/*
  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, -6, 3, 10, -3, -3, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([37,37,-w^4 + 4*w^3 - w^2 - 7*w + 3])

primes_array = [
[3, 3, w - 1],\
[17, 17, -w^2 + 2*w + 1],\
[17, 17, -w^3 + w^2 + 4*w],\
[19, 19, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + w - 4],\
[19, 19, w^2 - w - 1],\
[37, 37, w^4 - 2*w^3 - 3*w^2 + 3*w + 2],\
[37, 37, w^4 - 4*w^3 + w^2 + 7*w - 3],\
[53, 53, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 9*w - 6],\
[53, 53, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 3*w + 2],\
[64, 2, -2],\
[71, 71, -w^5 + 4*w^4 - w^3 - 8*w^2 + 3*w - 1],\
[71, 71, 2*w^5 - 5*w^4 - 8*w^3 + 15*w^2 + 12*w - 6],\
[71, 71, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 9*w - 7],\
[73, 73, -2*w^5 + 6*w^4 + 5*w^3 - 16*w^2 - 7*w + 3],\
[73, 73, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 6*w + 6],\
[89, 89, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 8*w - 4],\
[89, 89, w^5 - w^4 - 9*w^3 + 7*w^2 + 16*w - 6],\
[89, 89, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + 3*w - 3],\
[89, 89, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 10*w - 6],\
[107, 107, w^5 - 2*w^4 - 7*w^3 + 10*w^2 + 12*w - 6],\
[107, 107, 2*w^5 - 5*w^4 - 7*w^3 + 14*w^2 + 9*w - 5],\
[107, 107, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 6*w - 3],\
[107, 107, w^4 - 2*w^3 - 3*w^2 + 4*w + 1],\
[109, 109, 2*w^5 - 5*w^4 - 8*w^3 + 15*w^2 + 13*w - 6],\
[109, 109, -w^4 + 4*w^3 - w^2 - 9*w + 3],\
[127, 127, w^4 - 3*w^3 + 5*w - 4],\
[127, 127, -w^5 + 3*w^4 + w^3 - 6*w^2 + w + 3],\
[127, 127, 2*w^5 - 5*w^4 - 8*w^3 + 16*w^2 + 11*w - 6],\
[127, 127, w^5 - 4*w^4 + 2*w^3 + 7*w^2 - 7*w],\
[163, 163, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 5],\
[163, 163, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 2*w + 5],\
[163, 163, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 8*w + 4],\
[163, 163, w^5 - 4*w^4 + 10*w^2 - w - 4],\
[179, 179, -w^5 + 3*w^4 + 3*w^3 - 7*w^2 - 7*w - 1],\
[179, 179, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 5*w - 7],\
[179, 179, -2*w^5 + 5*w^4 + 10*w^3 - 19*w^2 - 18*w + 10],\
[179, 179, -2*w^5 + 5*w^4 + 9*w^3 - 17*w^2 - 14*w + 9],\
[197, 197, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 3*w - 7],\
[197, 197, -w^5 + 4*w^4 - w^3 - 7*w^2 + 3*w - 2],\
[197, 197, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 3],\
[197, 197, -2*w^5 + 8*w^4 - 2*w^3 - 17*w^2 + 10*w + 1],\
[197, 197, -w^4 + 3*w^3 + 2*w^2 - 6*w - 2],\
[197, 197, w^5 - 4*w^4 + w^3 + 9*w^2 - 4*w - 2],\
[199, 199, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 10*w + 4],\
[199, 199, -w^4 + 3*w^3 + 2*w^2 - 9*w + 1],\
[233, 233, 2*w^5 - 4*w^4 - 12*w^3 + 17*w^2 + 19*w - 9],\
[233, 233, -w^5 + w^4 + 8*w^3 - 6*w^2 - 13*w + 6],\
[233, 233, 2*w^5 - 6*w^4 - 6*w^3 + 19*w^2 + 9*w - 7],\
[233, 233, w^5 - 4*w^4 + 11*w^2 - 4*w - 2],\
[251, 251, -2*w^5 + 5*w^4 + 10*w^3 - 18*w^2 - 18*w + 7],\
[251, 251, w^3 - w^2 - 3*w - 2],\
[251, 251, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 3*w - 3],\
[251, 251, -w^5 + 4*w^4 - 11*w^2 + 4*w + 3],\
[269, 269, w^5 - 3*w^4 - w^3 + 6*w^2 - 2*w - 3],\
[269, 269, 2*w^5 - 7*w^4 - 3*w^3 + 20*w^2 + 3*w - 8],\
[271, 271, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 2*w - 7],\
[271, 271, -w^4 + 4*w^3 - w^2 - 10*w + 6],\
