/*
  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, 3, 3, -4, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([89,89,-w^4 + 2*w^2 + w + 2])

primes_array = [
[11, 11, w^4 + w^3 - 4*w^2 - 3*w + 2],\
[23, 23, -w^4 + 3*w^2 + 1],\
[23, 23, -w^4 + 3*w^2 + w - 2],\
[23, 23, w^4 - w^3 - 3*w^2 + 3*w + 2],\
[23, 23, -w^4 + w^3 + 4*w^2 - 3*w - 1],\
[23, 23, -w^2 + w + 3],\
[32, 2, 2],\
[43, 43, -2*w^4 + w^3 + 6*w^2 - 2*w - 1],\
[43, 43, -w^4 + 2*w^2 + w + 1],\
[43, 43, w^3 + w^2 - 4*w - 2],\
[43, 43, 2*w^4 - w^3 - 7*w^2 + 3*w + 3],\
[43, 43, w^4 - w^3 - 4*w^2 + 4*w + 2],\
[67, 67, 2*w^4 - 7*w^2 + 2],\
[67, 67, w^4 - 2*w^3 - 3*w^2 + 6*w + 2],\
[67, 67, 2*w^4 - 7*w^2 - w + 4],\
[67, 67, w^4 - 2*w^3 - 4*w^2 + 6*w + 2],\
[67, 67, -w^4 + w^3 + 5*w^2 - 3*w - 3],\
[89, 89, w^3 + w^2 - 4*w - 1],\
[89, 89, -2*w^4 + w^3 + 7*w^2 - 3*w - 2],\
[89, 89, -w^4 + w^3 + 4*w^2 - 4*w - 3],\
[89, 89, -w^4 + 2*w^2 + w + 2],\
[89, 89, 2*w^4 - w^3 - 6*w^2 + 2*w + 2],\
[109, 109, -w^3 + 2*w^2 + 3*w - 3],\
[109, 109, w^4 - 4*w^2 - 2*w + 3],\
[109, 109, -2*w^4 + 2*w^3 + 7*w^2 - 4*w - 3],\
[109, 109, 2*w^3 - 5*w - 1],\
[109, 109, -w^4 - w^3 + 5*w^2 + 2*w - 4],\
[131, 131, w^4 - 3*w^3 - 2*w^2 + 7*w],\
[131, 131, -w^4 + 2*w^3 + 5*w^2 - 7*w - 5],\
[131, 131, 2*w^4 - 2*w^3 - 6*w^2 + 3*w + 2],\
[131, 131, w^4 - 2*w^3 - 3*w^2 + 7*w],\
[131, 131, 2*w^4 + w^3 - 8*w^2 - 3*w + 4],\
[197, 197, 3*w^4 - w^3 - 10*w^2 + w + 5],\
[197, 197, 2*w^4 - 7*w^2 + w + 1],\
[197, 197, -3*w^4 + 2*w^3 + 10*w^2 - 5*w - 5],\
[197, 197, -2*w^4 - w^3 + 9*w^2 + 2*w - 5],\
[197, 197, -3*w^4 + 3*w^3 + 10*w^2 - 6*w - 5],\
[199, 199, w^4 + 2*w^3 - 5*w^2 - 5*w + 3],\
[199, 199, -2*w^4 + w^3 + 7*w^2 - 4*w - 2],\
[199, 199, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],\
[199, 199, 2*w^4 - 8*w^2 + w + 4],\
[199, 199, 3*w^4 - w^3 - 10*w^2 + 2*w + 4],\
[241, 241, w^4 - 2*w^3 - w^2 + 3*w - 3],\
[241, 241, -w^4 + 3*w^3 + 2*w^2 - 9*w],\
[241, 241, 3*w^4 - 11*w^2 - 2*w + 6],\
[241, 241, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],\
[241, 241, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6],\
[243, 3, -3],\
[263, 263, -w^4 + 4*w^2 + w + 1],\
[263, 263, 2*w^4 - w^3 - 8*w^2 + w + 2],\
