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

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

primes_array = [
[3, 3, w],\
[3, 3, w - 2],\
[3, 3, w - 1],\
[16, 2, 2],\
[23, 23, -w^3 + 4*w + 1],\
[25, 5, w^3 - 2*w^2 - 2*w + 2],\
[25, 5, w^3 - w^2 - 3*w + 1],\
[29, 29, w^3 - 2*w^2 - 4*w + 4],\
[29, 29, w^3 - w^2 - 5*w + 1],\
[43, 43, -w^3 + 2*w^2 + 3*w - 2],\
[43, 43, w^3 - w^2 - 4*w + 2],\
[61, 61, -w^2 + 2*w + 4],\
[61, 61, w^2 - 5],\
[79, 79, 3*w^3 - 7*w^2 - 8*w + 16],\
[79, 79, -w^3 + w^2 + 6*w - 4],\
[101, 101, -w^3 + 3*w^2 + w - 7],\
[101, 101, w^3 - 4*w - 4],\
[103, 103, 2*w^3 - w^2 - 8*w - 1],\
[103, 103, 2*w^3 - 5*w^2 - 4*w + 8],\
[107, 107, 2*w^2 - 3*w - 4],\
[107, 107, 3*w - 5],\
[107, 107, w^3 - 4*w^2 + 10],\
[107, 107, 2*w^2 - w - 5],\
[113, 113, -w^3 + 4*w^2 + w - 11],\
[113, 113, -2*w^3 + 4*w^2 + 5*w - 5],\
[113, 113, 2*w^3 - 2*w^2 - 7*w + 2],\
[113, 113, w^3 + w^2 - 6*w - 7],\
[121, 11, w^3 - 3*w^2 - 4*w + 8],\
[121, 11, w^3 - 7*w - 2],\
[127, 127, w^2 - 4*w + 2],\
[127, 127, w^3 - w^2 - 2*w - 2],\
[127, 127, -w^3 + 2*w^2 + w - 4],\
[127, 127, -2*w^3 + 5*w^2 + 3*w - 8],\
[131, 131, 2*w^3 - 2*w^2 - 9*w + 2],\
[131, 131, -w^3 - w^2 + 7*w + 5],\
[157, 157, w^3 - w^2 - 7*w + 8],\
[157, 157, w^3 - w^2 - 5*w + 4],\
[169, 13, 2*w^2 - 2*w - 5],\
[173, 173, -w^3 + 7*w - 1],\
[173, 173, -w^3 - w^2 + 6*w + 4],\
[179, 179, -2*w^3 + 4*w^2 + 5*w - 8],\
[179, 179, -2*w^3 + 2*w^2 + 7*w + 1],\
[181, 181, -w^3 + 4*w^2 - 7],\
[181, 181, w^2 - 4*w + 5],\
[191, 191, 3*w^3 - 7*w^2 - 9*w + 20],\
[191, 191, -w^3 + 4*w + 5],\
[191, 191, -w^3 + 3*w^2 + w - 8],\
[191, 191, 3*w^3 - 3*w^2 - 12*w + 2],\
[199, 199, 2*w^2 - 7],\
[199, 199, 3*w^3 - 6*w^2 - 9*w + 13],\
[233, 233, w^3 - 3*w^2 - 4*w + 11],\
[233, 233, 3*w - 7],\
[251, 251, 3*w^3 - 7*w^2 - 7*w + 13],\
[251, 251, 4*w^3 - w^2 - 20*w - 11],\
[251, 251, -w^3 + w^2 + 5*w - 7],\
[251, 251, -2*w^3 + 4*w^2 + 8*w - 11],\
[257, 257, w^2 - 2*w - 7],\
[257, 257, w^2 - 8],\
