/*
  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, 0, -4, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([25, 5, w^3 - 5*w - 1])

primes_array = [
[2, 2, w^3 - 4*w + 1],\
[9, 3, -w^2 + 2],\
[23, 23, w^3 - w^2 - 4*w + 1],\
[23, 23, w^2 - w - 3],\
[23, 23, -w^2 - w + 3],\
[23, 23, -w^3 - w^2 + 4*w + 1],\
[25, 5, w^3 - 5*w + 1],\
[25, 5, w^3 - 5*w - 1],\
[47, 47, -3*w^3 + 2*w^2 + 12*w - 8],\
[47, 47, -2*w^3 - w^2 + 6*w],\
[47, 47, 2*w^3 - w^2 - 6*w],\
[47, 47, w^2 + w - 5],\
[49, 7, 2*w^3 - 6*w - 1],\
[49, 7, -2*w^3 + 6*w - 1],\
[71, 71, 2*w^3 - w^2 - 7*w + 1],\
[71, 71, 3*w^3 - w^2 - 11*w + 2],\
[71, 71, 4*w^3 - 2*w^2 - 14*w + 5],\
[71, 71, -3*w^3 + 2*w^2 + 10*w - 4],\
[73, 73, -w^3 - 2*w^2 + 3*w + 5],\
[73, 73, -w^3 + 2*w^2 + 3*w - 3],\
[73, 73, w^3 + 2*w^2 - 3*w - 3],\
[73, 73, w^3 - 2*w^2 - 3*w + 5],\
[97, 97, 2*w^2 + w - 4],\
[97, 97, w^3 + 2*w^2 - 4*w - 4],\
[97, 97, -w^3 + 2*w^2 + 4*w - 4],\
[97, 97, 2*w^2 - w - 4],\
[121, 11, 2*w^3 - 5*w],\
[121, 11, 3*w^3 - 10*w],\
[167, 167, -2*w^3 + 3*w^2 + 5*w - 7],\
[167, 167, -2*w^3 - w^2 + 10*w - 2],\
[167, 167, 3*w^3 - 11*w + 3],\
[167, 167, 3*w^3 - 2*w^2 - 10*w + 8],\
[169, 13, 2*w^3 - 9*w],\
[169, 13, w^3 - 6*w],\
[191, 191, w^2 - 2*w - 4],\
[191, 191, 2*w^3 + w^2 - 8*w],\
[191, 191, -2*w^3 + w^2 + 8*w],\
[191, 191, w^2 + 2*w - 4],\
[193, 193, -w - 4],\
[193, 193, -w^3 + 4*w - 4],\
[193, 193, w^3 - 4*w - 4],\
[193, 193, w - 4],\
[239, 239, 4*w^3 - 3*w^2 - 16*w + 12],\
[239, 239, -5*w^3 + w^2 + 19*w - 6],\
[239, 239, 4*w^3 - 16*w + 3],\
[239, 239, w^3 - 2*w^2 - 5*w + 9],\
[241, 241, 2*w^3 + w^2 - 10*w - 4],\
[241, 241, 2*w^3 - w^2 - 10*w],\
[241, 241, -2*w^3 - w^2 + 10*w],\
[241, 241, -2*w^3 + w^2 + 10*w - 4],\
[263, 263, -2*w^3 + 2*w^2 + 6*w - 7],\
[263, 263, -4*w^3 + 3*w^2 + 13*w - 9],\
[263, 263, -w^3 + 2*w^2 + 2*w - 6],\
[263, 263, -3*w^3 + 3*w^2 + 9*w - 8],\
[289, 17, 3*w^3 - 9*w + 1],\
[289, 17, -3*w^3 + 9*w + 1],\
[311, 311, 2*w^3 + w^2 - 8*w + 2],\
[311, 311, -w^2 - 2*w + 6],\
