/*
  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([-1, 4, 5, -4, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([15, 15, w^4 - 2*w^3 - 3*w^2 + 3*w + 1])

primes_array = [
[3, 3, w - 1],\
[5, 5, w^2 - w - 2],\
[7, 7, w^4 - 2*w^3 - 3*w^2 + 4*w + 2],\
[13, 13, w^3 - 2*w^2 - 2*w + 2],\
[29, 29, -w^4 + 3*w^3 + 2*w^2 - 7*w - 1],\
[29, 29, -w^2 + 2*w + 3],\
[31, 31, w^4 - 2*w^3 - 3*w^2 + 5*w],\
[31, 31, w^3 - 2*w^2 - 3*w + 2],\
[32, 2, 2],\
[43, 43, -w^2 - w + 4],\
[53, 53, -w^4 + w^3 + 6*w^2 - 2*w - 5],\
[53, 53, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],\
[73, 73, w^4 - w^3 - 6*w^2 + 2*w + 6],\
[73, 73, w^3 - w^2 - 4*w + 2],\
[81, 3, 2*w^4 - 5*w^3 - 4*w^2 + 9*w + 2],\
[83, 83, w^3 - w^2 - 5*w],\
[89, 89, -w^4 + 2*w^3 + 4*w^2 - 4*w - 2],\
[97, 97, w^4 - 3*w^3 - 2*w^2 + 6*w + 2],\
[101, 101, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],\
[103, 103, -w^4 + w^3 + 6*w^2 - w - 7],\
[103, 103, 2*w^4 - 4*w^3 - 6*w^2 + 8*w + 1],\
[103, 103, w^4 - w^3 - 6*w^2 + 3*w + 4],\
[107, 107, -w^4 + 2*w^3 + 3*w^2 - 5*w + 2],\
[109, 109, w^3 - 4*w - 1],\
[109, 109, -w^2 + 2*w + 4],\
[131, 131, w^4 - w^3 - 6*w^2 + w + 6],\
[137, 137, w^3 - w^2 - 5*w + 3],\
[149, 149, w^4 - 3*w^3 - 3*w^2 + 9*w + 6],\
[149, 149, -w^4 + 6*w^2 + 3*w],\
[163, 163, 2*w^2 - w - 5],\
[167, 167, -2*w^4 + 3*w^3 + 8*w^2 - 4*w - 3],\
[169, 13, w^4 - 3*w^3 + 6*w - 3],\
[169, 13, w^3 - 4*w - 4],\
[173, 173, w^4 - w^3 - 4*w^2 + 2*w + 1],\
[173, 173, -w^4 + 3*w^3 + 3*w^2 - 7*w - 5],\
[173, 173, w^4 - w^3 - 4*w^2 + w - 1],\
[179, 179, w^2 - 3*w - 2],\
[191, 191, -w^4 + 3*w^3 + w^2 - 6*w - 1],\
[193, 193, -w^3 + 2*w^2 + w - 4],\
[193, 193, -w^4 + 3*w^3 + 3*w^2 - 8*w - 1],\
[199, 199, w^4 - 3*w^3 - w^2 + 7*w],\
[199, 199, -2*w^4 + 5*w^3 + 5*w^2 - 10*w - 2],\
[199, 199, w^4 - 3*w^3 + 5*w - 4],\
[211, 211, 2*w^3 - 2*w^2 - 7*w - 1],\
[211, 211, -2*w^3 + 3*w^2 + 5*w - 2],\
[223, 223, -2*w^4 + 3*w^3 + 10*w^2 - 7*w - 9],\
[227, 227, w^3 - w^2 - 4*w + 3],\
[229, 229, w^4 - 4*w^3 + 10*w - 5],\
[229, 229, -2*w^4 + 4*w^3 + 7*w^2 - 9*w - 8],\
[239, 239, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 3],\
