/* 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([-80, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([5, 5, w]) primes_array = [ [2, 2, w],\ [2, 2, w + 1],\ [3, 3, -2*w + 19],\ [5, 5, w],\ [5, 5, w + 4],\ [13, 13, w + 1],\ [13, 13, w + 11],\ [17, 17, w + 3],\ [17, 17, w + 13],\ [19, 19, w + 6],\ [19, 19, w + 12],\ [37, 37, w + 2],\ [37, 37, w + 34],\ [49, 7, -7],\ [59, 59, -4*w - 33],\ [59, 59, 4*w - 37],\ [61, 61, w + 28],\ [61, 61, w + 32],\ [71, 71, w + 22],\ [71, 71, w + 48],\ [79, 79, 2*w - 21],\ [79, 79, 2*w + 19],\ [107, 107, 12*w - 113],\ [113, 113, w + 17],\ [113, 113, w + 95],\ [121, 11, -11],\ [131, 131, w + 18],\ [131, 131, w + 112],\ [151, 151, w + 43],\ [151, 151, w + 107],\ [163, 163, 2*w - 23],\ [163, 163, -2*w - 21],\ [167, 167, w + 66],\ [167, 167, w + 100],\ [173, 173, w + 59],\ [173, 173, w + 113],\ [179, 179, w + 60],\ [179, 179, w + 118],\ [191, 191, w + 21],\ [191, 191, w + 169],\ [193, 193, 32*w - 303],\ [193, 193, 8*w - 77],\ [199, 199, w + 45],\ [199, 199, w + 153],\ [223, 223, w + 52],\ [223, 223, w + 170],\ [241, 241, w + 85],\ [241, 241, w + 155],\ [257, 257, 2*w - 9],\ [257, 257, -2*w - 7],\ [269, 269, w + 101],\ [269, 269, w + 167],\ [271, 271, w + 104],\ [271, 271, w + 166],\ [281, 281, -10*w + 93],\ [281, 281, 38*w - 359],\ [283, 283, w + 98],\ [283, 283, w + 184],\ [311, 311, w + 26],\ [311, 311, w + 284],\ [313, 313, w + 36],\ [313, 313, w + 276],\ [317, 317, 2*w - 3],\ [317, 317, -2*w - 1],\ [331, 331, w + 131],\ [331, 331, w + 199],\ [337, 337, w + 127],\ [337, 337, w + 209],\ [347, 347, w + 46],\ [347, 347, w + 300],\ [353, 353, 34*w - 321],\ [353, 353, -14*w + 131],\ [359, 359, w + 156],\ [359, 359, w + 202],\ [373, 373, w + 67],\ [373, 373, w + 305],\ [389, 389, 6*w - 53],\ [389, 389, 6*w + 47],\ [397, 397, 4*w + 39],\ [397, 397, -4*w + 43],\ [401, 401, -18*w + 169],\ [401, 401, 30*w - 283],\ [419, 419, w + 126],\ [419, 419, w + 292],\ [421, 421, w + 65],\ [421, 421, w + 355],\ [439, 439, 10*w - 97],\ [439, 439, -10*w - 87],\ [443, 443, 4*w - 31],\ [443, 443, -4*w - 27],\ [449, 449, w + 182],\ [449, 449, w + 266],\ [457, 457, w + 135],\ [457, 457, w + 321],\ [463, 463, 2*w - 29],\ [463, 463, -2*w - 27],\ [479, 479, w + 193],\ [479, 479, w + 285],\ [491, 491, w + 99],\ [491, 491, w + 391],\ [521, 521, w + 33],\ [521, 521, w + 487],\ [529, 23, -23],\ [547, 547, w + 74],\ [547, 547, w + 472],\ [557, 557, w + 