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

NN = ZF.ideal([131,131,-2*w^3 + 3*w^2 + 2*w - 4])

primes_array = [
[11, 11, -w^3 + 2*w^2 + w - 3],\
[11, 11, w^3 - 3*w],\
[16, 2, 2],\
[19, 19, -w^3 + 2*w + 2],\
[19, 19, 2*w^3 - 3*w^2 - 4*w + 2],\
[25, 5, 2*w^3 - 2*w^2 - 4*w + 1],\
[29, 29, w^3 - w^2 - 4*w + 1],\
[31, 31, w^3 - 4*w + 1],\
[31, 31, -w^2 + 2*w + 3],\
[41, 41, 2*w^2 - w - 3],\
[41, 41, -w^3 + 3*w^2 + w - 4],\
[49, 7, 2*w^3 - 3*w^2 - 5*w + 2],\
[49, 7, w^2 + w - 3],\
[61, 61, 2*w^3 - 3*w^2 - 4*w],\
[61, 61, -3*w^3 + 4*w^2 + 7*w - 3],\
[79, 79, 2*w^3 - 4*w^2 - 3*w + 2],\
[79, 79, w^3 + w^2 - 3*w - 5],\
[81, 3, -3],\
[89, 89, -3*w^3 + 4*w^2 + 5*w - 3],\
[89, 89, 3*w^3 - 2*w^2 - 7*w],\
[101, 101, -3*w^3 + 5*w^2 + 7*w - 5],\
[101, 101, -2*w^2 - w + 4],\
[109, 109, 3*w^3 - 5*w^2 - 5*w + 3],\
[109, 109, -2*w^3 + 7*w + 1],\
[109, 109, 4*w^3 - 5*w^2 - 9*w + 3],\
[109, 109, -2*w^3 + 5*w + 4],\
[121, 11, 3*w^3 - 3*w^2 - 6*w + 1],\
[131, 131, -3*w^3 + 2*w^2 + 8*w - 2],\
[131, 131, w^3 + w^2 - 5*w],\
[139, 139, -w^3 + 2*w^2 + 2*w - 6],\
[139, 139, 2*w^3 - 5*w^2 - 2*w + 6],\
[139, 139, 3*w^3 - 5*w^2 - 6*w + 4],\
[139, 139, -2*w^3 + 2*w^2 + 7*w - 2],\
[149, 149, -3*w^3 + 4*w^2 + 7*w - 2],\
[149, 149, w^3 - w^2 - 3*w - 3],\
[149, 149, 3*w^3 - 2*w^2 - 9*w + 2],\
[149, 149, w - 4],\
[169, 13, -4*w^3 + 5*w^2 + 8*w - 4],\
[169, 13, 3*w^3 - 2*w^2 - 6*w],\
[179, 179, -4*w^3 + 3*w^2 + 10*w - 3],\
[179, 179, -4*w^3 + 4*w^2 + 9*w - 3],\
[179, 179, -2*w^3 + w^2 + 8*w - 1],\
[179, 179, 4*w^3 - 3*w^2 - 8*w - 2],\
[191, 191, -w^3 + 4*w^2 - 6],\
[191, 191, -4*w^3 + 4*w^2 + 9*w - 2],\
[211, 211, 2*w^3 - w^2 - 8*w],\
[211, 211, -w^3 + 4*w^2 + w - 5],\
[229, 229, w^3 - 3*w^2 + w + 5],\
[229, 229, 2*w^3 - 7*w],\
[251, 251, 3*w^3 - 4*w^2 - 7*w + 1],\
[251, 251, w^3 - w - 4],\
[269, 269, 3*w^3 - 6*w^2 - 3*w + 7],\
[269, 269, 3*w^3 - 9*w - 2],\
[271, 271, -3*w^3 + w^2 + 10*w],\
[271, 271, 4*w^3 - 5*w^2 - 9*w + 2],\
[281, 281, -3*w^3 + w^2 + 11*w - 1],\
[281, 281, -2*w^2 + 5*w + 4],\
[281, 281, -3*w^3 + 3*w^2 + 10*w - 1],\
[281, 281, 3*w^3 - 4*w^2 - 7*w - 1],\
[311, 311, -w^3 + 7*w + 2],\
[311, 311, 3*w^2 - 7],\
[331, 331, w^3 + 2*w^2 - 3*w - 7],\
[331, 331, w^3 - w^2 - w + 5],\
[359, 359, -4*w^3 + 5*w^2 + 6*w - 4],\
[359, 359, -5*w^3 + 4*w^2 + 12*w - 2],\
[361, 19, -4*w^3 + 4*w^2 + 8*w - 3],\
[379, 379, -3*w^3 + 10*w + 2],\
[379, 379, 2*w^3 - 5*w^2 + 6],\
[389, 389, -5*w^3 + 6*w^2 + 10*w - 3],\
[389, 389, w^3 + 3*w^2 - 6*w - 7],\
[409, 409, -4*w^3 + 4*w^2 + 7*w],\
[409, 409, -4*w^2 + 4*w + 5],\
[421, 421, -2*w^3 + 2*w^2 + 8*w - 3],\
