/*
  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, 0, 8, 4, -6, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([55, 55, -w^3 + 5*w + 1])

primes_array = [
[5, 5, w - 1],\
[11, 11, w^3 - 3*w],\
[19, 19, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w],\
[25, 5, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 8*w],\
[29, 29, -w^4 + 4*w^2 + 1],\
[31, 31, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 3*w - 4],\
[31, 31, -w^2 + 2],\
[59, 59, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w - 2],\
[59, 59, w^4 - w^3 - 5*w^2 + 3*w + 4],\
[59, 59, -w^4 - w^3 + 4*w^2 + 5*w],\
[61, 61, w^5 - w^4 - 4*w^3 + 5*w^2 - 4],\
[64, 2, 2],\
[71, 71, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w],\
[71, 71, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 + w + 4],\
[71, 71, 2*w^5 - 3*w^4 - 9*w^3 + 13*w^2 + 4*w - 4],\
[71, 71, w^3 - 5*w],\
[79, 79, w^5 - w^4 - 4*w^3 + 4*w^2 + w - 2],\
[89, 89, w^3 - w^2 - 4*w + 1],\
[101, 101, -2*w^5 + 3*w^4 + 11*w^3 - 13*w^2 - 12*w + 2],\
[101, 101, 2*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 20*w + 4],\
[101, 101, w^5 - w^4 - 5*w^3 + 5*w^2 + 2*w - 3],\
[101, 101, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 1],\
[109, 109, -w^5 + w^4 + 7*w^3 - 4*w^2 - 11*w - 1],\
[121, 11, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 4],\
[131, 131, w^4 - 4*w^2 + w - 1],\
[131, 131, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 7*w + 2],\
[139, 139, w^4 + w^3 - 3*w^2 - 4*w - 4],\
[149, 149, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 8*w + 2],\
[149, 149, 2*w^5 - 11*w^3 - 2*w^2 + 11*w + 6],\
[151, 151, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 16*w - 6],\
[169, 13, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 6*w - 3],\
[179, 179, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 12*w - 1],\
[179, 179, 2*w^5 - 4*w^4 - 9*w^3 + 18*w^2 + 4*w - 7],\
[179, 179, -3*w^5 + 3*w^4 + 15*w^3 - 13*w^2 - 12*w + 3],\
[179, 179, -w^5 + 3*w^4 + 5*w^3 - 14*w^2 - 5*w + 6],\
[181, 181, w^3 + w^2 - 4*w - 6],\
[191, 191, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 3],\
[191, 191, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 3],\
[191, 191, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 3],\
[191, 191, -w^5 + w^4 + 7*w^3 - 4*w^2 - 12*w - 2],\
[199, 199, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w - 1],\
[199, 199, w^4 + w^3 - 5*w^2 - 3*w + 3],\
[199, 199, -w^5 + w^4 + 3*w^3 - 4*w^2 + 3*w + 2],\
[199, 199, w^2 + w - 3],\
[211, 211, 3*w^5 - 4*w^4 - 16*w^3 + 17*w^2 + 16*w - 3],\
