/*
  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.<x> = PolynomialRing(QQ)
g = P([2, -4, -6, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([41, 41, w^3 - 3*w^2 - 2*w + 7])

primes_array = [
[2, 2, -w],\
[7, 7, -w^3 + w^2 + 5*w + 1],\
[7, 7, -w + 1],\
[17, 17, -w^3 + 2*w^2 + 4*w - 3],\
[23, 23, -2*w^3 + 2*w^2 + 9*w + 1],\
[23, 23, -w^3 + w^2 + 3*w + 1],\
[31, 31, w^2 - w - 1],\
[31, 31, w^3 - 2*w^2 - 3*w + 5],\
[41, 41, w^3 - 3*w^2 - 2*w + 7],\
[41, 41, w^3 - 3*w^2 - 2*w + 5],\
[49, 7, 2*w^3 - 2*w^2 - 8*w - 1],\
[71, 71, -2*w^3 + 3*w^2 + 8*w - 1],\
[71, 71, w^2 - 5],\
[73, 73, -3*w^3 + 4*w^2 + 11*w - 3],\
[73, 73, 2*w^3 - w^2 - 9*w - 3],\
[79, 79, 3*w^3 - 4*w^2 - 13*w + 3],\
[79, 79, w^2 + w - 3],\
[81, 3, -3],\
[89, 89, -w^3 + 3*w^2 + 3*w - 11],\
[89, 89, 3*w^3 - 2*w^2 - 14*w - 3],\
[89, 89, -w^3 + 6*w + 7],\
[89, 89, 3*w^3 - 3*w^2 - 13*w + 1],\
[97, 97, -w^3 + 2*w^2 + 3*w - 7],\
[97, 97, 2*w^3 - 3*w^2 - 8*w + 7],\
[103, 103, -2*w^3 + 3*w^2 + 7*w - 7],\
[103, 103, w^3 - 5*w - 1],\
[103, 103, 4*w^3 - 4*w^2 - 18*w + 3],\
[103, 103, 2*w^3 - 3*w^2 - 7*w + 5],\
[113, 113, -4*w^3 + 5*w^2 + 15*w - 5],\
[113, 113, w^2 - 3*w - 1],\
[127, 127, -w^3 + 2*w^2 + 2*w - 5],\
[127, 127, 2*w^2 - w - 9],\
[127, 127, w^3 - 6*w - 1],\
[127, 127, -4*w^3 + 6*w^2 + 18*w - 9],\
[151, 151, -3*w^3 + 4*w^2 + 12*w - 1],\
[151, 151, 2*w^3 - 3*w^2 - 9*w + 1],\
[151, 151, w^3 - 4*w - 5],\
[151, 151, w^3 - 2*w^2 - 5*w + 5],\
[167, 167, 5*w^3 - 5*w^2 - 23*w + 3],\
[167, 167, 4*w^3 - 6*w^2 - 16*w + 5],\
[169, 13, -2*w^3 + 4*w^2 + 7*w - 5],\
[169, 13, -w^3 + 3*w^2 + 3*w - 7],\
[193, 193, w^2 - 3*w - 3],\
[193, 193, w^3 - 7*w - 3],\
[199, 199, 3*w^3 - 5*w^2 - 11*w + 9],\
[199, 199, 2*w^2 - w - 3],\
[233, 233, -2*w^3 + 5*w^2 + 7*w - 13],\
[233, 233, w^3 + w^2 - 8*w - 7],\
[241, 241, -2*w^3 + 2*w^2 + 11*w + 1],\
[241, 241, 5*w^3 - 5*w^2 - 24*w + 1],\
[263, 263, -2*w^3 + w^2 + 10*w + 1],\
[263, 263, -2*w^3 + 3*w^2 + 9*w + 1],\
[263, 263, -w^3 + 2*w^2 + 5*w - 7],\