[421, 421, w^3 + 3*w^2 - 6*w - 6],\
[431, 431, 4*w^3 - 6*w^2 - 10*w + 5],\
[431, 431, 3*w^3 - 5*w^2 - 6*w + 2],\
[431, 431, -4*w^3 + 3*w^2 + 8*w - 2],\
[431, 431, -5*w^3 + 6*w^2 + 10*w - 6],\
[449, 449, 4*w^3 - 5*w^2 - 6*w + 2],\
[449, 449, -w^3 + 2*w^2 + w - 7],\
[461, 461, 5*w^3 - 4*w^2 - 12*w + 1],\
[461, 461, w^3 + 2*w^2 - 2*w - 6],\
[479, 479, w^3 + 3*w^2 - 5*w - 7],\
[479, 479, 2*w^3 - 6*w^2 - w + 6],\
[491, 491, 3*w^2 - 5*w - 6],\
[491, 491, -2*w^3 - w^2 + 9*w + 1],\
[499, 499, 4*w^3 - 3*w^2 - 12*w + 4],\
[499, 499, 4*w^3 - 5*w^2 - 10*w + 2],\
[499, 499, -5*w^3 + 5*w^2 + 11*w - 4],\
[499, 499, -4*w^3 + 4*w^2 + 7*w - 3],\
[509, 509, -5*w^3 + 6*w^2 + 10*w - 4],\
[509, 509, -4*w^2 + 3*w + 8],\
[509, 509, -3*w^3 + w^2 + 10*w - 1],\
[509, 509, 5*w^3 - 6*w^2 - 12*w + 3],\
[521, 521, -5*w^3 + 6*w^2 + 10*w - 5],\
[521, 521, 2*w^3 - 6*w^2 - w + 7],\
[521, 521, w^3 - 5*w^2 + 9],\
[521, 521, w^3 + 3*w^2 - 5*w - 6],\
[529, 23, 3*w^3 - 5*w^2 - 6*w + 1],\
[529, 23, -2*w^3 - w^2 + 8*w + 1],\
[541, 541, 3*w^3 - 5*w^2 - 8*w + 3],\
[541, 541, -w^3 + 3*w^2 + 4*w - 7],\
[569, 569, -w^3 + 7*w - 1],\
[569, 569, -3*w^3 + 2*w^2 + 11*w],\
[571, 571, w^3 - 4*w - 6],\
[571, 571, 4*w^3 - 5*w^2 - 11*w + 2],\
[571, 571, -5*w^3 + 5*w^2 + 11*w - 3],\
[571, 571, -2*w^3 + 6*w^2 + 2*w - 9],\
[599, 599, 3*w^3 - 6*w^2 - 6*w + 4],\
[599, 599, 3*w^2 - 8],\
[601, 601, -4*w^3 + 2*w^2 + 12*w - 1],\
[601, 601, -5*w^3 + 7*w^2 + 11*w - 5],\
[619, 619, -w^3 + 5*w^2 - w - 6],\
[619, 619, 4*w^2 - 3*w - 7],\
[631, 631, 2*w^3 - 3*w^2 - 5*w - 3],\
[631, 631, 5*w^3 - 3*w^2 - 13*w + 2],\
[631, 631, 2*w^3 + w^2 - 11*w - 2],\
[631, 631, 6*w^3 - 6*w^2 - 14*w + 3],\
[641, 641, 2*w^2 - w - 9],\
[641, 641, -w^3 + 3*w^2 + 2*w - 10],\
[659, 659, -4*w^3 + w^2 + 14*w + 1],\
[659, 659, 5*w^3 - 6*w^2 - 11*w + 1],\
[709, 709, 6*w^3 - 7*w^2 - 14*w + 3],\
[709, 709, 6*w^3 - 8*w^2 - 15*w + 7],\
[709, 709, 2*w^3 - 5*w^2 - 5*w + 2],\
[709, 709, -4*w^3 + 3*w^2 + 11*w - 5],\
[739, 739, 5*w^3 - 7*w^2 - 7*w + 5],\
[739, 739, -w^3 - 3*w^2 + 6*w + 1],\
[751, 751, 5*w^3 - 3*w^2 - 11*w + 1],\
[751, 751, 5*w^3 - 6*w^2 - 13*w + 3],\
[769, 769, -w^2 + 2*w + 7],\
[769, 769, -4*w^3 + 6*w^2 + 11*w - 4],\
[809, 809, -5*w^3 + 4*w^2 + 13*w - 5],\
[809, 809, 2*w^3 + w^2 - 8*w],\
[811, 811, -w^3 - 4*w^2 + 7*w + 6],\
[811, 811, 3*w^3 - 5*w^2 - 7*w],\
[811, 811, -2*w^3 - w^2 + 9*w],\
[811, 811, -5*w^3 + 5*w^2 + 9*w - 5],\
[821, 821, -w^3 + 3*w^2 + 4*w - 8],\
[821, 821, -3*w^3 + 7*w + 6],\
[821, 821, -3*w^3 + w^2 + 6*w + 5],\
[821, 821, 5*w^3 - 8*w^2 - 9*w + 5],\
[829, 829, -w^3 + w^2 + w - 6],\
[829, 829, 5*w^3 - 6*w^2 - 9*w + 3],\
