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

NN = ZF.ideal([8, 2, w^3 - w^2 - 5*w + 2])

primes_array = [
[2, 2, -w],\
[3, 3, w + 1],\
[4, 2, -w^2 - w + 1],\
[13, 13, -w^2 + 3],\
[17, 17, -w + 3],\
[27, 3, w^3 - 2*w^2 - 3*w + 5],\
[31, 31, -w^2 - 2*w + 1],\
[41, 41, -w^2 + 5],\
[43, 43, w^3 - w^2 - 5*w - 1],\
[43, 43, w^3 - w^2 - 3*w + 1],\
[59, 59, -w - 3],\
[59, 59, -2*w^3 + 2*w^2 + 9*w - 5],\
[59, 59, -w^3 - 3*w^2 - w + 3],\
[59, 59, w^3 - 7*w + 1],\
[61, 61, -w^3 + w^2 + 2*w + 5],\
[61, 61, -w^3 - w^2 + 4*w + 3],\
[67, 67, 4*w^3 - 4*w^2 - 18*w + 7],\
[73, 73, -2*w^3 + 6*w - 3],\
[73, 73, -2*w + 3],\
[97, 97, w^3 - 2*w^2 - 5*w + 3],\
[101, 101, w^3 + w^2 - 5*w - 7],\
[101, 101, -w^3 - w^2 + 4*w + 5],\
[103, 103, 2*w^3 - 8*w - 1],\
[103, 103, w^2 - 2*w - 7],\
[109, 109, -w^3 + 2*w^2 + 3*w - 7],\
[113, 113, -w^3 + w^2 + 6*w - 3],\
[127, 127, -w^3 + w^2 + 6*w - 1],\
[131, 131, -2*w^3 + 3*w^2 + 10*w - 11],\
[137, 137, w^3 + w^2 - 6*w - 5],\
[139, 139, -3*w^3 + 2*w^2 + 15*w - 1],\
[149, 149, w^3 - 3*w^2 - 4*w + 11],\
[149, 149, -w^3 + 5*w - 1],\
[157, 157, -2*w^3 + 3*w^2 + 6*w - 1],\
[163, 163, 2*w^2 - 3],\
[163, 163, -2*w^3 + 2*w^2 + 7*w - 5],\
[167, 167, -2*w^3 + 2*w^2 + 8*w - 3],\
[167, 167, -2*w^2 + 2*w + 9],\
[173, 173, 2*w^3 - w^2 - 8*w + 3],\
[173, 173, 2*w^3 - 3*w^2 - 6*w + 3],\
[181, 181, 2*w^3 - 3*w^2 - 8*w + 5],\
[191, 191, -2*w^3 + w^2 + 10*w - 1],\
[193, 193, -w^3 + 3*w^2 + 2*w - 7],\
[193, 193, w^3 - 3*w - 3],\
[193, 193, 2*w^2 - 2*w - 3],\
[193, 193, w^3 - 7*w - 1],\
[197, 197, w^2 - 2*w - 5],\
[211, 211, w^2 - 4*w - 1],\
[229, 229, -w^3 + w^2 + 6*w - 7],\
[229, 229, -4*w^3 + 6*w^2 + 16*w - 13],\
[233, 233, 3*w^2 + 2*w - 5],\
[233, 233, 2*w^3 - 2*w^2 - 11*w + 7],\
[239, 239, -w^3 + w^2 + 7*w + 1],\
[241, 241, w^3 - 3*w^2 - 2*w + 9],\
[241, 241, w^3 + w^2 - 9*w - 5],\
[251, 251, 2*w^3 - 6*w + 1],\
[271, 271, 2*w^3 - w^2 - 8*w - 1],\
[271, 271, -2*w^2 + w + 7],\
[281, 281, 2*w^3 + 2*w^2 - 5*w - 1],\
[293, 293, w^3 + w^2 - 5*w - 1],\
[293, 293, -3*w + 7],\
[307, 307, -w^3 - 2*w^2 + 3*w + 5],\
[311, 311, -w^3 + 3*w^2 + 4*w - 7],\
[313, 313, 2*w^2 - 2*w - 5],\
[317, 317, -2*w^3 + 4*w^2 + 7*w - 13],\
[317, 317, -2*w^3 + 3*w^2 + 8*w - 11],\