[289, 17, -w^5 + 3*w^4 + w^3 - 5*w^2 - 2],\
[289, 17, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 6*w - 4],\
[307, 307, 2*w^5 - 5*w^4 - 10*w^3 + 19*w^2 + 16*w - 8],\
[307, 307, w^5 - 4*w^4 + 2*w^3 + 7*w^2 - 6*w - 1],\
[307, 307, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 3*w + 2],\
[307, 307, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 7*w - 9],\
[361, 19, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 8*w - 5],\
[361, 19, w^5 - 3*w^4 - w^3 + 5*w^2 + 3],\
[379, 379, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 9*w - 5],\
[379, 379, -2*w + 3],\
[397, 397, -w^5 + 5*w^4 - 6*w^3 - 6*w^2 + 14*w - 5],\
[397, 397, -w^5 + 3*w^4 + w^3 - 4*w^2 - w - 5],\
[397, 397, 3*w^5 - 9*w^4 - 7*w^3 + 23*w^2 + 9*w - 6],\
[397, 397, -w^4 + 5*w^3 - 2*w^2 - 10*w + 1],\
[431, 431, -3*w^5 + 8*w^4 + 11*w^3 - 24*w^2 - 18*w + 9],\
[431, 431, 2*w^5 - 6*w^4 - 3*w^3 + 13*w^2 + w - 3],\
[431, 431, -3*w^5 + 8*w^4 + 10*w^3 - 23*w^2 - 14*w + 8],\
[431, 431, -2*w^5 + 6*w^4 + 4*w^3 - 14*w^2 - 5*w],\
[433, 433, -3*w^5 + 9*w^4 + 8*w^3 - 25*w^2 - 11*w + 9],\
[433, 433, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 - 3*w + 7],\
[433, 433, 2*w^5 - 5*w^4 - 7*w^3 + 13*w^2 + 11*w - 4],\
[433, 433, -w^4 + 4*w^3 + w^2 - 10*w - 2],\
[449, 449, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 + w + 3],\
[449, 449, -w^4 + 4*w^3 - 2*w^2 - 5*w + 5],\
[449, 449, -w^5 + 4*w^4 - w^3 - 8*w^2 + 3*w - 2],\
[449, 449, -w^3 + 5*w + 3],\
[449, 449, w^3 - 5*w],\
[449, 449, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - w - 6],\
[467, 467, -2*w^4 + 7*w^3 - 16*w + 7],\
[467, 467, -2*w^4 + 6*w^3 + w^2 - 10*w + 3],\
[487, 487, -w^5 + 2*w^4 + 6*w^3 - 10*w^2 - 8*w + 9],\
[487, 487, 2*w^5 - 6*w^4 - 5*w^3 + 15*w^2 + 7*w - 3],\
[487, 487, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 8*w - 6],\
[487, 487, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 8*w + 1],\
[503, 503, 2*w^5 - 7*w^4 - 3*w^3 + 19*w^2 + 3*w - 6],\
[503, 503, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 4*w - 6],\
[503, 503, 2*w^5 - 5*w^4 - 9*w^3 + 17*w^2 + 13*w - 7],\
[503, 503, -w^5 + 4*w^4 - 10*w^2 + 3*w - 1],\
[521, 521, w^5 - 4*w^4 + 11*w^2 - w - 6],\
[521, 521, w^5 - w^4 - 9*w^3 + 8*w^2 + 15*w - 6],\
[521, 521, -w^5 + 4*w^4 - 2*w^3 - 8*w^2 + 9*w],\
[521, 521, -2*w^5 + 4*w^4 + 11*w^3 - 15*w^2 - 17*w + 9],\
[523, 523, 3*w^5 - 8*w^4 - 10*w^3 + 24*w^2 + 12*w - 10],\
[523, 523, -w^4 + 3*w^3 - w^2 - 4*w + 5],\
[523, 523, -2*w^5 + 4*w^4 + 11*w^3 - 14*w^2 - 17*w + 7],\
[523, 523, -w^5 + 5*w^4 - 4*w^3 - 9*w^2 + 9*w + 1],\
[577, 577, -3*w^5 + 8*w^4 + 10*w^3 - 22*w^2 - 14*w + 7],\
[577, 577, -2*w^5 + 7*w^4 + w^3 - 15*w^2 + 2*w + 2],\
[593, 593, 2*w^5 - 5*w^4 - 9*w^3 + 17*w^2 + 15*w - 10],\
[593, 593, w^3 - w^2 - 2*w - 2],\
[593, 593, -3*w^5 + 9*w^4 + 9*w^3 - 27*w^2 - 14*w + 10],\
[593, 593, 2*w^5 - 6*w^4 - 6*w^3 + 19*w^2 + 8*w - 7],\
[613, 613, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 12*w - 5],\
[613, 613, w^5 - w^4 - 9*w^3 + 8*w^2 + 13*w - 7],\
[613, 613, w^5 - 5*w^4 + 3*w^3 + 11*w^2 - 6*w - 2],\
[613, 613, -w^4 + 3*w^3 + w^2 - 5*w + 4],\
[613, 613, w^5 - w^4 - 9*w^3 + 7*w^2 + 14*w - 4],\
[613, 613, w^5 - 2*w^4 - 7*w^3 + 11*w^2 + 12*w - 8],\
[631, 631, 2*w^5 - 7*w^4 + 15*w^2 - 6*w - 2],\
[631, 631, -3*w^5 + 8*w^4 + 10*w^3 - 23*w^2 - 12*w + 10],\