[263, 263, 2*w^3 - 3*w^2 - 6*w + 5],\
[289, 17, -3*w^3 + 3*w^2 + 12*w - 1],\
[311, 311, 3*w^3 - 5*w^2 - 12*w + 3],\
[311, 311, -w^3 + 3*w^2 + 4*w - 9],\
[313, 313, 2*w^3 - 5*w^2 - 5*w + 9],\
[313, 313, w^3 - w^2 - 7*w - 1],\
[337, 337, -4*w^3 + 5*w^2 + 15*w - 1],\
[337, 337, w^3 - w^2 - 3*w + 5],\
[361, 19, -3*w^2 + 2*w + 11],\
[361, 19, 3*w^3 - w^2 - 14*w - 7],\
[367, 367, 4*w^3 - 6*w^2 - 14*w + 7],\
[367, 367, -2*w^3 + 10*w + 5],\
[383, 383, -2*w^3 + 4*w^2 + 6*w - 9],\
[383, 383, w^2 + w - 5],\
[383, 383, 2*w^2 - 2*w - 3],\
[383, 383, 3*w^3 - 4*w^2 - 13*w + 1],\
[401, 401, 3*w^3 - 3*w^2 - 11*w - 1],\
[401, 401, -4*w^3 + 4*w^2 + 17*w + 1],\
[431, 431, 2*w^2 - 3*w - 3],\
[431, 431, w^3 - 3*w^2 - w + 9],\
[433, 433, -4*w^3 + 5*w^2 + 15*w - 3],\
[433, 433, 3*w^3 - 2*w^2 - 13*w - 3],\
[433, 433, 3*w^3 - w^2 - 17*w - 9],\
[433, 433, -w^3 + 7*w + 9],\
[439, 439, 5*w^3 - 7*w^2 - 17*w + 7],\
[439, 439, -3*w^3 + 4*w^2 + 15*w - 7],\
[449, 449, 5*w^3 - 7*w^2 - 21*w + 13],\
[449, 449, 8*w^3 - 9*w^2 - 37*w + 7],\
[457, 457, 3*w^3 - 3*w^2 - 11*w + 1],\
[457, 457, -4*w^3 + 4*w^2 + 17*w - 1],\
[457, 457, w^3 - w^2 - 5*w - 5],\
[457, 457, w - 5],\
[463, 463, w^3 - w^2 - 2*w - 3],\
[463, 463, 3*w^3 - 3*w^2 - 14*w - 3],\
[463, 463, -3*w^3 + 2*w^2 + 12*w + 7],\
[463, 463, 5*w^3 - 6*w^2 - 20*w - 1],\
[479, 479, 4*w^3 - 3*w^2 - 19*w - 1],\
[479, 479, -2*w^3 + 4*w^2 + 6*w - 11],\
[487, 487, 5*w^3 - 6*w^2 - 22*w + 3],\
[487, 487, 4*w^3 - 5*w^2 - 16*w - 3],\
[503, 503, -2*w^2 + 4*w + 11],\
[503, 503, w^3 + 2*w^2 - 6*w - 13],\
[521, 521, -5*w^3 + 4*w^2 + 25*w + 3],\
[521, 521, -2*w^3 + w^2 + 12*w + 5],\
[529, 23, w^3 - w^2 - 4*w - 5],\
[569, 569, -4*w^3 + 6*w^2 + 16*w - 13],\
[569, 569, 2*w^3 - w^2 - 10*w - 9],\
[569, 569, -2*w^3 + 3*w^2 + 7*w - 9],\
[569, 569, 9*w^3 - 9*w^2 - 41*w + 3],\
[593, 593, -4*w^3 + 5*w^2 + 20*w - 3],\
[593, 593, -7*w^3 + 8*w^2 + 33*w - 5],\
[593, 593, -7*w^3 + 9*w^2 + 29*w - 15],\
[593, 593, -5*w^3 + 6*w^2 + 20*w - 11],\
[599, 599, -8*w^3 + 9*w^2 + 36*w - 5],\
