/*
  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, -5, 2, 8, -4, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([59, 59, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5])

primes_array = [
[9, 3, -w^2 + 2],\
[19, 19, -w^3 + 4*w],\
[19, 19, w^3 - 3*w - 1],\
[29, 29, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 3],\
[29, 29, w^4 - w^3 - 4*w^2 + 2*w + 2],\
[31, 31, -w^4 + 4*w^2 + w - 3],\
[41, 41, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 4],\
[49, 7, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 11*w + 6],\
[59, 59, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5],\
[59, 59, w^5 - w^4 - 4*w^3 + 3*w^2 + 3*w - 3],\
[59, 59, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 9*w - 4],\
[61, 61, -w^5 + w^4 + 5*w^3 - 3*w^2 - 7*w + 1],\
[64, 2, -2],\
[71, 71, -2*w^5 + 2*w^4 + 10*w^3 - 8*w^2 - 11*w + 6],\
[79, 79, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 9*w + 5],\
[79, 79, -w^4 - w^3 + 5*w^2 + 4*w - 3],\
[79, 79, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 8*w + 4],\
[81, 3, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 13*w - 8],\
[89, 89, 2*w^5 - 2*w^4 - 9*w^3 + 6*w^2 + 8*w - 1],\
[101, 101, -w^4 - w^3 + 5*w^2 + 3*w - 3],\
[101, 101, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 4],\
[109, 109, -2*w^5 + 4*w^4 + 8*w^3 - 14*w^2 - 6*w + 5],\
[109, 109, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 7],\
[125, 5, w^5 - w^4 - 5*w^3 + 4*w^2 + 5*w],\
[131, 131, -w^2 - w + 3],\
[131, 131, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 3*w - 4],\
[131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 5],\
[131, 131, -2*w^5 + 4*w^4 + 8*w^3 - 15*w^2 - 6*w + 7],\
[139, 139, w^4 - 3*w^3 - 3*w^2 + 9*w],\
[149, 149, 2*w^5 - 3*w^4 - 9*w^3 + 12*w^2 + 7*w - 5],\
[149, 149, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 2*w - 2],\
[151, 151, 3*w^5 - 4*w^4 - 13*w^3 + 14*w^2 + 10*w - 5],\
[151, 151, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 12*w + 6],\
[151, 151, 4*w^5 - 5*w^4 - 18*w^3 + 18*w^2 + 15*w - 9],\
[151, 151, 2*w^5 - 2*w^4 - 9*w^3 + 7*w^2 + 8*w - 5],\
[169, 13, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 13*w + 1],\
[179, 179, w^3 - w^2 - 2*w + 3],\
[181, 181, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 3],\
[181, 181, -2*w^4 + w^3 + 8*w^2 - 3*w - 2],\
[191, 191, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 4],\
[191, 191, -w^3 + 2*w^2 + 3*w - 4],\
[191, 191, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 12*w],\
[191, 191, w^4 - 2*w^3 - 4*w^2 + 7*w],\
[199, 199, -3*w^5 + 4*w^4 + 15*w^3 - 14*w^2 - 16*w + 6],\
[199, 199, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 12*w + 7],\
[211, 211, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 8*w + 5],\
[211, 211, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 8],\
[229, 229, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 6],\
[229, 229, w^4 - 4*w^2 - 1],\
[229, 229, w^5 - w^4 - 3*w^3 + 3*w^2 - 3*w - 2],\
[239, 239, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 8*w + 8],\
[241, 241, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 3*w - 2],\
[241, 241, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w + 1],\
[251, 251, -3*w^5 + 3*w^4 + 15*w^3 - 10*w^2 - 17*w + 3],\
[271, 271, -2*w^5 + 4*w^4 + 9*w^3 - 14*w^2 - 8*w + 2],\
[271, 271, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 4*w],\
[271, 271, w^5 - 2*w^4 - 3*w^3 + 7*w^2 - 2*w - 4],\
[289, 17, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 12*w - 9],\