[311, 311, 3*w^3 - 2*w^2 - 13*w + 9],\
[311, 311, -2*w^3 + w^2 + 8*w + 2],\
[313, 313, -w^3 - w^2 + 4*w - 3],\
[313, 313, w^2 - w - 7],\
[313, 313, w^2 + w - 7],\
[313, 313, w^3 - w^2 - 4*w - 3],\
[337, 337, 3*w^3 - 8*w - 2],\
[337, 337, -4*w^3 + 13*w - 2],\
[337, 337, 4*w^3 - 13*w - 2],\
[337, 337, 3*w^3 - 8*w + 2],\
[359, 359, w^3 + 2*w^2 - 6*w - 6],\
[359, 359, -2*w^3 + 2*w^2 + 9*w - 2],\
[359, 359, 2*w^3 + 2*w^2 - 9*w - 2],\
[359, 359, -w^3 + 2*w^2 + 6*w - 6],\
[361, 19, -w^3 + 5*w - 5],\
[361, 19, w^3 - 5*w - 5],\
[383, 383, -w^3 + w^2 - 3],\
[383, 383, 4*w^3 + w^2 - 15*w - 1],\
[383, 383, -4*w^3 + w^2 + 15*w - 1],\
[383, 383, w^3 + w^2 - 3],\
[409, 409, 3*w^3 + w^2 - 8*w - 1],\
[409, 409, 4*w^3 - w^2 - 13*w + 3],\
[409, 409, 4*w^3 + w^2 - 13*w - 3],\
[409, 409, -3*w^3 + w^2 + 8*w - 1],\
[431, 431, -3*w^3 - 2*w^2 + 11*w + 3],\
[431, 431, -w^3 + 2*w^2 + w - 5],\
[431, 431, w^3 + 2*w^2 - w - 5],\
[431, 431, 3*w^3 - 2*w^2 - 11*w + 3],\
[433, 433, 3*w^3 - 13*w + 1],\
[433, 433, w^3 + w^2 - 4*w - 7],\
[433, 433, -w^3 + w^2 + 4*w - 7],\
[433, 433, 3*w^3 - 13*w - 1],\
[457, 457, 2*w^3 - 9*w - 6],\
[457, 457, w^3 - 6*w - 6],\
[457, 457, -w^3 + 6*w - 6],\
[457, 457, -2*w^3 + 9*w - 6],\
[479, 479, -2*w^3 + w^2 + 4*w - 4],\
[479, 479, 4*w^3 + w^2 - 14*w],\
[479, 479, -4*w^3 + w^2 + 14*w],\
[479, 479, 2*w^3 + w^2 - 4*w - 4],\
[503, 503, -3*w^3 - 2*w^2 + 10*w + 2],\
[503, 503, 2*w^3 + 2*w^2 - 5*w - 6],\
[503, 503, -2*w^3 + 2*w^2 + 5*w - 6],\
[503, 503, 3*w^3 - 2*w^2 - 10*w + 2],\
[577, 577, 3*w^3 + w^2 - 11*w - 8],\
[577, 577, -w^3 + w^2 + 6*w - 9],\
[577, 577, -4*w^3 + 4*w^2 + 15*w - 12],\
[577, 577, -4*w^3 + w^2 + 12*w],\
[599, 599, 2*w^3 + 2*w^2 - 8*w - 1],\
[599, 599, 2*w^2 - 2*w - 7],\
[599, 599, 2*w^2 + 2*w - 7],\
[599, 599, -2*w^3 + 2*w^2 + 8*w - 1],\
[601, 601, -w^3 + 3*w^2 + 4*w - 7],\
[601, 601, 3*w^2 - w - 5],\
[601, 601, 3*w^2 + w - 5],\
[601, 601, w^3 + 3*w^2 - 4*w - 7],\
[647, 647, -6*w^3 + 3*w^2 + 21*w - 7],\
[647, 647, -5*w^3 + 3*w^2 + 17*w - 6],\