[631, 631, w^5 - 3*w^4 + w^3 + 3*w^2 - 6*w + 3],\
[631, 631, -3*w^5 + 8*w^4 + 13*w^3 - 29*w^2 - 21*w + 15],\
[683, 683, -w^5 + 4*w^4 - w^3 - 8*w^2 + 2*w - 1],\
[683, 683, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 11*w - 4],\
[701, 701, -w^5 + 3*w^4 + w^3 - 7*w^2 + 3*w + 2],\
[701, 701, 2*w^2 - 3*w - 4],\
[719, 719, 2*w^5 - 5*w^4 - 8*w^3 + 16*w^2 + 13*w - 8],\
[719, 719, w^5 - 5*w^4 + 4*w^3 + 10*w^2 - 10*w - 1],\
[719, 719, w^5 - w^4 - 9*w^3 + 6*w^2 + 16*w - 2],\
[719, 719, 2*w^5 - 5*w^4 - 9*w^3 + 18*w^2 + 15*w - 10],\
[739, 739, -2*w^5 + 9*w^4 - 6*w^3 - 15*w^2 + 16*w - 7],\
[739, 739, 4*w^5 - 9*w^4 - 19*w^3 + 30*w^2 + 28*w - 14],\
[757, 757, 2*w^5 - 5*w^4 - 9*w^3 + 16*w^2 + 17*w - 7],\
[757, 757, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 6*w - 6],\
[757, 757, 2*w^5 - 6*w^4 - 7*w^3 + 20*w^2 + 11*w - 6],\
[757, 757, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 1],\
[773, 773, 2*w^5 - 8*w^4 + 3*w^3 + 16*w^2 - 11*w],\
[773, 773, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 12*w - 6],\
[809, 809, 2*w^5 - 6*w^4 - 3*w^3 + 13*w^2 + 2*w - 3],\
[809, 809, -2*w^3 + 5*w^2 + 4*w - 6],\
[811, 811, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],\
[811, 811, 2*w^5 - 7*w^4 - w^3 + 17*w^2 - 3*w - 6],\
[827, 827, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 4],\
[827, 827, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 5],\
[829, 829, 2*w^5 - 4*w^4 - 11*w^3 + 15*w^2 + 18*w - 9],\
[829, 829, 2*w^5 - 6*w^4 - 7*w^3 + 19*w^2 + 13*w - 7],\
[829, 829, -2*w^5 + 9*w^4 - 5*w^3 - 18*w^2 + 15*w],\
[829, 829, 3*w^5 - 9*w^4 - 9*w^3 + 28*w^2 + 11*w - 11],\
[829, 829, w^3 - 2*w^2 - 3*w - 1],\
[829, 829, -2*w^5 + 4*w^4 + 12*w^3 - 17*w^2 - 20*w + 10],\
[863, 863, -w^4 + 4*w^3 + w^2 - 11*w],\
[863, 863, w^4 - 4*w^3 - w^2 + 11*w + 1],\
[919, 919, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 4*w + 1],\
[919, 919, -w^5 + 4*w^4 - w^3 - 8*w^2 + 6*w - 4],\
[919, 919, -w^4 + 5*w^3 - w^2 - 12*w + 1],\
[919, 919, -2*w^5 + 5*w^4 + 7*w^3 - 13*w^2 - 11*w + 6],\
[937, 937, 2*w^5 - 6*w^4 - 3*w^3 + 12*w^2 + 3*w],\
[937, 937, 2*w^5 - 4*w^4 - 10*w^3 + 12*w^2 + 16*w - 5],\
[937, 937, w^5 - 4*w^4 + 8*w^2 + w + 1],\
[937, 937, 2*w^4 - 8*w^3 + 2*w^2 + 16*w - 5],\
[953, 953, -w^5 + 4*w^4 - 11*w^2 + 3*w + 7],\
[953, 953, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 13*w - 7],\
[953, 953, -w^5 + 3*w^4 + w^3 - 7*w^2 + 4*w + 1],\
[953, 953, 2*w^5 - 8*w^4 + 3*w^3 + 16*w^2 - 11*w - 1],\
[971, 971, w^4 - w^3 - 5*w^2 + 2*w + 4],\
[971, 971, -w^4 + w^3 + 4*w^2 - w + 1],\
[971, 971, -2*w^5 + 5*w^4 + 10*w^3 - 20*w^2 - 15*w + 9],\
[971, 971, w^4 - 2*w^3 - 4*w^2 + 6*w + 1],\
[991, 991, -w - 3],\
[991, 991, -2*w^5 + 7*w^4 + 3*w^3 - 19*w^2 - 5*w + 6],\
[991, 991, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 5*w + 7],\
[991, 991, 2*w^5 - 7*w^4 - 2*w^3 + 16*w^2 + 2*w],\
[1009, 1009, 2*w^4 - 6*w^3 - 2*w^2 + 11*w],\
[1009, 1009, -w^5 + w^4 + 9*w^3 - 7*w^2 - 15*w + 6],\
[1061, 1061, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 4*w + 2],\
[1061, 1061, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + 2*w - 7],\
[1063, 1063, 2*w^5 - 5*w^4 - 10*w^3 + 18*w^2 + 18*w - 9],\
[1063, 1063, -2*w^5 + 5*w^4 + 8*w^3 - 15*w^2 - 14*w + 8],\
[1063, 1063, -w^5 + 4*w^4 - 11*w^2 + 4*w + 5],\
[1063, 1063, w^5 - 2*w^4 - 7*w^3 + 10*w^2 + 14*w - 5],\
