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

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

primes_array = [
[5, 5, w],\
[5, 5, w^3 - 2*w^2 - 2*w + 3],\
[11, 11, -w^2 + 4],\
[11, 11, w^2 - 2*w - 3],\
[16, 2, 2],\
[29, 29, w^3 - 4*w - 1],\
[29, 29, w^3 - 3*w^2 - w + 4],\
[41, 41, w^3 - 2*w^2 - w + 4],\
[41, 41, -w^3 + w^2 + 2*w + 2],\
[59, 59, -2*w^3 + 4*w^2 + 4*w - 7],\
[59, 59, -3*w^2 + 2*w + 7],\
[61, 61, -w^3 + 4*w^2 - 6],\
[61, 61, w^3 + w^2 - 5*w - 3],\
[71, 71, w^3 + w^2 - 4*w - 6],\
[71, 71, 3*w^2 - 2*w - 8],\
[71, 71, 3*w^2 - 4*w - 7],\
[71, 71, w^3 - 4*w^2 + w + 8],\
[79, 79, -2*w^3 + 3*w^2 + 5*w - 2],\
[79, 79, 2*w^2 - 3*w - 7],\
[79, 79, 2*w^2 - w - 8],\
[79, 79, -2*w^3 + 3*w^2 + 5*w - 4],\
[81, 3, -3],\
[89, 89, -w - 3],\
[89, 89, w - 4],\
[101, 101, w^3 - 6*w - 2],\
[101, 101, 2*w^3 - 3*w^2 - 5*w + 3],\
[101, 101, -w^3 + 3*w^2 + 3*w - 7],\
[109, 109, -3*w^2 + 5*w + 4],\
[109, 109, -2*w^3 + w^2 + 6*w + 4],\
[121, 11, 3*w^2 - 3*w - 8],\
[139, 139, -w^3 - w^2 + 6*w + 3],\
[139, 139, w^3 - 4*w^2 - w + 7],\
[149, 149, -2*w^3 + 6*w^2 + w - 11],\
[149, 149, 2*w^3 - 7*w - 6],\
[151, 151, -w^3 + 2*w^2 + 4*w - 4],\
[151, 151, w^3 - w^2 - 5*w + 1],\
[169, 13, -4*w^2 + 5*w + 11],\
[169, 13, 3*w^3 - 6*w^2 - 7*w + 8],\
[181, 181, w^3 + w^2 - 4*w - 7],\
[181, 181, -2*w^2 + 5*w + 1],\
[181, 181, 3*w^3 - 3*w^2 - 9*w + 2],\
[181, 181, -w^3 + 4*w^2 - w - 9],\
[191, 191, w^3 - 5*w + 1],\
[191, 191, -w^3 + 3*w^2 + 2*w - 3],\
[199, 199, -2*w^3 + 2*w^2 + 5*w + 4],\
[199, 199, -2*w^3 + 9*w + 4],\
[229, 229, 2*w^2 - 5*w - 4],\
[229, 229, -4*w^2 + 5*w + 8],\
[241, 241, -3*w^3 + 7*w^2 + 5*w - 12],\
[241, 241, -3*w^3 + 2*w^2 + 10*w + 3],\
[269, 269, -w^3 + 6*w^2 - 3*w - 13],\
[269, 269, w^3 + 3*w^2 - 6*w - 11],\
[271, 271, 2*w^3 - 2*w^2 - 7*w + 3],\
[271, 271, -2*w^3 + 4*w^2 + 5*w - 4],\
[289, 17, w^3 - 6*w^2 + 2*w + 11],\
[289, 17, w^3 + 3*w^2 - 7*w - 8],\
[311, 311, -2*w^3 + w^2 + 5*w + 4],\
[311, 311, 2*w^3 - 5*w^2 - w + 8],\
[331, 331, 3*w^3 - 3*w^2 - 11*w - 3],\
[331, 331, -3*w^3 + 6*w^2 + 8*w - 14],\
[349, 349, -3*w^3 + 6*w^2 + 6*w - 8],\
[349, 349, 3*w^3 - 3*w^2 - 9*w + 1],\
