/*
  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, -2, 5, 4, -5, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([25, 5, w^4 - w^3 - 4*w^2 + 2*w + 2])

primes_array = [
[7, 7, w^5 - 5*w^3 + 4*w],\
[13, 13, -w^3 + 3*w],\
[25, 5, w^4 - w^3 - 4*w^2 + 2*w + 2],\
[31, 31, -w^5 + 5*w^3 - 5*w + 1],\
[31, 31, -w^4 + w^3 + 5*w^2 - 3*w - 3],\
[37, 37, w^4 - 5*w^2 + w + 3],\
[43, 43, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 6*w],\
[47, 47, w^5 - w^4 - 4*w^3 + 4*w^2 - 2],\
[49, 7, -w^4 + 4*w^2 + w - 3],\
[53, 53, -w^4 + w^3 + 4*w^2 - 3*w - 4],\
[59, 59, w^4 - 5*w^2 - w + 5],\
[61, 61, w^3 - w^2 - 3*w],\
[64, 2, -2],\
[67, 67, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 7*w],\
[67, 67, 2*w^5 - w^4 - 10*w^3 + 4*w^2 + 9*w - 1],\
[73, 73, w^5 - w^4 - 4*w^3 + 5*w^2 + w - 3],\
[73, 73, 3*w^5 - 3*w^4 - 13*w^3 + 11*w^2 + 7*w - 3],\
[83, 83, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 6*w + 2],\
[97, 97, -2*w^4 + w^3 + 9*w^2 - 3*w - 4],\
[101, 101, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 6*w + 4],\
[101, 101, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + w - 4],\
[107, 107, -2*w^5 + w^4 + 10*w^3 - 4*w^2 - 9*w + 2],\
[107, 107, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 2],\
[113, 113, w^5 - w^4 - 4*w^3 + 3*w^2 + 2*w + 2],\
[121, 11, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 2],\
[127, 127, 2*w^5 - 3*w^4 - 9*w^3 + 12*w^2 + 5*w - 4],\
[131, 131, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 1],\
[131, 131, 2*w^5 - 3*w^4 - 7*w^3 + 12*w^2 - w - 4],\
[149, 149, -2*w^5 + 2*w^4 + 8*w^3 - 8*w^2 - 3*w + 4],\
[151, 151, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 3*w - 3],\
[157, 157, w^5 - w^4 - 4*w^3 + 5*w^2 - w - 2],\
[163, 163, -2*w^5 + 2*w^4 + 9*w^3 - 6*w^2 - 6*w],\
[163, 163, w^5 - 2*w^4 - 4*w^3 + 8*w^2 - 4],\
[163, 163, -2*w^5 + w^4 + 9*w^3 - 4*w^2 - 7*w + 4],\
[173, 173, 2*w^4 - w^3 - 8*w^2 + 2*w + 4],\
[173, 173, -w^5 + 3*w^4 + 4*w^3 - 14*w^2 - w + 9],\
[179, 179, -w^5 + 2*w^4 + 4*w^3 - 8*w^2 + 6],\
[191, 191, 2*w^4 - 2*w^3 - 9*w^2 + 6*w + 5],\
[193, 193, -w^5 + w^4 + 5*w^3 - 4*w^2 - 6*w + 2],\
[199, 199, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 6*w + 3],\
[211, 211, -w^5 + 5*w^3 - 2*w - 2],\
[211, 211, -2*w^5 + w^4 + 9*w^3 - 4*w^2 - 5*w + 1],\
[211, 211, w^5 - 5*w^3 - 2*w^2 + 4*w + 4],\
[223, 223, w^5 - 5*w^3 - w^2 + 7*w + 1],\
[227, 227, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 1],\
[227, 227, w^3 - 2*w^2 - 4*w + 3],\
[229, 229, -w^5 + 6*w^3 - 7*w + 1],\
