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

NN = ZF.ideal([27,9,w^3 - 2*w^2 - 4*w + 2])

primes_array = [
[3, 3, -w^3 + w^2 + 5*w],\
[3, 3, w - 1],\
[9, 3, w^3 - w^2 - 4*w],\
[13, 13, -2*w^3 + w^2 + 11*w + 3],\
[13, 13, w^3 - 2*w^2 - 4*w + 3],\
[16, 2, 2],\
[23, 23, -w^2 + 2],\
[23, 23, -w^3 + 2*w^2 + 4*w - 4],\
[49, 7, -w^3 + 2*w^2 + 5*w - 2],\
[49, 7, 2*w^3 - 2*w^2 - 9*w - 1],\
[53, 53, w^3 - 3*w^2 + 3],\
[53, 53, 2*w^3 - w^2 - 8*w - 3],\
[53, 53, -w^3 + w^2 + 6*w - 1],\
[61, 61, -w - 3],\
[61, 61, -w^3 + w^2 + 5*w - 4],\
[79, 79, w^3 - 2*w^2 - 4*w + 1],\
[79, 79, w^2 - 5],\
[101, 101, w^3 - 6*w],\
[101, 101, w^2 - 2*w - 4],\
[103, 103, 4*w^3 - 3*w^2 - 20*w - 3],\
[103, 103, w^3 - 2*w^2 - 1],\
[113, 113, 5*w^3 - 3*w^2 - 25*w - 6],\
[113, 113, 2*w - 3],\
[113, 113, w^3 - 3*w^2 - w + 4],\
[113, 113, 2*w^3 - 2*w^2 - 10*w - 1],\
[121, 11, -w^2 + 2*w - 2],\
[121, 11, -w^3 + 6*w + 6],\
[127, 127, 2*w^3 - 3*w^2 - 6*w + 3],\
[127, 127, -3*w^3 + 2*w^2 + 14*w + 1],\
[139, 139, 3*w^3 - 4*w^2 - 12*w + 6],\
[139, 139, 2*w^3 - w^2 - 8*w],\
[157, 157, -w^3 + 7*w + 5],\
[157, 157, w^3 - 7*w + 2],\
[173, 173, w^2 - 3*w - 3],\
[173, 173, -2*w^3 + w^2 + 11*w],\
[179, 179, -2*w^3 + w^2 + 8*w + 1],\
[179, 179, -3*w^3 + 4*w^2 + 12*w - 5],\
[181, 181, -2*w^3 + w^2 + 11*w + 6],\
[181, 181, w^2 - 3*w + 3],\
[191, 191, w^2 + w - 4],\
[191, 191, 2*w^3 - 3*w^2 - 9*w + 3],\
[199, 199, -w^3 - w^2 + 7*w + 6],\
[199, 199, -2*w^3 + 3*w^2 + 8*w - 2],\
[199, 199, -3*w^3 + 3*w^2 + 13*w],\
[199, 199, w^3 - 4*w - 4],\
[211, 211, -2*w^3 + 3*w^2 + 7*w - 6],\
[211, 211, w^3 - w^2 - 7*w + 2],\
[211, 211, 2*w^3 - w^2 - 9*w + 1],\
[211, 211, 2*w^3 - 2*w^2 - 11*w + 1],\
[233, 233, w^2 - 8],\
[233, 233, w^3 - 2*w^2 - 4*w - 2],\
[251, 251, 2*w^2 - 3*w - 4],\
[251, 251, 2*w^3 - 3*w^2 - 11*w + 4],\
[257, 257, -4*w^3 + 2*w^2 + 22*w + 5],\
[257, 257, 4*w^3 - 5*w^2 - 17*w + 8],\
[263, 263, 3*w^3 - 2*w^2 - 15*w],\
[263, 263, -w^3 + 2*w^2 + w - 3],\
[269, 269, -w^3 + w^2 + 2*w - 4],\
[269, 269, -w^3 - w^2 + 5*w + 4],\
[269, 269, 2*w^3 - 4*w^2 - 7*w + 7],\
[269, 269, -2*w^3 + 4*w^2 + 5*w],\
[277, 277, -4*w^3 + 3*w^2 + 19*w - 1],\
[277, 277, 2*w^3 - 3*w^2 - 5*w + 4],\
[283, 283, 3*w^3 - 3*w^2 - 13*w + 3],\
[283, 283, 2*w^3 - 2*w^2 - 7*w + 2],\
[289, 17, -w^3 + w^2 + 4*w - 5],\
[289, 17, w^3 - w^2 - 4*w - 4],\
[313, 313, -2*w^3 + 3*w^2 + 7*w - 9],\
[313, 313, -2*w^3 + w^2 + 9*w - 4],\
[337, 337, -2*w^3 + 3*w^2 + 8*w - 1],\
[337, 337, w^3 - 4*w - 5],\
[373, 373, -2*w^3 + w^2 + 10*w - 1],\
