/*
  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([3, 2, -5, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([11, 11, -w^3 + 3*w^2 + 2*w - 5])

primes_array = [
[3, 3, w],\
[4, 2, -w^3 + 3*w^2 + w - 2],\
[11, 11, -w^3 + 3*w^2 + 2*w - 5],\
[13, 13, w^3 - 2*w^2 - 4*w + 2],\
[13, 13, -w + 2],\
[17, 17, w^3 - 3*w^2 - 2*w + 2],\
[19, 19, -w^2 + w + 4],\
[19, 19, w^3 - 2*w^2 - 3*w + 2],\
[23, 23, w^2 - 2*w - 1],\
[27, 3, w^3 - 2*w^2 - 5*w - 1],\
[29, 29, -w^3 + 2*w^2 + 5*w - 1],\
[37, 37, -w^3 + 2*w^2 + 5*w - 4],\
[41, 41, 2*w^3 - 5*w^2 - 6*w + 4],\
[53, 53, w^3 - 2*w^2 - 3*w - 2],\
[59, 59, w - 4],\
[67, 67, 2*w^2 - 3*w - 8],\
[73, 73, 2*w^3 - 5*w^2 - 6*w + 5],\
[89, 89, -2*w^3 + 6*w^2 + 5*w - 7],\
[97, 97, -2*w^3 + 6*w^2 + 3*w - 7],\
[97, 97, -w^3 + 3*w^2 + 3*w - 1],\
[103, 103, w^2 - 4*w - 4],\
[113, 113, 2*w^3 - 5*w^2 - 4*w + 1],\
[113, 113, -w^3 + 3*w^2 + w - 5],\
[139, 139, 2*w^2 - 3*w - 4],\
[139, 139, w^3 - w^2 - 6*w - 1],\
[149, 149, 2*w^3 - 5*w^2 - 5*w + 2],\
[149, 149, w^3 - 4*w^2 + 10],\
[151, 151, -3*w - 1],\
[157, 157, w^2 - w - 8],\
[157, 157, w^3 - 7*w - 8],\
[163, 163, -3*w^3 + 8*w^2 + 8*w - 10],\
[163, 163, w^3 - 3*w^2 - 4*w + 7],\
[163, 163, -w^3 + 4*w^2 + w - 8],\
[163, 163, -2*w^3 + 5*w^2 + 7*w - 4],\
[167, 167, -2*w^3 + 5*w^2 + 8*w - 4],\
[169, 13, -2*w^3 + 4*w^2 + 7*w - 5],\
[173, 173, 2*w^3 - 4*w^2 - 9*w + 4],\
[173, 173, -w^3 + 5*w^2 - 2*w - 5],\
[181, 181, 3*w^3 - 7*w^2 - 12*w + 7],\
[191, 191, w^3 - 3*w^2 - 3*w - 1],\
[199, 199, 2*w^3 - 3*w^2 - 10*w - 1],\
[199, 199, 2*w^3 - 6*w^2 - 6*w + 13],\
[223, 223, -2*w^3 + 6*w^2 + 4*w - 11],\
[223, 223, 3*w^3 - 7*w^2 - 12*w + 8],\
[223, 223, -2*w^3 + 5*w^2 + 6*w - 1],\
[229, 229, -w^3 + 3*w^2 + 2*w + 1],\
[229, 229, 3*w^3 - 4*w^2 - 15*w - 5],\
[233, 233, -3*w^3 + 7*w^2 + 11*w - 5],\
[233, 233, 2*w^3 - 5*w^2 - 4*w + 5],\
[241, 241, w^3 - 3*w^2 - 5*w + 7],\
[257, 257, w^3 - 3*w^2 - 4*w + 8],\
[263, 263, w^2 - 4*w - 5],\
[271, 271, -2*w^3 + 6*w^2 + 5*w - 13],\
[277, 277, -2*w^3 + 6*w^2 + 3*w - 8],\
[277, 277, w^3 - 2*w^2 - 2*w - 2],\
[281, 281, 2*w^3 - 5*w^2 - 4*w + 2],\
[283, 283, -3*w^2 + 7*w + 10],\
[283, 283, -w^3 + 4*w^2 - w - 7],\
[293, 293, 2*w^3 - 5*w^2 - 3*w + 4],\
[293, 293, w^3 - 8*w - 4],\
