/* 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([-4, -7, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 16, -2*w^2 + w + 12]) primes_array = [ [2, 2, -w - 2],\ [2, 2, -w - 1],\ [5, 5, -w^2 + w + 7],\ [5, 5, -w^2 + w + 5],\ [17, 17, -w^2 + 3*w + 1],\ [23, 23, -w^2 + w + 3],\ [27, 3, 3],\ [29, 29, -w^2 + 3*w - 1],\ [37, 37, w^2 + w + 1],\ [41, 41, w^2 - w - 9],\ [43, 43, -3*w^2 + 3*w + 17],\ [47, 47, 2*w^2 - 2*w - 13],\ [47, 47, -2*w + 5],\ [53, 53, 3*w^2 - w - 19],\ [59, 59, 2*w - 1],\ [67, 67, w^2 + w - 5],\ [71, 71, -3*w^2 + w + 21],\ [79, 79, 3*w^2 - w - 23],\ [89, 89, 2*w^2 - 4*w - 7],\ [89, 89, -2*w^2 - 2*w + 5],\ [89, 89, 2*w - 3],\ [97, 97, 2*w^2 - 9],\ [103, 103, -2*w^2 + 2*w + 3],\ [103, 103, -5*w^2 + 5*w + 27],\ [103, 103, -2*w^2 + 6*w + 3],\ [107, 107, w^2 - 3*w - 5],\ [109, 109, -w^2 + w - 1],\ [113, 113, -4*w^2 + 8*w + 9],\ [113, 113, 4*w^2 - 2*w - 29],\ [113, 113, -2*w^2 + 15],\ [127, 127, -w^2 - w + 9],\ [131, 131, 2*w^2 - 2*w - 1],\ [149, 149, 3*w^2 - 3*w - 19],\ [149, 149, -4*w^2 + 4*w + 21],\ [149, 149, -3*w^2 + 5*w + 9],\ [157, 157, -2*w^2 - 2*w + 1],\ [163, 163, 2*w^2 - 2*w - 7],\ [167, 167, 2*w^2 - 4*w - 1],\ [173, 173, 5*w^2 - 13*w - 5],\ [179, 179, -4*w^2 + 6*w + 17],\ [179, 179, 4*w + 5],\ [179, 179, -2*w - 7],\ [181, 181, w^2 - 3*w - 7],\ [191, 191, w^2 - w - 11],\ [197, 197, -2*w^2 - 4*w + 3],\ [199, 199, w^2 - 5*w + 5],\ [199, 199, -4*w^2 + 33],\ [199, 199, w^2 - 3*w - 9],\ [211, 211, w^2 - 3*w + 3],\ [211, 211, -5*w^2 + 9*w + 15],\ [211, 211, w^2 - 5*w - 5],\ [223, 223, w^2 + 7*w + 5],\ [227, 227, 4*w^2 - 6*w - 15],\ [229, 229, -w^2 - w + 13],\ [239, 239, 2*w^2 - 4*w - 9],\ [251, 251, -4*w^2 + 2*w + 23],\ [271, 271, -2*w^2 + 8*w - 7],\ [277, 277, 2*w^2 - 5],\ [281, 281, -8*w^2 + 6*w + 51],\ [289, 17, 3*w^2 + w - 15],\ [331, 331, -5*w^2 + 5*w + 29],\ [337, 337, 3*w^2 - 3*w - 13],\ [343, 7, -7],\ [347, 347, 7*w^2 - 3*w - 51],\ [347, 347, -4*w^2 - 4*w + 11],\ [347, 347, -2*w^2 + 19],\ [349, 349, 2*w^2 - 6*w - 7],\ [353, 353, 2*w^2 + 1],\ [367, 367, 4*w - 1],\ [373, 373, 3*w^2 - 9*w - 1],\ [373, 373, 4*w^2 - 4*w - 27],\ [373, 373, -3*w^2 - 5*w + 1],\ [379, 379, -6*w^2 + 4*w + 43],\ [389, 389, w^2 - 5*w + 3],\ [397, 397, 2*w^2 - 4*w - 11],\ [409, 409, -6*w^2 + 2*w + 43],\ [421, 421, -w^2 + 5*w + 15],\ [431, 431, 8*w^2 - 6*w - 53],\ [439, 439, -9*w^2 + 19*w + 19],\ [443, 443, -3*w^2 - w + 11],\ [449, 449, 4*w^2 - 23],\ [457, 457, 8*w^2 - 4*w - 57],\ [461, 461, w^2 - 5*w - 7],\ [463, 463, -6*w^2 + 12*w + 17],\ [467, 467, -4*w^2 + 8*w + 13],\ [479, 479, 