/* 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([5, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([8, 2, 2]) primes_array = [ [3, 3, w - 2],\ [5, 5, w],\ [5, 5, -w + 3],\ [8, 2, 2],\ [9, 3, w^2 + w - 4],\ [13, 13, w + 3],\ [17, 17, -w^2 + w + 3],\ [23, 23, w^2 - 2],\ [23, 23, w^2 - 3],\ [23, 23, -w^2 + 8],\ [29, 29, w - 4],\ [37, 37, w^2 + w - 8],\ [41, 41, w^2 + 2*w - 4],\ [47, 47, 2*w^2 + w - 8],\ [59, 59, -2*w^2 - 3*w + 6],\ [61, 61, -2*w - 1],\ [67, 67, -2*w - 3],\ [79, 79, 2*w^2 - 9],\ [109, 109, w^2 + 2*w - 6],\ [109, 109, 2*w^2 + w - 14],\ [109, 109, -w^2 - w - 1],\ [113, 113, 2*w^2 + w - 12],\ [127, 127, w^2 - 3*w - 2],\ [131, 131, -4*w^2 - 5*w + 9],\ [137, 137, w^2 + 2*w - 7],\ [137, 137, 3*w + 8],\ [137, 137, 2*w^2 + w - 17],\ [139, 139, 3*w^2 + 2*w - 11],\ [149, 149, w - 6],\ [151, 151, w^2 + 2*w - 9],\ [157, 157, -w^2 - 2*w + 13],\ [157, 157, 2*w^2 - 7],\ [163, 163, w^2 + 3*w + 3],\ [163, 163, 2*w^2 - 3*w - 3],\ [163, 163, 2*w^2 - w - 7],\ [167, 167, 3*w^2 + w - 13],\ [169, 13, 2*w^2 + 5*w - 1],\ [173, 173, w^2 + 2*w + 2],\ [179, 179, 4*w^2 + 2*w - 19],\ [181, 181, w^2 + 3*w - 6],\ [191, 191, 3*w^2 + 2*w - 14],\ [193, 193, w^2 - 3*w - 3],\ [197, 197, 3*w^2 - 2*w - 13],\ [211, 211, -w - 6],\ [223, 223, -w^2 + w - 3],\ [223, 223, -3*w - 2],\ [223, 223, 2*w^2 - 2*w - 3],\ [229, 229, 2*w^2 - w - 4],\ [233, 233, 2*w + 7],\ [239, 239, -3*w - 4],\ [239, 239, w^2 - 11],\ [239, 239, 4*w^2 - w - 21],\ [241, 241, 3*w^2 + w - 19],\ [251, 251, -w^2 - 4*w - 1],\ [257, 257, w - 7],\ [263, 263, 3*w^2 + w - 12],\ [269, 269, 3*w^2 - 4*w - 11],\ [271, 271, 4*w^2 + 6*w - 11],\ [277, 277, -4*w^2 - 3*w + 17],\ [283, 283, 5*w^2 + 4*w - 18],\ [289, 17, 3*w^2 + 2*w - 9],\ [307, 307, -2*w^2 - w + 18],\ [307, 307, w^2 + 2*w + 3],\ [307, 307, -w^2 - 3],\ [313, 313, 3*w^2 - 13],\ [317, 317, 3*w^2 - 2*w - 12],\ [331, 331, w^2 + 3*w - 9],\ [331, 331, w^2 + 3*w - 11],\ [331, 331, w^2 + 3*w + 4],\ [343, 7, -7],\ [347, 347, 2*w^2 + 2*w - 13],\ [353, 353, 4*w^2 + 5*w - 13],\ [359, 359, 3*w^2 + 6*w - 1],\ [379, 379, w^2 - 3*w - 6],\ [383, 383, 3*w^2 - 23],\ [383, 383, w^2 - 12],\ [383, 383, 3*w^2 + 3*w - 13],\ [401, 401, -6*w^2 - 7*w + 16],\ [409, 409, -4*w^2 + 3*w + 21],\ [421, 421, w^2 - 3*w - 9],\ [433, 433, w^2 - 3*w - 8],\ [433, 433, w^2 + 4*w - 8],\ [433, 433, 3*w^2 - w - 12],\ [439, 439, -4*w - 11],\ [449, 449, 2*w^2 + 3*w - 19],\ [457, 457, w^2 - 4*w - 3],\ [461, 461, 5*w^2 + 6*w - 16],\ [461, 461, -2*w^2 + 3*w + 16],\ [461, 461, 2*w^2 + 4*w - 9],\ [467, 467, 3*w^2 - 4*w - 12],\ [479, 479, 3*w^2 + w - 9],\ [487, 487, 5*w^2 + 2*w - 22],\ [491, 491, 3*w^2 - 11],\ [499, 499, 5*w^2 + w - 24],\ [503, 503, 3*w^2 + 2*w - 17],\ [503, 503, 3*w^2 + w - 7],\ [503, 503, 2*w^2 - 4*w - 7],\ [509, 509, 3*w - 11],\ [521, 521, 3*w^2 - 3*w - 16],\ [523, 523, -w - 8],\ [541, 541, 6*w^2 + 7*w - 19],\ [547, 547, 3*w^2 + 4*w - 12],\ [557, 557, 2*w^2 - 3*w - 12],\ [563, 563, 4*w^2 + 3*w - 12],\ [563, 563, 4*w^2 + w - 17],\ [563, 563, -w^2 - 5*w - 3],\ [569, 569, w^2 - 4*w - 4],\ [577, 577, w^2 - w - 13],\ [587, 587, 4*w^2 + 3*w - 8],\ [587, 587, 2*w^2 - 3*w - 13],\ [587, 587, 2*w^2 + 3*w - 18],\ [593, 593, 4*w - 13],\ [599, 599, w - 9],\ [613, 613, -w^2 + 3*w - 7],\ [613, 613, w^2 + 5*w + 2],\ [613, 613, 3*w^2 - 3*w - 17],\ [617, 617, w^2 + 5*w + 8],\ [641, 641, 4*w^2 + 3*w - 11],\ [641, 641, w^2 + w - 14],\ [641, 641, 3*w^2 - 2*w - 9],\ [647, 647, 2*w^2 + 3*w - 13],\ [653, 653, 3*w^2 - 8],\ [659, 659, 5*w^2 + 4*w - 21],\ [659, 659, 2*w^2 - 4*w - 19],\ [659, 659, -2*w^2 - 6*w + 9],\ [677, 677, -6*w - 13],\ [691, 691, 6*w - 11],\ [701, 701, 3*w^2 - 3*w - 19],\ [719, 719, 2*w^2 + 3*w - 16],\ [727, 727, 3*w^2 + 4*w - 13],\ [733, 733, 3*w^2 - w - 8],\ [739, 739, 2*w^2 + 4*w - 11],\ [739, 739, 4*w^2 - 3*w - 24],\ [739, 739, 3*w^2 - w - 6],\ [743, 743, 4*w^2 - 2*w - 17],\ [751, 751, -w - 9],\ [761, 761, 5*w^2 + 5*w - 19],\ [769, 769, w^2 - 4*w - 6],\ [769, 769, -5*w - 1],\ [769, 769, 2*w^2 - 19],\ [773, 773, 2*w^2 - 6*w - 3],\ [797, 797, 4*w^2 - 17],\ [809, 809, -7*w^2 - 4*w + 29],\ [811, 811, 6*w^2 + 3*w - 29],\ [823, 823, 4*w^2 + 2*w - 13],\ [823, 823, -w^2 - 4*w - 7],\ [823, 823, 4*w^2 + 4*w - 17],\ [827, 827, -2*w^2 - 3*w - 2],\ [829, 829, -w^2 + w - 6],\ [829, 829, -w^2 + 4*w - 9],\ [829, 829, -2*w^2 + 3*w + 21],\ [839, 839, 6*w^2 - 2*w - 31],\ [841, 29, w^2 + 3*w + 6],\ [853, 853, -7*w^2 - 6*w + 27],\ [859, 859, 6*w - 1],\ [881, 881, w^2 + 5*w - 16],\ [907, 907, 2*w^2 + 7*w + 7],\ [919, 919, w^2 - 4*w - 9],\ [953, 953, -4*w^2 - w + 32],\ [967, 967, 5*w^2 - 5*w - 18],\ [971, 971, 3*w^2 - 4*w - 24],\ [977, 977, 2*w^2 + 4*w + 3],\ [997, 997, w^2 + 5*w - 12],\ [997, 997, 5*w^2 - 3*w - 22],\ [997, 997, 3*w^2 + 