[307, 307, -w^3 + 2*w^2 + 6*w - 2],\
[311, 311, w^2 - 5],\
[317, 317, -w^3 + 5*w^2 - 2*w - 11],\
[331, 331, -w^3 + 3*w^2 - 5],\
[349, 349, -2*w^2 + 5*w + 2],\
[353, 353, -2*w^3 + 5*w^2 + 4*w - 4],\
[353, 353, 3*w^3 - 7*w^2 - 10*w + 2],\
[353, 353, -2*w^3 + 7*w^2 + 2*w - 10],\
[353, 353, w^3 + w^2 - 9*w - 13],\
[361, 19, -w^3 + w^2 + 5*w - 1],\
[373, 373, -w^3 + 3*w^2 + 4*w - 11],\
[373, 373, 2*w^3 - 5*w^2 - 5*w - 2],\
[383, 383, w^3 - 5*w^2 + w + 13],\
[389, 389, 3*w^3 - 10*w^2 - 4*w + 16],\
[397, 397, -w^3 + 2*w^2 + 3*w + 5],\
[397, 397, -3*w^3 + 7*w^2 + 10*w - 7],\
[419, 419, -w^3 + 2*w^2 + 4*w + 4],\
[419, 419, 3*w^3 - 7*w^2 - 12*w + 5],\
[421, 421, -w^2 + 5*w - 2],\
[421, 421, 3*w^3 - 7*w^2 - 10*w + 5],\
[431, 431, w^2 - 10],\
[439, 439, w^3 - w^2 - 7*w - 7],\
[461, 461, -2*w^3 + 7*w^2 + 5*w - 10],\
[467, 467, -3*w^3 + 6*w^2 + 14*w - 4],\
[479, 479, 3*w^2 - 6*w - 5],\
[479, 479, -4*w^3 + 11*w^2 + 10*w - 10],\
[491, 491, -w^3 + w^2 + 6*w - 1],\
[499, 499, -4*w^3 + 9*w^2 + 15*w - 10],\
[499, 499, 2*w^3 - 5*w^2 - 7*w + 2],\
[503, 503, -2*w^3 + 7*w^2 - 8],\
[521, 521, 2*w^2 - w - 7],\
[521, 521, -3*w^3 + 7*w^2 + 11*w - 11],\
[569, 569, 2*w^2 - 5*w - 1],\
[569, 569, -4*w - 1],\
[571, 571, -w^3 + 3*w^2 - 7],\
[577, 577, -w^3 + w^2 + 8*w - 4],\
[587, 587, -3*w^3 + 8*w^2 + 7*w - 8],\
[593, 593, 3*w^2 - 4*w - 11],\
[593, 593, -3*w^3 + 9*w^2 + 6*w - 14],\
[601, 601, -3*w^3 + 9*w^2 + 4*w - 10],\
[601, 601, 5*w^3 - 9*w^2 - 24*w - 2],\
[607, 607, 2*w^2 - 7*w - 5],\
[613, 613, w^3 - 6*w - 2],\
[613, 613, -w^2 + 3*w + 8],\
[617, 617, -w^2 + 4*w - 5],\
[619, 619, 3*w^3 - 6*w^2 - 13*w + 5],\
[625, 5, -5],\
[631, 631, -2*w^3 + 4*w^2 + 8*w - 7],\
[647, 647, -2*w^3 + 4*w^2 + 5*w - 1],\
[647, 647, w^3 - 3*w^2 - 2*w - 2],\
[653, 653, w^3 - 4*w^2 - w + 13],\
[653, 653, w^3 - 2*w^2 - 7*w + 5],\
[659, 659, -4*w^3 + 10*w^2 + 15*w - 14],\
[659, 659, 5*w^3 - 12*w^2 - 17*w + 14],\
[661, 661, w^2 - 5*w - 2],\
[661, 661, -w^3 + 5*w^2 - 2*w - 7],\
[673, 673, -w^3 + w^2 + 6*w - 2],\
[683, 683, 2*w^3 - 8*w^2 + w + 8],\
[683, 683, 5*w^3 - 12*w^2 - 19*w + 14],\
[701, 701, -2*w^3 + 6*w^2 + 5*w - 4],\
[709, 709, -w^2 + 3*w - 4],\
[709, 709, -2*w^3 + 5*w^2 + 9*w - 8],\
[719, 719, w^2 - 7],\
[719, 719, -2*w^3 + 3*w^2 + 8*w + 5],\
