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

NN = ZF.ideal([62, 62, w^3 + w^2 - 5*w - 2])

primes_array = [
[2, 2, w],\
[17, 17, -w^2 - w + 3],\
[17, 17, -w^3 - w^2 + 3*w + 1],\
[17, 17, w^3 - w^2 - 3*w + 1],\
[17, 17, w^2 - w - 3],\
[31, 31, w^3 + w^2 - 2*w - 3],\
[31, 31, -w^3 + w^2 + 4*w - 1],\
[31, 31, w^3 + w^2 - 4*w - 1],\
[31, 31, -w^3 + w^2 + 2*w - 3],\
[47, 47, -2*w^3 + w^2 + 5*w - 1],\
[47, 47, 2*w^3 + w^2 - 6*w - 1],\
[47, 47, -2*w^3 + w^2 + 6*w - 1],\
[47, 47, 2*w^3 + w^2 - 5*w - 1],\
[49, 7, w^2 + 1],\
[49, 7, -2*w^2 + 3],\
[79, 79, -w^3 - w^2 + 4*w - 1],\
[79, 79, -w^3 + w^2 + 2*w - 5],\
[79, 79, w^3 + w^2 - 2*w - 5],\
[79, 79, w^3 - w^2 - 4*w - 1],\
[81, 3, -3],\
[97, 97, -2*w^3 + 8*w + 1],\
[97, 97, -2*w^3 + 4*w - 1],\
[97, 97, -2*w^3 + 4*w + 1],\
[97, 97, 2*w^3 - 8*w + 1],\
[113, 113, w^3 - w^2 - w + 3],\
[113, 113, 2*w^3 - w^2 - 7*w + 1],\
[113, 113, 2*w^3 + w^2 - 7*w - 1],\
[113, 113, w^3 + w^2 - w - 3],\
[127, 127, 2*w^3 + 2*w^2 - 6*w - 3],\
[127, 127, 2*w^2 - 2*w - 5],\
[127, 127, 2*w^2 + 2*w - 5],\
[127, 127, -2*w^3 + 2*w^2 + 6*w - 3],\
[191, 191, 2*w^3 - 9*w - 3],\
[191, 191, -w^3 + 3*w^2 + 3*w - 5],\
[191, 191, -w^3 - 3*w^2 + 3*w + 5],\
[191, 191, -2*w^3 + 9*w - 3],\
[193, 193, -4*w^3 - w^2 + 12*w + 1],\
[193, 193, 3*w^3 - w^2 - 11*w + 1],\
[193, 193, -3*w^3 - w^2 + 11*w + 1],\
[193, 193, -w^3 + 2*w^2 + 5*w - 7],\
[223, 223, 3*w^3 - w^2 - 10*w + 1],\
[223, 223, w^3 - 5*w - 5],\
[223, 223, -w^3 + 5*w - 5],\
[223, 223, -3*w^3 - w^2 + 10*w + 1],\
[239, 239, -2*w^2 + w - 1],\
[239, 239, -w^3 - 2*w^2 + 3*w + 9],\
[239, 239, w^3 - 2*w^2 - 3*w + 9],\
[239, 239, -2*w^2 - w - 1],\
[241, 241, -w^3 + w^2 + 2*w - 7],\
[241, 241, -w^3 - 2*w^2 + w + 5],\
[241, 241, w^3 - 2*w^2 - w + 5],\
[241, 241, w^3 + w^2 - 2*w - 7],\
[257, 257, -w^3 - 3*w^2 + w + 7],\
[257, 257, -2*w^2 + 3*w + 5],\
[257, 257, -2*w^2 - 3*w + 5],\
[257, 257, w^3 - 3*w^2 - w + 7],\
[271, 271, 3*w^3 - 10*w + 3],\
[271, 271, w^3 + 3],\
[271, 271, -w^3 + 3],\
[271, 271, -3*w^3 + 10*w + 3],\
[337, 337, -w^3 + w^2 + w - 5],\
[337, 337, -2*w^3 + w^2 + 7*w + 1],\
[337, 337, 2*w^3 + w^2 - 7*w + 1],\
[337, 337, w^3 + w^2 - w - 5],\
[353, 353, -w^3 + 6*w - 1],\
[353, 353, 3*w^3 - 8*w + 1],\
[353, 353, 3*w^3 - 8*w - 1],\
[353, 353, w^3 - 6*w - 1],\
[367, 367, -2*w^3 + w^2 + 8*w + 1],\
[367, 367, -2*w^3 + w^2 + 4*w - 5],\
