*w^2 + 6*w - 3],\ [229, 229, -8*w^4 + 4*w^3 + 49*w^2 + 9*w - 31],\ [229, 229, -w^3 + 2*w^2 + 2*w + 1],\ [229, 229, -w^4 + 7*w^2 + 5*w - 5],\ [233, 233, 2*w^4 - 13*w^2 - 7*w + 9],\ [239, 239, -2*w^4 + 2*w^3 + 12*w^2 - 3*w - 9],\ [239, 239, 5*w^4 - 3*w^3 - 31*w^2 - 4*w + 25],\ [239, 239, w^3 - 3*w^2 - 5*w + 5],\ [243, 3, -3],\ [251, 251, w^4 - 6*w^2 - 5*w + 5],\ [257, 257, -w^4 + 6*w^2 + 2*w - 1],\ [263, 263, 3*w^4 - 2*w^3 - 17*w^2 - w + 9],\ [271, 271, w^4 - 6*w^2 - 4*w + 5],\ [277, 277, 2*w^4 + w^3 - 14*w^2 - 11*w + 5],\ [283, 283, -2*w^4 + 2*w^3 + 11*w^2 - 2*w - 3],\ [283, 283, -2*w^2 + w + 9],\ [289, 17, w^2 - 5],\ [293, 293, 3*w^4 - 2*w^3 - 17*w^2 - 2*w + 9],\ [307, 307, 3*w^4 - 2*w^3 - 19*w^2 - w + 17],\ [307, 307, 4*w^4 - 3*w^3 - 23*w^2 - w + 13],\ [311, 311, -2*w^3 + 3*w^2 + 10*w - 7],\ [311, 311, w^4 + w^3 - 9*w^2 - 7*w + 9],\ [311, 311, -w^4 - w^3 + 7*w^2 + 8*w - 1],\ [313, 313, -2*w^4 + 2*w^3 + 11*w^2 - 4*w - 9],\ [331, 331, -w^3 + 2*w^2 + 5*w - 1],\ [337, 337, 2*w^4 - 13*w^2 - 9*w + 7],\ [337, 337, w^3 - 3*w^2 - 3*w + 5],\ [347, 347, 2*w^4 - 2*w^3 - 10*w^2 + 1],\ [349, 349, 3*w^4 - 4*w^3 - 14*w^2 + 6*w + 1],\ [349, 349, w^4 - 2*w^3 - 5*w^2 + 8*w - 1],\ [373, 373, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3],\ [373, 373, -w^4 - 2*w^3 + 9*w^2 + 12*w - 5],\ [383, 383, 2*w^4 - 13*w^2 - 5*w + 5],\ [389, 389, -2*w^4 + w^3 + 14*w^2 + w - 15],\ [389, 389, -2*w^3 + 3*w^2 + 7*w - 5],\ [397, 397, 4*w^4 - 2*w^3 - 24*w^2 - 5*w + 11],\ [409, 409, -3*w^4 + 20*w^2 + 9*w - 11],\ [419, 419, -w^4 + w^3 + 3*w^2 + w + 1],\ [419, 419, 5*w^4 - w^3 - 33*w^2 - 13*w + 25],\ [443, 443, -2*w^4 + w^3 + 10*w^2 + 6*w - 3],\ [443, 443, w^4 - 7*w^2 - 5*w + 7],\ [463, 463, w^3 - 5*w - 1],\ [467, 467, 2*w^4 - w^3 - 12*w^2 - w + 11],\ [479, 479, 2*w^4 - 14*w^2 - 7*w + 9],\ [479, 479, w^4 + w^3 - 7*w^2 - 10*w + 1],\ [479, 479, w^4 - w^3 - 5*w^2 - w - 1],\ [487, 487, -3*w^4 + 2*w^3 + 17*w^2 + 4*w - 11],\ [491, 491, -w^4 + w^3 + 6*w^2 - 3*w - 5],\ [499, 499, 2*w^3 - 3*w^2 - 9*w + 3],\ [503, 503, w^4 - w^3 - 7*w^2 + w + 3],\ [523, 523, -2*w^4 + w^3 + 13*w^2 + w - 7],\ [523, 523, 2*w^4 - w^3 - 12*w^2 - w + 5],\ [523, 523, -4*w^4 + 2*w^3 + 25*w^2 + 5*w - 15],\ [541, 541, -w^4 + w^3 + 5*w^2 + 3*w - 3],\ [557, 557, -3*w^4 + 2*w^3 + 17*w^2 - 9],\ [569, 569, -2*w^4 + 3*w^3 + 9*w^2 - 4*w - 1],\ [571, 571, w^3 - 2*w^2 - 3*w - 1],\ [577, 577, -6*w^4 + 2*w^3 + 40*w^2 + 10*w - 33],\ [577, 577, -2*w^4 + w^3 + 13*w^2 + 3*w - 13],\ [587, 587, 2*w^4 + w^3 - 15*w^2 - 12*w + 11],\ [593, 593, 5*w^4 - 3*w^3 - 30*w^2 - 5*w + 19],\ [599, 599, w^4 - 2*w^3 - 3*w^2 + 6*w - 5],\ [601, 601, -w^4 + 2*w^3 + 6*w^2 - 8*w - 9],\ [607, 607, 2*w^4 - 2*w^3 - 13*w^2 + 5*w + 15],\ [607, 607, -w^4 + 5*w^2 + 5*w - 1],\ [617, 617, w^3 - w^2 - w - 1],\ [631, 631, -2*w^4 + w^3 + 13*w^2 + 2*w - 7],\ [631, 631, 3*w^4 - 20*w^2 - 10*w + 13],\ [641, 641, 4*w^4 - w^3 - 27*w^2 - 8*w + 