2*w^2 - 6*w + 3],\ [491, 491, 4*w^2 - 6*w - 7],\ [499, 499, 2*w^2 - 8*w + 5],\ [499, 499, -4*w^2 + 2*w + 31],\ [499, 499, 4*w^2 - 4*w - 19],\ [503, 503, -2*w^2 - 2*w + 21],\ [509, 509, -2*w - 9],\ [529, 23, 3*w^2 - w - 13],\ [547, 547, -2*w^2 - 6*w + 1],\ [557, 557, 6*w + 7],\ [557, 557, 6*w^2 - 16*w - 3],\ [557, 557, -5*w^2 + 9*w + 13],\ [563, 563, -8*w^2 + 20*w + 11],\ [569, 569, 4*w^2 - 2*w - 21],\ [571, 571, 2*w^2 + 4*w - 5],\ [577, 577, 3*w^2 - 5*w - 15],\ [577, 577, 6*w + 11],\ [577, 577, -w^2 + 3*w - 5],\ [587, 587, -5*w^2 + w + 37],\ [607, 607, 3*w^2 - 3*w - 7],\ [613, 613, -6*w - 1],\ [617, 617, -w^2 - w - 5],\ [619, 619, -2*w^2 + 8*w + 3],\ [641, 641, -7*w^2 + w + 55],\ [643, 643, -6*w^2 + 2*w + 41],\ [643, 643, 6*w^2 - 4*w - 35],\ [643, 643, 3*w^2 + w - 17],\ [653, 653, -w^2 + 7*w + 7],\ [659, 659, -3*w^2 - w + 3],\ [673, 673, 2*w^2 + 2*w - 11],\ [673, 673, w^2 - 7*w + 13],\ [673, 673, -7*w^2 + 7*w + 37],\ [683, 683, w^2 - 5*w - 13],\ [683, 683, 2*w^2 - 6*w - 9],\ [683, 683, w^2 + w - 15],\ [691, 691, -6*w^2 + 6*w + 31],\ [691, 691, 4*w - 5],\ [691, 691, -4*w^2 + 35],\ [701, 701, 2*w^2 + 4*w + 5],\ [719, 719, -w^2 - 3*w - 7],\ [727, 727, 4*w^2 - 6*w - 11],\ [733, 733, w^2 + 3*w - 7],\ [739, 739, 3*w^2 - w - 11],\ [743, 743, 5*w^2 - 5*w - 33],\ [773, 773, -6*w^2 + 10*w + 23],\ [787, 787, -3*w^2 + 5*w + 1],\ [809, 809, 4*w^2 - 12*w - 1],\ [811, 811, -5*w^2 + 11*w + 5],\ [821, 821, -7*w^2 + 5*w + 41],\ [829, 829, w^2 + 7*w + 1],\ [841, 29, 6*w^2 - 16*w - 9],\ [853, 853, -11*w^2 + 9*w + 65],\ [857, 857, -9*w^2 + 17*w + 25],\ [857, 857, 4*w^2 - 4*w - 17],\ [857, 857, 9*w^2 - 7*w - 59],\ [859, 859, w^2 + 3*w - 13],\ [859, 859, -3*w^2 - w + 25],\ [859, 859, 6*w^2 - 8*w - 29],\ [877, 877, 3*w^2 - w - 9],\ [877, 877, 5*w^2 + 9*w - 7],\ [877, 877, -10*w^2 + 20*w + 27],\ [881, 881, -3*w^2 - w + 5],\ [887, 887, 3*w^2 + w - 7],\ [911, 911, 2*w^2 + 2*w - 13],\ [937, 937, 2*w^2 - 8*w - 1],\ [937, 937, -w^2 + 7*w + 1],\ [937, 937, 2*w^2 - 10*w + 11],\ [953, 953, -6*w^2 + 8*w + 25],\ [953, 953, -5*w^2 + w + 41],\ [953, 953, w^2 + 3*w - 11],\ [977, 977, 4*w^2 + 2*w - 19],\ [983, 983, 3*w^2 + 11*w + 3],\ [991, 991, 2*w - 11],\ [997, 997, 4*w^2 - 27],\ [997, 997, -w^2 - 9*w - 17],\ [997, 997, -4*w - 13]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 - x^2 - 4*x + 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, e^2 - 3, e^2 - 3, e^2 - 4*e - 3, 2*e^2 - 2*e - 4, 2*e^2 - 2*e - 8, -e^2 + 2*e + 1, -e^2 + 2*e + 9, -e^2 - 2*e + 1, 2*e^2 + 2*e - 12, 4*e^2 - 4*e - 8, -2*e^2 - 2*e + 8, 3*e^2 + 