/*
  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([4, 3, -5, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([16, 16, w^3 - 4*w])

primes_array = [
[2, 2, w^3 - 4*w - 2],\
[2, 2, w - 1],\
[11, 11, -w^3 + 5*w + 1],\
[17, 17, -w^3 + 3*w + 1],\
[23, 23, w^3 + w^2 - 4*w - 5],\
[29, 29, -w^3 + w^2 + 2*w - 1],\
[29, 29, w^3 + w^2 - 6*w - 5],\
[29, 29, w^3 - w^2 - 4*w + 1],\
[47, 47, w^3 - 3*w - 3],\
[53, 53, 2*w^2 - 2*w - 5],\
[53, 53, -w^3 + 2*w^2 + 5*w - 7],\
[59, 59, 2*w^3 - 8*w - 3],\
[59, 59, w^3 - w^2 - 2*w + 3],\
[67, 67, 2*w - 1],\
[71, 71, 2*w^2 - 5],\
[73, 73, -w^3 + w^2 + 4*w + 1],\
[73, 73, -2*w^3 + 8*w + 1],\
[73, 73, -w^3 + 5*w - 1],\
[79, 79, w^2 - w + 1],\
[79, 79, w^3 + w^2 - 4*w - 7],\
[81, 3, -3],\
[83, 83, 2*w^3 + w^2 - 9*w - 5],\
[83, 83, w^2 + w - 5],\
[97, 97, 4*w^3 + w^2 - 19*w - 11],\
[97, 97, -2*w^3 + 3*w^2 + 7*w - 7],\
[101, 101, -w^3 + 2*w^2 + 3*w - 3],\
[103, 103, w^3 + w^2 - 4*w - 1],\
[103, 103, -2*w^3 + w^2 + 7*w - 5],\
[109, 109, w^3 - w^2 - 2*w - 3],\
[113, 113, 2*w^2 - 9],\
[131, 131, -2*w^2 + 2*w + 7],\
[137, 137, -2*w^3 + w^2 + 9*w - 3],\
[137, 137, 2*w^3 - w^2 - 9*w - 1],\
[149, 149, 2*w^3 - w^2 - 7*w - 1],\
[157, 157, w^3 - 5*w + 3],\
[163, 163, -w^3 + 3*w^2 + 4*w - 7],\
[173, 173, w^3 + 2*w^2 - 3*w - 7],\
[179, 179, w^3 - w^2 - 2*w + 5],\
[179, 179, w^3 - 5*w - 5],\
[191, 191, 2*w^2 + 2*w - 7],\
[197, 197, 3*w^2 - w - 5],\
[197, 197, -2*w^3 + w^2 + 7*w - 7],\
[223, 223, 2*w^3 - 2*w^2 - 8*w + 3],\
[227, 227, w^2 + 3*w - 5],\
[233, 233, -3*w^3 - w^2 + 12*w + 9],\
[233, 233, 3*w^2 - w - 11],\
[239, 239, w^3 - 5*w + 5],\
[239, 239, w^3 + 2*w^2 - 5*w - 3],\
[241, 241, -w^3 + w^2 + 6*w - 5],\
[241, 241, w^3 + 2*w^2 - 5*w - 7],\
[251, 251, 3*w^3 - 13*w - 3],\
[251, 251, w^3 - 2*w^2 - 5*w + 3],\
[257, 257, -2*w^3 + w^2 + 7*w - 1],\
[271, 271, -2*w^3 + 2*w^2 + 6*w - 5],\
[277, 277, w^3 - w - 3],\
[277, 277, -w^3 - w^2 + 6*w + 1],\
[283, 283, -w^3 + 4*w^2 + w - 7],\
[293, 293, 4*w^3 - 18*w - 9],\
[293, 293, 3*w^3 - w^2 - 12*w - 1],\
[307, 307, 3*w^3 - 15*w - 7],\
[313, 313, 2*w^3 - 6*w - 1],\
[313, 313, w^3 + w^2 - 4*w - 9],\
[317, 317, -3*w^2 - w + 5],\
[317, 317, 2*w^3 - w^2 - 7*w - 5],\
[331, 331, 2*w^2 - 2*w - 9],\
[337, 337, w^3 - w^2 - 6*w + 1],\
[347, 347, -2*w^3 + 2*w^2 + 6*w - 3],\
[349, 349, 2*w^3 - 2*w^2 - 4*w + 3],\
[373, 373, 6*w^3 + 2*w^2 - 26*w - 15],\
[379, 379, 3*w^3 - w^2 - 12*w - 5],\
[389, 389, -2*w^3 + 4*w^2 + 4*w - 9],\
[389, 389, 3*w^3 + 2*w^2 - 13*w - 13],\
[397, 397, -3*w^3 + 2*w^2 + 11*w - 1],\
[401, 401, 2*w^3 - w^2 - 9*w - 3],\