[359, 359, -2*w^3 - w^2 + 7*w + 8],\
[359, 359, -2*w^3 + 6*w^2 + w - 12],\
[359, 359, 2*w^3 - 7*w - 7],\
[359, 359, -2*w^3 + 7*w^2 - w - 12],\
[361, 19, 4*w^2 - 4*w - 11],\
[361, 19, -4*w^2 + 4*w + 9],\
[389, 389, -w^3 + 4*w^2 - w - 11],\
[389, 389, w^3 + w^2 - 4*w - 9],\
[401, 401, w^2 - 8],\
[401, 401, w^2 - 2*w - 7],\
[409, 409, -3*w^2 + 5*w + 9],\
[409, 409, -3*w^3 + w^2 + 12*w + 4],\
[409, 409, -w^3 + w^2 + 6*w + 1],\
[409, 409, 3*w^2 - w - 11],\
[419, 419, w^3 - 7*w - 3],\
[419, 419, -w^3 + 3*w^2 + 4*w - 9],\
[431, 431, -2*w^3 + 2*w^2 + 7*w - 4],\
[431, 431, w^3 + 2*w^2 - 7*w - 4],\
[439, 439, -2*w^3 + 5*w^2 + 2*w - 11],\
[439, 439, 2*w^3 - w^2 - 6*w - 6],\
[449, 449, w^3 + 3*w^2 - 7*w - 9],\
[449, 449, w^3 - 6*w^2 + 5*w + 11],\
[449, 449, 3*w^3 - 7*w^2 - 3*w + 8],\
[449, 449, -w^3 + 6*w^2 - 2*w - 12],\
[461, 461, -3*w^3 + 8*w^2 + 4*w - 18],\
[461, 461, 3*w^3 - 4*w^2 - 5*w - 1],\
[479, 479, -w^3 + 5*w^2 - 2*w - 16],\
[479, 479, 2*w^3 - 4*w^2 - w - 1],\
[491, 491, -w^3 + 7*w^2 - w - 18],\
[491, 491, w^3 + 2*w^2 - 5*w - 11],\
[491, 491, -w^3 + 5*w^2 - 2*w - 13],\
[491, 491, w^3 + 4*w^2 - 10*w - 13],\
[499, 499, 3*w^3 - 4*w^2 - 8*w + 3],\
[499, 499, -w^3 + 4*w^2 + 2*w - 6],\
[499, 499, -w^3 - w^2 + 7*w + 1],\
[499, 499, 3*w^3 - 5*w^2 - 7*w + 6],\
[529, 23, 4*w^3 - 3*w^2 - 13*w - 1],\
[529, 23, w^3 + 4*w^2 - 9*w - 8],\
[569, 569, -3*w^3 + 7*w^2 + 4*w - 14],\
[569, 569, 3*w^3 - 4*w^2 - 7*w + 1],\
[569, 569, 3*w^3 - 5*w^2 - 6*w + 7],\
[569, 569, w^3 - 4*w^2 + 4*w + 7],\
[571, 571, -w^3 + 7*w^2 - w - 17],\
[571, 571, w^3 + 4*w^2 - 10*w - 12],\
[599, 599, 3*w^3 - 6*w^2 - 5*w + 7],\
[599, 599, 3*w^3 - 3*w^2 - 8*w + 1],\
[619, 619, -w^3 + 3*w^2 + 4*w - 7],\
[619, 619, 2*w^3 - 6*w^2 - w + 13],\
[619, 619, -w^3 + 3*w^2 - 2*w - 6],\
[619, 619, w^3 - 7*w - 1],\
[641, 641, 3*w^3 - 3*w^2 - 7*w + 1],\
[641, 641, -3*w^3 + 6*w^2 + 4*w - 6],\
[659, 659, -2*w^3 + 5*w^2 + 6*w - 7],\
[659, 659, -2*w^3 + w^2 + 10*w - 2],\
[661, 661, 2*w^3 - 4*w^2 - w + 6],\
[661, 661, w^3 + 2*w^2 - 5*w - 12],\
[701, 701, -w^3 + 3*w^2 - w - 9],\
[701, 701, -2*w^3 + w^2 + 9*w - 1],\
[701, 701, 2*w^3 - 8*w - 1],\