[239, 239, -3*w^5 + 3*w^4 + 16*w^3 - 12*w^2 - 16*w + 3],\
[239, 239, 4*w^5 - 6*w^4 - 21*w^3 + 25*w^2 + 19*w - 5],\
[239, 239, -w^5 + w^4 + 7*w^3 - 5*w^2 - 10*w + 2],\
[241, 241, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 4*w + 2],\
[251, 251, 3*w^5 - 4*w^4 - 16*w^3 + 18*w^2 + 15*w - 5],\
[251, 251, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 2],\
[251, 251, -3*w^5 + 4*w^4 + 16*w^3 - 17*w^2 - 15*w + 2],\
[251, 251, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 4],\
[269, 269, w^5 - w^4 - 6*w^3 + 6*w^2 + 7*w - 5],\
[271, 271, 3*w^5 - 3*w^4 - 15*w^3 + 12*w^2 + 11*w - 1],\
[271, 271, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 2*w],\
[281, 281, w^2 - w - 4],\
[281, 281, -3*w^5 + 4*w^4 + 16*w^3 - 18*w^2 - 14*w + 4],\
[281, 281, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w],\
[281, 281, -3*w^5 + 3*w^4 + 14*w^3 - 13*w^2 - 8*w + 4],\
[289, 17, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 10*w - 1],\
[311, 311, -w^5 - w^4 + 6*w^3 + 6*w^2 - 8*w - 5],\
[311, 311, -2*w^5 + 3*w^4 + 10*w^3 - 13*w^2 - 8*w + 7],\
[331, 331, 2*w^5 - 3*w^4 - 11*w^3 + 12*w^2 + 12*w - 4],\
[331, 331, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 12*w + 3],\
[331, 331, -w^5 + w^4 + 4*w^3 - 3*w^2 + w - 1],\
[331, 331, -2*w^5 + 5*w^4 + 10*w^3 - 21*w^2 - 8*w + 5],\
[359, 359, 2*w^4 - 2*w^3 - 9*w^2 + 7*w + 3],\
[361, 19, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],\
[379, 379, 3*w^5 - 2*w^4 - 15*w^3 + 8*w^2 + 11*w - 1],\
[379, 379, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 13*w - 3],\
[389, 389, w^5 + w^4 - 5*w^3 - 5*w^2 + 3*w + 4],\
[389, 389, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 16*w - 5],\
[389, 389, -w^5 + 4*w^4 + 5*w^3 - 17*w^2 - 4*w + 4],\
[389, 389, 3*w^5 - 4*w^4 - 15*w^3 + 18*w^2 + 10*w - 6],\
[401, 401, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 8*w + 3],\
[401, 401, -2*w^5 + 4*w^4 + 11*w^3 - 17*w^2 - 13*w + 3],\
[409, 409, w^3 + w^2 - 3*w - 5],\
[419, 419, -w^5 + w^4 + 4*w^3 - 6*w^2 + 4],\
[419, 419, -w^4 + 2*w^3 + 5*w^2 - 8*w - 6],\
[421, 421, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 6],\
[431, 431, 2*w^3 + w^2 - 9*w - 3],\
[439, 439, -w^5 + 6*w^3 + w^2 - 6*w - 4],\
[439, 439, w^5 - 2*w^4 - 4*w^3 + 9*w^2 - 7],\
[439, 439, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 14*w - 4],\
[439, 439, -2*w^5 + 3*w^4 + 11*w^3 - 11*w^2 - 13*w - 1],\
[439, 439, w^4 - w^3 - 3*w^2 + 3*w - 3],\
[439, 439, -3*w^5 + 4*w^4 + 16*w^3 - 16*w^2 - 17*w + 2],\
[449, 449, -w^3 - 2*w^2 + 5*w + 6],\
[461, 461, w^5 - w^4 - 6*w^3 + 3*w^2 + 10*w + 1],\