21],\ [643, 643, -5*w^4 + w^3 + 34*w^2 + 11*w - 27],\ [647, 647, -3*w^4 + 2*w^3 + 17*w^2 + 3*w - 13],\ [647, 647, -5*w^4 + 2*w^3 + 31*w^2 + 8*w - 21],\ [653, 653, -w^4 + 2*w^3 + 3*w^2 - 3*w + 3],\ [659, 659, -4*w^4 + 3*w^3 + 23*w^2 - w - 13],\ [661, 661, -4*w^4 + 3*w^3 + 22*w^2 + 2*w - 7],\ [661, 661, -3*w^4 + 3*w^3 + 17*w^2 - 2*w - 9],\ [719, 719, -2*w^4 - w^3 + 14*w^2 + 11*w - 9],\ [719, 719, -4*w^4 + 2*w^3 + 25*w^2 + 3*w - 15],\ [727, 727, w^4 + 2*w^3 - 9*w^2 - 13*w + 5],\ [739, 739, -w^3 + 2*w^2 + 2*w - 5],\ [739, 739, -2*w^4 - w^3 + 14*w^2 + 11*w - 11],\ [739, 739, 6*w^4 - 5*w^3 - 34*w^2 + 2*w + 19],\ [743, 743, -5*w^4 + 2*w^3 + 33*w^2 + 6*w - 29],\ [787, 787, -3*w - 1],\ [787, 787, 2*w^4 - w^3 - 12*w^2 - 5*w + 11],\ [797, 797, -2*w^4 + 2*w^3 + 9*w^2 + w + 1],\ [797, 797, 3*w^4 - 4*w^3 - 15*w^2 + 6*w + 7],\ [811, 811, w^4 - 3*w^3 - 3*w^2 + 9*w - 1],\ [811, 811, 2*w^4 - 14*w^2 - 4*w + 13],\ [823, 823, -3*w^4 + 2*w^3 + 18*w^2 - w - 13],\ [823, 823, -6*w^4 + 2*w^3 + 38*w^2 + 11*w - 25],\ [823, 823, w^4 - 9*w^2 - w + 15],\ [827, 827, -w^4 + 9*w^2 + 2*w - 9],\ [829, 829, -2*w^4 + 4*w^3 + 8*w^2 - 10*w - 1],\ [841, 29, 3*w^4 - 2*w^3 - 16*w^2 - 4*w + 9],\ [853, 853, 2*w^3 - w^2 - 9*w - 1],\ [859, 859, w^3 - 6*w - 1],\ [863, 863, 4*w^4 - w^3 - 26*w^2 - 10*w + 19],\ [863, 863, -2*w^4 + 12*w^2 + 6*w - 3],\ [877, 877, -w^3 + 6*w - 1],\ [877, 877, -4*w^4 + w^3 + 26*w^2 + 10*w - 15],\ [877, 877, 2*w^4 - w^3 - 10*w^2 - 4*w + 5],\ [881, 881, 2*w^3 - 4*w^2 - 8*w + 5],\ [883, 883, -w^4 + 5*w^2 + 6*w - 1],\ [887, 887, -2*w^4 + w^3 + 12*w^2 + 5*w - 7],\ [907, 907, -2*w^4 + 2*w^3 + 9*w^2 - w - 3],\ [929, 929, -w^3 + 2*w^2 + 6*w - 7],\ [947, 947, -6*w^4 + 4*w^3 + 37*w^2 - 27],\ [953, 953, -w^4 + 2*w^3 + 5*w^2 - 5*w - 5],\ [961, 31, -2*w^4 + 12*w^2 + 9*w - 7],\ [967, 967, -4*w^4 + 2*w^3 + 23*w^2 + 6*w - 11],\ [977, 977, 4*w^4 - 2*w^3 - 24*w^2 - 7*w + 11],\ [991, 991, 2*w^4 - 15*w^2 - 5*w + 17],\ [991, 991, -2*w^4 + w^3 + 11*w^2 + 5*w - 1],\ [997, 997, -5*w^4 + w^3 + 33*w^2 + 13*w - 23]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-1, 1, -2, 7, 4, 8, 8, 6, -8, 0, 8, -4, 0, -12, -4, 4, -6, 10, -14, 12, 12, 4, 18, 6, -6, -4, -4, -22, 14, -22, -4, -24, -14, 2, 0, -2, 6, 14, -18, 0, -4, -20, -20, 24, -24, 10, 6, 14, -18, 8, 16, 8, -4, -28, 2, -8, 16, -18, 4, 4, -18, -6, -28, 20, 24, -24, -8, -26, 4, -18, 2, 20, 14, -30, -30, 10, 32, 26, 26, -10, -38, -12, 4, -12, -12, -16, 4, -8, 24, 16, 8, -12, 36, -8, -4, -36, 12, -22, 2, 26, 36, -34, 2, -20, -14, 24, 6, -8, -40, 42, -24, 16, -2, -20, 40, 24, 6, 36, -6, 22, 48, 0, -40, 4, 4, -28, 8, -20, 44, -46, -6, -4, 44, -16, 8, 8, 36, 14, 2, -46, -20, 32, 32, 2, -50, 22, 14, 12, -8, 12, -30, 36, 46, 10, -8, 2, 48, -32, -22] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, w])] = 1 AL_eigenvalues[ZF.ideal([7, 7, w^4 - w^3 - 6*w^2 + w + 5])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]