[367, 367, 2*w^3 + w^2 - 4*w - 5],\
[367, 367, 2*w^3 + w^2 - 8*w + 1],\
[383, 383, -w^3 - w^2 + 3*w - 3],\
[383, 383, w^2 - w - 7],\
[383, 383, w^2 + w - 7],\
[383, 383, w^3 - w^2 - 3*w - 3],\
[401, 401, 3*w^3 - w^2 - 9*w - 1],\
[401, 401, w^3 - w^2 - 4*w + 7],\
[401, 401, w^3 + w^2 - 4*w - 7],\
[401, 401, -3*w^3 - w^2 + 9*w - 1],\
[431, 431, 2*w^2 + 2*w - 7],\
[431, 431, 2*w^3 - 2*w^2 - 6*w + 1],\
[431, 431, -2*w^3 - 2*w^2 + 6*w + 1],\
[431, 431, 2*w^2 - 2*w - 7],\
[433, 433, w^3 - 2*w - 5],\
[433, 433, w^3 - 4*w - 5],\
[433, 433, -w^3 + 4*w - 5],\
[433, 433, -w^3 + 2*w - 5],\
[449, 449, 3*w^3 + 3*w^2 - 9*w - 5],\
[449, 449, w^2 + 4*w - 1],\
[449, 449, -w^2 + 4*w + 1],\
[449, 449, -3*w^3 + 3*w^2 + 9*w - 5],\
[463, 463, -w^3 + w^2 + 3*w - 7],\
[463, 463, -w^2 - w - 3],\
[463, 463, -w^2 + w - 3],\
[463, 463, w^3 + w^2 - 3*w - 7],\
[479, 479, 2*w^3 - w^2 - 9*w + 3],\
[479, 479, 3*w^3 - w^2 - 7*w + 1],\
[479, 479, -3*w^3 - w^2 + 7*w + 1],\
[479, 479, -2*w^3 - w^2 + 9*w + 3],\
[529, 23, -w^2 - 3],\
[529, 23, w^2 - 7],\
[577, 577, -3*w^3 + 11*w - 5],\
[577, 577, 3*w^3 + w^2 - 12*w - 3],\
[577, 577, -3*w^3 + w^2 + 12*w - 3],\
[577, 577, 3*w^3 - 11*w - 5],\
[593, 593, -2*w^3 - 3*w^2 + 5*w + 1],\
[593, 593, 2*w^3 + 4*w^2 - 6*w - 9],\
[593, 593, -2*w^3 + 4*w^2 + 6*w - 9],\
[593, 593, 2*w^3 - 3*w^2 - 5*w + 1],\
[607, 607, -w^3 + 2*w^2 + 7*w - 3],\
[607, 607, 4*w^3 + 2*w^2 - 11*w - 5],\
[607, 607, -4*w^3 + 2*w^2 + 11*w - 5],\
[607, 607, -3*w^2 - 2*w + 9],\
[625, 5, -5],\
[641, 641, w^3 - 4*w^2 - 4*w + 7],\
[641, 641, 3*w^3 - w^2 - 9*w - 3],\
[641, 641, -3*w^3 - w^2 + 9*w - 3],\
[641, 641, -w^3 - 4*w^2 + 4*w + 7],\
[673, 673, 2*w^3 - 2*w^2 - 7*w + 1],\
[673, 673, -2*w^3 + w^2 + w - 3],\
[673, 673, -2*w^3 - w^2 + w + 3],\
[673, 673, 2*w^3 + 2*w^2 - 7*w - 1],\
[719, 719, -w^3 - 4*w^2 + 4*w + 3],\
[719, 719, w^3 - w^2 + 5],\
[719, 719, -w^3 - w^2 + 5],\
[719, 719, -w^3 + 4*w^2 + 4*w - 3],\
[751, 751, 3*w^3 - 3*w^2 - 8*w + 3],\
[751, 751, w^3 + 3*w^2 - 6*w - 9],\
[751, 751, -w^3 + 3*w^2 + 6*w - 9],\
[751, 751, -3*w^3 - 3*w^2 + 8*w + 3],\
[769, 769, w^3 - 4*w^2 - 4*w + 9],\
[769, 769, w^3 + 4*w^2 - 2*w - 7],\
[769, 769, -w^3 + 4*w^2 + 2*w - 7],\
[769, 769, w^3 + 4*w^2 - 4*w - 9],\
[863, 863, -w^3 + 4*w^2 + 3*w - 9],\
[863, 863, 4*w^2 - w - 7],\
[863, 863, 4*w^2 + w - 7],\
[863, 863, w^3 + 4*w^2 - 3*w - 9],\