[241, 241, w^3 - w^2 - 2*w - 2],\
[257, 257, -w^4 + w^3 + 6*w^2 - 4*w - 4],\
[263, 263, 2*w^3 - 4*w^2 - 5*w + 6],\
[269, 269, w^3 - 2*w^2 - 4*w + 4],\
[269, 269, -2*w^4 + 2*w^3 + 11*w^2 - 3*w - 10],\
[271, 271, w^3 - 5*w],\
[281, 281, w^4 - w^3 - 5*w^2 + 3*w + 3],\
[281, 281, -2*w^4 + 4*w^3 + 6*w^2 - 6*w - 3],\
[281, 281, w^3 - 5*w - 1],\
[283, 283, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 3],\
[289, 17, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 2],\
[293, 293, 2*w^4 - 5*w^3 - 7*w^2 + 14*w + 9],\
[311, 311, -w^4 + 2*w^3 + 2*w^2 - 4*w + 3],\
[313, 313, w^3 - 6*w],\
[331, 331, w^4 - 2*w^3 - 2*w^2 + w - 2],\
[359, 359, w^4 - w^3 - 5*w^2 + 2*w - 1],\
[373, 373, -w^4 + 3*w^3 + 2*w^2 - 8*w - 3],\
[373, 373, 2*w^3 - 4*w^2 - 3*w + 3],\
[379, 379, -w^4 + 4*w^3 - w^2 - 9*w + 3],\
[379, 379, -w^4 + 3*w^3 + w^2 - 6*w - 2],\
[379, 379, w^4 + w^3 - 7*w^2 - 6*w + 3],\
[383, 383, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],\
[383, 383, w^4 - 5*w^2 - 3*w + 3],\
[383, 383, 2*w^4 - 5*w^3 - 3*w^2 + 8*w],\
[389, 389, 3*w^4 - 7*w^3 - 8*w^2 + 16*w + 1],\
[397, 397, -2*w^4 + 3*w^3 + 8*w^2 - 3*w - 7],\
[397, 397, -w^4 + 2*w^3 + 3*w^2 - 4*w - 5],\
[397, 397, -2*w^4 + 3*w^3 + 7*w^2 - 4*w - 3],\
[409, 409, -w^4 + w^3 + 7*w^2 - 5*w - 9],\
[419, 419, w^4 - 2*w^3 - w^2 + 2*w - 2],\
[419, 419, 2*w^4 - 3*w^3 - 7*w^2 + 5*w + 2],\
[419, 419, -w^4 + 4*w^3 - 9*w - 2],\
[433, 433, -w^4 + 3*w^3 - 8*w + 4],\
[433, 433, w^4 - 7*w^2 - w + 6],\
[439, 439, 3*w^4 - 6*w^3 - 10*w^2 + 11*w + 7],\
[449, 449, -2*w^4 + 5*w^3 + 5*w^2 - 11*w - 4],\
[457, 457, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 2],\
[463, 463, 2*w^4 - 5*w^3 - 4*w^2 + 8*w + 1],\
[467, 467, 2*w^4 - 4*w^3 - 6*w^2 + 6*w + 1],\
[479, 479, 3*w^4 - 4*w^3 - 12*w^2 + 4*w + 7],\
[479, 479, 2*w^4 - 6*w^3 - 4*w^2 + 14*w + 1],\
[491, 491, w^4 - 6*w^2 - w + 4],\
[503, 503, 2*w^4 - w^3 - 11*w^2 - 3*w + 8],\
[503, 503, 2*w^3 - 5*w^2 - 2*w + 6],\
[509, 509, w^4 - 5*w^2 - 6*w],\
[509, 509, w^4 - 3*w^3 - 3*w^2 + 9*w + 1],\
[523, 523, 3*w^4 - 7*w^3 - 7*w^2 + 12*w + 3],\