[373, 373, -w^3 + 2*w^2 + 2*w - 5],\
[389, 389, w^3 + w^2 - 7*w - 11],\
[389, 389, -2*w^2 + 3*w - 2],\
[419, 419, -w^3 + 3*w^2 + 4*w - 7],\
[419, 419, -w^3 + 3*w^2 + 4*w - 5],\
[433, 433, -2*w^3 + 2*w^2 + 6*w - 5],\
[433, 433, -4*w^3 + 4*w^2 + 18*w - 7],\
[433, 433, 2*w^3 - 3*w^2 - 11*w + 1],\
[433, 433, 7*w^3 - 4*w^2 - 35*w - 8],\
[443, 443, -3*w^3 + w^2 + 15*w + 3],\
[443, 443, -2*w^3 + 4*w^2 + 5*w - 6],\
[467, 467, 6*w^3 - 4*w^2 - 30*w - 3],\
[467, 467, -2*w^3 + 4*w^2 + 2*w - 3],\
[467, 467, 3*w - 4],\
[467, 467, 3*w^3 - 3*w^2 - 15*w - 1],\
[491, 491, -w^3 + 4*w^2 - 3*w + 2],\
[491, 491, -5*w^3 + 2*w^2 + 27*w + 13],\
[503, 503, 3*w^3 - 18*w - 11],\
[503, 503, -3*w^2 + 6*w + 1],\
[523, 523, w^3 - 3*w^2 - 3*w + 6],\
[523, 523, -3*w^3 + 16*w + 9],\
[523, 523, 2*w^2 - w - 5],\
[523, 523, 2*w^3 - 5*w^2 - 4*w + 5],\
[529, 23, 2*w^3 - 2*w^2 - 8*w - 5],\
[547, 547, -w^3 + 2*w^2 + w - 4],\
[547, 547, -2*w^3 + w^2 + 11*w - 2],\
[547, 547, -3*w^3 + 2*w^2 + 15*w - 1],\
[547, 547, w^2 - 3*w - 5],\
[563, 563, -3*w^3 + 4*w^2 + 10*w - 7],\
[563, 563, -3*w^3 + 3*w^2 + 15*w + 4],\
[569, 569, 2*w^3 - w^2 - 11*w - 7],\
[569, 569, w^2 - 3*w + 4],\
[571, 571, -3*w^3 + 2*w^2 + 12*w + 5],\
[571, 571, 4*w^3 - 5*w^2 - 16*w + 1],\
[601, 601, -5*w^3 + 4*w^2 + 26*w + 4],\
[601, 601, -w^2 + 6*w - 4],\
[607, 607, 2*w^3 - 4*w^2 - 6*w + 1],\
[607, 607, 2*w^3 - 2*w^2 - 11*w - 5],\
[607, 607, w^3 - w^2 - 7*w + 8],\
[607, 607, 2*w^3 - 10*w - 9],\
[625, 5, -5],\
[641, 641, -4*w^3 + 5*w^2 + 19*w - 13],\
[641, 641, -4*w^3 + 4*w^2 + 21*w - 1],\
[653, 653, 3*w^3 - 2*w^2 - 14*w + 2],\
[653, 653, 4*w^3 - 7*w^2 - 17*w + 10],\
[653, 653, -2*w^3 + w^2 + 14*w + 6],\
[653, 653, -2*w^3 + 3*w^2 + 6*w - 6],\
[659, 659, 3*w - 5],\
[659, 659, 3*w^3 - 3*w^2 - 15*w - 2],\
[673, 673, 2*w^3 - 4*w^2 - 9*w + 4],\
[673, 673, 3*w^3 - 3*w^2 - 16*w + 3],\
[673, 673, -w^3 + 2*w^2 + 4*w - 9],\
[673, 673, -w^3 + 3*w^2 + 5*w - 9],\
[677, 677, -3*w^3 + 4*w^2 + 15*w - 3],\
[677, 677, -w^3 + 2*w^2 + 7*w - 6],\
[701, 701, w^2 - 3*w - 6],\
[701, 701, 2*w^3 - w^2 - 11*w + 3],\
[719, 719, w^3 - w^2 - 8*w + 7],\
[719, 719, -4*w^3 + 5*w^2 + 21*w - 11],\
[797, 797, -2*w^3 + w^2 + 13*w + 2],\
[797, 797, -2*w^3 + w^2 + 13*w - 1],\
[809, 809, -w^3 + w^2 + 6*w - 7],\
[809, 809, w^3 - w^2 - 6*w - 5],\
[823, 823, -w^3 + 2*w^2 + 6*w - 8],\
[823, 823, 2*w^3 - 3*w^2 - 10*w],\
[823, 823, -7*w^3 + 5*w^2 + 34*w + 6],\
[823, 823, 6*w^3 - 6*w^2 - 27*w + 1],\
[829, 829, -w^3 + 4*w^2 + w - 11],\
[829, 829, 4*w^3 - 3*w^2 - 18*w - 3],\
