/* 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, -1, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([13, 13, -w^2 + 3]) primes_array = [ [5, 5, w],\ [5, 5, -w^2 + w + 2],\ [7, 7, -w^2 + 2],\ [13, 13, -w^2 + 3],\ [13, 13, w^2 - w - 4],\ [16, 2, 2],\ [19, 19, -w^2 + w + 1],\ [23, 23, -w^3 + 4*w + 2],\ [25, 5, -w^3 + w^2 + 3*w - 1],\ [29, 29, w^3 - w^2 - 4*w + 1],\ [29, 29, -w + 3],\ [31, 31, -w^3 + w^2 + 5*w - 2],\ [37, 37, w^3 - 4*w - 1],\ [37, 37, w^3 - 3*w + 1],\ [53, 53, -w^3 + 2*w^2 + 4*w - 6],\ [59, 59, w^2 + w - 4],\ [61, 61, -2*w^2 + w + 8],\ [73, 73, -w^3 + 5*w - 1],\ [79, 79, 2*w^2 + w - 6],\ [79, 79, 2*w^3 - w^2 - 9*w + 3],\ [81, 3, -3],\ [83, 83, -w^3 + w^2 + 3*w - 4],\ [97, 97, -w^3 + 6*w + 2],\ [97, 97, -2*w^3 + w^2 + 10*w - 1],\ [97, 97, 3*w^3 - 4*w^2 - 15*w + 13],\ [97, 97, w^3 - w^2 - 6*w + 2],\ [103, 103, -2*w^3 + 3*w^2 + 11*w - 9],\ [109, 109, w^2 + 3*w - 3],\ [137, 137, -w^2 + 7],\ [149, 149, w^3 - 2*w^2 - 5*w + 4],\ [149, 149, w^3 + 2*w^2 - 6*w - 7],\ [151, 151, w^3 + w^2 - 5*w - 3],\ [163, 163, w^3 - 3*w - 4],\ [163, 163, -w^3 + 6*w - 3],\ [167, 167, w^3 - 2*w^2 - 6*w + 4],\ [169, 13, w^2 + 2*w - 4],\ [191, 191, w^3 - w^2 - 5*w - 2],\ [191, 191, 2*w^3 - 9*w - 2],\ [197, 197, w^3 - 4*w + 4],\ [211, 211, -w^3 + 2*w^2 + 6*w - 8],\ [223, 223, -w^3 + 3*w^2 + 4*w - 6],\ [229, 229, w^3 - w^2 - 5*w - 1],\ [251, 251, 2*w^3 - 2*w^2 - 7*w + 3],\ [257, 257, 3*w^2 - 14],\ [263, 263, w^2 - 3*w - 1],\ [263, 263, w^2 - 2*w - 7],\ [269, 269, -w^3 + w^2 + 2*w - 4],\ [269, 269, 2*w^3 - 11*w - 1],\ [271, 271, -w^3 + w^2 + 4*w - 7],\ [277, 277, w^3 - w^2 - 3*w + 8],\ [311, 311, w^3 + 2*w^2 - 7*w - 11],\ [311, 311, -3*w^2 - 2*w + 8],\ [317, 317, -2*w^3 + 2*w^2 + 7*w - 4],\ [331, 331, w^2 - w - 8],\ [331, 331, 3*w^3 - 2*w^2 - 12*w + 9],\ [337, 337, -2*w^3 + 2*w^2 + 11*w - 8],\ [343, 7, w^3 - 3*w^2 - 4*w + 11],\ [347, 347, -2*w^3 + 2*w^2 + 9*w - 4],\ [349, 349, -w^3 + 2*w^2 + 2*w - 6],\ [353, 353, 2*w^2 - w - 12],\ [353, 353, 2*w^3 - w^2 - 9*w - 1],\ [367, 367, -2*w^3 + w^2 + 10*w - 6],\ [367, 367, 2*w^3 - 11*w - 3],\ [379, 379, -3*w - 1],\ [379, 379, w^3 - 2*w - 3],\ [383, 383, -w^3 + 2*w^2 + 4*w - 11],\ [383, 383, -w^3 - w^2 + 7*w + 4],\ [389, 389, 2*w^3 - w^2 - 11*w + 4],\ [401, 401, w^3 + 2*w^2 - 7*w - 6],\ [401, 401, 2*w^3 - 11*w - 6],\ [419, 419, -2*w^3 + w^2 + 8*w - 1],\ [421, 421, -w^3 + 3*w^2 + 3*w - 12],\ [421, 421, -w^3 - 2*w^2 + 2*w + 6],\ [433, 433, w^3 - 2*w^2 - 6*w + 11],\ [433, 433, w^2 - 8],\ [439, 439, -w^3 + 3*w^2 + 3*w - 7],\ [443, 443, w^3 - w^2 - 2*w - 4],\ [443, 443, 2*w^3 + w^2 - 7*w - 2],\ [443, 443, 2*w^3 - w^2 - 8*w + 2],\ [443, 443, -w^2 - 2],\ [467, 467, 3*w^3 - 5*w^2 - 15*w + 19],\ [467, 467, 2*w^3 - 3*w^2 - 8*w + 13],\ [467, 467, -w^3 + 2*w^2 + 3*w - 9],\ [467, 467, -w^3 + 3*w^2 + 3*w - 8],\ [479, 479, -w^3 - 3*w^2 + 5*w + 13],\ [479, 479, -w^3 + w^2 + w - 4],\ [487, 487, 3*w^3 - w^2 - 13*w - 2],\ [491, 491, w^2 + w - 9],\ [499, 499, -3*w^3 + 3*w^2 + 13*w - 7],\ [503, 503, 2*w^2 + w - 9],\ [509, 509, -2*w^2 - 3*w + 6],\ [509, 509, -w^3 + w^2 - 4],\ [521, 521, -3*w^2 + w + 12],\ [523, 523, 2*w^3 - 8*w - 3],\ [529, 23, 2*w^3 - w^2 - 6*w - 1],\ [541, 541, -w^3 + w^2 + 7*w - 4],\ [547, 547, 2*w^3 - 3*w^2 - 9*w + 7],\ [557, 557, 3*w^2 - 2*w - 9],\ [557, 557, w^3 - 6*w - 7],\ [563, 563, -2*w^3 + w^2 + 6*w - 6],\ [577, 577, -w^3 + 4*w^2 + 2*w - 13],\ [587, 587, w^2 - 2*w + 3],\ [587, 587, w^2 + 2*w - 6],\ [593, 593, -2*w^3 + 2*w^2 + 11*w - 3],\ [599, 599, w^3 + 3*w^2 - 6*w - 12],\ [601, 601, -2*w^3 + w^2 + 9*w + 2],\ [607, 607, -w^3 + 3*w^2 + 4*w - 9],\ [617, 617, -2*w^3 + w^2 + 7*w - 3],\ [619, 619, -2*w^3 + 4*w^2 + 9*w - 11],\ [631, 631, 3*w^2 - w - 7],\ [631, 631, -w^3 - w^2 + 6*w + 1],\ [641, 641, -w^3 + 2*w^2 + 5*w - 1],\ [641, 641, -w^3 + 2*w^2 + 2*w - 7],\ [643, 643, -3*w^3 + w^2 + 14*w - 4],\ [647, 647, -3*w^3 + 5*w^2 + 14*w - 18],\ [647, 647, -w^3 + 7*w + 2],\ [647, 647, -2*w^3 + 7*w - 1],\ [647, 647, 2*w^3 - w^2 - 10*w - 1],\ [653, 653, -2*w^3 - w^2 + 11*w + 4],\ [659, 659, -w^3 - w^2 + 5*w - 1],\ [673, 673, -w^3 + w^2 + 7*w - 2],\ [673, 673, -w^3 + w^2 + 2*w - 8],\ [683, 683, 2*w^3 - 11*w - 8],\ [683, 683, 2*w^3 - 7*w - 3],\ [701, 701, 2*w^3 - 4*w^2 - 10*w + 17],\ [727, 727, -w^3 + w^2 + 4*w - 8],\ [727, 727, 3*w^2 - 11],\ [739, 739, 