[523, 523, 3*w^4 - 8*w^3 - 7*w^2 + 17*w + 3],\
[523, 523, w^4 - 7*w^2 - 2*w + 4],\
[529, 23, -w^4 + w^3 + 7*w^2 - 2*w - 9],\
[541, 541, w^4 - 3*w^3 - 3*w^2 + 5*w + 2],\
[547, 547, -w^3 + 3*w^2 + 3*w - 4],\
[557, 557, 2*w^3 - 4*w^2 - 6*w + 9],\
[577, 577, 2*w^4 - 2*w^3 - 11*w^2 + 5*w + 10],\
[577, 577, w^4 - 4*w^3 - w^2 + 9*w],\
[577, 577, 2*w^4 - 3*w^3 - 9*w^2 + 4*w + 8],\
[577, 577, w^4 - 7*w^2 - 3*w + 7],\
[577, 577, -3*w^4 + 3*w^3 + 13*w^2 - w - 4],\
[593, 593, 3*w - 2],\
[601, 601, 2*w^4 - 4*w^3 - 7*w^2 + 5*w + 6],\
[601, 601, -w^4 + 3*w^3 + w^2 - 7*w - 1],\
[607, 607, w^3 - w^2 - 2*w - 3],\
[625, 5, 2*w^4 - 5*w^3 - 8*w^2 + 14*w + 11],\
[631, 631, -w^2 + w - 2],\
[631, 631, w^4 - 7*w^2 - w + 8],\
[643, 643, -2*w^4 + 4*w^3 + 9*w^2 - 10*w - 8],\
[647, 647, 3*w^4 - 6*w^3 - 10*w^2 + 12*w + 8],\
[653, 653, -3*w^4 + 6*w^3 + 11*w^2 - 12*w - 9],\
[653, 653, w^4 - w^3 - 2*w^2 - 3*w - 3],\
[653, 653, -2*w^4 + 4*w^3 + 7*w^2 - 6*w - 8],\
[701, 701, w^4 - w^3 - 2*w^2 - w - 5],\
[719, 719, 2*w^4 - 5*w^3 - 6*w^2 + 14*w + 2],\
[727, 727, 2*w^4 - 5*w^3 - 5*w^2 + 9*w + 3],\
[733, 733, 2*w^4 - 5*w^3 - 5*w^2 + 13*w + 3],\
[739, 739, -2*w^3 + 4*w^2 + 4*w - 5],\
[743, 743, w^4 - 2*w^3 - 3*w^2 + 6*w - 3],\
[743, 743, 2*w^4 - 2*w^3 - 11*w^2 + 2*w + 10],\
[757, 757, -w^4 + 2*w^3 + 5*w^2 - 6*w - 10],\
[757, 757, 2*w^4 - 6*w^3 - 2*w^2 + 11*w - 4],\
[761, 761, 3*w^3 - 4*w^2 - 9*w + 3],\
[769, 769, 3*w^3 - 5*w^2 - 9*w + 7],\
[769, 769, -2*w^4 + 4*w^3 + 7*w^2 - 11*w - 2],\
[773, 773, 2*w^4 - 5*w^3 - 6*w^2 + 12*w + 1],\
[773, 773, 3*w^4 - 4*w^3 - 12*w^2 + 3*w + 5],\
[773, 773, 2*w^3 - w^2 - 10*w - 4],\
[797, 797, -w^4 + 4*w^3 + 2*w^2 - 12*w - 3],\
[811, 811, w^4 - 2*w^3 - 6*w^2 + 8*w + 4],\
[823, 823, w^4 - 4*w^3 - w^2 + 10*w + 1],\
[823, 823, 2*w^4 - 4*w^3 - 4*w^2 + 4*w - 3],\
[827, 827, -w^4 + 2*w^3 + 2*w^2 - w + 3],\
[827, 827, 2*w^4 - 4*w^3 - 5*w^2 + 7*w + 2],\
[829, 829, -w^4 + 4*w^3 + 2*w^2 - 10*w - 2],\
[829, 829, 3*w^3 - 3*w^2 - 11*w - 2],\
[829, 829, w^4 - 2*w^3 - 3*w^2 + 2*w - 3],\
[839, 839, -w^4 + 5*w^3 - 3*w^2 - 10*w + 5],\
[839, 839, -w^4 + w^3 + 5*w^2 - 9],\