[269, 269, 2*w^3 - 3*w^2 - 8*w + 2],\
[269, 269, 2*w^3 - 3*w^2 - 8*w + 7],\
[283, 283, -2*w^3 + 4*w^2 + 4*w - 7],\
[283, 283, 2*w^3 - 2*w^2 - 6*w - 1],\
[289, 17, -w^2 + w - 2],\
[289, 17, w^2 - w - 7],\
[311, 311, -w^3 + 2*w^2 + 3*w - 8],\
[311, 311, -w^3 + w^2 + 4*w + 4],\
[313, 313, -w^3 + 2*w^2 + 5*w - 5],\
[313, 313, w^3 - w^2 - 6*w + 1],\
[337, 337, 3*w - 1],\
[337, 337, 3*w - 2],\
[347, 347, -2*w^3 - w^2 + 11*w + 11],\
[347, 347, -2*w^3 + 7*w^2 + 3*w - 19],\
[367, 367, 4*w^3 - 8*w^2 - 13*w + 19],\
[367, 367, 4*w^3 - 4*w^2 - 17*w - 2],\
[373, 373, w^2 - 3*w - 5],\
[373, 373, w^2 + w - 7],\
[433, 433, 2*w^3 - 3*w^2 - 5*w + 1],\
[433, 433, -2*w^3 + 3*w^2 + 5*w - 5],\
[443, 443, w^3 - 7*w + 2],\
[443, 443, 3*w^3 - 6*w^2 - 12*w + 20],\
[491, 491, -2*w^3 + 3*w^2 + 8*w - 8],\
[491, 491, 2*w^3 - 3*w^2 - 11*w + 14],\
[521, 521, w^3 + w^2 - 8*w - 7],\
[521, 521, w^3 - 5*w^2 + 2*w + 10],\
[521, 521, w^3 + 2*w^2 - 5*w - 8],\
[521, 521, w^3 - 4*w^2 - 3*w + 13],\
[523, 523, 3*w^3 - 5*w^2 - 11*w + 11],\
[523, 523, w^3 + w^2 - 6*w - 1],\
[529, 23, 3*w^2 - 3*w - 10],\
[547, 547, w^3 - 3*w^2 - 4*w + 14],\
[547, 547, -2*w^3 + 2*w^2 + 7*w - 5],\
[547, 547, w^3 - 4*w + 4],\
[547, 547, 3*w^3 - 9*w^2 - 6*w + 22],\
[563, 563, w^3 - 2*w^2 - 6*w + 8],\
[563, 563, -3*w^3 + 4*w^2 + 11*w - 7],\
[563, 563, 3*w^3 - 5*w^2 - 10*w + 5],\
[563, 563, 3*w^3 - 8*w^2 - 3*w + 10],\
[571, 571, 2*w^3 - 3*w^2 - 5*w + 4],\
[571, 571, -2*w^3 + 3*w^2 + 5*w - 2],\
[599, 599, 2*w^3 - 2*w^2 - 5*w + 1],\
[599, 599, -2*w^3 + 4*w^2 + 3*w - 4],\
[641, 641, -2*w^2 + 7*w - 7],\
[641, 641, 2*w^2 + w - 8],\
[641, 641, 2*w^2 - 5*w - 5],\
[641, 641, w^3 - 4*w^2 - 2*w + 16],\
[647, 647, -w^3 + 3*w^2 - 2*w - 4],\
[647, 647, w^3 - w - 4],\
[653, 653, -2*w^3 + 4*w^2 + 9*w - 13],\
[653, 653, 3*w^3 - 7*w^2 - 6*w + 11],\
[673, 673, w^3 - 2*w^2 - 5*w + 11],\
[673, 673, -2*w^3 + 2*w^2 + 10*w - 5],\
[673, 673, 2*w^3 - 4*w^2 - 8*w + 5],\
[673, 673, 2*w^3 - 7*w^2 - w + 13],\