2*e - 11, 4*e^2 - 8, -2*e^2 + 2*e, -2*e^2 + 6*e + 8, 4*e^2 + 4*e - 16, e^2 - 7, -3*e^2 - 4*e + 17, 2, 10, 2*e^2 - 6*e - 8, 2*e^2 - 6*e - 8, -6*e^2 + 6*e + 12, 2*e^2 - 6*e - 12, 7*e^2 - 6*e - 15, -e^2 + 2*e - 3, -8*e + 2, e^2 - 4*e + 5, 4*e^2 - 8*e - 12, -2*e^2 - 2*e + 12, 8*e - 2, -3*e^2 + 1, 3*e^2 - 2*e - 15, 3*e^2 - 2*e - 7, 4*e^2 - 24, 6*e^2 - 2*e - 8, -5*e^2 + 2*e + 5, -6*e^2 + 2*e + 12, 4*e + 8, -4*e^2 - 4*e + 20, 6, -4*e^2 + 20, -8*e - 2, -4*e^2 + 12, 2*e^2 - 2*e - 4, -2*e^2 - 10*e + 8, -2*e^2 - 6*e + 8, 2*e^2 + 2*e - 4, 8*e - 12, -4*e^2 + 8*e + 12, -2*e^2 - 2*e + 4, -3*e^2 + 4*e + 5, 6*e^2 - 6*e - 4, -4*e, 6*e^2 + 2*e - 12, -7*e^2 + 8*e + 13, -5*e^2 - 2*e + 13, e^2 + 17, -2*e^2 + 2*e, 8*e^2 - 8*e - 22, -2*e^2 + 10*e + 12, 4*e + 16, -8*e^2 + 8*e + 12, -4*e^2 - 8*e + 16, e^2 - 11, 7*e^2 - 2*e - 23, 2*e^2 - 2*e + 4, -3*e^2 - 8*e + 25, -9*e^2 - 6*e + 41, -8*e - 2, 8*e - 4, -9*e^2 + 14*e + 29, 8*e^2 - 26, 3*e^2 + 6*e + 5, 8*e^2 - 8*e - 26, 4*e + 20, 4*e^2 + 4*e - 24, -6*e^2 + 6*e + 24, -8*e^2 + 8*e + 18, -e^2 + 6*e + 1, 13*e^2 - 8*e - 31, -8*e^2 - 4*e + 36, 2*e^2 + 6*e + 16, 8*e^2 - 24, -2*e^2 + 2*e + 16, 10*e^2 - 2*e - 24, -8*e^2 + 8*e + 36, -8*e^2 + 12, 2*e^2 + 10*e - 8, -e^2 + 18*e + 1, -3*e^2 - 4*e + 17, 10*e^2 - 18*e - 32, -11*e^2 - 4*e + 29, 8*e^2 + 8*e - 18, -7*e^2 + 12*e + 41, 6*e^2 + 6*e - 28, 9*e^2 - 39, -12*e + 8, -15*e^2 + 8*e + 33, -5*e^2 - 6*e + 33, -8*e + 26, 12*e^2 - 12*e - 28, -2*e^2 + 14*e + 24, 5*e^2 + 12*e - 11, 8*e^2 - 8*e - 6, 2*e^2 - 14*e - 20, -8*e + 2, -28, -6*e^2 - 10*e + 40, -4*e^2 + 4*e + 20, 16*e - 2, 2*e^2 + 2*e - 12, 3*e^2 + 2*e - 7, -3*e^2 + 8*e + 21, 8*e^2 - 14, 8*e^2 - 52, -6*e^2 - 6*e + 12, -14*e^2 + 6*e + 32, 2*e^2 + 10*e - 20, -2*e^2 + 10*e + 16, 10*e^2 - 2*e - 40, -11*e^2 + 12*e + 45, 2*e^2 - 10*e - 20, 2*e^2 - 10*e - 28, -3*e^2 - 8*e + 33, 8*e^2 - 4*e - 32, 6*e^2 + 6*e - 32, -8*e^2 + 16*e + 14, -8*e^2 + 8*e + 12, -8*e^2 + 8*e + 26, -18*e^2 + 14*e + 52, 3*e^2 - 14*e + 5, -3*e^2 + 16*e + 25, 9*e^2 - 4*e - 43, 7*e^2 - 22*e - 23, 8*e + 2, -e^2 + 6*e - 39, -9*e^2 + 6*e + 33, 16*e^2 - 36, -4*e^2 - 8*e + 32, -2*e^2 - 2*e - 12, 19*e^2 - 6*e - 59, 9*e^2 + 8*e - 59, -13*e^2 + 14*e + 25, e^2 + 4*e + 5, -8*e^2 + 4*e - 12, -12*e + 20, -3*e^2 - 16*e + 37, e^2 + 8*e - 15, -8*e^2 + 42, 3*e^2 - 2*e - 3, 8*e^2 + 16*e - 38, 11*e^2 - 10*e - 11, -9*e^2 - 6*e + 29, -18*e^2 + 10*e + 44, 2*e^2 - 6*e - 16, 7*e^2 - 14*e - 31, 11*e^2 - 6*e - 43, 9*e^2 - 24*e - 35] 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 - 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]