[263, 263, 3*w^4 - 3*w^3 - 9*w^2 + 7*w + 2],\
[263, 263, -2*w^3 + w^2 + 4*w - 4],\
[263, 263, -3*w^2 + w + 5],\
[307, 307, 2*w^4 - w^3 - 6*w^2 + 3*w - 2],\
[307, 307, w^4 - 2*w^2 - 2*w - 4],\
[307, 307, -3*w^4 + 2*w^3 + 11*w^2 - 3*w - 6],\
[307, 307, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 6],\
[307, 307, -w^4 + 3*w^3 + 4*w^2 - 7*w - 3],\
[331, 331, -w^4 - 2*w^3 + 3*w^2 + 7*w - 1],\
[331, 331, -w^4 + 3*w^3 + 4*w^2 - 8*w - 4],\
[331, 331, w^4 - w^2 - 4],\
[331, 331, 3*w^4 - 2*w^3 - 9*w^2 + 5*w + 2],\
[331, 331, -3*w^4 + 2*w^3 + 11*w^2 - 6*w - 5],\
[353, 353, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],\
[353, 353, -2*w^4 - w^3 + 7*w^2 + 4*w - 4],\
[353, 353, -w^4 + 3*w^3 + 3*w^2 - 9*w - 3],\
[353, 353, -3*w^4 + 11*w^2 + w - 7],\
[353, 353, w^4 - w^3 - 6*w^2 + 3*w + 4],\
[373, 373, w^3 + 2*w^2 - 5*w - 3],\
[373, 373, 2*w^4 - 3*w^3 - 6*w^2 + 6*w],\
[373, 373, -w^4 + w^3 + 3*w^2 - 3*w + 3],\
[373, 373, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 5],\
[373, 373, -w^4 + w^3 + 4*w^2 - 3*w - 6],\
[397, 397, -w^4 - w^3 + 6*w^2 + 3*w - 5],\
[397, 397, 2*w^4 - 2*w^3 - 7*w^2 + 3*w + 3],\
[397, 397, -w^4 + 5*w^2 + 2*w - 5],\
[397, 397, w^4 + 2*w^3 - 5*w^2 - 5*w + 4],\
[397, 397, -w^4 + 3*w^3 + 3*w^2 - 7*w - 2],\
[419, 419, 3*w^3 - w^2 - 8*w],\
[419, 419, 2*w^4 - 3*w^3 - 6*w^2 + 6*w + 1],\
[419, 419, w^4 - 2*w^3 - 5*w^2 + 7*w + 4],\
[419, 419, -2*w^4 - w^3 + 9*w^2 + 3*w - 7],\
[419, 419, -w^4 + 3*w^3 + w^2 - 8*w + 1],\
[439, 439, -w^4 + w^3 + 5*w^2 - 5*w - 5],\
[439, 439, 2*w^4 - 9*w^2 - 2*w + 8],\
[439, 439, -3*w^4 + 2*w^3 + 11*w^2 - 5*w - 3],\
[439, 439, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 4],\
[439, 439, -w^4 + 3*w^3 + 3*w^2 - 6*w - 3],\
[461, 461, -w^4 + 5*w^2 + 2*w - 6],\
[461, 461, -w^4 + 3*w^2 - w + 3],\
[461, 461, -w^4 - 2*w^3 + 5*w^2 + 5*w - 5],\
[461, 461, -2*w^4 + 7*w^2 + w - 6],\
[461, 461, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],\
[463, 463, 2*w^4 - 2*w^3 - 8*w^2 + 5*w + 3],\
[463, 463, w^4 + w^3 - 3*w^2 - 2*w - 2],\
[463, 463, 2*w^4 - 7*w^2 - 2*w + 5],\
[463, 463, w^4 - 2*w^3 - 2*w^2 + 6*w + 1],\
[463, 463, -w^3 + 2*w^2 + w - 5],\
