/*
  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, -8, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([15, 15, w + 3])

primes_array = [
[3, 3, w],\
[3, 3, w + 1],\
[5, 5, w^2 - w - 7],\
[8, 2, 2],\
[11, 11, w - 1],\
[13, 13, w^2 - 2*w - 8],\
[17, 17, w^2 - 2*w - 7],\
[19, 19, -w^2 + 2*w + 4],\
[19, 19, -2*w^2 + 3*w + 16],\
[23, 23, -w^2 + 2*w + 2],\
[31, 31, 2*w^2 - 3*w - 13],\
[37, 37, w^2 - w - 10],\
[43, 43, 3*w^2 - 5*w - 19],\
[43, 43, w^2 - w - 4],\
[43, 43, 2*w^2 - 2*w - 17],\
[47, 47, w^2 - 2],\
[67, 67, w^2 - 3*w - 5],\
[79, 79, -w^2 + 3*w - 1],\
[83, 83, w^2 + w - 4],\
[97, 97, w^2 - 11],\
[101, 101, -3*w^2 + 4*w + 20],\
[103, 103, -6*w^2 + 9*w + 44],\
[109, 109, 2*w - 5],\
[113, 113, -w - 5],\
[113, 113, 2*w^2 - 4*w - 13],\
[113, 113, w^2 - 4*w - 4],\
[121, 11, w^2 - 8],\
[127, 127, 3*w^2 - 6*w - 17],\
[131, 131, w^2 - 2*w - 11],\
[137, 137, 5*w^2 - 6*w - 40],\
[149, 149, -5*w^2 + 7*w + 35],\
[151, 151, w^2 + w - 5],\
[169, 13, -2*w^2 + 7*w + 5],\
[181, 181, -5*w - 1],\
[191, 191, -3*w^2 + 3*w + 25],\
[193, 193, 2*w^2 - 4*w - 7],\
[197, 197, -2*w^2 + w + 13],\
[211, 211, 2*w^2 - w - 20],\
[223, 223, 3*w^2 - 9*w - 2],\
[229, 229, 2*w^2 - 4*w - 19],\
[229, 229, 3*w - 2],\
[229, 229, 2*w^2 - 7],\
[233, 233, -w^2 + 2*w - 2],\
[239, 239, -w^2 + w - 2],\
[241, 241, -2*w - 7],\
[251, 251, -5*w^2 + 6*w + 37],\
[263, 263, 7*w^2 - 11*w - 49],\
[269, 269, 3*w^2 - 6*w - 14],\
[269, 269, w^2 - 4*w - 7],\
[269, 269, w^2 + 2*w - 4],\
[281, 281, w^2 - 4*w - 10],\
[281, 281, 7*w^2 - 11*w - 47],\
[281, 281, w^2 + w - 17],\
[283, 283, 2*w^2 - 5*w - 11],\
[283, 283, 2*w^2 - w - 14],\
[283, 283, 2*w^2 - 3*w - 4],\
[289, 17, w^2 - 5*w - 5],\
[293, 293, 3*w^2 - 7*w - 5],\
[293, 293, 3*w^2 - 4*w - 26],\
[293, 293, 2*w^2 - w - 8],\
[307, 307, -5*w^2 + 7*w + 34],\
[311, 311, -7*w^2 + 9*w + 56],\
[313, 313, -3*w^2 + 3*w + 28],\
[313, 313, w^2 + w - 14],\
[313, 313, 2*w^2 - 3*w - 19],\
[331, 331, w^2 - w - 13],\
[337, 337, 3*w^2 - 2*w - 16],\
[343, 7, -7],\
[347, 347, -5*w^2 + 8*w + 32],\
[353, 353, 3*w - 4],\
[359, 359, 4*w^2 - 7*w - 28],\
[367, 367, w^2 + w - 8],\
[367, 367, 4*w^2 - 5*w - 34],\
[367, 367, -3*w^2 + 2*w + 31],\
[373, 373, 4*w^2 - 8*w - 23],\
[379, 379, -3*w^2 + 5*w + 16],\