[1117, 1117, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 7*w + 5],\
[1117, 1117, -w^5 + 2*w^4 + 3*w^3 - 2*w^2 - 3*w - 3],\
[1117, 1117, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 5*w - 8],\
[1117, 1117, w^5 - 4*w^4 + 3*w^3 + 4*w^2 - 7*w + 4],\
[1151, 1151, -w^4 + 4*w^3 - 10*w - 1],\
[1151, 1151, w^5 - 3*w^4 - w^3 + 8*w^2 - 3*w - 3],\
[1151, 1151, 3*w^5 - 8*w^4 - 8*w^3 + 19*w^2 + 8*w - 3],\
[1151, 1151, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 12*w - 5],\
[1171, 1171, w^5 - 2*w^4 - 5*w^3 + 5*w^2 + 7*w + 1],\
[1171, 1171, 3*w^5 - 10*w^4 - 4*w^3 + 25*w^2 + 3*w - 7],\
[1171, 1171, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 8*w + 3],\
[1171, 1171, -2*w^5 + 5*w^4 + 7*w^3 - 16*w^2 - 7*w + 6],\
[1187, 1187, -3*w^5 + 8*w^4 + 12*w^3 - 27*w^2 - 18*w + 11],\
[1187, 1187, w^4 - 2*w^3 - 2*w^2 + 4*w - 3],\
[1259, 1259, 2*w^5 - 6*w^4 - 5*w^3 + 16*w^2 + 9*w - 5],\
[1259, 1259, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 9*w - 7],\
[1277, 1277, -3*w^5 + 11*w^4 + w^3 - 25*w^2 + 2*w + 3],\
[1277, 1277, -w^4 + w^3 + 5*w^2 - w - 6],\
[1277, 1277, 2*w^5 - 6*w^4 - 4*w^3 + 13*w^2 + 5*w + 1],\
[1277, 1277, w^4 - 5*w^3 + 3*w^2 + 9*w - 7],\
[1277, 1277, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - 4*w - 1],\
[1277, 1277, 3*w^5 - 9*w^4 - 6*w^3 + 22*w^2 + 5*w - 4],\
[1279, 1279, -2*w^5 + 4*w^4 + 12*w^3 - 17*w^2 - 18*w + 11],\
[1279, 1279, -3*w^5 + 11*w^4 - 23*w^2 + 5*w + 2],\
[1279, 1279, w^3 + w^2 - 5*w - 2],\
[1279, 1279, -2*w^5 + 6*w^4 + 7*w^3 - 21*w^2 - 9*w + 9],\
[1279, 1279, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 9*w + 4],\
[1279, 1279, -2*w^3 + 5*w^2 + 3*w - 8],\
[1297, 1297, -2*w^5 + 4*w^4 + 11*w^3 - 14*w^2 - 19*w + 7],\
[1297, 1297, -w^5 + w^4 + 10*w^3 - 9*w^2 - 19*w + 7],\
[1331, 11, -3*w^5 + 7*w^4 + 16*w^3 - 26*w^2 - 28*w + 9],\
[1331, 11, w^3 - w^2 - w - 3],\
[1367, 1367, 2*w^5 - 6*w^4 - 4*w^3 + 13*w^2 + 5*w],\
[1367, 1367, w^3 - 6*w],\
[1367, 1367, 3*w^5 - 9*w^4 - 6*w^3 + 22*w^2 + 5*w - 5],\
[1367, 1367, w^3 - 6*w - 2],\
[1369, 37, 2*w^5 - 6*w^4 - 7*w^3 + 20*w^2 + 10*w - 8],\
[1369, 37, 3*w^5 - 9*w^4 - 8*w^3 + 25*w^2 + 10*w - 8],\
[1423, 1423, 3*w^5 - 10*w^4 - 5*w^3 + 26*w^2 + 6*w - 3],\
[1423, 1423, -2*w^4 + 7*w^3 - w^2 - 13*w + 4],\
[1459, 1459, -w^5 + 4*w^4 - 3*w^3 - 4*w^2 + 8*w - 5],\
[1459, 1459, 2*w^5 - 5*w^4 - 6*w^3 + 11*w^2 + 8*w],\
[1459, 1459, -2*w^5 + 7*w^4 + 4*w^3 - 22*w^2 - 2*w + 7],\
[1459, 1459, -3*w^5 + 8*w^4 + 14*w^3 - 29*w^2 - 24*w + 11],\
[1493, 1493, -3*w^5 + 9*w^4 + 9*w^3 - 25*w^2 - 16*w + 6],\
[1493, 1493, -2*w^5 + 3*w^4 + 15*w^3 - 14*w^2 - 27*w + 9],\
[1493, 1493, w^5 - 4*w^4 + 2*w^3 + 5*w^2 - 5*w + 3],\
[1493, 1493, -3*w^5 + 8*w^4 + 9*w^3 - 21*w^2 - 11*w + 7],\
[1531, 1531, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 + w + 4],\
[1531, 1531, 2*w^5 - 6*w^4 - 6*w^3 + 17*w^2 + 11*w - 7],\
[1549, 1549, w^5 - 4*w^4 - 3*w^3 + 14*w^2 + 7*w - 4],\
[1549, 1549, -4*w^5 + 10*w^4 + 18*w^3 - 34*w^2 - 28*w + 15],\
[1567, 1567, -3*w^5 + 6*w^4 + 17*w^3 - 23*w^2 - 25*w + 9],\
[1567, 1567, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 12*w + 4],\
[1583, 1583, w^5 - 2*w^4 - 8*w^3 + 11*w^2 + 15*w - 7],\
[1583, 1583, w^5 - 4*w^4 - w^3 + 13*w^2 - w - 7],\
[1601, 1601, -w^4 + 4*w^3 - 3*w^2 - 6*w + 7],\
[1601, 1601, 3*w^5 - 8*w^4 - 9*w^3 + 22*w^2 + 10*w - 8],\