[479, 479, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 15*w - 2],\
[491, 491, w^5 - w^4 - 6*w^3 + 3*w^2 + 10*w + 2],\
[499, 499, w^5 - 5*w^3 - 2*w^2 + 4*w + 4],\
[509, 509, -w^5 + 2*w^4 + 5*w^3 - 10*w^2 - 4*w + 4],\
[509, 509, -w^5 + w^4 + 6*w^3 - 6*w^2 - 7*w + 4],\
[521, 521, -3*w^5 + 5*w^4 + 16*w^3 - 22*w^2 - 14*w + 5],\
[541, 541, -w^4 + w^3 + 5*w^2 - 2*w - 5],\
[541, 541, 2*w^5 - 3*w^4 - 9*w^3 + 14*w^2 + 4*w - 5],\
[541, 541, -2*w^5 + 3*w^4 + 9*w^3 - 13*w^2 - 2*w + 4],\
[569, 569, 2*w^5 - w^4 - 11*w^3 + 5*w^2 + 11*w - 3],\
[571, 571, w^5 - w^4 - 6*w^3 + 2*w^2 + 8*w + 4],\
[571, 571, 3*w^5 - 5*w^4 - 15*w^3 + 22*w^2 + 10*w - 7],\
[599, 599, -w^4 + w^3 + 3*w^2 - 4*w],\
[599, 599, -w^4 + 6*w^2 + w - 4],\
[601, 601, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 7*w + 4],\
[619, 619, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 11*w + 2],\
[631, 631, w^4 + 2*w^3 - 5*w^2 - 8*w + 2],\
[631, 631, -w^5 + 4*w^4 + 5*w^3 - 18*w^2 - 4*w + 6],\
[631, 631, w^4 + 2*w^3 - 4*w^2 - 8*w + 1],\
[641, 641, -w^5 - w^4 + 5*w^3 + 5*w^2 - 5*w - 2],\
[641, 641, -w^4 + 6*w^2 + w - 5],\
[641, 641, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + 2*w - 5],\
[659, 659, -w^4 - 2*w^3 + 5*w^2 + 8*w - 1],\
[661, 661, 2*w^5 - 2*w^4 - 9*w^3 + 8*w^2 + 3*w + 1],\
[661, 661, 2*w^5 - 3*w^4 - 11*w^3 + 12*w^2 + 10*w - 1],\
[661, 661, w^5 - 3*w^4 - 6*w^3 + 12*w^2 + 9*w],\
[661, 661, -2*w^5 + 3*w^4 + 12*w^3 - 13*w^2 - 14*w + 1],\
[691, 691, w^5 - 6*w^3 - w^2 + 9*w + 1],\
[691, 691, 3*w^5 - 5*w^4 - 15*w^3 + 21*w^2 + 13*w - 5],\
[691, 691, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 6*w + 2],\
[691, 691, 2*w^5 - 3*w^4 - 12*w^3 + 12*w^2 + 16*w - 2],\
[701, 701, -w^5 - w^4 + 6*w^3 + 4*w^2 - 8*w - 2],\
[709, 709, -3*w^5 + 3*w^4 + 14*w^3 - 12*w^2 - 8*w + 3],\
[709, 709, w^5 + w^4 - 6*w^3 - 4*w^2 + 8*w + 1],\
[709, 709, 2*w^5 - 4*w^4 - 10*w^3 + 16*w^2 + 9*w - 2],\
[709, 709, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 14*w - 3],\
[719, 719, w^4 - w^3 - 6*w^2 + 3*w + 4],\
[729, 3, -3],\
[739, 739, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w - 5],\
[751, 751, -2*w^5 + 3*w^4 + 11*w^3 - 13*w^2 - 12*w + 1],\
[751, 751, -w^5 + w^4 + 4*w^3 - 4*w^2 - 2*w - 1],\
[761, 761, -2*w^5 + 2*w^4 + 12*w^3 - 9*w^2 - 15*w],\
[769, 769, -3*w^5 + 5*w^4 + 17*w^3 - 21*w^2 - 19*w + 3],\
[769, 769, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 5*w - 1],\
[769, 769, -2*w^5 + 4*w^4 + 10*w^3 - 18*w^2 - 9*w + 7],\
[769, 769, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 14*w + 2],\
[811, 811, -3*w^5 + 2*w^4 + 16*w^3 - 9*w^2 - 15*w + 2],\