8*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 13*x^2 - 2*x + 37 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, e^3 - 3*e^2 - 7*e + 18, e^3 - 3*e^2 - 7*e + 18, 1, e^3 - 3*e^2 - 6*e + 18, e^3 - 3*e^2 - 5*e + 18, e^3 - 2*e^2 - 7*e + 14, -3*e^3 + 6*e^2 + 20*e - 37, e^3 - e^2 - 7*e + 4, e^3 - e^2 - 7*e + 4, 2*e^3 - 4*e^2 - 14*e + 22, -2*e^3 + 3*e^2 + 13*e - 11, 2*e^3 - 4*e^2 - 11*e + 22, -2*e^3 + 5*e^2 + 13*e - 33, e^3 - 6*e^2 - 5*e + 35, 3*e^3 - 5*e^2 - 19*e + 26, -2*e^3 + 6*e^2 + 13*e - 29, -e^3 + 4*e^2 + 8*e - 29, -9*e^3 + 20*e^2 + 62*e - 113, 4*e^3 - 11*e^2 - 27*e + 61, -3*e^3 + 7*e^2 + 19*e - 40, 3*e^3 - 7*e^2 - 22*e + 45, 3*e^3 - 6*e^2 - 22*e + 43, -e^2 - 2*e + 2, 4*e^3 - 10*e^2 - 23*e + 57, e^3 + 2*e^2 - 5*e - 15, -3*e^3 + 10*e^2 + 19*e - 60, -5*e^3 + 16*e^2 + 35*e - 90, e^3 - 3*e^2 - 9*e + 16, -4*e^3 + 7*e^2 + 27*e - 43, 8*e^3 - 21*e^2 - 55*e + 125, -5*e^3 + 12*e^2 + 32*e - 73, -7*e^3 + 11*e^2 + 48*e - 64, 4*e^3 - 6*e^2 - 27*e + 35, -3*e^3 + 4*e^2 + 23*e - 31, 7*e^3 - 18*e^2 - 44*e + 107, 16, -7*e^3 + 20*e^2 + 46*e - 113, -3*e^3 + 5*e^2 + 24*e - 35, 5*e^3 - 17*e^2 - 31*e + 104, -6*e^3 + 12*e^2 + 38*e - 76, -6*e^3 + 11*e^2 + 40*e - 66, -e^2 - 3*e - 5, -3*e^3 + 9*e^2 + 20*e - 51, e^3 - 6*e^2 - 8*e + 57, -9*e^3 + 25*e^2 + 61*e - 152, -3*e^3 + 10*e^2 + 18*e - 47, -4*e^3 + 8*e^2 + 26*e - 54, 12*e^3 - 26*e^2 - 85*e + 145, -9*e^3 + 27*e^2 + 59*e - 160, 3*e^3 - e^2 - 23*e + 2, 9*e^3 - 25*e^2 - 59*e + 146, -e^3 + 2*e^2 + 13*e - 9, -2*e^3 + 2*e^2 + 10*e, 7*e^3 - 21*e^2 - 52*e + 123, 6*e^3 - 17*e^2 - 45*e + 101, -5*e^3 + 8*e^2 + 30*e - 43, 12*e^3 - 30*e^2 - 84*e + 166, e^3 + 2*e^2 - 8*e - 17, 6*e^3 - 19*e^2 - 36*e + 117, 11*e^3 - 28*e^2 - 73*e + 172, 5*e^3 - 2*e^2 - 35*e + 6, -8*e^3 + 12*e^2 + 58*e - 66, -8*e^3 + 20*e^2 + 57*e - 113, -3*e^3 + 4*e^2 + 21*e - 27, -5*e^3 + 12*e^2 + 36*e - 83, 4*e^3 - 11*e^2 - 32*e + 72, -12*e^3 + 32*e^2 + 80*e - 194, -6*e^3 + 18*e^2 + 39*e - 113, -5*e^3 + 16*e^2 + 30*e - 97, e^3 + e^2 - 8*e - 8, 6*e^3 - 9*e^2 - 42*e + 51, 6*e^3 - 16*e^2 - 40*e + 84, -7*e^3 + 15*e^2 + 54*e - 90, -8*e^3 + 24*e^2 + 50*e - 150, 8*e^3 - 13*e^2 - 59*e + 85, -3*e^3 + 7*e^2 + 15*e - 58, 4*e^3 - 11*e^2 - 30*e + 57, -7*e^3 + 18*e^2 + 51*e - 96, 8*e^3 - 11*e^2 - 51*e + 59, 2*e^3 - 12*e^2 - 14*e + 92, 3*e^3 - 13*e^2 - 24*e + 84, -6*e^3 + 14*e^2 + 51*e - 72, -6*e + 2, 6*e^3 - 10*e^2 - 40*e + 