[647, 647, 4*w^3 - 2*w^2 - 14*w + 3],\
[647, 647, 5*w^3 - 2*w^2 - 18*w + 4],\
[673, 673, -5*w^3 + 2*w^2 + 16*w - 6],\
[673, 673, 3*w^3 - w^2 - 14*w + 5],\
[673, 673, -3*w^3 - w^2 + 14*w + 5],\
[673, 673, 5*w^3 + 2*w^2 - 16*w - 6],\
[719, 719, 3*w^3 - w^2 - 13*w - 2],\
[719, 719, 6*w^3 - w^2 - 23*w + 7],\
[719, 719, -5*w^3 + 20*w - 4],\
[719, 719, -3*w^3 - w^2 + 13*w - 2],\
[743, 743, -3*w^3 + w^2 + 11*w + 2],\
[743, 743, w^3 + w^2 - w - 6],\
[743, 743, -w^3 + w^2 + w - 6],\
[743, 743, 3*w^3 + w^2 - 11*w + 2],\
[769, 769, -4*w^3 + 17*w - 2],\
[769, 769, -2*w^3 + w^2 + 8*w - 8],\
[769, 769, 2*w^3 + w^2 - 8*w - 8],\
[769, 769, 4*w^3 - 17*w - 2],\
[839, 839, w^3 + 3*w^2 - 6*w - 9],\
[839, 839, -4*w^3 - 2*w^2 + 16*w + 3],\
[839, 839, 4*w^3 - 2*w^2 - 16*w + 3],\
[839, 839, -w^3 + 3*w^2 + 6*w - 9],\
[841, 29, -4*w^3 + w^2 + 12*w - 2],\
[841, 29, 4*w^3 + w^2 - 12*w - 2],\
[863, 863, 2*w^2 + 3*w - 6],\
[863, 863, 3*w^3 - 2*w^2 - 12*w + 2],\
[863, 863, 3*w^3 + 2*w^2 - 12*w - 2],\
[863, 863, 2*w^2 - 3*w - 6],\
[887, 887, w^3 + 2*w^2 - 6*w - 8],\
[887, 887, 2*w^2 - 2*w - 9],\
[887, 887, 2*w^2 + 2*w - 9],\
[887, 887, -w^3 + 2*w^2 + 6*w - 8],\
[911, 911, -3*w^3 - 3*w^2 + 11*w + 4],\
[911, 911, w^3 + 3*w^2 - w - 8],\
[911, 911, -w^3 + 3*w^2 + w - 8],\
[911, 911, 3*w^3 - 3*w^2 - 11*w + 4],\
[937, 937, -w^3 - w^2 + 8*w + 1],\
[937, 937, 4*w^3 - w^2 - 17*w + 3],\
[937, 937, -4*w^3 - w^2 + 17*w + 3],\
[937, 937, w^3 - w^2 - 8*w + 1],\
[961, 31, -4*w^3 + 12*w - 1],\
[961, 31, 4*w^3 - 12*w - 1],\
[983, 983, w^2 + 3*w - 7],\
[983, 983, 3*w^3 - w^2 - 12*w - 3],\
[983, 983, 3*w^3 + w^2 - 12*w + 3],\
[983, 983, w^2 - 3*w - 7]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 7*x^2 + 8