[571, 571, w^4 - w^3 - 3*w^2 + 2*w - 3],\
[571, 571, 3*w^4 - 2*w^3 - 10*w^2 + 6*w + 2],\
[571, 571, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 6],\
[571, 571, 2*w^4 - 5*w^2 - 2*w - 2],\
[571, 571, 3*w^4 - w^3 - 9*w^2 + 2*w + 3],\
[593, 593, 3*w^4 - w^3 - 9*w^2 + w + 3],\
[593, 593, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3],\
[593, 593, -2*w^4 + 2*w^3 + 7*w^2 - 7*w - 4],\
[593, 593, w^4 - w^2 - 2*w - 4],\
[593, 593, 3*w^4 - w^3 - 10*w^2 + 3*w + 2],\
[617, 617, -w^4 - w^3 + 6*w^2 + 4*w - 5],\
[617, 617, 3*w^3 - w^2 - 7*w],\
[617, 617, 2*w^4 - w^3 - 8*w^2 + 6],\
[617, 617, 2*w^4 - 3*w^3 - 7*w^2 + 6*w + 3],\
[617, 617, -w^4 - 2*w^3 + 6*w^2 + 5*w - 6],\
[659, 659, -w^4 + w^3 + 4*w^2 - 5*w - 4],\
[659, 659, 2*w^3 + w^2 - 7*w],\
[659, 659, -3*w^4 + 2*w^3 + 9*w^2 - 4*w - 4],\
[659, 659, 3*w^4 - w^3 - 11*w^2 + 3*w + 3],\
[659, 659, w^4 - w^2 - w - 5],\
[661, 661, -5*w^4 + 2*w^3 + 18*w^2 - 4*w - 9],\
[661, 661, -w^4 + 3*w^3 + 6*w^2 - 9*w - 8],\
[661, 661, w^4 + 2*w^3 - 2*w^2 - 7*w - 2],\
[661, 661, -w^4 + w^3 + 5*w^2 - 2*w - 9],\
[661, 661, -3*w^4 + 2*w^3 + 12*w^2 - 2*w - 8],\
[683, 683, -w^4 + w^3 + 4*w^2 - 5*w - 3],\
[683, 683, 2*w^3 + w^2 - 7*w - 1],\
[683, 683, w^4 - w^2 - w - 4],\
[683, 683, 3*w^4 - 2*w^3 - 9*w^2 + 4*w + 3],\
[683, 683, 3*w^4 - w^3 - 11*w^2 + 3*w + 4],\
[727, 727, -5*w^4 + 3*w^3 + 19*w^2 - 7*w - 9],\
[727, 727, -2*w^4 - 2*w^3 + 7*w^2 + 9*w - 4],\
[727, 727, 3*w^4 + 2*w^3 - 13*w^2 - 6*w + 6],\
[727, 727, w^4 - 2*w^2 + 3*w - 4],\
[727, 727, -2*w^4 + 5*w^3 + 6*w^2 - 14*w - 4],\
[769, 769, -2*w^4 - w^3 + 9*w^2 + 2*w - 7],\
[769, 769, -w^4 + 4*w^2 + 3*w - 4],\
[769, 769, -w^3 + 3*w^2 + 3*w - 4],\
[769, 769, 3*w^4 - 3*w^3 - 10*w^2 + 6*w + 3],\
[769, 769, 3*w^3 - 8*w - 2],\
[857, 857, w^4 - 5*w^3 - 2*w^2 + 14*w - 1],\
[857, 857, 4*w^4 - w^3 - 13*w^2 + 4],\
[857, 857, -2*w^4 + 4*w^3 + 6*w^2 - 10*w - 5],\
[857, 857, 4*w^4 + w^3 - 14*w^2 - 5*w + 6],\
[857, 857, 4*w^4 + w^3 - 16*w^2 - 4*w + 8],\
[859, 859, 3*w^4 - w^3 - 11*w^2 + 5*w + 5],\
[859, 859, -2*w^4 + 5*w^3 + 6*w^2 - 13*w - 1],\
[859, 859, -2*w^4 - 2*w^3 + 5*w^2 + 7*w + 1],\
[859, 859, -w^3 + w^2 + 6*w + 1],\
[859, 859, 4*w^4 - w^3 - 13*w^2 + 3*w + 4],\