[853, 853, 2*w^4 - 6*w^3 - 5*w^2 + 15*w + 4],\
[859, 859, -3*w^4 + 8*w^3 + 6*w^2 - 18*w + 5],\
[863, 863, w^4 - w^3 - 6*w^2 + 5*w + 2],\
[877, 877, 3*w^3 - 4*w^2 - 8*w + 1],\
[881, 881, 3*w^4 - 7*w^3 - 9*w^2 + 15*w + 3],\
[881, 881, -w^3 + 7*w + 1],\
[883, 883, -2*w^4 + 6*w^3 + 3*w^2 - 15*w],\
[887, 887, w^4 - w^3 - 5*w^2 + w - 1],\
[907, 907, 3*w^4 - 5*w^3 - 11*w^2 + 9*w + 5],\
[919, 919, w^4 - 2*w^3 - 6*w^2 + 8*w + 7],\
[937, 937, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 3],\
[937, 937, -3*w^4 + 8*w^3 + 6*w^2 - 16*w + 1],\
[937, 937, 3*w^4 - 5*w^3 - 11*w^2 + 7*w + 4],\
[947, 947, w^3 - w^2 - 7*w + 2],\
[947, 947, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],\
[967, 967, -w^4 + 4*w^3 - 11*w],\
[971, 971, 3*w^4 - 6*w^3 - 8*w^2 + 10*w],\
[971, 971, w^4 - 8*w^2 + 2*w + 9],\
[971, 971, w^4 - 4*w^3 + 2*w^2 + 7*w - 4],\
[977, 977, -w^4 + 4*w^3 - 12*w + 5],\
[983, 983, -w^4 + w^3 + 4*w^2 + 3*w - 2],\
[983, 983, -2*w^4 + 3*w^3 + 8*w^2 - 6*w - 4],\
[997, 997, 2*w^4 - 4*w^3 - 5*w^2 + 5*w - 3],\
[997, 997, w^4 + w^3 - 5*w^2 - 9*w - 2]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 - 3*x^2 - 9*x + 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [-1, -1, -1, e, -e^2 + 2*e + 5, e^2 - 4*e - 5, -e - 2, e^2 - e - 9, e^2 - 5*e - 5, e^2 - 2*e - 8, -2*e^2 + 6*e + 9, e^2 - 13, -2*e^2 + 6*e + 4, -2*e^2 + 4*e + 14, e^2 - 6*e - 3, -e^2 + 2*e + 9, -e^2 + 4*e + 9, 2*e - 10, -e^2 + 6*e + 1, -2*e^2 + 4*e + 10, -2*e^2 + 9*e + 8, e^2 - 2*e - 5, -e^2 + e + 7, 2*e^2 - 10*e - 11, e^2 - 3*e + 9, -2*e^2 + 10*e + 17, e^2 - 6*e - 8, 2*e^2 - 6*e - 10, -2*e^2 + 10*e + 8, e^2 - 10*e, 3*e^2 - 6*e - 22, -2*e^2 + e + 24, -2*e^2 + 10*e + 12, e^2 - 3*e - 1, e^2 - 2*e + 3, -2*e^2 + 4*e + 17, -e^2 + 2*e + 9, 2*e^2 - 10*e - 4, -e^2 + 4*e + 13, 3*e^2 - 11*e - 13, e^2 - 2*e - 19, -5*e^2 + 14*e + 21, 3*e^2 - 13*e - 19, -3*e^2 + 14*e + 21, -2*e^2 + 6*e + 4, e^2 + 4*e - 19, e^2 - 6*e + 11, -e^2 + 4*e + 17, -e^2 + 4*e + 5, e^2 - 6*e - 13, -2*e + 9, e^2 - 2*e - 18, -3*e^2 + 14*e + 15, e^2 - 4*e - 17, 2*e^2 - 12*e - 19, -3*e^2 + 13*e + 23, e^2 - 4*e - 