[677, 677, -w^3 + 2*w^2 + 6*w - 2],\
[677, 677, 3*w^3 - 7*w^2 - 8*w + 13],\
[677, 677, 3*w^3 - 2*w^2 - 13*w - 1],\
[677, 677, -w^3 + w^2 + 7*w - 5],\
[701, 701, w^2 - 5*w - 1],\
[701, 701, -w^3 + 4*w^2 + 3*w - 7],\
[701, 701, -w^3 - w^2 + 8*w + 1],\
[701, 701, w^2 + 3*w - 5],\
[719, 719, -2*w^3 + w^2 + 7*w + 5],\
[719, 719, 2*w^3 - 5*w^2 - 3*w + 11],\
[727, 727, 2*w^2 - 4*w - 11],\
[727, 727, -3*w^3 + w^2 + 10*w + 5],\
[751, 751, 4*w^3 - 12*w^2 - 7*w + 28],\
[751, 751, 2*w^3 - 6*w^2 - 5*w + 20],\
[757, 757, -w^3 + 2*w^2 - 5],\
[757, 757, w^3 - w^2 - w - 4],\
[797, 797, -4*w^3 + 7*w^2 + 13*w - 11],\
[797, 797, 4*w^3 - 5*w^2 - 17*w + 2],\
[797, 797, -4*w^3 + 7*w^2 + 15*w - 16],\
[797, 797, 4*w^3 - 5*w^2 - 15*w + 5],\
[809, 809, -w^3 + 4*w^2 + 3*w - 16],\
[809, 809, 3*w^3 - 8*w^2 - 10*w + 26],\
[841, 29, 3*w^2 - 3*w - 8],\
[857, 857, 4*w^3 - 2*w^2 - 21*w - 7],\
[857, 857, -w^3 + 2*w^2 - w + 4],\
[881, 881, -w^3 + 2*w^2 + 7*w - 10],\
[881, 881, w^3 - 5*w^2 - w + 13],\
[881, 881, w^3 + 2*w^2 - 8*w - 8],\
[881, 881, -w^3 + w^2 + 8*w + 2],\
[887, 887, w^3 - w^2 - 5*w - 5],\
[887, 887, -w^3 + 2*w^2 + 4*w - 10],\
[907, 907, -3*w^3 + 3*w^2 + 9*w - 5],\
[907, 907, 3*w^3 - 6*w^2 - 6*w + 4],\
[919, 919, 4*w^3 - 3*w^2 - 16*w - 2],\
[919, 919, -4*w^3 + 9*w^2 + 10*w - 17],\
[937, 937, -3*w^3 + w^2 + 12*w + 7],\
[937, 937, -3*w^3 + 8*w^2 + 5*w - 17],\
[953, 953, -2*w^3 + 6*w^2 + 2*w - 13],\
[953, 953, -w^3 + 3*w^2 + 4*w - 2],\
[953, 953, w^3 - 7*w + 4],\
[953, 953, 2*w^3 - 8*w - 7],\
[961, 31, 3*w^3 - 4*w^2 - 13*w + 1],\
[961, 31, 3*w^3 - 5*w^2 - 12*w + 13],\
[991, 991, 3*w^3 - w^2 - 15*w - 4],\
[991, 991, -w^3 - 3*w^2 + 7*w + 5],\
[991, 991, w^3 - 6*w^2 + 2*w + 8],\
[991, 991, 3*w^3 - 8*w^2 - 8*w + 17]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 6*x^2 + 2