[839, 839, -4*w^3 + 2*w^2 + 14*w - 3],\
[839, 839, 5*w^3 - 7*w^2 - 7*w + 4],\
[841, 29, -w^2 + 6*w + 1],\
[859, 859, -4*w^3 + 2*w^2 + 13*w - 2],\
[859, 859, 5*w^3 - 6*w^2 - 12*w + 2],\
[881, 881, -2*w^3 + 4*w^2 + w - 9],\
[881, 881, -6*w^3 + 9*w^2 + 12*w - 7],\
[911, 911, 5*w^3 - 6*w^2 - 11*w],\
[911, 911, -w^3 + 4*w^2 + 4*w - 8],\
[929, 929, 5*w^3 - 8*w^2 - 11*w + 7],\
[929, 929, -4*w^3 + 13*w + 3],\
[929, 929, 3*w^3 - 7*w^2 - w + 8],\
[929, 929, -6*w^3 + 5*w^2 + 16*w - 5],\
[941, 941, -4*w^3 + w^2 + 11*w],\
[941, 941, -2*w^3 - w^2 + 10*w - 1],\
[941, 941, -5*w^3 + 5*w^2 + 9*w],\
[941, 941, -5*w^2 + 5*w + 6],\
[961, 31, -5*w^3 + 5*w^2 + 10*w - 3],\
[971, 971, -2*w^3 - 3*w^2 + 8*w + 7],\
[971, 971, 3*w^3 - 8*w^2 - 2*w + 9]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 - 4*x - 8
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [-e + 2, e, -1/2*e + 1, -e - 1, 1/2*e + 3, 3/2*e - 3, -e - 2, e - 2, e - 2, -3/2*e + 5, -e - 1, 2*e - 7, e - 6, -3*e + 10, 7, 2*e + 1, -2*e - 6, 4*e - 10, 2*e, -2*e + 4, -5/2*e + 11, e + 2, -e + 12, 2*e - 2, e + 10, -e - 10, e, -1/2*e - 5, -1, -e - 13, -2*e - 12, -9/2*e + 7, -3*e, -3*e - 1, 2*e - 6, 4*e - 8, 4*e - 8, -2*e - 4, -4*e - 2, e - 8, 5/2*e + 7, 3*e - 21, 3*e - 10, -4*e + 9, -4*e - 2, 3/2*e + 7, -e + 4, -3*e - 9, -3*e + 13, 1/2*e + 15, -3*e + 2, 5*e - 10, -3/2*e - 9, 3/2*e + 1, -9/2*e + 7, -5/2*e - 7, 2*e - 6, 2*e + 16, -e + 8, 7*e - 14, -3*e + 18, 4*e - 13, -e - 8, -14, -7/2*e - 5, -5*e + 15, 2*e + 4, 3*e + 14, -15/2*e + 7, 16, -21/2*e + 19, 5*e - 2, 2*e + 24, 3*e - 32, -3*e + 6, 7*e - 26, -e + 26, -8, 3*e + 18, 4*e + 6, 2*e - 2, e - 1, -6*e + 24, 7*e - 11, -3*e + 22, -4*e + 1, e + 4, 5*e - 22, -2*e + 18, -17/2*e + 19, -e - 6, 6*e - 24, -4*e + 8, e - 8, 5/2*e + 19, -3*e - 3, e + 4, -6, 11/2*e - 9, -4*e + 6, 6*e - 36, 2*e + 12, -6*e + 4, -2*e + 22, -2*e - 20, 3*e - 14, -5*e + 16, 3*e - 14, -1/2*e - 21, 5/2*e + 31, 5*e - 8, -3*e + 22, 7*e - 14, -6*e + 21, 4*e - 10, -5*e - 1, 2*e + 3, -13/2*e - 5, -4*e + 30, 4, -e - 10, -2*e - 20, 5*e + 12, -4*e + 32, -6*e + 34, 9/2*e - 37, 9*e - 28, -e + 26, -7*e + 11, 3*e - 10, -22, 44, 5/2*e - 23, 9/2*e - 3, 9/2*e - 23, -9*e - 4, 7*e + 2, 2*e + 7, 8*e - 22, -4*e + 34, e + 18, 15/2*e - 27, -11*e + 16, 27, 6*e - 13, 13*e - 20, -6*e + 12, 9*e - 7, -7*e + 42, 1/2*e + 51, -2*e + 26, -4*e - 8, 7*e - 8, 6, 4*e - 20, -6*e - 10, -5*e - 22, -5/2*e - 7, 1/2*e - 21, 14*e - 40, 4*e + 3, 6*e + 10, -2*e - 5, -21/2*e + 31]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([131,131,-2*w^3 + 3*w^2 + 2*w - 4])] = 1

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