[389, 389, 3*w - 7],\
[397, 397, 3*w^2 - 6*w - 13],\
[397, 397, w^2 - 2*w - 13],\
[397, 397, 2*w^2 - w - 4],\
[401, 401, w^2 - 5*w - 13],\
[401, 401, 2*w^2 - 2*w - 5],\
[401, 401, 3*w^2 - 3*w - 17],\
[409, 409, -2*w^2 + w + 17],\
[419, 419, 2*w^2 - w - 5],\
[421, 421, -7*w^2 + 11*w + 50],\
[433, 433, 3*w^2 - 6*w - 23],\
[439, 439, 2*w^2 - 5*w - 14],\
[443, 443, 2*w^2 - 7*w - 7],\
[461, 461, w^2 + w - 10],\
[463, 463, -9*w^2 + 12*w + 64],\
[463, 463, -w^2 + 3*w - 4],\
[463, 463, 7*w^2 - 10*w - 55],\
[467, 467, -3*w^2 + 6*w + 22],\
[479, 479, -7*w^2 + 11*w + 52],\
[487, 487, -6*w^2 + 10*w + 41],\
[491, 491, w^2 - 3*w - 14],\
[503, 503, -w^2 + 6*w - 1],\
[523, 523, 8*w^2 - 12*w - 55],\
[529, 23, 4*w^2 - 5*w - 25],\
[541, 541, w^2 - 5*w - 8],\
[541, 541, -6*w^2 + 7*w + 44],\
[541, 541, 4*w^2 - 10*w - 7],\
[547, 547, w^2 - 8*w - 5],\
[557, 557, -5*w^2 + 16*w + 10],\
[577, 577, -w^2 - 4],\
[577, 577, 4*w^2 - 9*w - 20],\
[577, 577, w^2 - 5*w - 11],\
[587, 587, 8*w^2 - 10*w - 65],\
[593, 593, -4*w^2 + 4*w + 31],\
[599, 599, w^2 - 2*w - 14],\
[599, 599, -4*w - 11],\
[599, 599, w^2 - 5*w - 10],\
[601, 601, 2*w^2 - 7*w - 8],\
[607, 607, -6*w^2 + 7*w + 47],\
[617, 617, 3*w^2 - 3*w - 29],\
[631, 631, -9*w^2 + 27*w + 11],\
[631, 631, 2*w^2 - 9*w - 1],\
[631, 631, -2*w^2 + 4*w - 1],\
[641, 641, 8*w^2 - 10*w - 61],\
[641, 641, -8*w^2 + 11*w + 56],\
[641, 641, w^2 + 2*w - 7],\
[647, 647, w^2 - 5*w - 17],\
[653, 653, 2*w^2 - 6*w - 19],\
[659, 659, 4*w^2 - 5*w - 35],\
[659, 659, -w^2 + 6*w - 4],\
[659, 659, -5*w^2 + 13*w + 8],\
[661, 661, -3*w - 10],\
[673, 673, 3*w^2 - 7*w - 26],\
[691, 691, -6*w^2 + 8*w + 49],\
[691, 691, -5*w^2 + 9*w + 28],\
[691, 691, 3*w^2 - 7*w - 17],\
[701, 701, -8*w^2 + 13*w + 55],\
[719, 719, -10*w^2 + 14*w + 71],\
[751, 751, 3*w^2 - 5*w - 13],\
[761, 761, 5*w^2 - 7*w - 32],\
[787, 787, 5*w - 2],\
[809, 809, 4*w^2 - 6*w - 23],\
[811, 811, 5*w^2 - 10*w - 29],\
[821, 821, -9*w^2 + 14*w + 67],\
[827, 827, 5*w^2 - 9*w - 34],\
[827, 827, 2*w^2 - 6*w - 13],\
[827, 827, -2*w^2 + w - 2],\
[829, 829, w^2 - 6*w - 8],\
[877, 877, 3*w^2 - 10],\
[881, 881, 3*w^2 - 4*w - 5],\
[887, 887, 3*w^2 - 3*w - 14],\
[887, 887, 2*w^2 - 9*w - 7],\
[887, 887, -3*w^2 + 2*w + 25],\
[907, 907, -6*w^2 + 9*w + 49],\