58],\ [557, 557, w + 498],\ [563, 563, w + 253],\ [563, 563, w + 309],\ [571, 571, w + 241],\ [571, 571, w + 329],\ [577, 577, w + 96],\ [577, 577, w + 480],\ [593, 593, w + 248],\ [593, 593, w + 344],\ [617, 617, w + 61],\ [617, 617, w + 555],\ [641, 641, w + 143],\ [641, 641, w + 497],\ [643, 643, w + 51],\ [643, 643, w + 591],\ [647, 647, 8*w - 71],\ [647, 647, 8*w + 63],\ [659, 659, 4*w - 27],\ [659, 659, -4*w - 23],\ [661, 661, 44*w - 417],\ [661, 661, 20*w - 191],\ [691, 691, w + 91],\ [691, 691, w + 599],\ [701, 701, w + 38],\ [701, 701, w + 662],\ [719, 719, w + 303],\ [719, 719, w + 415],\ [727, 727, w + 295],\ [727, 727, w + 431],\ [773, 773, -6*w - 43],\ [773, 773, 6*w - 49],\ [809, 809, w + 220],\ [809, 809, w + 588],\ [811, 811, -10*w - 89],\ [811, 811, 10*w - 99],\ [821, 821, w + 41],\ [821, 821, w + 779],\ [827, 827, w + 177],\ [827, 827, w + 649],\ [841, 29, -29],\ [859, 859, w + 318],\ [859, 859, w + 540],\ [863, 863, w + 42],\ [863, 863, w + 820],\ [883, 883, w + 173],\ [883, 883, w + 709],\ [887, 887, w + 73],\ [887, 887, w + 813],\ [911, 911, 8*w - 69],\ [911, 911, 8*w + 61],\ [929, 929, w + 251],\ [929, 929, w + 677],\ [937, 937, w + 372],\ [937, 937, w + 564],\ [947, 947, 12*w + 97],\ [947, 947, 12*w - 109],\ [953, 953, 6*w - 47],\ [953, 953, -6*w - 41],\ [961, 31, -31],\ [967, 967, w + 388],\ [967, 967, w + 578],\ [971, 971, w + 176],\ [971, 971, w + 794],\ [983, 983, w + 140],\ [983, 983, w + 842],\ [997, 997, w + 399],\ [997, 997, w + 597]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 - 3*x + 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -e^2 - e + 2, -e^2 - e + 1, 1, 3*e^2 + e - 5, -e^2 - e + 3, -e^2 - 3*e + 4, -2*e^2 + 2*e + 3, -e^2 - e + 2, -2*e - 1, 4*e^2 + 4*e - 8, 5*e - 2, -8*e^2 - 3*e + 13, e^2 - 7, -5*e^2 - 2*e + 5, 5*e^2 + 6*e - 9, -2*e^2 + 4*e + 7, -2*e^2 - 7*e + 5, -e^2 - 3*e - 3, 2*e^2 - 4*e - 1, -e^2 + e - 6, e^2 - e - 4, -13*e^2 - 5*e + 20, 6*e^2 + 4*e - 20, 10*e^2 + 7*e - 15, -2*e^2 - e + 6, 3*e^2 + 4, -5*e^2 - e + 5, 5*e - 11, -2*e^2 - e + 6, 3*e^2 - 13, -13*e^2 - 6*e + 21, -e^2 + 8*e + 3, 6*e^2 + 15*e - 18, -4*e^2 - 5*e + 8, 4*e^2 + 5*e - 14, 5*e^2 + 7*e - 9, 8*e^2 + 5*e - 11, -4*e^2 - 7*e + 5, -e^2 + 5*e + 14, -12*e^2 - 3*e + 13, 11*e^2 + 4*e - 21, 6*e^2 + 3*e - 6, -4*e^2 - 11*e + 14, -5*e^2 - 10*e + 11, 10*e^2 - 27, -14*e^2 - e + 24, 7*e^2 + e - 7, 4*e^2 + 6*e - 14, 6*e^2 + 8*e - 23, 8*e^2 - 2*e - 22, 