78, -2*e^3 + 4*e^2 + 17*e - 8, e^3 - 12*e^2 - 2*e + 59, -7*e^2 - e + 39, 4*e^2 - 6*e - 40, -4*e^3 + 5*e^2 + 30*e - 15, -5*e^2 - 7*e + 49, 2*e^3 - 15*e^2 - 7*e + 89, 3*e^3 - 2*e^2 - 23*e + 39, -e^3 + 5*e^2 + 10*e - 18, 9*e^3 - 11*e^2 - 67*e + 64, -9*e^3 + 32*e^2 + 56*e - 195, -16*e^3 + 50*e^2 + 104*e - 304, -5*e^3 + 11*e^2 + 27*e - 70, -12*e^3 + 28*e^2 + 81*e - 175, 14*e^3 - 39*e^2 - 94*e + 243, 11*e^3 - 28*e^2 - 70*e + 181, -4*e^3 + 10*e^2 + 25*e - 72, 16*e^3 - 43*e^2 - 105*e + 257, e^3 - 10*e^2 - 9*e + 80, -14*e^3 + 33*e^2 + 104*e - 177, e^3 - 11*e^2 + 60, -5*e^3 + 5*e^2 + 32*e - 11, -2*e^3 + 4*e^2 + 14*e - 30, e^3 - 2*e^2 - 15*e, 2*e^3 - 8*e^2 - 16*e + 36, -3*e^3 + 13*e^2 + 26*e - 71, -2*e^3 + e^2 + 8*e - 19, 9*e^2 - 7*e - 55, 4*e^3 - 18*e^2 - 22*e + 118, 4*e^3 - 9*e^2 - 23*e + 55, -5*e^3 + 8*e^2 + 40*e - 47, 2*e^3 - 9*e^2 - 16*e + 82, -9*e^3 + 13*e^2 + 64*e - 93, -10*e^3 + 30*e^2 + 73*e - 183, -2*e^3 + 8*e^2 + 18*e - 46, -14*e^3 + 24*e^2 + 92*e - 132, -4*e^3 + 5*e^2 + 37*e - 11, -6*e^3 + 13*e^2 + 38*e - 86, 5*e^3 - 12*e^2 - 37*e + 64, 9*e^3 - 27*e^2 - 62*e + 148, 2*e^3 - 9*e^2 - 11*e + 45, -e^3 - 13*e^2 + 8*e + 85, 14*e^3 - 42*e^2 - 94*e + 238, 3*e^3 + e^2 - 23*e - 12, 3*e^3 - 8*e^2 - 18*e + 43, -14*e^3 + 36*e^2 + 88*e - 222, 2*e^3 + 5*e^2 - 14*e - 28, -3*e^3 + 3*e^2 + 18*e - 35, 2*e^3 - 10*e^2 - 18*e + 74, 3*e^3 + 3*e^2 - 23*e - 20, -2*e^3 + 8*e^2 + 6*e - 42, -3*e^3 + 8*e^2 + 29*e - 60, -8*e^3 + 30*e^2 + 52*e - 176, -4*e^3 + 7*e^2 + 28*e - 45, -16*e^3 + 46*e^2 + 105*e - 286, 18*e^3 - 43*e^2 - 121*e + 255, 11*e^3 - 21*e^2 - 73*e + 106, 15*e^3 - 40*e^2 - 105*e + 217, -6*e^3 + 6*e^2 + 51*e - 28, -16*e^3 + 41*e^2 + 105*e - 229, e^3 - 13*e^2 - 13*e + 96, 2*e^3 + 7*e^2 - 17*e - 41, 14*e^3 - 26*e^2 - 103*e + 144, -4*e^3 - 5*e^2 + 35*e + 25, 5*e^3 - 20*e^2 - 38*e + 135, e^3 - 3*e^2 + e + 40, -13*e^3 + 26*e^2 + 96*e - 129, -7*e^3 + 26*e^2 + 47*e - 161, 5*e^3 - 7*e^2 - 45*e + 56, 5*e^3 - 9*e^2 - 40*e + 74, 5*e^3 - 5*e^2 - 36*e + 16, 3*e^3 - 18*e^2 - 29*e + 119, -2*e^3 + 17*e^2 + 9*e - 107, 14*e^3 - 38*e^2 - 84*e + 232, 5*e^3 - 13*e^2 - 35*e + 72, -22*e^3 + 45*e^2 + 152*e - 257, -18*e^3 + 36*e^2 + 128*e - 190, -18*e^3 + 35*e^2 + 117*e - 187, 17*e^3 - 36*e^2 - 104*e + 209, 5*e^3 - 4*e^2 - 26*e - 1] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([8, 2, 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]