[1619, 1619, 2*w^5 - 5*w^4 - 11*w^3 + 19*w^2 + 21*w - 6],\
[1619, 1619, w^5 - 2*w^4 - 8*w^3 + 13*w^2 + 12*w - 12],\
[1619, 1619, -4*w^5 + 9*w^4 + 20*w^3 - 32*w^2 - 31*w + 13],\
[1619, 1619, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 10*w - 12],\
[1637, 1637, -w^5 + 2*w^4 + 4*w^3 - 4*w^2 - 5*w - 3],\
[1637, 1637, w^5 - 4*w^4 + 2*w^3 + 6*w^2 - 5*w + 4],\
[1637, 1637, 4*w^5 - 10*w^4 - 17*w^3 + 31*w^2 + 29*w - 12],\
[1637, 1637, -w^5 + 5*w^4 - 5*w^3 - 8*w^2 + 15*w - 1],\
[1657, 1657, w^4 - 2*w^3 - w^2 + w - 4],\
[1657, 1657, 2*w^5 - 5*w^4 - 8*w^3 + 17*w^2 + 9*w - 7],\
[1693, 1693, w^5 - 2*w^4 - 9*w^3 + 14*w^2 + 15*w - 8],\
[1693, 1693, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + w],\
[1709, 1709, 2*w^5 - 7*w^4 - 3*w^3 + 18*w^2 + 4*w - 3],\
[1709, 1709, 3*w^5 - 8*w^4 - 11*w^3 + 25*w^2 + 14*w - 9],\
[1709, 1709, -w^5 + w^4 + 11*w^3 - 11*w^2 - 20*w + 9],\
[1709, 1709, -w^5 + w^4 + 7*w^3 - 3*w^2 - 12*w + 1],\
[1747, 1747, -w^5 + 5*w^4 - 3*w^3 - 9*w^2 + 7*w - 4],\
[1747, 1747, -3*w^5 + 11*w^4 - 24*w^2 + 5*w + 3],\
[1783, 1783, -w^4 + 3*w^3 + 2*w^2 - 5*w - 4],\
[1783, 1783, -2*w^5 + 7*w^4 + 2*w^3 - 18*w^2 - w + 4],\
[1871, 1871, -w^5 + 3*w^4 + 4*w^3 - 11*w^2 - 9*w + 6],\
[1871, 1871, 2*w^5 - 8*w^4 + w^3 + 17*w^2 - 3*w - 2],\
[1871, 1871, 2*w^5 - 6*w^4 - 5*w^3 + 16*w^2 + 9*w - 6],\
[1871, 1871, -3*w^5 + 11*w^4 - w^3 - 23*w^2 + 12*w - 3],\
[1873, 1873, -w^5 + w^4 + 8*w^3 - 5*w^2 - 11*w + 4],\
[1873, 1873, 3*w^5 - 6*w^4 - 15*w^3 + 19*w^2 + 22*w - 9],\
[1889, 1889, 2*w^5 - 5*w^4 - 8*w^3 + 14*w^2 + 13*w - 5],\
[1889, 1889, -2*w^5 + 5*w^4 + 8*w^3 - 15*w^2 - 12*w + 8],\
[1889, 1889, 2*w^5 - 7*w^4 - w^3 + 16*w^2 - 3*w - 5],\
[1889, 1889, w^5 - 4*w^4 + w^3 + 8*w^2 - 4*w - 3],\
[1907, 1907, -3*w^5 + 6*w^4 + 18*w^3 - 25*w^2 - 28*w + 12],\
[1907, 1907, 2*w^5 - 5*w^4 - 7*w^3 + 14*w^2 + 8*w - 1],\
[1979, 1979, 2*w^4 - 9*w^3 + 5*w^2 + 18*w - 11],\
[1979, 1979, 4*w^5 - 10*w^4 - 14*w^3 + 27*w^2 + 20*w - 8],\
[1997, 1997, 3*w^5 - 8*w^4 - 11*w^3 + 25*w^2 + 14*w - 10],\
[1997, 1997, -4*w^5 + 13*w^4 + 5*w^3 - 32*w^2 + w + 4],\
[1997, 1997, 2*w^5 - 7*w^4 - 3*w^3 + 18*w^2 + 4*w - 4],\
[1997, 1997, -4*w^5 + 11*w^4 + 15*w^3 - 35*w^2 - 24*w + 11]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^10 - 109*x^8 + 3826*x^6 - 45740*x^4 + 69883*x^2 - 2588
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [-1014876/33569667991*e^8 + 121920306/33569667991*e^6 - 4667507452/33569667991*e^4 + 58383994609/33569667991*e^2 - 76713410693/33569667991, -29126661/67139335982*e^9 + 3196022099/67139335982*e^7 - 56599327316/33569667991*e^5 + 684415270194/33569667991*e^3 - 2142147259903/67139335982*e, e, -3876972/33569667991*e^8 + 364535597/33569667991*e^6 - 10436115901/33569667991*e^4 + 98713192037/33569667991*e^2 - 192555228656/33569667991, -2057103/33569667991*e^8 + 213188752/33569667991*e^6 - 6520734762/33569667991*e^4 + 49241762880/33569667991*e^2 + 53785671563/33569667991, 7039019/33569667991*e^8 - 699350737/33569667991*e^6 + 21533824816/33569667991*e^4 - 221741115385/33569667991*e^2 + 233196667521/33569667991, 1, 33222936/33569667991*e^9 - 3635130417/33569667991*e^7 + 128232417578/33569667991*e^5 - 1543312101017/33569667991*e^3 + 2409861365990/33569667991*e, 33933151/67139335982*e^9 - 3673811397/67139335982*e^7 + 63990908962/33569667991*e^5 - 765773149718/33569667991*e^3 + 2679156685595/67139335982*e, -452736/33569667991*e^8 + 122263873/33569667991*e^6 - 7962314010/33569667991*e^4 + 147459615985/33569667991*e^2 - 301718185285/33569667991, -8448624/33569667991*e^9 + 904216451/33569667991*e^7 - 31028906075/33569667991*e^5 + 365677072043/33569667991*e^3 - 638586888902/33569667991*e, -72802027/67139335982*e^9 + 7900278731/67139335982*e^7 - 137717489742/33569667991*e^5 + 1626302446331/33569667991*e^3 - 4643550922929/67139335982*e, 27770353/67139335982*e^9 - 3039435373/67139335982*e^7 + 53624653468/33569667991*e^5 - 649675583698/33569667991*e^3 + 2364294630789/67139335982*e, 2996041/33569667991*e^8 - 308521407/33569667991*e^6 + 10045795433/33569667991*e^4 - 121716138422/33569667991*e^2 + 190184137809/33569667991, 10159585/33569667991*e^8 - 1091699306/33569667991*e^6 + 36678793953/33569667991*e^4 - 382802718279/33569667991*e^2 + 128232160966/33569667991, 17741408/33569667991*e^9 - 1913019998/33569667991*e^7 + 66084399480/33569667991*e^5 - 767405615508/33569667991*e^3 + 882278721453/33569667991*e, 61500727/67139335982*e^9 - 6735525943/67139335982*e^7 + 118907739120/33569667991*e^5 - 1427535215861/33569667991*e^3 + 4225043512283/67139335982*e, -26127315/67139335982*e^9 + 2786878507/67139335982*e^7 - 47194779356/33569667991*e^5 + 525077147677/33569667991*e^3 - 758824729815/67139335982*e, -28669215/67139335982*e^9 + 3082718749/67139335982*e^7 - 52823666991/33569667991*e^5 + 600175093071/33569667991*e^3 - 1319986912603/67139335982*e, -1836395/67139335982*e^9 + 248198739/67139335982*e^7 - 5548690782/33569667991*e^5 + 91491215315/33569667991*e^3 - 913018029171/67139335982*e, 78217839/67139335982*e^9 - 8544546681/67139335982*e^7 + 150027178494/33569667991*e^5 - 1777810122045/33569667991*e^3 + 4593978522851/67139335982*e, -318791/67139335982*e^9 + 56357757/67139335982*e^7 - 1842563513/33569667991*e^5 + 49821171491/33569667991*e^3 - 898769225259/67139335982*e, -8663217/67139335982*e^9 + 987749695/67139335982*e^7 - 18018773240/33569667991*e^5 + 210212351187/33569667991*e^3 - 25455264363/67139335982*e, 23206221/33569667991*e^8 - 2271628580/33569667991*e^6 + 67896935708/33569667991*e^4 - 634389952967/33569667991*e^2 + 321210308237/33569667991, -2549420/33569667991*e^8 + 67714303/33569667991*e^6 + 7501805784/33569667991*e^4 - 214231775644/33569667991*e^2 + 129509443761/33569667991, 13507883/33569667991*e^8 - 1307383379/33569667991*e^6 + 38010149063/33569667991*e^4 - 348146675264/33569667991*e^2 + 354173942971/33569667991, -14345928/33569667991*e^8 + 1530513706/33569667991*e^6 - 51622023600/33569667991*e^4 + 572686444767/33569667991*e^2 - 682217310716/33569667991, 274005/33569667991*e^8 - 13269001/33569667991*e^6 - 1030913239/33569667991*e^4 + 59284084084/33569667991*e^2 - 305696440476/33569667991, 187242/33569667991*e^8 - 114185104/33569667991*e^6 + 7822920547/33569667991*e^4 - 122273856669/33569667991*e^2 - 204501123644/33569667991, -5554467/33569667991*e^8 + 701213543/33569667991*e^6 - 28485571128/33569667991*e^4 + 371853362692/33569667991*e^2 - 108806397900/33569667991, -6541992/33569667991*e^8 + 853785709/33569667991*e^6 - 35967358722/33569667991*e^4 + 497763583639/33569667991*e^2 - 795568317434/33569667991, -1207242/33569667991*e^8 - 115753992/33569667991*e^6 + 14684133560/33569667991*e^4 - 287788269835/33569667991*e^2 - 210468689557/33569667991, -10009114/33569667991*e^8 + 928941296/33569667991*e^6 - 25219606544/33569667991*e^4 + 201474443803/33569667991*e^2 - 159315075961/33569667991, 6692877/33569667991*e^9 - 673644840/33569667991*e^7 + 20662977932/33569667991*e^5 - 189624998741/33569667991*e^3 - 89093716888/33569667991*e, 13680499/33569667991*e^9 - 1534921482/33569667991*e^7 + 56588663519/33569667991*e^5 - 743378673414/33569667991*e^3 + 1785235996801/33569667991*e, -59129543/33569667991*e^9 + 6457018911/33569667991*e^7 - 227198623092/33569667991*e^5 + 2727207064121/33569667991*e^3 - 4303338136494/33569667991*e, -821519/33569667991*e^9 + 126278433/33569667991*e^7 - 6429874112/33569667991*e^5 + 124598436021/33569667991*e^3 - 735595614505/33569667991*e, 45682477/67139335982*e^9 - 4969788581/67139335982*e^7 + 86987313269/33569667991*e^5 - 1027759819050/33569667991*e^3 + 2476104103619/67139335982*e, 62771677/33569667991*e^9 - 6883446064/33569667991*e^7 + 243444365875/33569667991*e^5 - 2930168377116/33569667991*e^3 + 4304206595731/33569667991*e, 11259819/67139335982*e^9 - 1222286217/67139335982*e^7 + 20833380733/33569667991*e^5 - 217784070243/33569667991*e^3 - 297782890621/67139335982*e, 63760607/33569667991*e^9 - 6926435522/33569667991*e^7 + 241751859622/33569667991*e^5 - 2846569561721/33569667991*e^3 + 3579739589199/33569667991*e, 10900456/33569667991*e^9 - 1200350611/33569667991*e^7 + 42377214644/33569667991*e^5 - 500470796972/33569667991*e^3 + 678490352090/33569667991*e, 32994213/33569667991*e^9 - 3578478742/33569667991*e^7 + 124456757253/33569667991*e^5 - 1459071923894/33569667991*e^3 + 1998781192340/33569667991*e, 484797/33569667991*e^8 - 82651490/33569667991*e^6 + 4737040508/33569667991*e^4 - 100954127908/33569667991*e^2 + 44263498504/33569667991, -24045671/33569667991*e^8 + 2385176199/33569667991*e^6 - 72334516398/33569667991*e^4 + 682990354119/33569667991*e^2 - 412963129601/33569667991, -23683498/33569667991*e^9 + 2582406009/33569667991*e^7 - 91393557270/33569667991*e^5 + 1133341415298/33569667991*e^3 - 2365362484939/33569667991*e, 16353039/33569667991*e^9 - 1796045655/33569667991*e^7 + 63360325286/33569667991*e^5 - 744431730593/33569667991*e^3 + 791196423273/33569667991*e, -30487679/33569667991*e^9 + 3343648302/33569667991*e^7 - 118737008968/33569667991*e^5 + 1459330651641/33569667991*e^3 - 2675429726969/33569667991*e, -17485334/33569667991*e^9 + 1887020183/33569667991*e^7 - 65123019297/33569667991*e^5 + 750691664723/33569667991*e^3 - 913398716689/33569667991*e, -44967552/33569667991*e^9 + 4940068124/33569667991*e^7 - 174636219552/33569667991*e^5 + 2079834369411/33569667991*e^3 - 2626317325747/33569667991*e, 21761745/33569667991*e^9 - 2325540665/33569667991*e^7 + 79975715645/33569667991*e^5 - 955977557070/33569667991*e^3 + 1988642586039/33569667991*e, -98790687/67139335982*e^9 + 10630211645/67139335982*e^7 - 182979172286/33569667991*e^5 + 2116960588376/33569667991*e^3 - 5344705858739/67139335982*e, -53900973/33569667991*e^9 + 5926936065/33569667991*e^7 - 210402166135/33569667991*e^5 + 2546465569362/33569667991*e^3 - 3913421736991/33569667991*e, -67888943/33569667991*e^9 + 7325931457/33569667991*e^7 - 253560349066/33569667991*e^5 + 2974545634273/33569667991*e^3 - 4294962036316/33569667991*e, 36186916/33569667991*e^9 - 3983264207/33569667991*e^7 + 141503486513/33569667991*e^5 - 1711533727516/33569667991*e^3 + 2588942846652/33569667991*e, 19796115/33569667991*e^8 - 2002475287/33569667991*e^6 + 64190153737/33569667991*e^4 - 679059255716/33569667991*e^2 + 78228675424/33569667991, -9162144/33569667991*e^8 + 1043519616/33569667991*e^6 - 39541606908/33569667991*e^4 + 517301709083/33569667991*e^2 - 825373217128/33569667991, 8932512/33569667991*e^8 - 1009979219/33569667991*e^6 + 35417626601/33569667991*e^4 - 373689548020/33569667991*e^2 - 162859646299/33569667991, 13500363/33569667991*e^8 - 1535509318/33569667991*e^6 + 56769730117/33569667991*e^4 - 679004673705/33569667991*e^2 - 29383806192/33569667991, 11424420/33569667991*e^8 - 1358162658/33569667991*e^6 + 51892968795/33569667991*e^4 - 646388961754/33569667991*e^2 + 191800972708/33569667991, 1551979/33569667991*e^8 - 19534613/33569667991*e^6 - 4638992829/33569667991*e^4 + 79732784840/33569667991*e^2 + 366213604455/33569667991, -19003847/33569667991*e^8 + 2055749887/33569667991*e^6 - 72883593817/33569667991*e^4 + 893403736311/33569667991*e^2 - 934115854531/33569667991, -4260876/33569667991*e^8 + 574984759/33569667991*e^6 - 25259970275/33569667991*e^4 + 351732713906/33569667991*e^2 - 85904609596/33569667991, 7423914/33569667991*e^8 - 1143116070/33569667991*e^6 + 53946953542/33569667991*e^4 - 788288128277/33569667991*e^2 + 374568273506/33569667991, 5401682/33569667991*e^8 - 671149519/33569667991*e^6 + 25852357584/33569667991*e^4 - 273522804636/33569667991*e^2 - 704517850366/33569667991, -18136551/33569667991*e^8 + 1653066527/33569667991*e^6 - 44148405088/33569667991*e^4 + 363490718183/33569667991*e^2 - 390946080805/33569667991, 14278501/33569667991*e^8 - 1509116287/33569667991*e^6 + 49309270117/33569667991*e^4 - 502306982300/33569667991*e^2 + 238975967936/33569667991, -25568894/33569667991*e^8 + 2544945380/33569667991*e^6 - 79684097558/33569667991*e^4 + 846723163968/33569667991*e^2 - 1021261765440/33569667991, -4086855/33569667991*e^8 + 457029364/33569667991*e^6 - 15855749666/33569667991*e^4 + 199579420089/33569667991*e^2 - 1140300857544/33569667991, -41634294/33569667991*e^8 + 4268132173/33569667991*e^6 - 135785702334/33569667991*e^4 + 1385635079562/33569667991*e^2 - 1019032290644/33569667991, 11877570/33569667991*e^8 - 1137527652/33569667991*e^6 + 33091714606/33569667991*e^4 - 304381718365/33569667991*e^2 + 177054726522/33569667991, -122585489/67139335982*e^9 + 13300215107/67139335982*e^7 - 232016187804/33569667991*e^5 + 2751813414826/33569667991*e^3 - 8277077642551/67139335982*e, 75711305/33569667991*e^9 - 8309716241/33569667991*e^7 + 294334608703/33569667991*e^5 - 3555878971837/33569667991*e^3 + 5539184401408/33569667991*e, -78272541/67139335982*e^9 + 8483242961/67139335982*e^7 - 147212898352/33569667991*e^5 + 1710283895707/33569667991*e^3 - 4112414200971/67139335982*e, -130713917/67139335982*e^9 + 14257656509/67139335982*e^7 - 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3653950802146/33569667991*e^3 + 6045126462946/33569667991*e, -44694952/33569667991*e^9 + 4817216415/33569667991*e^7 - 166492838944/33569667991*e^5 + 1929609417153/33569667991*e^3 - 2158608531986/33569667991*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([37,37,-w^4 + 4*w^3 - w^2 - 7*w + 3])] = -1

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