[599, 599, -8*w^3 + 10*w^2 + 36*w - 9],\
[599, 599, -w^3 - 2*w^2 + 5*w + 11],\
[599, 599, 3*w^3 - 5*w^2 - 12*w + 1],\
[601, 601, -6*w^3 + 8*w^2 + 23*w - 5],\
[601, 601, 3*w^3 - w^2 - 13*w - 7],\
[607, 607, 5*w^3 - 7*w^2 - 19*w + 9],\
[607, 607, -2*w^3 + 9*w + 3],\
[617, 617, -w^3 + 2*w^2 + 5*w + 3],\
[617, 617, -2*w^3 + 3*w^2 + 9*w - 9],\
[625, 5, -5],\
[641, 641, -3*w^3 + 4*w^2 + 13*w - 9],\
[641, 641, 10*w^3 - 10*w^2 - 47*w + 3],\
[673, 673, 4*w^3 - 4*w^2 - 20*w + 1],\
[673, 673, -4*w^3 + 3*w^2 + 21*w + 5],\
[719, 719, -3*w^3 + 4*w^2 + 13*w + 1],\
[719, 719, w^2 + w - 7],\
[743, 743, 6*w^3 - 6*w^2 - 27*w + 5],\
[743, 743, w^3 - 4*w^2 - w + 5],\
[751, 751, 4*w^3 - 6*w^2 - 13*w + 7],\
[751, 751, 3*w^3 - w^2 - 15*w - 5],\
[809, 809, w^2 - 2*w - 9],\
[809, 809, w^2 - 2*w + 3],\
[823, 823, -2*w^3 + 3*w^2 + 12*w + 3],\
[823, 823, -4*w^3 + 5*w^2 + 20*w - 9],\
[839, 839, 3*w^3 - 2*w^2 - 14*w + 1],\
[839, 839, 3*w^3 - 4*w^2 - 10*w + 7],\
[857, 857, w^3 - 5*w - 9],\
[857, 857, 2*w^3 - 3*w^2 - 7*w - 3],\
[863, 863, -2*w^3 - w^2 + 10*w + 13],\
[863, 863, 2*w^2 + w - 7],\
[863, 863, -5*w^3 + 7*w^2 + 21*w - 5],\
[863, 863, w^3 + 3*w^2 - 10*w - 19],\
[881, 881, -4*w^3 + 3*w^2 + 17*w + 1],\
[881, 881, -5*w^3 + 6*w^2 + 19*w - 5],\
[887, 887, -3*w^3 + 5*w^2 + 8*w - 7],\
[887, 887, -3*w^3 + w^2 + 16*w + 5],\
[911, 911, 3*w^3 - 2*w^2 - 9*w - 3],\
[911, 911, 8*w^3 - 9*w^2 - 35*w + 3],\
[929, 929, -4*w^3 + 7*w^2 + 15*w - 9],\
[929, 929, -w^3 + 4*w^2 + 3*w - 9],\
[953, 953, -w^3 + 2*w^2 + 4*w - 9],\
[953, 953, w^3 - 2*w^2 - 4*w - 3],\
[953, 953, 3*w^3 - 6*w^2 - 11*w + 11],\
[953, 953, -2*w^3 + 5*w^2 + 7*w - 7],\
[961, 31, -4*w^3 + 4*w^2 + 16*w + 1],\
[977, 977, -w^3 + 3*w^2 - w - 5],\
[977, 977, -3*w^3 + 3*w^2 + 16*w - 1],\
[977, 977, -2*w^3 + 13*w + 7],\
[977, 977, w^3 - w^2 - 8*w - 1],\
[983, 983, -3*w^3 + 2*w^2 + 15*w - 1],\
[983, 983, -2*w^3 + 3*w^2 + 5*w - 7],\
[991, 991, -3*w^3 + 5*w^2 + 13*w + 1],\
[991, 991, 3*w^3 - 4*w^2 - 10*w + 9]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^8 - 28*x^6 + 270*x^4 - 1038*x^2 + 1376