[311, 311, -2*w^5 + 3*w^4 + 7*w^3 - 9*w^2 - 3*w + 2],\
[331, 331, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 14*w - 12],\
[349, 349, -4*w^5 + 6*w^4 + 18*w^3 - 21*w^2 - 17*w + 7],\
[349, 349, -w^4 + w^3 + 5*w^2 - 3*w - 3],\
[359, 359, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 10*w + 9],\
[361, 19, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 6*w - 4],\
[361, 19, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 9],\
[379, 379, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 14*w + 7],\
[379, 379, 2*w^5 - 4*w^4 - 10*w^3 + 15*w^2 + 11*w - 5],\
[389, 389, -5*w^5 + 7*w^4 + 22*w^3 - 25*w^2 - 17*w + 11],\
[389, 389, w^5 - 3*w^4 - 3*w^3 + 12*w^2 - 2*w - 5],\
[401, 401, -4*w^5 + 6*w^4 + 18*w^3 - 23*w^2 - 15*w + 10],\
[401, 401, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - w + 4],\
[401, 401, 4*w^5 - 5*w^4 - 20*w^3 + 19*w^2 + 21*w - 11],\
[401, 401, -3*w^5 + 3*w^4 + 16*w^3 - 12*w^2 - 18*w + 6],\
[409, 409, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 2],\
[409, 409, -w^5 + 3*w^4 + 5*w^3 - 13*w^2 - 5*w + 9],\
[419, 419, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 10*w - 6],\
[419, 419, -w^5 + 7*w^3 - w^2 - 10*w + 4],\
[419, 419, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 9],\
[421, 421, 3*w^5 - 3*w^4 - 16*w^3 + 12*w^2 + 18*w - 7],\
[431, 431, 4*w^5 - 7*w^4 - 17*w^3 + 25*w^2 + 13*w - 10],\
[431, 431, -2*w^5 + 3*w^4 + 7*w^3 - 10*w^2 - w + 5],\
[431, 431, 4*w^5 - 6*w^4 - 19*w^3 + 23*w^2 + 17*w - 10],\
[439, 439, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 11*w + 4],\
[461, 461, w^5 - 2*w^4 - 3*w^3 + 6*w^2 + 2],\
[461, 461, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 7],\
[479, 479, -2*w^5 + w^4 + 9*w^3 - 2*w^2 - 8*w],\
[479, 479, 2*w^5 - 2*w^4 - 10*w^3 + 8*w^2 + 12*w - 7],\
[491, 491, -3*w^5 + 3*w^4 + 14*w^3 - 12*w^2 - 12*w + 7],\
[491, 491, -3*w^5 + 3*w^4 + 15*w^3 - 12*w^2 - 14*w + 8],\
[491, 491, -2*w^2 + w + 6],\
[499, 499, -3*w^5 + 3*w^4 + 14*w^3 - 10*w^2 - 13*w + 6],\
[509, 509, -2*w^5 + 2*w^4 + 8*w^3 - 6*w^2 - 5*w],\
[521, 521, 3*w^5 - 4*w^4 - 14*w^3 + 15*w^2 + 15*w - 9],\
[521, 521, -3*w^5 + 5*w^4 + 14*w^3 - 19*w^2 - 15*w + 9],\
[521, 521, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 16*w + 6],\
[529, 23, w^5 - 3*w^4 - 5*w^3 + 11*w^2 + 7*w - 5],\
[541, 541, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 14*w + 10],\
[569, 569, -w^5 - w^4 + 6*w^3 + 4*w^2 - 6*w - 1],\
[571, 571, -2*w^4 + w^3 + 9*w^2 - 3*w - 4],\
[599, 599, -3*w^5 + 3*w^4 + 14*w^3 - 11*w^2 - 11*w + 4],\
[601, 601, w^4 - 2*w^2 - w - 2],\
[619, 619, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 10*w - 5],\
[619, 619, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 3*w - 3],\
[619, 619, -3*w^5 + 5*w^4 + 13*w^3 - 20*w^2 - 10*w + 10],\
[619, 619, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 8],\
[619, 619, -4*w^5 + 6*w^4 + 19*w^3 - 23*w^2 - 17*w + 11],\
[619, 619, -4*w^5 + 6*w^4 + 17*w^3 - 22*w^2 - 13*w + 10],\
[631, 631, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 5*w - 6],\
[641, 641, -w^3 + w^2 + 3*w - 5],\
[641, 641, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + 3*w + 4],\
[659, 659, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 5*w + 5],\
[661, 661, -2*w^5 + 3*w^4 + 8*w^3 - 12*w^2 - 3*w + 8],\
[661, 661, 2*w^5 - 3*w^4 - 8*w^3 + 10*w^2 + 6*w - 1],\
[691, 691, -w^3 - w^2 + 5*w + 1],\
[691, 691, -3*w^5 + 4*w^4 + 13*w^3 - 13*w^2 - 10*w + 6],\
[691, 691, 3*w^5 - 4*w^4 - 15*w^3 + 15*w^2 + 14*w - 6],\