[337, 337, 3*w^3 + 2*w^2 - 7*w + 1],\
[347, 347, 2*w^3 - 2*w^2 - 11*w + 1],\
[349, 349, -5*w^3 + 6*w^2 + 23*w - 13],\
[359, 359, 2*w^3 - 2*w^2 - 12*w + 9],\
[359, 359, -2*w^3 + 8*w + 5],\
[367, 367, -4*w^3 + 4*w^2 + 19*w - 5],\
[367, 367, 2*w^3 + w^2 - 10*w - 7],\
[383, 383, 2*w^3 - 2*w^2 - 7*w + 1],\
[389, 389, 2*w^2 - w - 5],\
[397, 397, 3*w^3 - 4*w^2 - 11*w + 9],\
[397, 397, 3*w^3 - 3*w^2 - 12*w + 7],\
[397, 397, 2*w^3 - 4*w^2 - 8*w + 15],\
[397, 397, -w^3 + 3*w^2 + w - 7],\
[409, 409, -w^3 - w^2 + 9*w - 1],\
[419, 419, 2*w^2 - 3*w - 7],\
[419, 419, 4*w^3 - 3*w^2 - 18*w + 3],\
[431, 431, -3*w^3 + w^2 + 17*w - 1],\
[431, 431, 3*w^3 - w^2 - 16*w - 1],\
[433, 433, -2*w^3 + 5*w^2 + 8*w - 19],\
[443, 443, 3*w - 5],\
[449, 449, -2*w^3 + 2*w^2 + 9*w - 9],\
[449, 449, -5*w^3 + 8*w^2 + 19*w - 17],\
[457, 457, -3*w^3 + 3*w^2 + 13*w - 9],\
[457, 457, -3*w^3 + 2*w^2 + 13*w + 1],\
[457, 457, -2*w^3 + w^2 + 8*w + 7],\
[457, 457, 3*w^3 - 3*w^2 - 15*w + 7],\
[461, 461, -2*w^3 + 2*w^2 + 9*w + 1],\
[479, 479, -w^3 + w^2 + 7*w - 9],\
[487, 487, -w^3 - w^2 + w - 3],\
[499, 499, -2*w^3 + 7*w - 3],\
[499, 499, -5*w^3 + 7*w^2 + 20*w - 15],\
[541, 541, -3*w^3 + 4*w^2 + 13*w - 7],\
[547, 547, -w^3 - w^2 + 3*w + 5],\
[557, 557, 4*w - 1],\
[563, 563, 2*w^3 - w^2 - 6*w - 1],\
[571, 571, w^3 - w^2 - w - 3],\
[587, 587, -2*w^3 + 2*w^2 + 10*w + 1],\
[593, 593, 2*w^3 - 2*w^2 - 6*w + 3],\
[613, 613, -5*w^3 + 5*w^2 + 23*w - 7],\
[613, 613, -w^3 + w^2 + 5*w - 7],\
[617, 617, -w - 5],\
[619, 619, -3*w^3 + 4*w^2 + 11*w - 7],\
[625, 5, -5],\
[643, 643, -3*w^3 + 3*w^2 + 14*w - 9],\
[643, 643, -3*w^2 - 4*w + 1],\
[653, 653, -w^3 + w^2 + 9*w + 3],\
[653, 653, -4*w^2 + 6*w + 11],\
[659, 659, -w^2 - 4*w + 1],\
[661, 661, -3*w^3 + 3*w^2 + 15*w - 5],\
[661, 661, w^3 + w^2 - 7*w - 11],\
[673, 673, 2*w^3 - 3*w^2 - 12*w + 1],\
[673, 673, -2*w^3 + w^2 + 10*w - 3],\
[683, 683, -w^3 - 2*w^2 + 5*w + 11],\
[691, 691, 4*w^3 - 3*w^2 - 18*w + 5],\
[709, 709, w^3 - 3*w - 7],\
[719, 719, 2*w^3 - 8*w + 1],\
[727, 727, -3*w^3 + 6*w^2 + 13*w - 19],\
[733, 733, w^3 - w^2 + 3],\
[739, 739, -w^3 + w^2 + 3*w - 7],\
[751, 751, -w^2 - 3],\
[773, 773, 2*w^3 - 2*w^2 - 11*w + 3],\
[787, 787, w^3 - w^2 - 3*w - 5],\
[797, 797, -2*w^3 + 2*w^2 + 5*w - 1],\