1, 4*e^2 - 16*e - 26, e^2 - 7*e - 21, e^2 - 10*e + 3, -4, -2*e^2 + 5*e + 24, 2*e^2 - 6*e - 15, 2*e - 24, -4*e^2 + 14*e + 16, 4*e^2 - 10*e - 20, e^2 - 2*e + 11, -5*e^2 + 13*e + 25, 4*e^2 - 14*e - 22, -e^2 + 2*e + 27, 4*e^2 - 15*e - 36, 2*e - 7, 5*e^2 - 18*e - 27, -2*e^2 + 12*e + 26, -3*e^2 + 11*e + 13, -4*e^2 + 10*e + 16, -5*e^2 + 16*e + 11, e^2 - e - 7, 4*e - 24, 5*e^2 - 24*e - 33, -2*e^2 + 11*e - 4, -e^2 - 4*e + 24, 6*e - 16, 3*e^2 - 18*e - 21, e^2 + 2*e - 34, -e^2 + 6*e - 9, e^2 - 2*e - 21, -e, -3*e^2 + 10*e + 2, 5*e^2 - 16*e - 15, e^2 + 4*e - 33, -e^2 + 4*e + 17, -e^2 + 5*e + 5, -3*e^2 + 10*e + 43, -2*e^2 + 8*e + 18, -2*e^2 + 16*e, 2*e^2 - 11*e - 8, e^2 - 2*e + 10, e^2 - 38, -e^2 + 8*e - 3, 3*e^2 - 14*e - 13, -5*e + 2, 5*e^2 - 10*e - 45, -3*e^2 + 18*e + 25, -2*e^2 + 10*e + 29, 3*e^2 - 12*e - 19, -e^2 - 4*e + 17, 2*e^2 - 5*e - 14, -e^2 - 4*e + 11, e^2 - 2*e - 29, 2*e^2 - 14*e - 8, -4*e^2 + 6*e + 40, 2*e^2 - 4*e - 40, e^2 + 4*e - 39, e^2 - 6, 2*e^2 - 4*e - 24, 2*e^2 + 4*e - 36, -3*e^2 + 6*e - 2, -e^2 + 2*e + 13, e^2 - 5*e - 19, -3*e^2 + 6*e + 27, 2*e^2 - 14*e - 8, -8*e^2 + 28*e + 32, -4*e, -7*e^2 + 32*e + 35, 6*e^2 - 22*e - 20, 5*e^2 - 26*e - 37, -5*e^2 + 23*e + 13, 10*e^2 - 33*e - 50, -5*e^2 + 17*e + 5, 2*e^2 - 9*e + 8, 2*e^2 - 2*e - 20, 6*e + 1, 2*e^2 - 4*e - 24, 7*e^2 - 32*e - 48, -e^2 - 4*e + 15, -4*e^2 + 26*e + 17, 8*e^2 - 25*e - 54, 2*e^2 - 6*e + 9, 5*e^2 - 17*e - 15, 2*e^2 - 12, e^2 - 6*e - 25, 3*e^2 - 8*e - 18, -6*e - 19, 7*e^2 - 21*e - 31, -3*e^2 + 12*e + 39, 3*e^2 - 2*e - 52, -4*e^2 + 4*e + 30, -e^2 - 6*e + 21, 5*e^2 - 12*e - 25, -e^2 + 4*e - 27, 8*e, -6*e^2 + 19*e + 22, -4*e^2 + 2*e + 38, 9*e^2 - 24*e - 55, -5*e^2 + 24*e + 49, -3*e^2 + 8*e + 11, -e^2 - 8*e + 18, 2*e^2 - 11*e + 6, -4*e^2 + 12*e - 2, -e^2 - 2*e + 33, -8*e - 21, -3*e^2 + 8*e + 19, 3*e^2 - 12*e - 43, -8*e - 5, -3*e^2 + 12*e - 5, -2*e^2 + e + 26, -6*e^2 + 10*e + 40, -e^2 - 8*e + 5, 2*e^2 - 6*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([3, 3, w - 1])] = 1
AL_eigenvalues[ZF.ideal([5, 5, w^2 - w - 2])] = 1

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