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1/2*e^6 - 12*e^4 + 88*e^2 - 182, e, 1/4*e^7 - 6*e^5 + 89/2*e^3 - 191/2*e, e^4 - 14*e^2 + 38, e^7 - 25*e^5 + 191*e^3 - 409*e, 1/4*e^7 - 6*e^5 + 89/2*e^3 - 187/2*e, -3/2*e^7 + 36*e^5 - 265*e^3 + 553*e, 3/4*e^7 - 18*e^5 + 265/2*e^3 - 555/2*e, -1, -e^4 + 15*e^2 - 42, -2*e^6 + 50*e^4 - 383*e^2 + 826, 3/2*e^7 - 37*e^5 + 279*e^3 - 588*e, -7/4*e^7 + 43*e^5 - 647/2*e^3 + 1377/2*e, -3*e^6 + 75*e^4 - 574*e^2 + 1234, -e^6 + 24*e^4 - 176*e^2 + 374, -e^7 + 25*e^5 - 192*e^3 + 416*e, 3/2*e^7 - 36*e^5 + 265*e^3 - 553*e, -2*e^6 + 47*e^4 - 340*e^2 + 706, 7*e^6 - 171*e^4 + 1280*e^2 - 2706, -2*e^6 + 49*e^4 - 368*e^2 + 782, -3*e^6 + 71*e^4 - 517*e^2 + 1078, e^6 - 25*e^4 + 193*e^2 - 422, -4*e^6 + 95*e^4 - 694*e^2 + 1450, e^6 - 21*e^4 + 136*e^2 - 270, -1/4*e^7 + 7*e^5 - 117/2*e^3 + 259/2*e, 3*e^7 - 74*e^5 + 559*e^3 - 1189*e, 1/4*e^7 - 6*e^5 + 89/2*e^3 - 195/2*e, 3*e^7 - 74*e^5 + 558*e^3 - 1181*e, 2*e^6 - 47*e^4 + 342*e^2 - 722, -3*e^6 + 69*e^4 - 488*e^2 + 998, e^5 - 14*e^3 + 36*e, -1/4*e^7 + 7*e^5 - 117/2*e^3 + 265/2*e, -7/4*e^7 + 42*e^5 - 617/2*e^3 + 1279/2*e, -5/2*e^7 + 61*e^5 - 457*e^3 + 971*e, -9/2*e^7 + 110*e^5 - 824*e^3 + 1742*e, 3/4*e^7 - 18*e^5 + 263/2*e^3 - 537/2*e, -3/2*e^7 + 36*e^5 - 265*e^3 + 554*e, -4*e^7 + 97*e^5 - 722*e^3 + 1523*e, 5/4*e^7 - 30*e^5 + 445/2*e^3 - 951/2*e, -3/4*e^7 + 18*e^5 - 267/2*e^3 + 565/2*e, -e^6 + 27*e^4 - 218*e^2 + 474, -3*e^6 + 69*e^4 - 487*e^2 + 990, -7*e^6 + 168*e^4 - 1239*e^2 + 2602, -e^6 + 24*e^4 - 174*e^2 + 366, 17/4*e^7 - 103*e^5 + 1529/2*e^3 - 3203/2*e, -13/4*e^7 + 79*e^5 - 1177/2*e^3 + 2479/2*e, -2*e^6 + 54*e^4 - 438*e^2 + 970, -e^6 + 22*e^4 - 146*e^2 + 282, -5*e^6 + 119*e^4 - 870*e^2 + 1810, -e^6 + 26*e^4 - 203*e^2 + 434, -23/4*e^7 + 142*e^5 - 2147/2*e^3 + 4569/2*e, -2*e^7 + 49*e^5 - 369*e^3 + 789*e, 11/4*e^7 - 69*e^5 + 1059/2*e^3 - 2285/2*e, 13/4*e^7 - 79*e^5 + 1181/2*e^3 - 2507/2*e, e^6 - 22*e^4 + 152*e^2 - 310, -4*e^7 + 99*e^5 - 751*e^3 + 1605*e, -3*e^7 + 73*e^5 - 543*e^3 + 1135*e, 9*e^6 - 223*e^4 + 1689*e^2 - 3590, -4*e^6 + 103*e^4 - 806*e^2 + 1742, -3*e^4 + 36*e^2 - 66, 4*e^6 - 96*e^4 + 703*e^2 - 1442, e^6 - 25*e^4 + 190*e^2 - 406, -3*e^6 + 73*e^4 - 542*e^2 + 1142, -17/4*e^7 + 103*e^5 - 1527/2*e^3 + 3189/2*e, 1/2*e^7 - 13*e^5 + 103*e^3 - 223*e, -4*e^7 + 98*e^5 - 736*e^3 + 1558*e, 2*e^5 - 30*e^3 + 90*e, -1/4*e^7 + 7*e^5 - 115/2*e^3 + 229/2*e, -1/2*e^7 + 10*e^5 - 61*e^3 + 117*e, -3*e^6 + 72*e^4 - 526*e^2 + 1090, -9*e^6 + 220*e^4 - 1648*e^2 + 3498, 11/4*e^7 - 68*e^5 + 1027/2*e^3 - 2171/2*e, -3*e^7 + 72*e^5 - 532*e^3 + 1126*e, 10*e^6 - 243*e^4 + 1806*e^2 - 3778, -13*e^6 + 312*e^4 - 2300*e^2 + 4822, 5*e^6 - 128*e^4 + 995*e^2 - 2146, -8*e^4 + 108*e^2 - 254, 13/2*e^7 - 159*e^5 + 1190*e^3 - 2512*e, 15/4*e^7 - 90*e^5 + 1327/2*e^3 - 2777/2*e, 4*e^4 - 63*e^2 + 194, 4*e^6 - 101*e^4 + 782*e^2 - 1698, -11*e^6 + 269*e^4 - 2017*e^2 + 4282, 7*e^6 - 177*e^4 + 1365*e^2 - 2942, 11*e^6 - 266*e^4 + 1974*e^2 - 4138, -e^6 + 23*e^4 - 164*e^2 + 334, 15/4*e^7 - 94*e^5 + 1439/2*e^3 - 3071/2*e, 11/2*e^7 - 133*e^5 + 985*e^3 - 2063*e, -35/4*e^7 + 214*e^5 - 3205/2*e^3 + 6771/2*e, 3/4*e^7 - 18*e^5 + 267/2*e^3 - 575/2*e, -5/4*e^7 + 31*e^5 - 465/2*e^3 + 949/2*e, -15/4*e^7 + 91*e^5 - 1355/2*e^3 + 2851/2*e, -3*e^7 + 74*e^5 - 560*e^3 + 1201*e, -e^7 + 24*e^5 - 178*e^3 + 391*e, -23/4*e^7 + 138*e^5 - 2035/2*e^3 + 4273/2*e, 5/4*e^7 - 29*e^5 + 413/2*e^3 - 851/2*e, 13*e^6 - 323*e^4 + 2454*e^2 - 5234, -12*e^6 + 287*e^4 - 2107*e^2 + 4398, 13*e^6 - 322*e^4 + 2446*e^2 - 5226, -8*e^6 + 201*e^4 - 1541*e^2 + 3290, 3*e^6 - 76*e^4 + 586*e^2 - 1238, 6*e^6 - 144*e^4 + 1060*e^2 - 2214, 4*e^6 - 103*e^4 + 806*e^2 - 1754, -6*e^6 + 158*e^4 - 1258*e^2 + 2750, -5*e^6 + 126*e^4 - 966*e^2 + 2062, 5*e^6 - 127*e^4 + 984*e^2 - 2130, -5*e^6 + 119*e^4 - 870*e^2 + 1826, 11/4*e^7 - 69*e^5 + 1059/2*e^3 - 2285/2*e, 7/2*e^7 - 86*e^5 + 646*e^3 - 1360*e, 11/2*e^7 - 136*e^5 + 1030*e^3 - 2200*e, -13/4*e^7 + 82*e^5 - 1261/2*e^3 + 2699/2*e, 10*e^6 - 