[233, 233, 3*w^5 - 2*w^4 - 15*w^3 + 8*w^2 + 13*w - 3],\
[233, 233, -w^5 + 4*w^3 + w^2 - 2*w - 3],\
[233, 233, -w^3 + w^2 + 4*w - 1],\
[257, 257, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 4*w - 2],\
[257, 257, w^5 + w^4 - 6*w^3 - 4*w^2 + 6*w + 1],\
[263, 263, 3*w^5 - 3*w^4 - 13*w^3 + 10*w^2 + 7*w - 1],\
[269, 269, w^5 - w^4 - 4*w^3 + 5*w^2 - 3],\
[283, 283, w^5 - w^4 - 5*w^3 + 5*w^2 + 5*w - 1],\
[293, 293, -2*w^4 + w^3 + 8*w^2 - 3*w - 4],\
[293, 293, 2*w^5 - 4*w^4 - 8*w^3 + 17*w^2 + 3*w - 8],\
[307, 307, w^5 - 7*w^3 + 11*w - 1],\
[311, 311, w^4 - 2*w^3 - 4*w^2 + 8*w + 1],\
[313, 313, -w^5 + 5*w^3 + w^2 - 3*w - 3],\
[317, 317, -w^5 + 4*w^3 - w^2 - w + 4],\
[317, 317, -3*w^5 + 4*w^4 + 13*w^3 - 16*w^2 - 6*w + 8],\
[331, 331, w^5 - 7*w^3 + 10*w],\
[331, 331, 3*w^4 - 2*w^3 - 13*w^2 + 7*w + 7],\
[331, 331, w^5 - 4*w^3 + w^2 + w - 3],\
[343, 7, -4*w^5 + 3*w^4 + 19*w^3 - 10*w^2 - 15*w + 2],\
[347, 347, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 7*w],\
[349, 349, 3*w^5 - 2*w^4 - 14*w^3 + 7*w^2 + 9*w + 1],\
[349, 349, -w^5 + 2*w^4 + 3*w^3 - 9*w^2 + w + 7],\
[359, 359, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 10*w - 4],\
[359, 359, w^5 - 7*w^3 + 11*w + 1],\
[373, 373, w^5 - 6*w^3 + 6*w - 2],\
[383, 383, -4*w^5 + 4*w^4 + 18*w^3 - 14*w^2 - 12*w + 3],\
[389, 389, w^5 + w^4 - 5*w^3 - 5*w^2 + 6*w + 4],\
[397, 397, 2*w^5 - w^4 - 9*w^3 + 5*w^2 + 6*w - 5],\
[397, 397, -2*w^5 + 3*w^4 + 8*w^3 - 10*w^2 - 3*w + 3],\
[409, 409, -w^5 + w^4 + 5*w^3 - 3*w^2 - 6*w - 3],\
[421, 421, -3*w^5 + w^4 + 14*w^3 - 3*w^2 - 10*w - 2],\
[431, 431, 2*w^4 - w^3 - 10*w^2 + 3*w + 7],\
[433, 433, -3*w^5 + 2*w^4 + 14*w^3 - 6*w^2 - 11*w + 2],\
[433, 433, 2*w^5 - 3*w^4 - 10*w^3 + 11*w^2 + 10*w - 2],\
[439, 439, -2*w^5 + 2*w^4 + 10*w^3 - 8*w^2 - 7*w + 3],\
[439, 439, 3*w^5 - 4*w^4 - 13*w^3 + 16*w^2 + 8*w - 5],\
[439, 439, -w^5 + w^4 + 5*w^3 - 6*w^2 - 4*w + 6],\
[461, 461, w^5 + w^4 - 5*w^3 - 4*w^2 + 6*w + 1],\
[467, 467, w^5 + w^4 - 4*w^3 - 6*w^2 + w + 5],\
[479, 479, w^5 - w^4 - 5*w^3 + 3*w^2 + 2*w + 1],\
[487, 487, 3*w^5 - 3*w^4 - 13*w^3 + 12*w^2 + 7*w - 3],\
[499, 499, 2*w^5 - 2*w^4 - 11*w^3 + 7*w^2 + 12*w - 1],\
[503, 503, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 5*w - 3],\
[509, 509, 3*w^5 - 3*w^4 - 15*w^3 + 11*w^2 + 13*w - 2],\
[509, 509, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 2],\
[521, 521, 3*w^5 - 3*w^4 - 14*w^3 + 10*w^2 + 11*w - 3],\
[521, 521, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 5*w],\