[929, 929, 2*w^2 - 7*w - 10],\
[941, 941, -4*w^2 + 6*w + 35],\
[941, 941, 4*w - 7],\
[941, 941, 6*w - 1],\
[947, 947, w^2 - 7*w - 7],\
[961, 31, 3*w^2 - 2*w - 4],\
[977, 977, w^2 + 2*w - 10],\
[977, 977, w^2 + 4*w - 4],\
[977, 977, w^2 - 17],\
[983, 983, 4*w^2 - 5*w - 23],\
[983, 983, -3*w - 11],\
[983, 983, 2*w^2 + 3*w - 7],\
[991, 991, -7*w^2 + 8*w + 56],\
[997, 997, 3*w^2 - 5*w - 10],\
[997, 997, 3*w^2 - 12*w - 1],\
[997, 997, w^2 + w - 19]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 + 4*x - 3
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1, -1, 1, e, e + 3, e + 2, e, -e - 7, -e - 4, -e - 3, -2*e - 4, -e + 2, -2*e - 4, -4*e - 7, -2*e - 1, -6, -e - 1, 3*e - 1, -2*e - 9, -e - 4, -2*e - 3, -3*e - 7, 3*e - 1, 0, 3*e - 3, -3*e - 12, 2*e - 4, 11, 5*e + 6, -3*e - 18, 3, -10, 4*e - 4, -2*e - 7, -3*e - 18, 5*e + 5, 9*e + 15, 3*e + 5, 4*e + 5, 4*e + 14, 3*e + 5, -2*e - 22, 12, 3*e + 9, 2*e + 17, 3*e, -4*e - 15, 5*e - 3, -3*e - 3, -4*e, 3*e + 15, -2*e - 6, -2*e - 3, 6*e + 14, 8, 4*e + 11, 4*e - 4, -3*e - 18, e - 24, -3*e + 6, -3*e + 5, 3*e + 6, e - 16, 2*e + 20, -e + 11, -7*e - 7, 7*e + 11, 6*e + 29, -6, -7*e, 3*e + 9, -2*e - 10, 4*e + 2, -5*e + 5, -3*e + 23, 12*e + 20, 6*e - 3, 2*e - 4, 20, 8*e + 32, 6, -e - 27, -2*e + 18, -8*e - 34, 4*e, -7*e - 7, -3*e - 10, 2*e + 2, -2*e - 36, -4*e + 12, 4*e + 26, -8*e - 1, 5*e + 17, -4*e + 6, -6*e - 12, 3*e + 23, 7*e + 12, 4*e + 27, 9*e + 11, -6*e - 31, e - 1, -8*e - 37, 2*e + 14, -2*e - 28, 15*e + 24, -9*e - 37, -5*e + 8, -3*e + 8, 8*e + 30, 9*e + 21, 9*e + 15, 9*e + 24, -7*e + 9, -3*e - 31, 10*e + 20, -e - 15, -8*e + 8, -e + 20, -9*e - 4, -12*e - 27, 7*e + 3, -2*e + 9, -3*e - 15, -7*e + 3, 9*e + 42, -5*e - 36, -6*e + 12, 2*e + 35, 5*e - 22, 9*e + 44, 4*e + 20, -3*e - 16, 9*e + 33, -6*e - 6, -6*e + 8, -12*e - 21, 14, -10*e - 21, e - 28, -4*e - 36, -2*e - 15, 6*e + 27, 5*e + 3, 9*e + 11, -8*e + 14, 42, -8*e + 6, -8*e - 15, -33, -4*e - 34, 9*e + 33, -9*e - 15, 2*e + 9, 18*e + 33, 3*e - 39, 2*e - 1, 5*e + 12, -10*e - 12, -12*e - 30, -15, -12*e - 18, 14*e + 18, 3*e - 34, 12*e + 14, 18*e + 44, -5*e - 1]
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
AL_eigenvalues[ZF.ideal([5, 5, w^2 - w - 7])] = -1

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