/*
  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([13, 13, -w + 2])

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

hecke_eigenvalues_array = [e, -2*e^2 - 2*e + 3, 2*e^2 + 6*e - 2, -e^2 - e + 6, -1, e^2 + 4*e, 2*e^2 + e - 2, 3*e^2 + 5*e - 6, -2*e^2 - 2*e + 1, -3*e^2 + 6, 2*e^2 + 8*e + 1, -2*e^2 - 7*e, -6*e - 4, 4*e^2 + 2*e - 11, -e^2 - e + 10, -4*e^2 - 10*e + 4, -6*e^2 - 10*e + 7, 3*e + 8, -5*e^2 + 10, -2*e + 8, -4*e^2 + 7, 2*e^2 + 4*e + 10, 6*e^2 - 10, -2*e^2 - 4*e + 6, -12*e^2 - 24*e + 8, -4*e^2 + 4*e + 16, -e^2 + e - 6, e^2 + 6*e, 2*e - 4, 2*e^2 - 4*e - 12, 7*e^2 + 17*e - 2, -4*e - 2, 4*e^2 + 22*e + 5, -4*e^2 - 12*e - 4, -2*e^2 + 7*e + 4, -11*e^2 - 24*e + 8, -6*e^2 - 10*e + 2, 12*e^2 + 22*e - 1, -5*e^2 - 10*e - 6, e^2 + 13*e + 12, -8*e^2 - 12*e + 12, -4*e^2 - 4*e + 14, 9*e^2 + 22*e + 6, -2*e^2 + 6*e + 1, 6*e^2 + 16*e - 6, 13*e^2 + 23*e - 14, 12*e^2 + 16*e + 2, 5*e + 4, -12*e^2 - 18*e + 8, 18*e^2 + 28*e - 4, -6*e^2 - 5*e + 10, -12*e^2 - 18*e + 24, 2*e^2 + 12*e + 10, 2*e^2 + 12*e - 12, -12*e^2 - 20*e + 18, -6*e^2 - 10*e + 5, 12*e + 12, -3*e^2 - 12*e + 16, -19*e^2 - 23*e + 18, e^2 + 9*e - 10, 6*e^2 + 12*e - 4, 2*e^2 - 10*e - 2, -20*e^2 - 21*e + 22, -11*e^2 - 7*e + 18, -6*e^2 - 18*e + 14, 11*e^2 - 5*e - 40, -10*e^2 - 14*e + 24, -14*e^2 - 30*e - 2, 4*e^2 + 8*e - 22, 8*e^2 + 26*e + 4, 16*e^2 + 20*e - 4, 6*e^2 + 10*e, 18*e^2 + 30*e - 10, -6*e^2 - 14*e - 4, -16*e^2 - 13*e + 34, 26*e^2 + 36*e - 15, 20*e^2 + 24*e - 24, 17*e^2 + 31*e - 18, 16*e^2 + 12*e - 22, -26*e^2 - 44*e + 20, -6*e^2 - 8*e + 28, 2*e^2 - 12, 4*e^2 + 22*e + 16, -16*e^2 - 16*e + 18, 18*e^2 + 33*e - 24, 14*e^2 + 9*e - 32, 27*e^2 + 44*e - 18, -e^2 - 16*e + 6, e^2 - e + 26, 24*e^2 + 36*e - 31, 4*e^2 + 14*e - 18, -4*e^2 + 2*e + 4, -10*e^2 - 18*e + 18, -7*e^2 - 20*e - 8, -11*e^2 - 23*e + 14, -2*e^2 + 36, -2*e^2 + 2*e + 24, 10*e^2 + 16*e - 8, 12*e^2 + 14*e - 18, -4*e^2 + 8*e + 4, -4*e^2 + 5*e - 12, -16*e^2 - 22*e + 14, 12*e^2 + 30*e - 6, -16*e^2 - 41*e - 2, -6*e^2 - 2*e + 22, -2*e^2 - 14*e - 20, -14*e^2 - 12*e + 16, 15*e^2 + 18*e - 16, -8*e^2 + 2*e - 2, -9*e^2 - 14*e - 6, -8*e^2 - 2*e + 27, -8*e^2 - 18*e + 14, -18*e^2 - 12*e + 44, -10*e^2 - 6*e - 10, 6*e^2 - 4*e - 6, -8*e^2 - 15*e + 10, -e^2 + 3*e - 8, 8*e^2 + 6*e, -3*e^2 - 24*e - 26, -10*e^2 - 23*e - 12, -12*e^2 + 10*e + 50, -8*e^2 - 4*e - 14, 9*e^2 - 2*e - 46, 14*e^2 + 18*e + 4, -17*e^2 - 34*e + 36, 16*e^2 + 21*e + 8, -19*e^2 - 13*e + 18, 5*e^2 + 24*e - 10, 10*e^2 + 12*e - 42, -20*e^2 - 55*e + 10, 12*e^2 + 24*e - 41, 6*e^2 - 6*e - 13, 24*e^2 + 45*e - 22, -19*e^2 - 21*e + 26, -24*e^2 - 44*e + 30, -20*e^2 - 26*e + 24, 24*e^2 + 20*e - 38, 16*e^2 + 4*e - 48, 4*e^2 + 2*e + 2, -21*e^2 - 42*e + 18, 10*e^2 + 22*e + 8, -14*e^2 - 28*e + 28, -32*e^2 - 44*e + 35, 22*e^2 + 46*e - 34, -7*e^2 + 14*e + 24, 28*e^2 + 40*e - 32, 14*e^2 + 7*e - 24, -4*e^2 - 23*e - 22, 2*e^2 + 24*e + 26, 22*e^2 + 12*e - 29, -6*e^2 - 16*e - 22, 8*e^2 + 8*e - 38, -2*e^2 + 2*e + 45, -28*e^2 - 38*e + 24, -33*e^2 - 48*e + 32, e^2 + 18, -20*e^2 - 2*e + 48, 4*e^2 + 26*e, 12*e^2 - 8*e - 30, 26*e^2 + 16*e - 58, -7*e^2 - 33*e + 8, -18*e^2 - 50*e + 7, -6*e^2 + 2*e + 20, -20*e^2 - 58*e + 8, 22*e^2 + 60*e - 30, 14*e^2 - 4*e - 34, 12*e^2 - 6*e - 30, -4*e^2 - 8*e - 46, -e^2 + e - 12, -25*e^2 - 21*e + 40, -14*e^2 + 8*e + 42, -2*e^2 + 10*e - 11, 6*e^2 - 4*e + 1, 7*e^2 + 18*e + 2, 14*e^2 + 28*e - 38, -26*e^2 - 52*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([13, 13, -w + 2])] = 1

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