[701, 701, 3*w^5 - 6*w^4 - 13*w^3 + 23*w^2 + 10*w - 10],\
[701, 701, -3*w^5 + 6*w^4 + 13*w^3 - 23*w^2 - 11*w + 9],\
[709, 709, 3*w^5 - 6*w^4 - 13*w^3 + 22*w^2 + 11*w - 10],\
[709, 709, -w^5 + 5*w^3 - 4*w - 2],\
[709, 709, -w^5 + w^4 + 7*w^3 - 6*w^2 - 11*w + 5],\
[719, 719, -2*w^5 + 3*w^4 + 9*w^3 - 12*w^2 - 7*w + 4],\
[751, 751, w^4 + w^3 - 5*w^2 - 6*w + 4],\
[769, 769, 4*w^5 - 5*w^4 - 19*w^3 + 18*w^2 + 18*w - 10],\
[769, 769, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 13*w - 11],\
[809, 809, w^5 - w^4 - 4*w^3 + 5*w^2 + w - 7],\
[809, 809, 2*w^5 - 5*w^4 - 9*w^3 + 19*w^2 + 10*w - 7],\
[809, 809, -w^5 + 5*w^3 + w^2 - 7*w],\
[809, 809, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 15*w + 11],\
[811, 811, -2*w^5 + 4*w^4 + 8*w^3 - 13*w^2 - 6*w + 5],\
[811, 811, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - w],\
[811, 811, 2*w^5 - 4*w^4 - 9*w^3 + 16*w^2 + 10*w - 9],\
[811, 811, 2*w^5 - 4*w^4 - 10*w^3 + 16*w^2 + 10*w - 7],\
[821, 821, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 5*w - 2],\
[829, 829, 3*w^5 - 5*w^4 - 13*w^3 + 19*w^2 + 12*w - 9],\
[839, 839, 2*w^5 - 4*w^4 - 7*w^3 + 14*w^2 + w - 6],\
[839, 839, 4*w^5 - 5*w^4 - 18*w^3 + 19*w^2 + 14*w - 10],\
[841, 29, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 10*w + 5],\
[841, 29, -w^5 + 3*w^4 + 6*w^3 - 12*w^2 - 10*w + 6],\
[859, 859, 3*w^5 - 3*w^4 - 15*w^3 + 10*w^2 + 14*w - 2],\
[881, 881, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 4],\
[911, 911, -4*w^5 + 5*w^4 + 19*w^3 - 20*w^2 - 17*w + 11],\
[919, 919, 2*w^5 - 3*w^4 - 10*w^3 + 13*w^2 + 9*w - 10],\
[929, 929, -w^5 + w^4 + 5*w^3 - 2*w^2 - 8*w - 1],\
[941, 941, -5*w^5 + 7*w^4 + 23*w^3 - 26*w^2 - 20*w + 11],\
[961, 31, -3*w^5 + 4*w^4 + 16*w^3 - 16*w^2 - 20*w + 8],\
[971, 971, w^5 - 5*w^3 + 5*w - 4],\
[991, 991, 3*w^5 - 5*w^4 - 12*w^3 + 18*w^2 + 8*w - 6],\
[991, 991, 3*w^5 - 5*w^4 - 14*w^3 + 18*w^2 + 12*w - 7]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [e, -2, -e - 3, 2*e, -4*e - 8, -2, -2*e, 2*e + 8, 1, -12, -4*e - 6, -4, 2*e - 3, 0, -4*e - 12, -5*e - 12, -e - 10, -2*e - 6, 7*e + 10, 4*e + 4, 6*e + 4, -2*e - 4, -e + 12, -6*e - 18, -10*e - 12, 8*e + 16, 4*e - 8, 5*e - 1, 6*e - 2, -e + 3, -3*e - 12, -e - 19, 12*e + 20, 7*e, 13*e + 19, -2*e - 18, 22, 7*e + 23, -2*e - 18, 7*e + 11, -6*e - 18, -12, -6*e + 6, -8*e - 16, -4*e - 12, -4, -5*e + 9, 4*e - 6, -5*e - 6, -5*e - 5, -5*e + 4, -6*e - 24, -3*e - 17, e - 3, -8*e, 13*e + 19, 15*e + 24, -e + 29, 9*e + 20, -9*e - 17, -2*e + 6, -8*e - 12, -3*e + 2, 8*e - 2, -2*e - 20, 2, -4*e - 24, 2*e - 24, 6*e + 6, 5*e + 26, -6*e - 36, 15*e + 23, -7*e - 21, 3*e + 34, 10*e + 12, 6*e + 4, -5*e - 6, -11*e - 19, 4*e + 28, e - 19, 4*e + 2, 8*e + 22, -11*e - 2, -3*e + 12, 16*e + 18, 3*e + 21, -13*e - 35, e + 25, 5*e + 29, 15*e + 23, -10*e - 6, 11*e + 33, e + 31, -6*e - 24, -e - 34, -12*e - 22, 6*e - 14, -6*e - 6, 32, -14*e - 8, 3*e + 40, 9*e - 8, e + 13, 14*e + 4, -6*e - 18, -22*e - 38, 4*e, 20*e + 38, -12*e - 26, 6*e + 6, 6*e - 18, -18*e - 28, 16*e + 44, 6*e + 18, 11*e + 16, 16*e + 20, 8*e + 8, -21*e - 41, 8*e - 6, -2*e + 12, -11*e - 16, 14*e + 8, -4*e - 2, 12*e - 10, 8*e + 26, -3*e - 27, -8*e - 30, 3*e + 42, 18, 8*e - 4, -10*e, 10*e + 24, -18*e - 36, 3*e + 13, -3*e + 35, 15*e + 3, 3*e - 37, 9*e + 3, -21*e - 40, -e - 9, 12, -3*e + 21, -10*e - 30, 6*e + 6, 15*e + 24, -21*e - 53, e + 34, 2*e + 12, -9*e - 2]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([59, 59, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5])] = -1

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