[797, 797, -7*w^3 + 8*w^2 + 31*w - 15],\
[809, 809, 4*w^3 - 6*w^2 - 17*w + 13],\
[809, 809, 3*w^2 - 4*w - 5],\
[811, 811, 2*w^2 + 4*w + 3],\
[811, 811, -3*w^3 + 3*w^2 + 12*w - 5],\
[821, 821, 2*w^3 - 10*w - 1],\
[821, 821, 6*w^3 - 10*w^2 - 24*w + 29],\
[823, 823, -5*w^3 + 3*w^2 + 24*w - 1],\
[823, 823, w^3 + w^2 + w + 3],\
[827, 827, -4*w^3 + 3*w^2 + 16*w - 7],\
[829, 829, 2*w^3 - 4*w^2 - 6*w + 11],\
[841, 29, w^3 - 3*w^2 - 5*w + 3],\
[841, 29, -3*w^3 + 3*w^2 + 13*w - 1],\
[857, 857, -4*w^3 + 6*w^2 + 13*w - 7],\
[857, 857, -3*w^3 + w^2 + 15*w + 1],\
[859, 859, -4*w^2 + 3*w + 17],\
[863, 863, -4*w^3 + 8*w^2 + 14*w - 21],\
[877, 877, -w^3 - w^2 + 6*w - 1],\
[881, 881, -5*w - 1],\
[883, 883, -4*w^3 + 4*w^2 + 21*w - 13],\
[883, 883, 2*w^3 - 11*w - 1],\
[887, 887, -3*w^3 - w^2 + 8*w - 5],\
[907, 907, w^3 - w^2 - w + 5],\
[911, 911, 2*w^2 - 5*w - 5],\
[919, 919, -w^3 + 3*w^2 + 6*w - 3],\
[919, 919, 5*w^2 + 4*w - 9],\
[929, 929, w^3 + 3*w^2 - 8*w - 17],\
[937, 937, -4*w^3 + 4*w^2 + 17*w - 7],\
[953, 953, -2*w^3 + 2*w^2 + 7*w + 5],\
[953, 953, 3*w^3 - 4*w^2 - 11*w + 3],\
[967, 967, 4*w^3 - 2*w^2 - 17*w + 3],\
[971, 971, w^3 - 3*w^2 - 8*w + 1],\
[983, 983, -2*w^3 - 2*w^2 + 12*w + 13],\
[991, 991, -2*w^3 - 2*w^2 + 8*w + 7],\
[991, 991, w^3 + w^2 - 10*w + 1]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 - x^2 - 5*x - 2
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1, e, 1, 3*e^2 - 6*e - 10, -2*e^2 + 4*e + 8, 2, -4*e^2 + 8*e + 12, -4*e^2 + 4*e + 12, -2*e, 2*e^2 - 2*e - 6, -2*e^2 + 14, 2*e^2 - 3*e - 12, -4*e^2 + 9*e + 10, -5*e^2 + 6*e + 12, e^2 - 6, 3*e^2 - 8*e - 6, -3*e + 2, -e, 2*e^2 - 8*e - 4, 6*e^2 - 8*e - 18, 7*e^2 - 10*e - 22, -2*e^2 + 6*e + 6, 6*e^2 - 13*e - 18, 2*e^2 - 4*e - 16, -4*e^2 + 9*e + 8, -2*e^2 + 16, 8*e^2 - 15*e - 18, 2*e^2 - 8*e + 2, -4*e^2 + 2*e + 12, 5*e^2 - 6*e - 20, 2*e^2 - 8*e - 6, 2*e^2 - 3*e, -2*e^2 + 6*e + 10, e^2 + 4*e + 2, -3*e^2 + 6*e + 6, -8*e^2 + 21*e + 30, -3*e^2 - 2*e + 16, -7*e^2 + 12*e + 30, -2*e^2 + 7*e - 4, -4*e^2 + 13*e + 12, 4*e^2 - 4*e - 12, e^2 - 2*e + 4, 10*e^2 - 22*e - 34, -2*e^2 - 2*e, -4*e^2 + 10*e + 10, -2*e^2 - 3*e + 8, 10*e^2 - 14*e - 22, 2*e^2 - 6*e + 14, 6*e^2 + 2*e - 30, 2*e^2 - 8*e, 2*e^2 - 6, 2*e^2 + 2*e - 