2*w^3 + 3*w^2 - 8*w - 13],\ [743, 743, 2*w^2 + w - 12],\ [743, 743, 3*w^2 - 2*w - 14],\ [751, 751, w^3 + w^2 - 3*w - 7],\ [757, 757, -2*w^3 + 10*w - 1],\ [761, 761, -2*w^3 + w^2 + 7*w - 1],\ [761, 761, 3*w^2 + w - 7],\ [773, 773, -2*w^3 + 2*w^2 + 8*w - 1],\ [797, 797, -2*w^3 + w^2 + 9*w + 3],\ [797, 797, 3*w^3 - 14*w - 8],\ [809, 809, 5*w^3 - 5*w^2 - 24*w + 16],\ [821, 821, -4*w^3 + 3*w^2 + 18*w - 9],\ [821, 821, w^3 - w - 3],\ [823, 823, 3*w^2 + w - 8],\ [829, 829, -2*w^3 + 3*w^2 + 10*w - 7],\ [829, 829, 3*w^2 - 2*w - 16],\ [841, 29, 2*w^2 + w - 11],\ [853, 853, 3*w^3 - 2*w^2 - 14*w + 3],\ [853, 853, 3*w^3 - 5*w^2 - 14*w + 17],\ [857, 857, 3*w^2 - w - 8],\ [857, 857, 3*w^3 - w^2 - 13*w + 3],\ [859, 859, w^3 - 3*w^2 - 6*w + 12],\ [877, 877, w^3 + 2*w^2 - 5*w - 4],\ [877, 877, w^2 - w - 9],\ [881, 881, w^3 + 2*w^2 - 4*w - 12],\ [881, 881, w^2 - 4*w - 4],\ [883, 883, -3*w^3 + 2*w^2 + 11*w - 8],\ [887, 887, -2*w^3 + w^2 + 6*w - 1],\ [911, 911, 3*w^2 - w - 17],\ [911, 911, 2*w^3 - 2*w^2 - 12*w + 3],\ [919, 919, 5*w^3 - 6*w^2 - 22*w + 24],\ [919, 919, -2*w^3 + w^2 + 6*w - 7],\ [929, 929, 2*w^3 - 7*w - 1],\ [937, 937, -w^3 - w^2 + 8*w - 2],\ [947, 947, 3*w^3 + w^2 - 15*w - 6],\ [947, 947, -2*w^3 + 9*w + 9],\ [953, 953, w^3 - 2*w^2 - 6*w + 1],\ [977, 977, w^3 - w^2 - 3*w + 9],\ [977, 977, w^3 + w^2 - 3*w - 8],\ [983, 983, w^3 - 5*w - 7],\ [991, 991, -w^3 + 2*w^2 + 2*w - 8],\ [991, 991, w^2 + 4*w - 1],\ [997, 997, 2*w^3 + w^2 - 12*w - 8]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 7*x^3 + 9*x^2 + 23*x - 43 K. = NumberField(heckePol) hecke_eigenvalues_array = [-e^3 + 4*e^2 + 4*e - 16, e, e^3 - 4*e^2 - 3*e + 17, -1, 0, 2*e - 2, 2*e^3 - 9*e^2 - 3*e + 35, -2*e^3 + 8*e^2 + 4*e - 24, 4*e^3 - 18*e^2 - 8*e + 68, 3*e^3 - 12*e^2 - 7*e + 38, 2, 2*e^3 - 10*e^2 - e + 36, -5*e^3 + 21*e^2 + 15*e - 85, -3*e^2 + 6*e + 12, -5*e^3 + 22*e^2 + 9*e - 76, -4*e^3 + 17*e^2 + 11*e - 62, 4*e^3 - 18*e^2 - 8*e + 62, 2*e^3 - 8*e^2 - 10*e + 32, -3*e^3 + 14*e^2 + 5*e - 51, e^3 - 6*e^2 - 3*e + 28, 5*e^3 - 24*e^2 - 9*e + 95, 2*e^3 - 10*e^2 - 4*e + 36, 4*e^3 - 17*e^2 - 7*e + 56, -2*e^3 + 12*e^2 - 2*e - 44, 2*e^3 - 12*e^2 - 2*e + 54, 2*e^2 - 2*e - 18, -6*e^3 + 28*e^2 + 8*e - 104, 5*e^3 - 21*e^2 - 13*e + 81, 7*e^3 - 34*e^2 - 8*e + 122, -6*e^3 + 27*e^2 + 15*e - 113, 2*e^3 - 9*e^2 + e + 29, e^3 - 4*e^2 + e + 11, -6*e^3 + 28*e^2 + 6*e - 102, -2*e^3 + 7*e^2 + 9*e - 32, -6*e^3 + 28*e^2 + 16*e - 116, 10*e^3 - 46*e^2 - 26*e + 186, 2*e^2 - 6*e - 12, -e^3 + 6*e^2 + 4*e - 28, -14*e^3 + 64*e^2 + 24*e - 234, -4*e^3 + 18*e^2 + 4*e - 72, -8*e^3 + 39*e^2 + 19*e - 168, -4*e^2 + 38, 3*e^3 - 18*e^2 + e + 78, -5*e^3 + 22*e^2 + 12*e - 100, 7*e^3 - 29*e^2 - 23*e + 123, -8*e^3 + 38*e^2 + 18*e - 156, 8*e^3 - 32*e^2 - 22*e + 106, 12*e^3 - 52*e^2 - 30*e + 220, 4*e^3 - 16*e^2 - 18*e + 78, -4*e^3 + 24*e^2 - 2*e - 88, 8*e^3 - 35*e^2 - 21*e + 138, -16*e^3 + 70*e^2 + 44*e - 290, 10*e^3 - 46*e^2 - 24*e + 182, -11*e^3 + 54*e^2 + 19*e - 211, 2*e^2 + 2*e - 18, 4*e^3 - 17*e^2 - 16*e + 84, 12*e^3 - 54*e^2 - 30*e + 212, 8*e^3 - 30*e^2 - 20*e + 86, -10*e^3 + 42*e^2 + 24*e - 154, -6*e^3 + 22*e^2 + 20*e - 88, 3*e^3 - 15*e^2 - 7*e + 71, -e^3 + 7*e^2 - e - 33, -3*e^3 + 14*e^2 + e - 54, -14*e^3 + 62*e^2 + 36*e - 248, 5*e^3 - 22*e^2 - 12*e + 66, -6*e^3 + 33*e^2 + 6*e - 146, 14*e^3 - 58*e^2 - 32*e + 200, 16*e^3 - 74*e^2 - 36*e + 290, -2*e^3 + 10*e^2 - 2*e - 32, -2*e^3 + 10*e^2 + 4*e - 26, -6*e^3 + 30*e^2 + 14*e - 134, -14*e^3 + 66*e^2 + 16*e - 244, -6*e^3 + 26*e^2 + 10*e - 90, -8*e^2 + 18*e + 40, 16*e^3 - 70*e^2 - 38*e + 272, 16*e^3 - 70*e^2 - 42*e + 274, -12*e^3 + 54*e^2 + 30*e - 222, -2*e^3 + 10*e^2 + 4*e - 40, -16*e^3 + 72*e^2 + 36*e - 280, -2*e^3 + 16*e^2 - 12*e - 62, 18*e^3 - 82*e^2 - 37*e + 314, -2*e^3 + 6*e^2 + 8*e - 8, 18*e^3 - 78*e^2 - 44*e + 310, 13*e^3 - 61*e^2 - 19*e + 229, 4*e^3 - 21*e^2 - 7*e + 85, 3*e^3 - 12*e^2 - 9*e + 45, -20*e^3 + 84*e^2 + 52*e - 320, 5*e^3 - 16*e^2 - 21*e + 58, 12*e^3 - 58*e^2 - 24*e + 224, 8*e^3 - 30*e^2 - 23*e + 106, 2*e^3 - 16*e^2 + 14*e + 66, 5*e^3 - 17*e^2 - 25*e + 59, 2*e^3 - 11*e^2 + 5*e + 22, -16*e^3 + 70*e^2 + 36*e - 268, -10*e^3 + 39*e^2 + 26*e - 130, 2*e^3 - 11*e^2 + 9*e + 46, -6*e^3 + 22*e^2 + 16*e - 80, 2*e^3 - 14*e^2 + 6*e + 62, -16*e^3 + 68*e^2 + 42*e - 268, -6*e^3 + 20*e^2 + 18*e - 46, -21*e^3 + 98*e^2 + 39*e - 370, -16*e^3 + 71*e^2 + 41*e - 305, 14*e^3 - 64*e^2 - 31*e + 264, -18*e^3 + 86*e^2 + 30*e - 320, 4*e^3 - 23*e^2 - 6*e + 110, -14*e^3 + 63*e^2 + 35*e - 252, -5*e^3 + 20*e^2 + 16*e - 66, -11*e^3 + 50*e^2 + 23*e - 197, 2*e^3 - 14*e^2 + 48, 10*e^3 - 40*e^2 - 36*e + 156, -2*e^3 + 8*e^2 + 2*e - 6, -10*e^3 + 42*e^2 + 22*e - 158, -14*e^3 + 64*e^2 + 32*e - 272, 4*e^2 - 4*e - 36, 2*e^3 - 13*e^2 + 9*e + 42, -12*e^3 + 61*e^2 + 11*e - 219, 2*e^3 - 10*e^2 + 44, 2*e^3 - 12*e^2 + 4*e + 64, 15*e^3 - 69*e^2 - 23*e + 253, -12*e^3 + 50*e^2 + 26*e - 160, -12*e^3 + 62*e^2 + 6*e - 234, 18*e^3 - 70*e^2 - 62*e + 280, -6*e^3 + 26*e^2 + 12*e - 64, 10*e^3 - 49*e^2 - 13*e + 198, 24*e^3 - 100*e^2 - 65*e + 390, -3*e^3 + 4*e^2 + 26*e - 16, 14*e^3 - 62*e^2 - 24*e + 206, 14*e^3 - 62*e^2 - 44*e + 268, -16*e^3 + 76*e^2 + 30*e - 298, -8*e^3 + 42*e^2 - 2*e - 148, -10*e^3 + 52*e^2 + 12*e - 194, -6*e^3 + 28*e^2 + 16*e - 142, -4*e^3 + 24*e^2 + 8*e - 112, -4*e^3 + 16*e^2 + 18*e - 38, -14*e^3 + 62*e^2 + 32*e - 264, 3*e^3 - 12*e^2 - 23*e + 62, 6*e^3 - 22*e^2 - 18*e + 94, -24*e^3 + 109*e^2 + 51*e - 427, -14*e^3 + 62*e^2 + 16*e - 218, -12*e^3 + 53*e^2 + 30*e - 200, -e^3 + 9*e^2 - 3*e - 29, 4*e^3 - 14*e^2 - 6*e + 30, 2*e^3 - 14*e^2 + 14*e + 48, 14*e^3 - 64*e^2 - 16*e + 232, 2*e^3 - 10*e^2 - 4*e + 50, 6*e^3 - 29*e^2 + 2*e + 114, 16*e^3 - 76*e^2 - 34*e + 314, -7*e^3 + 33*e^2 + 11*e - 109, -28*e^3 + 125*e^2 + 48*e - 456, -14*e^3 + 60*e^2 + 36*e - 266, -2*e^3 + 6*e^2 - 12, -2*e^3 + 7*e^2 + 5*e - 4, 8*e^3 - 30*e^2 - 26*e + 96, 10*e^3 - 45*e^2 - 26*e + 178, 16*e^3 - 70*e^2 - 46*e + 298, -30*e^3 + 138*e^2 + 55*e - 510, -14*e^3 + 68*e^2 + 26*e - 250, 16*e^3 - 78*e^2 - 33*e + 330, 6*e^3 - 24*e^2 - 8*e + 46, 24*e^3 - 108*e^2 - 52*e + 422, 10*e^3 - 34*e^2 - 50*e + 134, 20*e^3 - 84*e^2 - 41*e + 290, 9*e^3 - 43*e^2 - 23*e + 207, 2*e^3 - 10*e^2 + 3*e + 66, 8*e^3 - 34*e^2 - 30*e + 150, 22*e^3 - 93*e^2 - 51*e + 349, -8*e^3 + 39*e^2 + 19*e - 185, 13*e^3 - 54*e^2 - 31*e + 195, -11*e^3 + 54*e^2 + 21*e - 221, 6*e^3 - 26*e^2 - 8*e + 98] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([13, 13, -w^2 + 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]