239*e^4 + 1756*e^2 - 3690, 3*e^6 - 73*e^4 + 543*e^2 - 1110, 5/4*e^7 - 32*e^5 + 495/2*e^3 - 1041/2*e, -e^7 + 24*e^5 - 178*e^3 + 378*e, -2*e^6 + 57*e^4 - 488*e^2 + 1142, -8*e^6 + 194*e^4 - 1440*e^2 + 3034, -9*e^6 + 215*e^4 - 1582*e^2 + 3318, -13*e^6 + 322*e^4 - 2440*e^2 + 5202, -4*e^6 + 102*e^4 - 798*e^2 + 1762, 5*e^6 - 116*e^4 + 827*e^2 - 1666, -9*e^6 + 221*e^4 - 1660*e^2 + 3494, -5/2*e^7 + 62*e^5 - 473*e^3 + 1025*e, 31/4*e^7 - 188*e^5 + 2797/2*e^3 - 5907/2*e, -33/4*e^7 + 201*e^5 - 3001/2*e^3 + 6319/2*e, -1/4*e^7 + 6*e^5 - 89/2*e^3 + 211/2*e, 1/4*e^7 - 3*e^5 + 7/2*e^3 + 15/2*e, 2*e^5 - 24*e^3 + 32*e, 21*e^6 - 503*e^4 + 3699*e^2 - 7734, -9*e^6 + 229*e^4 - 1772*e^2 + 3790, -3/2*e^7 + 37*e^5 - 277*e^3 + 572*e, 25/4*e^7 - 156*e^5 + 2381/2*e^3 - 5099/2*e, -4*e^7 + 99*e^5 - 751*e^3 + 1601*e, -17/4*e^7 + 105*e^5 - 1589/2*e^3 + 3379/2*e, -7*e^6 + 159*e^4 - 1108*e^2 + 2234, 7*e^6 - 171*e^4 + 1284*e^2 - 2754, 39/4*e^7 - 237*e^5 + 3531/2*e^3 - 7435/2*e, -7/2*e^7 + 83*e^5 - 603*e^3 + 1237*e, 25/4*e^7 - 149*e^5 + 2185/2*e^3 - 4585/2*e, -7*e^7 + 171*e^5 - 1282*e^3 + 2724*e, 7*e^6 - 168*e^4 + 1242*e^2 - 2614, -4*e^6 + 101*e^4 - 782*e^2 + 1686, -1/4*e^7 + 6*e^5 - 89/2*e^3 + 191/2*e, -29/4*e^7 + 178*e^5 - 2673/2*e^3 + 5651/2*e, -3/4*e^7 + 19*e^5 - 287/2*e^3 + 559/2*e, -2*e^7 + 49*e^5 - 372*e^3 + 810*e, 3*e^6 - 72*e^4 + 533*e^2 - 1130, 2*e^6 - 44*e^4 + 298*e^2 - 614, -5*e^6 + 133*e^4 - 1066*e^2 + 2326, 24*e^6 - 585*e^4 + 4364*e^2 - 9198, 9*e^6 - 216*e^4 + 1586*e^2 - 3302, 13*e^6 - 319*e^4 + 2390*e^2 - 5022, -17*e^6 + 422*e^4 - 3204*e^2 + 6806, 9*e^6 - 214*e^4 + 1558*e^2 - 3210, 20*e^6 - 491*e^4 + 3688*e^2 - 7790, 3*e^6 - 71*e^4 + 526*e^2 - 1146, -3*e^6 + 64*e^4 - 418*e^2 + 810, -5/2*e^7 + 60*e^5 - 439*e^3 + 900*e, 23/4*e^7 - 139*e^5 + 2063/2*e^3 - 4333/2*e, -e^5 + 14*e^3 - 50*e, 3/4*e^7 - 17*e^5 + 235/2*e^3 - 455/2*e]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([41, 41, w^3 - 3*w^2 - 2*w + 7])] = 1

# EXAMPLE:
# pp = ZF.ideal(2).factor()[0][0]
# hecke_eigenvalues[pp]