[701, 701, w^3 - 2*w - 8],\
[709, 709, 2*w^3 - 2*w^2 - 9*w + 1],\
[709, 709, w^3 - w^2 + w - 2],\
[719, 719, -5*w^2 + 4*w + 14],\
[719, 719, -5*w^2 + 6*w + 13],\
[739, 739, 4*w^3 - 6*w^2 - 11*w + 7],\
[739, 739, -4*w^3 + 6*w^2 + 11*w - 6],\
[751, 751, -w^3 + 7*w^2 - 4*w - 16],\
[751, 751, -w^3 - 4*w^2 + 7*w + 14],\
[769, 769, w^3 - 3*w^2 - w + 11],\
[769, 769, -w^3 + 4*w + 8],\
[811, 811, -w^3 + 4*w^2 - 14],\
[811, 811, -w^3 - w^2 + 5*w + 11],\
[829, 829, -w^3 + 2*w^2 - w - 6],\
[829, 829, w^2 + 3*w - 1],\
[829, 829, 3*w^3 - 13*w - 6],\
[829, 829, w^3 - w^2 - 6],\
[841, 29, 5*w^2 - 5*w - 14],\
[859, 859, -w^3 + 6*w^2 - 11],\
[859, 859, -3*w^3 + 5*w^2 + 8*w - 2],\
[881, 881, -3*w^3 + w^2 + 11*w + 2],\
[881, 881, -3*w^3 + 8*w^2 + 4*w - 11],\
[911, 911, -2*w^2 - w + 11],\
[911, 911, 2*w^2 - 5*w - 8],\
[919, 919, w^3 - 5*w^2 - w + 8],\
[919, 919, -w^3 - 2*w^2 + 8*w + 3],\
[941, 941, 2*w^3 + 2*w^2 - 11*w - 7],\
[941, 941, -2*w^3 + 8*w^2 + w - 14],\
[961, 31, -2*w^2 + 2*w + 11],\
[961, 31, 5*w^2 - 5*w - 13],\
[971, 971, w^3 - 2*w^2 - 3*w - 3],\
[971, 971, -w^3 + w^2 + 4*w - 7],\
[991, 991, -w^3 + 4*w^2 + 2*w - 16],\
[991, 991, w^3 + 3*w^2 - 5*w - 13],\
[991, 991, -w^3 + 6*w^2 - 4*w - 14],\
[991, 991, 2*w^3 + 3*w^2 - 10*w - 12]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 - 2*x^2 - 4*x + 4
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -1/2*e^2 + e + 3, -e + 2, e^2 - e - 4, -1/2*e^2 + 2*e + 3, e^2 - 2*e + 2, -2*e^2 + 7, -2*e^2 + 3*e + 2, 2*e^2 - 6*e - 6, -1, 1/2*e^2 - 3*e - 1, -4*e^2 + 3*e + 10, e^2 - 8, 2*e^2 + e - 8, -5/2*e^2 + 5*e + 7, 2*e^2 - 3*e + 2, e^2 - 5*e + 2, 2*e^2 + 3*e - 9, 5*e^2 - 4*e - 14, e^2 + 3*e - 13, e^2 - 2*e - 3, -2*e^2 + 10, 2*e^2 - 4*e + 4, e - 4, -2*e^2 - e + 8, -5*e + 8, -3*e + 14, 3*e^2 - 3*e - 6, -e^2 - 5*e + 6, -1/2*e^2 - 4*e + 1, -2*e^2 - e + 16, e^2 - 2*e - 3, -e^2 + 1, -5/2*e^2 - 3*e + 17, 3*e^2 - 4*e - 4, 4*e^2 - 5*e - 16, -e^2 - e + 10, -7/2*e^2 + 9*e + 5, -5*e^2 + 7*e + 14, e^2 - 9*e - 2, 3*e^2 + 2*e - 13, 2*e^2 + e - 14, e^2 + 9*e - 10, 3*e^2 - 10*e - 1, -3*e^2 - e + 12, -5*e^2 + e + 16, -5*e^2 - e + 24, 4*e + 14, e^2 + 10*e - 8, 