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 0, -e, -e^2 - 1, -2*e^2 + 2, 3*e^3 - 14*e, -3*e^3 + 14*e, -2*e^3 + 11*e, 2*e^3 - 11*e, e^2 - 6, e^2 - 6, 2*e, -2*e, -e^3 + 9*e, e^3 - 9*e, -4*e^3 + 22*e, 4*e^3 - 22*e, e^2 - 12, e^2 - 12, 4*e^2 - 10, -5*e^3 + 26*e, 5*e^3 - 26*e, 4*e^2 - 10, 4*e^2 - 16, 3*e^3 - 9*e, -3*e^3 + 9*e, 4*e^2 - 16, -e^3 - e, e^3 + e, 3*e^3 - 16*e, e^3 - 11*e, -e^3 + 11*e, -3*e^3 + 16*e, -6*e^2 + 18, -6*e^2 + 18, -4*e^2 + 14, -4*e^2 + 14, -8, -5*e^2 + 14, -5*e^2 + 14, 4*e^3 - 28*e, -4*e^3 + 28*e, -2*e^2 - 2, -2*e^2 - 2, -e^2 - 8, 2*e^3 - 11*e, -2*e^3 + 11*e, -e^2 - 8, -2*e^3 + e, 2*e^3 - e, -4*e^3 + 28*e, 4*e^3 - 28*e, -e^3 + 7*e, -e^2 - 2, -e^2 - 2, e^3 - 7*e, e^2 - 28, e^2 - 28, 8*e^2 - 20, 8*e^2 - 20, 8*e^3 - 57*e, -8*e^3 + 57*e, e^2 + 4, -20, 6*e^2 - 30, 6*e^2 - 30, -9*e^3 + 47*e, 9*e^3 - 47*e, 2*e^3 - 15*e, -2*e^3 + 15*e, -2*e^2 + 8, -2*e^2 + 8, -10, -10, 4*e^3 - 11*e, -4*e^3 + 11*e, 9*e^3 - 50*e, -9*e^3 + 50*e, 12, 12, 9*e^2 - 30, 9*e^2 - 30, 8*e^3 - 29*e, -6*e^2 + 24, -6*e^2 + 24, -8*e^3 + 29*e, 6*e^2 - 4, 6*e^2 - 4, -44, -2*e^2 - 30, 6*e^2 - 34, 6*e^2 - 34, -2*e^2 - 30, -11*e^3 + 62*e, 6*e^2 - 18, 6*e^2 - 18, 11*e^3 - 62*e, -10*e^3 + 44*e, 10*e^3 - 44*e, 3*e^3 - 6*e, -3*e^3 + 6*e, -6*e^2 + 36, -14*e^3 + 77*e, 14*e^3 - 77*e, -6*e^2 + 36, -5*e^3 + 26*e, 5*e^3 - 26*e, 16*e^3 - 79*e, -16*e^3 + 79*e, 6*e^2 - 16, -10*e^2 + 24, -10*e^2 + 24, 6*e^2 - 16, -11*e^3 + 74*e, -7*e^2 + 28, -7*e^2 + 28, 11*e^3 - 74*e, -9*e^3 + 63*e, 6*e^2 - 42, 6*e^2 - 42, 9*e^3 - 63*e, -13*e^3 + 55*e, 13*e^3 - 55*e, 7*e^3 - 51*e, -7*e^3 + 51*e, 4, 4, -7*e^3 + 44*e, 7*e^3 - 44*e, -9*e^3 + 39*e, 12*e^2 - 42, 12*e^2 - 42, 9*e^3 - 39*e, e^3 + 11*e, -e^3 - 11*e, -e^2 + 48, 2*e^3 + 10*e, -2*e^3 - 10*e, 5*e^3 - 44*e, -12*e^2 + 48, -12*e^2 + 48, -5*e^3 + 44*e, -9*e^2 + 12, -9*e^2 + 12, -12*e^3 + 68*e, 12*e^3 - 68*e, 11*e^3 - 82*e, -11*e^3 + 82*e, -2*e^3 + 10*e, 2*e^3 - 10*e, -9*e^3 + 72*e, 4*e^2 + 8, 4*e^2 + 8, 9*e^3 - 72*e, 2*e^2 + 12, 2*e^2 + 12, 3*e^2 - 14, 6*e^2 - 22, 6*e^2 - 22, 3*e^2 - 14]
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 - 2])] = -1

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