[821, 821, 2*w^3 - w^2 - 8*w + 1],\
[829, 829, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 1],\
[829, 829, 2*w^5 - 2*w^4 - 10*w^3 + 9*w^2 + 8*w - 6],\
[839, 839, -w^4 - 2*w^3 + 5*w^2 + 6*w - 1],\
[841, 29, -2*w^5 + 2*w^4 + 12*w^3 - 9*w^2 - 16*w + 2],\
[859, 859, -w^5 + 2*w^4 + 4*w^3 - 10*w^2 - w + 4],\
[859, 859, 2*w^5 - 3*w^4 - 10*w^3 + 11*w^2 + 7*w + 1],\
[881, 881, -2*w^5 + 3*w^4 + 9*w^3 - 14*w^2 - 4*w + 4],\
[881, 881, 2*w^5 - 2*w^4 - 10*w^3 + 10*w^2 + 6*w - 5],\
[911, 911, w^5 - w^4 - 4*w^3 + 5*w^2 - 2*w - 5],\
[911, 911, w^3 - 4*w - 4],\
[929, 929, -w^5 + 3*w^4 + 6*w^3 - 12*w^2 - 9*w + 1],\
[941, 941, 2*w^4 - 10*w^2 + w + 6],\
[941, 941, 3*w^5 - 3*w^4 - 15*w^3 + 14*w^2 + 10*w - 5],\
[941, 941, -w^5 + 5*w^3 + w^2 - 2*w - 4],\
[961, 31, -w^4 - w^3 + 6*w^2 + 4*w - 4],\
[961, 31, 2*w^5 - 3*w^4 - 12*w^3 + 12*w^2 + 15*w - 3],\
[971, 971, -2*w^5 + 3*w^4 + 12*w^3 - 13*w^2 - 14*w],\
[991, 991, w^5 - 3*w^4 - 4*w^3 + 14*w^2 + w - 6]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 + 4*x - 16
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1, -1, e, -2, e - 2, -e - 4, 4, -e - 8, -4, e - 4, -2*e - 6, -e - 11, -2*e - 8, e + 12, -2*e - 4, e, 0, -14, 2*e - 6, -2*e - 2, 4*e + 10, -e + 2, -18, 10, 2*e + 4, -e - 12, -2*e - 12, -6, -2*e + 6, -2*e, -e - 18, 2*e + 4, -4, 2*e - 4, -2*e + 4, 2*e - 2, 2*e - 8, -4*e - 8, -4*e - 16, -4*e - 8, 3*e, 4, -e - 8, 2*e + 12, 2*e + 8, -4*e - 4, -24, -8, 3*e + 2, -3*e + 8, -2*e - 20, 2*e + 16, 4*e + 4, 2*e + 22, -e, 2*e, -4*e - 6, -2*e - 6, 4*e + 10, -3*e + 6, 2*e + 2, 16, 8, -2*e, 2*e - 12, 4*e - 4, -4*e - 12, 20, 4*e + 18, 7*e + 20, 3*e - 8, 2*e + 10, -3*e + 10, 6, 7*e + 18, 2, -2*e + 2, -4*e + 10, 2*e - 12, e + 16, -5*e - 18, -4*e, -6*e - 16, 3*e - 4, 16, -2*e - 20, -2*e - 16, 7*e + 12, -e - 30, -4*e + 14, -9*e - 20, -e - 20, 2*e + 20, -5*e - 10, 2*e + 10, 4*e + 18, 4*e + 18, 5*e + 22, 4*e + 14, 7*e + 14, e - 8, -8, 32, -44, e + 22, -e + 28, 2*e + 16, -4*e + 8, -2*e + 16, -3*e - 10, -2*e + 26, 4*e + 18, 6*e + 4, -4*e - 22, -26, -8*e - 18, 2*e + 22, -8*e - 8, 12, -2*e - 12, -2*e - 20, -6*e - 26, -2*e - 18, 4*e - 2, 3*e - 30, 2*e - 10, -3*e + 8, 5*e - 6, 2*e + 12, -e - 20, 6*e + 32, 3*e - 14, 2*e - 14, 3*e - 10, -9*e - 14, 2*e - 2, 2*e + 4, 4*e - 26, -3*e - 46, -2*e + 6, -4*e - 32, -3*e - 6, 10*e + 12, -4*e - 20, -30, -46, -6*e - 12, -16, -4*e + 10, -8*e - 34, 7*e + 26, e - 6, -e - 42, 6*e + 42, -2*e + 16, e + 40]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([5, 5, w - 1])] = -1
AL_eigenvalues[ZF.ideal([11, 11, w^3 - 3*w])] = 1

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