[881, 881, 2*w^4 + w^3 - 11*w^2 - 3*w + 11],\
[881, 881, w^4 - 6*w^2 - 3*w + 5],\
[881, 881, -2*w^4 + 5*w^3 + 6*w^2 - 12*w - 1],\
[881, 881, -2*w^4 - 3*w^3 + 9*w^2 + 8*w - 4],\
[881, 881, 3*w^4 + w^3 - 12*w^2 - 3*w + 7],\
[947, 947, 2*w^4 + 3*w^3 - 7*w^2 - 8*w],\
[947, 947, 2*w^4 - 3*w^3 - 6*w^2 + 4*w],\
[947, 947, 2*w^4 + 2*w^3 - 7*w^2 - 8*w + 4],\
[947, 947, -w^4 + 2*w^3 - 7*w + 3],\
[947, 947, -4*w^4 + 5*w^3 + 14*w^2 - 12*w - 5],\
[967, 967, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],\
[967, 967, -2*w^4 - w^3 + 7*w^2 + 4*w - 5],\
[967, 967, w^4 - 2*w^2 + w - 4],\
[967, 967, w^4 - w^3 - 2*w^2 - 3],\
[967, 967, -2*w^4 + 3*w^3 + 6*w^2 - 8*w - 4],\
[991, 991, -w^4 + 4*w^3 + 2*w^2 - 12*w - 1],\
[991, 991, w^4 + 3*w^3 - 3*w^2 - 11*w - 1],\
[991, 991, 2*w^3 - 2*w^2 - 5*w + 8],\
[991, 991, -2*w^4 + 3*w^3 + 4*w^2 - 5*w + 3],\
[991, 991, 4*w^4 - 4*w^3 - 11*w^2 + 9*w]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 - 16*x - 16
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 1/2*e^2 - 8, -1/4*e^2 + 1/2*e + 4, -1/4*e^2, 1/4*e^2 - 5/2*e - 2, -e^2 + 3*e + 12, 5/4*e^2 - 2*e - 13, -1/2*e^2 + e + 4, -1/2*e^2 - e + 8, 3/4*e^2 - 4, 1/4*e^2 - 1/2*e + 4, -e^2 + 8, 3/4*e^2 - 7/2*e - 8, 7/4*e^2 - 1/2*e - 18, -1/2*e^2 + e + 8, 1/2*e^2 - 2*e, 3/4*e^2 + 1/2*e - 6, 1/2*e^2 - 3*e - 6, -e^2 + 14, -1/4*e^2 + 1/2*e, 1, -5/4*e^2 + 1/2*e + 14, 2*e + 6, 2*e^2 - 2*e - 14, -9/4*e^2 + e + 22, -7/4*e^2 + 7/2*e + 20, -3/4*e^2 - e + 10, -1/2*e^2 + 3*e + 8, e, -1/4*e^2 - 2*e + 12, 3/2*e^2 - 5*e - 24, -1/4*e^2 + 3/2*e + 10, 2*e^2 - 4*e - 22, 3/4*e^2 + e - 2, -1/2*e^2 + 2*e + 6, 1/2*e^2 + 6, -7/4*e^2 + 11/2*e + 14, e - 12, 5/4*e^2 - 4*e - 32, 11/4*e^2 - 7/2*e - 34, 3/2*e^2 - 4*e - 8, -2*e^2 + 3*e + 12, 3/4*e^2 + 7/2*e - 8, 7/4*e^2 - 5*e - 30, e^2 - 5*e - 22, -7/2*e^2 + 6*e + 34, -e^2 - 3*e + 22, -11/4*e^2 + 9/2*e + 16, -15/4*e^2 + 9/2*e + 40, -7*e + 8, 5/4*e^2 + 7/2*e - 18, -9/4*e^2 + 6*e + 36, -11/4*e^2 + 3/2*e + 32, 1/2*e^2 - 4*e - 16, 7/4*e^2 - 5/2*e - 24, 7/4*e^2 - 11/2*e - 38, -3/4*e^2 + 1/2*e, 9/4*e^2 - 4*e - 8, 5/2*e^2 - e - 36, -13/4*e^2 + 6*e + 28, 3/4*e^2 - 2*e - 32, -3*e - 12, 3*e^2 - 7*e - 36, -13/4*e^2 + 5*e + 42, -5/2*e^2 + 9*e + 46, 5/4*e^2 + 3*e - 22, -e^2 + e - 14, -1/4*e^2 + 1/2*e + 14, -11/4*e^2 + 15/2*e + 30, 1/4*e^2 + 7/2*e - 14, 9/4*e^2 + 5/2*e - 30, -3/4*e^2 - 3/2*e + 20, -1/4*e^2 - 3*e + 6, 3*e^2 - e - 42, 7/2*e^2 - 9*e - 38, -7/4*e^2 + e + 18, 1/4*e^2 + 9/2*e - 2, 2*e^2 - 5*e - 10, 1/2*e^2 - 7*e - 8, -7/4*e^2 - 1/2*e + 28, -2*e^2 + 2*e + 24, 3/2*e^2 + 5*e - 16, 3/4*e^2 + 1/2*e + 16, 15/4*e^2 - 17/2*e - 38, 3/2*e^2 - 2*e + 8, 5/4*e^2 + 5/2*e - 30, -11/4*e^2 + 8*e + 20, -7/2*e^2 + 6*e + 52, 3/2*e^2 - 4*e - 6, -5/4*e^2 + 19/2*e + 8, -e^2 - e - 2, e^2 - e - 10, e^2 - 6*e - 10, 13/4*e^2 - 6*e - 52, -4*e^2 + 4*e + 32, 17/4*e^2 - 11/2*e - 32, -7/2*e^2 + e + 36, -3*e^2 + 11*e + 44, 11/4*e^2 - 23/2*e - 36, -3/2*e^2 - 3*e + 24, 15/4*e^2 - 15/2*e - 58, 2*e^2 - 2*e - 12, -7/4*e^2 + 1/2*e - 2, -11/4*e^2 + 9/2*e + 46, 8*e - 2, 11/4*e^2 + 7/2*e - 34, -3*e^2 + 8*e + 30, 9/2*e^2 - 7*e - 50, -7/4*e^2 - 3/2*e + 48, 9*e - 10, 7/4*e^2 - 11/2*e - 8, 7/4*e^2 + 3/2*e - 12, 5/4*e^2 - 11*e - 10, -1/2*e^2 + 12, 5/2*e^2 - 4*e - 12, -5*e^2 + 15*e + 60, 13/4*e^2 - 2*e - 24, -7/4*e^2 - 1/2*e + 18, 5/4*e^2 - 17/2*e - 26, 2*e^2 - 10*e - 26, -15/4*e^2 + 7*e + 38, 4*e^2 + 3*e - 50, 7/4*e^2 - 15/2*e - 40, 3*e^2 - 48, -3/2*e^2 + 11*e + 8, -36, -7/4*e^2 - 3/2*e + 34, -3/2*e^2 - 2*e + 24, -e^2 + 6*e + 20, 11/4*e^2 - 5/2*e - 48, 1/2*e^2 + e - 36, -3/2*e^2 - e - 4, 1/2*e^2 + 5*e - 24, -2*e^2 + e + 6, 7/2*e^2 + 5*e - 54, 11/4*e^2 - 1/2*e - 20, -e^2 + 8*e - 2, 5/2*e^2 - 5*e - 30, 2*e^2 + 3*e - 6, -3/4*e^2 + 1/2*e - 20, 5/2*e^2 - 5*e - 22, 3*e^2 + 3*e - 30, 3/2*e^2 + 3*e - 14, -1/2*e^2 + 5*e - 8, -21/4*e^2 + 29/2*e + 52, 5/2*e^2 - 12*e - 36, -3/2*e^2 + 2*e + 52, 2*e^2 + e, 13/4*e^2 + 7/2*e - 28, 2*e^2 - 6*e - 54, -13/2*e^2 + 8*e + 62, -e^2 - 3*e - 18, -3/2*e^2 + 8*e + 42, -19/4*e^2 + 8*e + 36, 9/4*e^2 - 6*e - 64, -13/4*e^2 - 3/2*e + 34, 4*e - 20, -e^2 + e, -7/2*e^2 - e + 24, 2*e^2 + 8*e - 16, 19/4*e^2 - 9/2*e - 52, -4*e^2 + 10*e + 44, 9/2*e^2 - 12*e - 36, -2*e^2 + 2*e + 36, e^2 - 2*e + 12, -3*e^2 - 6*e + 36, 7/4*e^2 - 2*e - 36, 4*e + 24]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([89,89,-w^4 + 2*w^2 + w + 2])] = -1

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