[409, 409, w^3 + w^2 - 2*w - 5],\
[419, 419, 3*w^3 - 2*w^2 - 11*w - 1],\
[421, 421, -w^3 + 3*w^2 + 2*w - 11],\
[439, 439, 2*w^3 - 6*w - 5],\
[449, 449, -w^3 + w^2 + 6*w - 3],\
[457, 457, w^3 - w^2 - 3],\
[461, 461, 2*w^3 - 2*w^2 - 10*w + 1],\
[461, 461, -2*w^3 - 2*w^2 + 6*w + 7],\
[463, 463, -2*w^3 + 3*w^2 + 5*w - 1],\
[467, 467, 3*w^3 - 6*w^2 - 13*w + 19],\
[487, 487, w^2 - 3*w - 5],\
[491, 491, -2*w^3 + w^2 + 5*w - 1],\
[499, 499, -w^3 + 3*w^2 + 6*w - 9],\
[503, 503, -2*w^3 + 3*w^2 + 7*w - 13],\
[503, 503, 4*w^3 + 2*w^2 - 18*w - 11],\
[509, 509, -3*w^3 + 3*w^2 + 12*w - 11],\
[523, 523, 3*w^3 - 11*w - 5],\
[541, 541, 3*w^2 - w - 9],\
[541, 541, -w^3 - 2*w^2 + 7*w + 9],\
[541, 541, -3*w^3 + 2*w^2 + 11*w - 5],\
[541, 541, -w^3 + 4*w^2 + 3*w - 11],\
[569, 569, -3*w^3 + 3*w^2 + 12*w - 13],\
[571, 571, 4*w^3 - w^2 - 17*w - 5],\
[571, 571, w^2 + 3*w - 3],\
[577, 577, -w^3 - 2*w^2 + 7*w + 3],\
[593, 593, -2*w^3 + w^2 + 9*w - 5],\
[601, 601, -2*w^3 + 6*w - 5],\
[607, 607, 2*w^3 - 2*w^2 - 10*w + 3],\
[607, 607, 2*w^3 - 3*w^2 - 5*w + 9],\
[613, 613, 2*w^3 - w^2 - 5*w - 5],\
[613, 613, w^2 + w - 9],\
[617, 617, 6*w^3 - 28*w - 13],\
[619, 619, -w^3 + 2*w^2 + 5*w + 1],\
[619, 619, 3*w^2 - 5*w - 7],\
[625, 5, -5],\
[653, 653, 2*w^3 - 5*w^2 - 9*w + 13],\
[653, 653, 2*w^3 + w^2 - 7*w + 1],\
[659, 659, -3*w^3 + 2*w^2 + 9*w - 7],\
[659, 659, -3*w^3 - 2*w^2 + 17*w + 11],\
[661, 661, -2*w^3 + 8*w - 5],\
[677, 677, -w^3 - 3*w^2 + 6*w + 7],\
[677, 677, 3*w^2 + w - 11],\
[677, 677, -w^3 - 2*w^2 + 9*w + 3],\
[677, 677, 2*w^3 - 4*w^2 - 1],\
[719, 719, -w^3 + 3*w^2 - 7],\
[719, 719, -w^2 + w - 3],\
[727, 727, -w^3 + 5*w^2 + 4*w - 13],\
[727, 727, 3*w^3 - 11*w - 3],\
[739, 739, 2*w^3 + 2*w^2 - 8*w - 11],\
[739, 739, -w^3 + 3*w^2 + 4*w - 5],\
[743, 743, w^3 + 3*w^2 - 4*w - 7],\
[751, 751, -2*w^3 + 2*w^2 + 8*w - 1],\
[757, 757, w^3 + 3*w^2 - 6*w - 11],\
[757, 757, -4*w - 1],\
[769, 769, 2*w^3 - 2*w^2 - 8*w + 11],\
[773, 773, -w^3 + 7*w - 1],\
[787, 787, w^2 - 3*w - 7],\
[797, 797, 4*w^3 - 16*w - 7],\
[797, 797, w^3 - 7*w - 1],\
[809, 809, 3*w^3 - 3*w^2 - 12*w + 5],\
[809, 809, w^3 - 5*w^2 - 2*w + 17],\
[811, 811, w^3 - 2*w^2 - 7*w + 9],\
[823, 823, -2*w^3 + 3*w^2 + w + 1],\
[827, 827, -3*w^3 + w^2 + 12*w - 3],\
[827, 827, 8*w^3 - 36*w - 15],\
[829, 829, -w^2 - w - 3],\
[853, 853, 4*w^3 + w^2 - 17*w - 7],\
[853, 853, 4*w - 3],\
[857, 857, 2*w^3 + 3*w^2 - 11*w - 11],\
[859, 859, w^3 - 5*w^2 - 4*w + 15],\
[877, 877, -4*w^3 + w^2 + 15*w - 3],\
[881, 881, -4*w^2 + 6*w + 9],\
[881, 881, -w^3 + 3*w - 5],\