[523, 523, -2*w^5 + 2*w^4 + 10*w^3 - 7*w^2 - 11*w + 1],\
[523, 523, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 11*w],\
[523, 523, -2*w^5 + w^4 + 10*w^3 - 4*w^2 - 10*w + 4],\
[547, 547, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 7],\
[557, 557, 3*w^5 - 3*w^4 - 14*w^3 + 10*w^2 + 9*w - 1],\
[569, 569, 3*w^5 - 4*w^4 - 13*w^3 + 14*w^2 + 8*w - 2],\
[571, 571, w^4 + w^3 - 5*w^2 - 2*w + 5],\
[577, 577, 3*w^4 - 2*w^3 - 13*w^2 + 7*w + 6],\
[577, 577, w^5 + w^4 - 5*w^3 - 4*w^2 + 4*w + 4],\
[587, 587, 2*w^5 - w^4 - 10*w^3 + 5*w^2 + 9*w - 4],\
[599, 599, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 9*w + 3],\
[599, 599, w^5 + 2*w^4 - 6*w^3 - 9*w^2 + 8*w + 6],\
[601, 601, -2*w^5 + 3*w^4 + 8*w^3 - 11*w^2 - 5*w + 4],\
[607, 607, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 3*w - 1],\
[613, 613, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 7],\
[613, 613, -w^5 + 2*w^4 + 3*w^3 - 7*w^2],\
[617, 617, -3*w^5 + 3*w^4 + 12*w^3 - 11*w^2 - 5*w + 4],\
[625, 5, -2*w^5 + 2*w^4 + 9*w^3 - 6*w^2 - 5*w + 1],\
[631, 631, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 8*w + 1],\
[641, 641, w^5 - 5*w^3 - w^2 + 5*w + 5],\
[659, 659, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 5*w + 3],\
[661, 661, 4*w^5 - 3*w^4 - 19*w^3 + 11*w^2 + 14*w - 2],\
[677, 677, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 6*w - 2],\
[683, 683, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 4],\
[683, 683, -2*w^4 + w^3 + 9*w^2 - 5*w - 4],\
[701, 701, -2*w^5 + 3*w^4 + 10*w^3 - 11*w^2 - 10*w + 4],\
[701, 701, -3*w^5 + w^4 + 15*w^3 - 3*w^2 - 14*w + 1],\
[709, 709, 2*w^4 - 2*w^3 - 9*w^2 + 5*w + 7],\
[719, 719, w^4 - 2*w^3 - 4*w^2 + 7*w + 4],\
[719, 719, w^3 + 2*w^2 - 4*w - 5],\
[727, 727, 2*w^5 - 3*w^4 - 8*w^3 + 11*w^2 + 4*w - 1],\
[729, 3, -3],\
[743, 743, 3*w^5 - 2*w^4 - 15*w^3 + 6*w^2 + 12*w],\
[743, 743, -w^4 + w^3 + 6*w^2 - 4*w - 3],\
[769, 769, 2*w^5 - 3*w^4 - 8*w^3 + 12*w^2 + 3*w - 2],\
[769, 769, -3*w^5 + 4*w^4 + 13*w^3 - 15*w^2 - 8*w + 3],\
[787, 787, 2*w^3 - w^2 - 5*w],\
[797, 797, 3*w^5 - w^4 - 15*w^3 + 4*w^2 + 12*w - 1],\
[827, 827, 3*w^4 - w^3 - 13*w^2 + 2*w + 8],\
[839, 839, -2*w^5 + w^4 + 10*w^3 - 4*w^2 - 11*w + 2],\
[853, 853, -2*w^4 + w^3 + 10*w^2 - 4*w - 6],\
[853, 853, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w - 3],\
[853, 853, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 8*w + 3],\
[857, 857, w^5 - 2*w^4 - 3*w^3 + 10*w^2 - 2*w - 7],\
[859, 859, -4*w^5 + 3*w^4 + 18*w^3 - 10*w^2 - 12*w],\