[841, 29, -3*w^3 + 3*w^2 + 12*w - 1],\
[841, 29, -3*w^3 + 3*w^2 + 12*w - 2],\
[857, 857, -3*w^3 + 2*w^2 + 12*w - 1],\
[857, 857, 2*w^2 - 9],\
[857, 857, -4*w^3 + 5*w^2 + 16*w - 7],\
[857, 857, 2*w^3 - 4*w^2 - 8*w + 3],\
[881, 881, -3*w^3 + 2*w^2 + 12*w + 2],\
[881, 881, 4*w^3 - 5*w^2 - 16*w + 4],\
[883, 883, -4*w^3 + 4*w^2 + 22*w + 1],\
[883, 883, -2*w^3 + 2*w^2 + 14*w - 7],\
[887, 887, 8*w^3 - 6*w^2 - 41*w - 4],\
[887, 887, -w^3 + 3*w^2 - 5*w + 1],\
[907, 907, -3*w^3 + 4*w^2 + 10*w - 4],\
[907, 907, 4*w^3 - 3*w^2 - 18*w],\
[907, 907, w^3 - w^2 - 5*w - 5],\
[907, 907, w - 6],\
[911, 911, -3*w^3 + 6*w^2 + 7*w],\
[911, 911, 5*w^3 - 2*w^2 - 25*w - 13],\
[911, 911, 2*w^3 - w^2 - 8*w - 6],\
[911, 911, -3*w^3 + 4*w^2 + 12*w],\
[919, 919, -2*w^3 + 3*w^2 + 11*w - 10],\
[919, 919, -w^3 - w^2 + 8*w + 10]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [-1, 0, e, -e, e, -5, -2*e - 8, 2*e + 8, -3*e - 8, -3*e - 8, e + 4, -e - 4, -6, 3*e + 4, -3*e - 4, 4*e + 12, -4*e - 12, -3*e - 12, 3*e + 12, 2*e + 12, 2*e + 12, e - 8, 3*e, -e + 8, 3*e, 4*e + 14, -4*e - 14, 6*e + 20, -6*e - 20, -4*e - 12, -4*e - 12, -3*e - 16, -3*e - 16, -3*e, -3*e, 8*e + 20, 8*e + 20, -3*e + 8, -3*e + 8, 2*e - 4, 2*e - 4, -6*e - 20, -6*e - 16, 6*e + 20, 6*e + 16, -2*e - 12, 20, 2*e + 12, -20, -e - 16, -e - 16, -2*e - 20, 2*e + 20, e + 4, -e - 4, 2*e - 4, 2*e - 4, -e - 16, -4*e + 2, -4*e + 2, e + 16, 3*e - 8, 3*e - 8, -16, 16, -4*e + 10, 4*e - 10, 3*e - 16, -3*e + 16, -2, 2, e, -e, e - 8, e - 8, -2*e - 8, 2*e + 8, -3*e - 20, 3*e + 20, -14, 14, -12, -12, -2*e - 32, -2*e - 32, -4*e - 40, 4*e + 40, 2*e - 4, -2*e + 4, 4*e + 16, 4*e + 16, -2*e - 16, 6*e + 16, -2*e - 16, 6*e + 16, -e + 40, 2*e - 8, 2*e - 24, 2*e - 8, 2*e - 24, 4*e - 20, -4*e + 20, e - 32, e - 32, 2*e - 12, -2*e + 12, 4*e - 10, -4*e + 10, 14*e + 36, -16, -16, 14*e + 36, -2, -11*e - 32, -11*e - 32, -8*e - 2, -e - 16, -e - 16, 8*e + 2, 6*e - 12, -6*e + 12, -4*e - 18, 7*e, 7*e, -4*e - 18, -4*e + 2, 4*e - 2, -8*e - 26, -8*e - 26, 8*e + 32, 8*e + 32, 11*e + 32, 11*e + 32, -8*e - 50, 8*e + 50, 8*e + 16, 8*e + 16, -8*e - 48, -8*e - 48, -3*e + 20, 3*e - 20, 12*e + 50, -12*e - 50, e + 16, 5*e + 44, e + 16, -5*e - 44, -3*e - 48, -3*e - 48, -10*e - 32, 10*e + 32, -2*e - 32, 2*e + 32, 14*e + 52, -14*e - 52, -6*e - 32, 6*e + 32, -6*e - 36, -6*e - 36, -4*e - 28, 4*e + 28, 8*e, -8*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([3,3,-w + 1])] = -1
AL_eigenvalues[ZF.ideal([3,3,w^3 - w^2 - 5*w])] = 1

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