9*e^2 - 12*e - 23, e - 14, -e^2 + e - 13, -3*e^2 + 11*e + 2, -1/2*e^2 + 4*e - 5, 7*e^2 - 7*e - 24, e^2 + 6*e - 10, -8*e^2 + 8*e + 24, -19/2*e^2 + 11*e + 21, e^2 - 5*e + 16, 4*e^2 + e - 16, 11/2*e^2 - 3*e - 21, -2*e^2 + e - 2, -5*e^2 + 9*e + 14, 2*e^2 - 9*e + 4, -3*e^2 - e + 12, 7*e^2 - 10*e - 27, 13/2*e^2 - 19*e - 25, 6*e^2 + 6*e - 32, -e^2 + 12*e + 8, 4*e^2 - 2*e - 6, 8*e^2 - 9*e - 14, 5/2*e^2 - 2*e - 19, -5*e^2 + 7*e + 12, -5*e^2 + e + 16, 4*e^2 - 4*e - 28, 5*e^2 - 3*e - 18, 11*e^2 - 10*e - 26, -5*e^2 + 14*e + 24, 4*e^2 - 7*e - 19, -3*e^2 + 7*e + 2, 6*e^2 - e - 32, -7*e^2 + 18*e + 20, 9*e^2 - 8*e - 12, -5/2*e^2 + 9*e - 1, -8*e^2 + 19*e + 17, -5*e - 20, 3*e^2 + 4*e - 16, -9/2*e^2 + 14*e + 19, 15/2*e^2 + e - 29, -9*e^2 + 15*e + 34, 1/2*e^2 + 2*e - 19, -6*e^2 + 2*e + 11, -2*e^2 + 8*e + 28, 4*e^2 - 3*e, -12*e^2 + 7*e + 44, -e^2 - 13*e + 18, -6*e^2 + 13*e + 4, e^2 - 6*e + 8, -e^2 + e + 2, -2*e^2 - 11*e + 6, 5*e^2 + 7*e - 33, -7/2*e^2 - 12*e + 19, 7*e + 22, -3/2*e^2 + 14*e - 7, -3*e^2 + 2*e - 8, -9/2*e^2 + 4*e + 23, 3*e^2 - 10*e - 38, 2*e^2 - 7*e - 4, 3*e^2 - 7*e + 10, -5*e^2 + 9*e + 4, -15*e + 10, e^2 + 7*e - 14, -2*e^2 + 19*e - 7, 2*e^2 - 6*e - 36, 6*e^2 - 8*e - 29, 3/2*e^2 - 4*e - 23, -7*e^2 - 5*e + 24, -e^2 - 7*e + 6, -e^2 + 17*e - 10, 6*e^2 - 19*e - 18, -4*e^2 + 2*e + 28, 7/2*e^2 + 3*e - 31, 9*e^2 - 17*e - 11, 11*e^2 - 25*e - 26, -2*e^2 + 15*e + 12, 3*e^2 - 2*e - 20, 8*e^2 - 21*e - 14, -13*e^2 + 15*e + 18, -5*e^2 + 18*e + 6, -e^2 + 10*e - 6, -6*e^2 + 6*e + 32, -2*e^2 + 6*e + 38, -10*e^2 + 18*e + 35, 3*e^2 - 4*e, e^2 + 2*e - 44, -2*e^2 + 2*e + 4, -7*e^2 + 9*e + 36, -5*e^2 + 17*e + 32, 3*e^2 - 3*e - 38, -8*e^2 + 15*e + 18, -9*e^2 + 12*e + 26, 9*e^2 - 46, -e^2 - 12*e + 32, 13/2*e^2 + 4*e - 17, 3*e^2 + e - 39, -9*e^2 - 3*e + 26, -2*e^2 + 15*e - 13, -10*e^2 + 18*e + 31, 2*e^2 - 9*e + 12, -8*e^2 + 25*e + 15, 17/2*e^2 - 31*e - 35, -12*e^2 + 16*e + 36, 4*e^2 - 18*e - 29, 4*e^2 + e + 4, -1/2*e^2 + 2*e - 23, -11/2*e^2 + 25*e + 35, -e^2 - 17*e + 6]
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, -2*w^3 + 4*w^2 + 4*w - 7])] = 1

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