8, -8*e^2 + 6*e + 34, -2*e^2 - 4*e + 4, -10*e^2 + 16*e + 36, -10*e^2 + 22*e + 32, 4*e^2 - 10*e - 12, -4*e^2 + 14*e + 4, 12*e^2 - 16*e - 42, -10*e^2 + 16*e + 26, -9*e^2 + 16*e + 24, -13*e^2 + 16*e + 40, -e^2 + 8*e + 12, 4*e^2 - 6, -4*e^2 + 7*e + 16, -17*e^2 + 30*e + 48, 10*e^2 - 20*e - 44, -2*e^2 + 3*e - 20, 6*e^2 - 8*e - 16, -4*e + 12, 4*e^2 - 17*e - 18, 10*e^2 - 22*e - 16, 6*e^2 - 11*e - 6, -12*e^2 + 15*e + 28, -6*e + 6, -6*e^2 + 14*e + 22, -4*e^2 + 12*e - 2, 8*e^2 - 8*e - 10, -2*e + 28, 15*e^2 - 30*e - 38, 8*e^2 - 14*e - 16, -7*e^2 + 8*e + 44, 4*e^2 + 6*e - 20, -6*e^2 + 10*e - 4, 2*e^2 - 6*e - 32, -4*e^2 + 12*e + 24, -10*e^2 + 19*e + 40, 6*e^2 - 17*e - 38, -4*e^2 + 13*e + 2, -5*e^2 - 4*e + 34, -4*e^2 + e + 24, 9*e^2 - 14*e - 42, 5*e^2 - 4*e, -11*e^2 + 14*e + 36, -2*e^2 + 10*e - 2, 2*e^2 - 4*e - 36, 10*e^2 - 22*e - 50, -e^2 + 10*e + 4, -10*e^2 + 24*e + 50, 14*e^2 - 27*e - 46, -4*e - 36, 7*e^2 - 12*e - 22, 10*e^2 - 18*e - 14, 10*e^2 - 12*e - 18, -4*e^2 - 3*e + 40, 6*e^2 - 14*e - 20, 4*e^2 + 10*e - 30, 14*e^2 - 20*e - 38, 5*e^2 - 20*e - 32, 14*e^2 - 23*e - 60, 19*e^2 - 30*e - 50, -8*e + 10, -e + 40, 6*e^2 - 8*e - 26, -15*e^2 + 30*e + 54, 16*e^2 - 27*e - 46, -20*e^2 + 26*e + 58, 4*e^2 - 15*e - 22, -4*e^2 + 22*e - 8, 16*e^2 - 32*e - 42, -5*e^2 - 4*e + 36, 11*e^2 - 8*e - 40, -12*e^2 + 18*e + 22, -20*e^2 + 43*e + 56, 11*e^2 - 22*e - 12, -10*e^2 + 32*e + 26, 10*e^2 - 8*e - 42, -16*e^2 + 28*e + 58, e^2 - 10*e - 10, -2*e^2 - 2*e + 18, -2*e + 12, 2*e^2 - 3*e - 14, 4*e^2 + 4*e + 4, -2*e^2 + e + 44, -e^2 + 6, -2*e^2 + 12*e - 24, -10*e^2 + 20*e + 20, 12*e^2 - 21*e - 34, 4*e^2 + 6*e - 38, 20*e^2 - 22*e - 66, -19*e^2 + 26*e + 54, 11*e^2 - 20*e - 44, -16*e^2 + 36*e + 60, -10*e^2 + 35*e + 22, -8*e^2 + 5*e + 10, -5*e^2 + 6*e - 2, -11*e^2 + 14*e + 66, -2*e^2 - 8*e + 12, -12*e + 22, 6*e - 20, -30*e^2 + 46*e + 86, 8*e - 4, -3*e^2 - 8*e + 12, 8*e + 4, 12*e^2 - 10*e - 54, -18*e^2 + 31*e + 62, 13*e^2 - 28*e - 52, 3*e^2 - 16*e - 18, 8*e^2 - 4*e - 28, -7*e^2 + 24*e + 26, 14*e^2 - 11*e - 54, -6*e^2 + 20*e + 20, 20*e^2 - 36*e - 68]
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])] = -1
AL_eigenvalues[ZF.ideal([4, 2, -w^2 - w + 1])] = -1

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