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -e^2 + 6, -e^3 + 4*e, -e^3 + 4*e, 2*e^3 - 10*e, 2*e^3 - 10*e, -3*e^2 + 14, -1, e^3 - 6*e, e^3 - 6*e, -2*e^3 + 8*e, -2*e^3 + 8*e, -2, -2, -2*e^3 + 16*e, -2*e^3 + 16*e, -3*e^3 + 12*e, -3*e^3 + 12*e, 4*e^2 - 14, -3*e^2 + 10, 4*e^2 - 14, -3*e^2 + 10, 8*e^2 - 26, 8*e^2 - 26, e^2 - 2, e^2 - 2, 4*e^2 - 18, 4*e^2 - 18, 2*e^3 - 14*e, 2*e^3 - 14*e, -e^3, -e^3, -3*e^2 + 2, -3*e^2 + 2, -3*e^3 + 26*e, e^3 - 10*e, -3*e^3 + 26*e, e^3 - 10*e, 2*e^2 - 6, 9*e^2 - 30, 2*e^2 - 6, 9*e^2 - 30, 2*e, 2*e, 2*e, 2*e, 8*e^2 - 38, 8*e^2 - 38, -6*e^2 + 10, -6*e^2 + 10, 2*e^3 - 8*e, 4*e^3 - 26*e, 2*e^3 - 8*e, 4*e^3 - 26*e, 2*e^2 - 14, 2*e^2 - 14, 5*e^3 - 32*e, 5*e^3 - 32*e, -e^3 - 4*e, -e^3 - 4*e, -e^2 - 2, -4*e^2 + 38, -e^2 - 2, -4*e^2 + 38, -e^2 + 22, -e^2 + 22, 2*e^2 - 18, 2*e^2 - 18, 8*e^3 - 30*e, 8*e^3 - 30*e, -4*e^3 + 26*e, -4*e^3 + 26*e, 4*e^2 + 14, 7*e^2 - 26, 4*e^3 - 12*e, 7*e^3 - 26*e, 4*e^3 - 12*e, 7*e^3 - 26*e, -7*e^2 + 18, -7*e^2 + 18, -7*e^2 + 18, -7*e^2 + 18, -4*e^3 + 24*e, 2*e^3 - 4*e, -4*e^3 + 24*e, 2*e^3 - 4*e, 2*e^2 - 26, -8*e^2 + 38, 2*e^2 - 26, -8*e^2 + 38, -4*e^2 + 26, -4*e^2 + 26, -4*e^2 + 26, -4*e^2 + 26, 3*e^3 - 10*e, -4*e^3 + 14*e, 3*e^3 - 10*e, -4*e^3 + 14*e, 6*e^3 - 42*e, 6*e^3 - 42*e, 6*e^3 - 16*e, 6*e^3 - 16*e, -16*e^2 + 46, -16*e^2 + 46, -5*e^2 + 38, -5*e^2 + 38, -2*e^3 + 10*e, -2*e^3 + 10*e, -2*e^3 + 10*e, -2*e^3 + 10*e, 2*e^2 - 14, 2*e^2 - 14, 2*e^2 - 14, 2*e^2 - 14, 4*e^3 - 2*e, 4*e^3 - 2*e, -6*e^3 + 36*e, -6*e^3 + 36*e, -2*e^2 - 10, 5*e^2 - 34, -2*e^2 - 10, 5*e^2 - 34, 2*e^3 - 12*e, 4*e^3 - 4*e, 2*e^3 - 12*e, 4*e^3 - 4*e, 2*e^3 - 4*e, -14*e^3 + 62*e, 2*e^3 - 4*e, -14*e^3 + 62*e, -4*e^2 + 26, -4*e^2 + 26, 13*e^2 - 62, 13*e^2 - 62, -12*e^3 + 50*e, -12*e^3 + 50*e, -12*e^3 + 50*e, -12*e^3 + 50*e, -20*e^2 + 86, 5*e^2 + 30, 4*e^3 - 34*e, -10*e^3 + 40*e, 4*e^3 - 34*e, -10*e^3 + 40*e, 3*e^3 - 4*e, 3*e^3 - 4*e, 4*e^3 - 26*e, 4*e^3 - 26*e, -6*e^3 + 20*e, -6*e^3 + 20*e, -12*e^3 + 48*e, -12*e^3 + 48*e, -10*e^2 + 14, -10*e^2 + 14, -3*e^2 - 10, -3*e^2 - 10, 9*e^2 - 42, 9*e^2 - 42, 4*e^3 - 46*e, 5*e^3 - 16*e, 4*e^3 - 46*e, 5*e^3 - 16*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([25,5,w^3-5*w-1])] = 1

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