[733, 733, -2*w^3 + 7*w^2 + 3*w - 10],\
[739, 739, 4*w^3 - 11*w^2 - 13*w + 14],\
[739, 739, -5*w^3 + 10*w^2 + 21*w - 4],\
[739, 739, 3*w^3 - 8*w^2 - 6*w + 8],\
[739, 739, -w^3 + w^2 + 8*w - 5],\
[743, 743, -w - 5],\
[743, 743, w^3 - w^2 - 8*w - 8],\
[751, 751, -2*w^3 + 5*w^2 + 8*w - 2],\
[751, 751, -2*w^3 + 6*w^2 - 1],\
[757, 757, -2*w^3 + 6*w^2 + 3*w - 10],\
[757, 757, 5*w^2 - 9*w - 22],\
[761, 761, -2*w^3 + 4*w^2 + 9*w - 7],\
[761, 761, 2*w^3 - 3*w^2 - 12*w - 1],\
[769, 769, 5*w^3 - 13*w^2 - 14*w + 10],\
[773, 773, -3*w^3 + 5*w^2 + 16*w + 1],\
[773, 773, w^3 - 11*w + 2],\
[787, 787, -w^3 + 4*w^2 - w - 11],\
[797, 797, -5*w^3 + 12*w^2 + 17*w - 11],\
[809, 809, -2*w^3 + 7*w^2 + 4*w - 11],\
[809, 809, 4*w^3 - 9*w^2 - 14*w + 4],\
[821, 821, -w^3 + 5*w^2 - 2*w - 17],\
[827, 827, -3*w^3 + 8*w^2 + 9*w - 7],\
[829, 829, 2*w^3 - 6*w^2 - 5*w + 2],\
[829, 829, -w^3 + 4*w^2 - w - 10],\
[839, 839, w^2 - 8],\
[853, 853, -5*w^3 + 12*w^2 + 18*w - 16],\
[853, 853, w^3 - 6*w - 8],\
[863, 863, 5*w^3 - 11*w^2 - 19*w + 7],\
[881, 881, -3*w^3 + 9*w^2 + 9*w - 11],\
[907, 907, -w^3 + 2*w^2 + 7*w - 4],\
[907, 907, -2*w^3 + 3*w^2 + 7*w + 4],\
[929, 929, -w^3 + 5*w^2 - 3*w - 11],\
[937, 937, 2*w^3 - 5*w^2 - 6*w + 11],\
[937, 937, -2*w^3 + 6*w^2 + 3*w - 11],\
[941, 941, -3*w^3 + 8*w^2 + 11*w - 8],\
[953, 953, -4*w^3 + 10*w^2 + 10*w - 13],\
[953, 953, w^3 - 5*w^2 - w + 13],\
[967, 967, 3*w^3 - 7*w^2 - 9*w + 7],\
[967, 967, -2*w^3 + 5*w^2 + 8*w + 1],\
[967, 967, 4*w^3 - 13*w^2 - 6*w + 20],\
[967, 967, -2*w^2 + 3*w + 13],\
[971, 971, -6*w^3 + 13*w^2 + 25*w - 8],\
[977, 977, w^3 - w^2 - 4*w - 10],\
[977, 977, 2*w^3 - 6*w^2 - 7*w + 13],\
[983, 983, -4*w^3 + 10*w^2 + 11*w - 13],\
[983, 983, -w^3 + 5*w^2 - w - 7],\
[983, 983, 3*w^3 - 9*w^2 - 7*w + 19],\
[983, 983, -2*w^3 + 3*w^2 + 10*w - 1],\
[997, 997, 5*w^3 - 12*w^2 - 17*w + 13],\
[997, 997, -2*w^3 + 3*w^2 + 13*w - 4],\
[997, 997, w^3 - 7*w - 2],\
[997, 997, w^3 - 9*w - 2]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 17*x^2 + 36