[881, 881, 3*w^2 - 3*w - 11],\
[881, 881, -w^3 + 4*w^2 + w - 13],\
[883, 883, 2*w^3 - 3*w^2 - 3*w - 3],\
[883, 883, w^3 - 5*w^2 + 13],\
[887, 887, -w^3 + 2*w^2 + w - 7],\
[911, 911, -4*w^3 + 4*w^2 + 14*w - 13],\
[911, 911, 2*w^2 + 2*w - 9],\
[919, 919, -w^2 - 5*w + 9],\
[919, 919, 3*w^3 - 15*w - 5],\
[937, 937, -3*w^3 + 3*w^2 + 10*w - 5],\
[953, 953, -8*w^3 - w^2 + 35*w + 17],\
[961, 31, 3*w^3 - w^2 - 14*w - 5],\
[961, 31, 2*w^3 - 5*w^2 - 5*w + 9],\
[967, 967, 5*w^2 + w - 9],\
[967, 967, -5*w^3 + 23*w + 13],\
[977, 977, -w^3 - 2*w^2 + 7*w + 11],\
[977, 977, -3*w^3 + 2*w^2 + 13*w - 7],\
[983, 983, 2*w^3 - 6*w - 7],\
[991, 991, 6*w^2 - 11],\
[991, 991, 3*w^3 - 5*w^2 - 12*w + 13],\
[997, 997, w^3 + w^2 - 8*w - 3],\
[997, 997, -3*w^3 + 4*w^2 + 9*w - 9]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [0, e, 2*e - 2, 0, -2*e - 2, -2*e - 4, 8, 2*e - 8, -4*e + 4, 6*e - 8, -6*e + 4, -4*e + 8, 4*e - 4, -4*e, 4*e, -2*e, -2*e, -6*e + 8, -2*e - 10, 4*e, 8, -4*e - 4, 2*e - 2, 4*e - 8, -6*e + 4, 8, -4*e + 16, 4*e + 4, 6*e - 4, 10, -2*e - 6, 8*e - 10, -6*e + 12, -6*e + 8, 2*e - 4, 16, 8*e - 8, 4*e - 4, 8*e, 12, -4*e, -2*e + 12, -4*e + 4, -8, 2*e + 8, 8*e - 18, 12, -12*e + 16, -8*e + 16, -2*e + 12, -4*e + 8, -4*e + 16, 2*e - 20, 4*e - 12, -12*e + 8, -16*e + 18, 2*e - 2, -12*e + 16, 10*e + 4, -4*e + 8, 8*e - 8, -10*e + 12, 14*e - 16, -2*e - 24, 4*e - 16, -4*e - 8, -8*e + 8, 4*e - 32, 10*e - 20, -12*e + 24, 2*e - 8, -2, -8*e - 6, 6*e - 20, 6*e + 4, -12*e + 20, -2, -4*e + 12, 8*e - 2, -10*e + 32, -16*e + 26, 16*e - 16, -2*e + 14, -2*e - 22, -4*e + 24, 2*e + 22, 20*e - 24, -14*e + 10, -4*e - 16, -8*e + 14, -28, 0, -6*e - 12, -8*e + 38, 10*e - 12, -4*e - 32, 12*e, -12*e + 16, -2*e - 4, 4*e + 32, 22, 14*e - 2, -4*e + 36, -10*e + 12, 18*e - 4, -12*e - 8, -8*e + 12, -20*e + 12, 8*e - 42, 6*e + 8, -4*e + 8, -12*e, 48, -18*e + 32, 26, -2*e - 44, -6*e + 16, -6, -28*e + 28, 20*e - 16, -4*e - 24, 12*e - 32, 16*e - 8, -6*e + 22, 24, 16*e - 20, 26, -14*e + 8, 12*e - 16, -6*e - 28, -8*e - 12, -8*e + 26, -2*e + 4, -18*e + 16, 4*e - 8, -16*e + 24, -10*e + 22, -8*e - 8, 40, -10*e - 16, -24*e + 16, 22*e - 12, 18, -16*e + 4, 8*e + 2, -6*e + 24, -8*e + 26, 16*e - 34, 10*e - 24, 8*e, -20*e + 24, -22*e + 2, -4*e - 12, 4*e - 16, 22*e - 2, -14*e + 34, -18*e + 28, 2*e - 24, 10*e - 4, 10*e - 40, 6*e + 38, -4*e - 32, -6*e - 4, -6*e + 12, -12, -6*e - 14, -6*e + 34, 12*e + 16, 14]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

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

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