[881, 881, 2*w^2 - 3*w - 7],\
[881, 881, 3*w^3 + 2*w^2 - 9*w - 1],\
[881, 881, 3*w^3 - 2*w^2 - 9*w + 1],\
[881, 881, 2*w^2 + 3*w - 7],\
[911, 911, -w^3 - w^2 + 7*w + 1],\
[911, 911, 4*w^3 - w^2 - 11*w + 3],\
[911, 911, 4*w^3 + w^2 - 11*w - 3],\
[911, 911, w^3 - w^2 - 7*w + 1],\
[929, 929, 2*w^3 - 10*w + 3],\
[929, 929, 4*w^3 - 10*w - 3],\
[929, 929, -4*w^3 + 10*w - 3],\
[929, 929, 2*w^3 - 10*w - 3],\
[977, 977, -4*w^3 + 3*w^2 + 11*w - 11],\
[977, 977, 4*w^3 + 3*w^2 - 16*w - 9],\
[977, 977, -2*w^3 + 5*w^2 + 5*w - 15],\
[977, 977, 4*w^3 + 3*w^2 - 11*w - 11],\
[991, 991, -4*w^3 + 11*w + 1],\
[991, 991, w^3 - 7*w - 1],\
[991, 991, -w^3 + 7*w - 1],\
[991, 991, 4*w^3 - 11*w + 1]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [1, e, -2*e + 4, e, -2*e + 4, 2*e - 4, 1, -e, 2*e - 4, -2, -2, -2, 3*e - 6, 3*e - 4, 0, -e - 2, -2*e + 16, -e - 2, -4*e + 2, -4*e + 4, -2, -2, -2, 6*e - 10, -5*e + 4, 4*e - 8, -2*e, 14, 5*e - 12, 2*e - 8, -2*e + 14, -4*e, -3*e - 4, 6*e - 16, 6*e - 16, -6*e, 3*e - 10, 6*e - 14, -4*e + 16, -6, -e - 8, -3*e + 28, -4*e - 4, 2*e - 12, -10*e + 20, -10*e + 20, 9*e - 22, -10, 2*e + 6, -2*e + 28, 2*e + 6, -4*e + 14, 10*e - 22, 4*e - 14, 2*e + 22, -2*e - 6, -e - 10, 4*e, 2*e - 14, -4*e - 6, e + 20, 3*e - 16, 2*e + 2, 3*e - 16, 10*e - 26, 4, -4*e + 26, 4*e - 18, 4*e - 4, -8*e + 12, 5*e - 22, 18, 4*e + 12, 2*e - 2, -16, -10*e + 14, -2*e - 12, -5*e - 8, 2, 6*e - 6, -4*e + 4, -4*e + 4, -3*e - 14, -5*e + 22, 3*e - 20, -2*e + 20, -4*e + 6, -6*e - 8, -10*e + 30, 4*e - 22, 2*e + 14, -5*e - 10, -4*e - 14, 5*e - 26, -9*e + 26, -8*e + 8, -6*e - 12, -11*e + 28, 7*e + 4, 4*e + 8, 7*e + 4, 8*e - 14, -11*e + 26, e + 10, -14*e + 30, 6*e - 30, -10*e + 24, -2*e - 20, 4*e + 22, -2*e - 20, -9*e + 20, 8, 10*e - 22, -2*e + 44, -4*e - 2, -13*e + 26, 7*e - 34, 42, -4*e + 14, -8*e + 18, 6*e - 34, -2*e + 10, -4*e - 4, 6*e + 12, 6*e + 12, 20, -2*e + 6, 12*e - 14, 8*e + 8, -3*e + 6, 2*e + 16, 13*e - 14, 10*e - 10, 10*e - 10, -8*e + 14, 4*e - 8, 2*e - 22, 4*e - 8, 3*e - 40, 2*e - 4, 2*e - 4, 4*e + 10, 2*e - 4, 2*e + 26, 12, 13*e - 22, -2*e - 2, 2*e - 6, 30, -2*e - 34, -16*e + 18, 12*e - 38, 4*e - 44, -5*e - 32, -8*e + 22, -6*e + 50, 6*e - 16, 6*e - 16, -22*e + 38]
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([31, 31, -w^3 + w^2 + 4*w - 1])] = -1

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