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, e, -1, 1/3*e^3 - 14/3*e, -4, 1/3*e^3 - 11/3*e, 4, 2*e, 1/3*e^3 - 14/3*e, -1/3*e^3 + 14/3*e, 2/3*e^3 - 22/3*e, -2*e, -e, -2*e^2 + 16, -2*e^2 + 16, -e^2 + 4, -1/3*e^3 + 23/3*e, -e^3 + 13*e, -e^2 + 18, 1/3*e^3 - 5/3*e, -e^2 + 8, 1/3*e^3 + 1/3*e, -e^2 + 2, e^2 - 4, -1/3*e^3 + 26/3*e, 0, 2*e^2 - 16, -2/3*e^3 + 22/3*e, 0, e^2 - 8, 0, -4, -e^2 + 20, -1/3*e^3 + 8/3*e, -2/3*e^3 + 16/3*e, 2/3*e^3 - 25/3*e, -2/3*e^3 + 22/3*e, e^2, 2*e, -3*e^2 + 24, e^3 - 14*e, e^2 + 8, -20, -e^3 + 14*e, -2/3*e^3 + 40/3*e, e^2 - 4, 2/3*e^3 - 16/3*e, -2/3*e^3 + 13/3*e, -1/3*e^3 + 35/3*e, -e^2 - 10, e^2 - 6, e^2 - 8, -2*e^2 + 16, -4, 2*e^2 - 28, 5/3*e^3 - 73/3*e, -2*e^2 + 28, e^2 - 20, 1/3*e^3 - 38/3*e, 5/3*e^3 - 52/3*e, -4/3*e^3 + 44/3*e, e^2 + 16, e^2 + 12, 3*e^2 - 32, -2*e^3 + 22*e, 5/3*e^3 - 67/3*e, 1/3*e^3 - 5/3*e, 30, 2*e^2 - 22, -2/3*e^3 + 43/3*e, e^2 - 16, -2*e^2 + 36, 16, -4*e^2 + 40, -2*e^2 + 24, -4*e, -3*e^2 + 28, -1/3*e^3 + 38/3*e, 2/3*e^3 - 52/3*e, -e^3 + 8*e, -4*e^2 + 44, -e^2, 2*e^2 - 36, 5/3*e^3 - 76/3*e, 2*e^3 - 24*e, e^3 - 6*e, -4/3*e^3 + 68/3*e, -2*e^3 + 18*e, 0, -3*e^2, -2*e^2 + 6, -2*e^3 + 21*e, -3*e, -5/3*e^3 + 55/3*e, -4*e^2 + 48, -2/3*e^3 + 25/3*e, 4/3*e^3 - 62/3*e, -4*e^2 + 22, 4*e^2 - 30, e^2 - 10, 1/3*e^3 - 35/3*e, e^3 - 10*e, -e^3 + 14*e, e^2 - 48, 2*e^2 + 6, 1/3*e^3 - 38/3*e, -3*e^2 + 14, -4*e, 2*e^3 - 18*e, 4*e^2 - 40, 7*e^2 - 68, -4/3*e^3 + 38/3*e, 4/3*e^3 - 56/3*e, 4/3*e^3 - 26/3*e, 2*e^3 - 24*e, -6*e^2 + 52, 1/3*e^3 - 5/3*e, 3*e^2, 2/3*e^3 - 46/3*e, -2*e^3 + 32*e, -8*e^2 + 68, 6*e, 3*e^2 - 32, -6*e^2 + 48, 2*e^2 - 20, -2*e^3 + 26*e, 5/3*e^3 - 46/3*e, -2*e^3 + 34*e, 2*e^3 - 16*e, -3*e^2 + 48, -16, -8*e, -2*e^2 + 32, -5*e^2 + 44, -4*e^2 + 20, -1/3*e^3 + 41/3*e, -5/3*e^3 + 31/3*e, -8/3*e^3 + 91/3*e, -2*e^3 + 38*e, 1/3*e^3 - 20/3*e, -2*e^2 + 32, 5/3*e^3 - 70/3*e, -4*e^2 + 30, -2*e^3 + 27*e, e^2 + 4, -2*e^3 + 36*e, 2*e^3 - 30*e, 20, 4*e^2 - 36, -7/3*e^3 + 80/3*e, 5*e^2 - 44, 2/3*e^3 - 40/3*e, 5/3*e^3 - 109/3*e, -1/3*e^3 - 10/3*e, -2*e^2 + 44, -4*e^2 + 14, -4*e^2 + 34, 8*e^2 - 74, -2*e^3 + 20*e, 2*e^3 - 31*e, 6*e^2 - 54, -4*e, -2*e^3 + 18*e, -3*e^2 + 64, 8, 2*e^3 - 32*e, -2*e^2 + 54, -46, 2*e^3 - 42*e, 2*e^2 - 40, 4*e^2 - 48, -2/3*e^3 + 52/3*e, -6*e, -4/3*e^3 + 68/3*e, -2/3*e^3 + 52/3*e, -12*e]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([11, 11, -w^3 + 3*w^2 + 2*w - 5])] = 1

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