[863, 863, w^5 - 3*w^4 - 3*w^3 + 14*w^2 - 2*w - 9],\
[877, 877, -w^5 + 7*w^3 - 12*w - 1],\
[881, 881, 2*w^4 - 2*w^3 - 9*w^2 + 8*w + 6],\
[881, 881, 3*w^5 - w^4 - 15*w^3 + 4*w^2 + 12*w - 2],\
[887, 887, -w^5 - w^4 + 5*w^3 + 5*w^2 - 2*w - 4],\
[919, 919, 4*w^5 - 4*w^4 - 18*w^3 + 14*w^2 + 11*w - 3],\
[919, 919, -w^5 + 4*w^3 - w - 3],\
[929, 929, -w^5 + 2*w^4 + 4*w^3 - 10*w^2 + 5],\
[937, 937, 3*w^5 - 4*w^4 - 12*w^3 + 15*w^2 + 5*w - 3],\
[947, 947, 3*w^5 - 3*w^4 - 13*w^3 + 10*w^2 + 8*w - 4],\
[953, 953, 2*w^5 - 9*w^3 - w^2 + 6*w + 4],\
[953, 953, -3*w^5 + 3*w^4 + 12*w^3 - 11*w^2 - 5*w + 2],\
[967, 967, -2*w^5 + 9*w^3 + 2*w^2 - 7*w - 4],\
[977, 977, 3*w^5 - w^4 - 14*w^3 + 3*w^2 + 11*w - 1],\
[977, 977, w^4 - w^3 - 6*w^2 + 3*w + 3],\
[997, 997, 3*w^5 - 2*w^4 - 15*w^3 + 7*w^2 + 13*w - 4]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 - 8*x + 10
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [2, e, -1, 2*e - 8, -2*e + 6, 2*e - 12, 2, -3*e + 10, -4*e + 14, 10, 6, 2*e - 6, -4*e + 21, 2*e - 8, 6*e - 24, e + 6, -2*e + 10, -2*e + 10, 2*e - 6, 3*e - 12, 2*e - 14, -3*e + 16, 0, 2*e - 2, 6, -18, -4*e + 22, -5*e + 20, -2, -e + 8, -4*e + 14, -2*e, 6*e - 16, -8*e + 32, 2, -4*e + 18, -4*e + 16, -5*e + 8, 7*e - 22, -e - 6, e + 10, -3*e + 4, -3*e + 4, 8*e - 28, 5*e - 20, -6, -4*e + 10, 6*e - 24, 2*e - 26, 3*e - 24, -2*e + 4, 4*e + 2, -10*e + 40, 2*e - 18, 5*e - 26, 3*e - 10, 2*e, e - 34, 8*e - 44, 4, -3*e + 20, -4*e + 18, -2*e + 2, 2*e + 6, -6*e + 40, -4, e - 10, -6*e + 4, -5*e + 16, 5*e - 14, 8*e - 30, -6*e + 20, -2*e - 16, 3*e - 14, -2*e + 10, -12*e + 42, -10, -26, 12*e - 46, 3*e + 16, -e - 6, 9*e - 46, -6*e + 16, -5*e + 22, -5*e + 18, 18, 8*e - 16, -4*e + 28, -e - 6, 2*e + 2, -8*e + 28, -10*e + 24, 10*e - 38, 2*e - 18, 14*e - 62, -2*e + 12, -2*e - 4, -2*e + 44, -8*e + 58, 4*e - 2, -10*e + 56, 6*e - 16, -8*e + 22, -2*e + 28, 24, 8*e - 24, -6*e + 26, -6*e + 40, -6*e + 6, -8*e + 46, 2*e + 16, -4*e - 6, -8*e + 24, 12*e - 42, 5*e - 20, 9*e - 50, 2*e - 34, -6*e - 6, 6*e + 8, 12*e - 46, 8*e - 38, -4*e + 26, e - 18, 8*e - 12, 14*e - 60, -2*e - 20, 3*e - 10, -14*e + 50, -6*e + 24, -12*e + 42, -2*e + 30, 10*e - 52, -10*e + 28, 2*e + 34, -6*e + 60, -6*e - 16, e - 52, 8*e - 6, -8*e + 64, 8*e - 48, 2*e, -8*e + 2, -10*e + 10, -16*e + 46, -40, 14*e - 48, -18, -2*e + 26, e - 22, -3*e - 12, -10*e + 56, 4*e - 8, -7*e + 8, 2*e + 18, -20*e + 86]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([25, 5, w^4 - w^3 - 4*w^2 + 2*w + 2])] = 1

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