2*e^2 - 4*e - 3, 13*e^2 + 3*e - 33, -5*e^2 - 9*e + 18, 10*e^2 - 8*e - 27, -6*e^2 - 6*e + 21, -12*e^2 - 7*e + 19, e^2 + 16*e - 1, 10*e - 4, e^2 + 6*e - 6, -12*e^2 - 5*e + 33, -10*e^2 - 4*e + 32, 4*e^2 + e - 34, 6*e^2 + 11*e - 32, 5*e^2 - 31, -14*e^2 - 15*e + 24, 6*e^2 - 11*e - 18, 6*e^2 - e - 14, 7*e^2 + 6*e - 22, 9*e^2 + 5*e - 1, -9*e^2 - 5*e + 7, 8*e^2 + 11*e - 21, 3*e^2 - 2*e + 11, -6*e^2 - 14*e + 11, 2*e^2 - e - 8, -9*e^2 - 8*e + 5, 4*e^2 - 15*e - 21, -5*e^2 + 3*e - 15, -6*e^2 - 8*e + 24, -e^2 - 7*e - 22, -2*e^2 - 5*e + 12, e^2 + 6*e - 25, 19*e^2 + 7*e - 36, -12*e^2 + 3*e + 27, 2*e^2 - 6*e + 13, 7*e^2 + 12*e + 2, -8*e^2 + e + 15, -3*e^2 + 7*e + 19, -9*e^2 + 8*e + 14, 7*e^2 - 5*e - 24, 3*e^2 - e - 18, -3*e^2 - 2*e + 17, -e^2 + 16*e + 4, 3*e^2 - 12*e - 4, 7*e^2 + 3*e - 24, 20*e^2 + 8*e - 29, -16*e^2 - 6*e + 47, -3*e^2 + e - 16, -18*e^2 - 16*e + 41, 10*e^2 + 20*e - 14, 8*e^2 + 14*e - 12, -17*e^2 - 15*e + 20, 10*e^2 + 14*e - 15, -3*e^2 - 6*e + 10, 15*e^2 - 5*e - 37, 17*e^2 + 10*e - 43, -3*e^2 - 13*e + 5, 7*e^2 + 11*e + 2, -20*e^2 - 19*e + 35, -6*e^2 - 3*e + 33, -6*e^2 + 18*e + 33, -15*e^2 - 25*e + 27, -13*e^2 - 15*e + 17, -e^2 + 7*e + 11, 14*e^2 - 8*e - 40, 14*e^2 + 15*e - 23, -16*e^2 - 25*e + 33, 3*e^2 - 6*e - 12, 11*e^2 + 16*e - 22, 17*e^2 + 17*e - 38, 6*e^2 + 7*e + 11, -10*e^2 - 13*e + 34, 11*e^2 + 13*e - 51, 23*e^2 + 20*e - 37, 7*e^2 - 21*e - 26, -11*e^2 - 23*e + 28, 3*e^2 - 8*e + 14, -24*e^2 - 19*e + 60, -4*e^2 + 7*e + 48, -26*e^2 - 4*e + 46, 26*e^2 + 20*e - 57, -5*e^2 - 9*e + 37, 7*e^2 + 9*e - 11, 21*e^2 + 19*e - 53, e^2 - 10*e - 35, -14*e^2 - 3*e + 42, -3*e^2 - 5*e - 19, -5*e^2 + 5*e + 1, -9*e^2 + 45, 6*e^2 + 22*e - 36, -5*e^2 - 5*e + 17, 3*e^2 + 9*e + 15, -19*e^2 - 19*e + 63, 18*e^2 + 9*e - 55, 5*e^2 - 5*e - 29, 12*e^2 + 11*e - 32, 6*e^2 + 16*e - 2, 15*e^2 - 9*e - 30, -6*e^2 - 3*e - 3, 18*e^2 + e - 44, -3*e^2 - 3*e + 8, -17*e^2 - 28*e + 63, -26*e^2 - 7*e + 45, -7*e^2 - 5*e + 31, 32*e^2 + 18*e - 56, -e^2 + 6*e - 23, 25*e^2 + 17*e - 29, -23*e^2 - 9*e + 50, 21*e^2 + 4*e - 63, 12*e^2 - 3*e - 31, -8*e^2 - 11, -7*e^2 - 19*e - 10, -2*e^2 + 3*e - 14, 3*e^2 + 13*e - 21, 26*e^2 + 22*e - 53, -19*e^2 - 24*e + 68, -12*e^2 - 6*e - 11, -20*e^2 - 16*e + 28, -11*e^2 - e + 4, 27*e^2 + 21*e - 71, -31*e^2 - 3*e + 53, -18*e^2 - 4*e + 16, -6*e^2 + 14*e + 28] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([5, 5, w])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]