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

NN = ZF.ideal([41, 41, -2*w^2 - w + 7])

primes_array = [
[3, 3, w],\
[5, 5, w + 1],\
[7, 7, -w^2 + 2],\
[8, 2, 2],\
[9, 3, -w^2 + w + 4],\
[19, 19, w^2 + w - 4],\
[25, 5, -w^2 + 2*w + 2],\
[37, 37, 2*w + 1],\
[41, 41, -2*w^2 - w + 7],\
[43, 43, -2*w^2 + 5],\
[47, 47, 3*w - 4],\
[49, 7, 2*w^2 - w - 5],\
[53, 53, -2*w^2 + 2*w + 7],\
[61, 61, -w^2 - 3*w + 4],\
[61, 61, 3*w^2 - w - 10],\
[61, 61, w^2 - 2*w - 4],\
[67, 67, 2*w^2 - w - 4],\
[67, 67, 2*w^2 - w - 2],\
[67, 67, w^2 + 2*w - 5],\
[71, 71, -2*w^2 - w + 10],\
[83, 83, w - 5],\
[97, 97, -w^2 - 2],\
[101, 101, w^2 - 3*w - 2],\
[103, 103, w^2 - 8],\
[107, 107, w^2 - w - 8],\
[109, 109, 4*w^2 + w - 14],\
[113, 113, 2*w^2 + 2*w - 7],\
[113, 113, -3*w - 1],\
[113, 113, -3*w^2 - w + 4],\
[127, 127, -w - 5],\
[131, 131, 2*w^2 - 5*w - 1],\
[149, 149, -3*w^2 + 3*w + 10],\
[151, 151, 3*w^2 - w - 8],\
[157, 157, 5*w^2 - 22],\
[157, 157, 3*w^2 + 4*w - 7],\
[157, 157, -3*w^2 - w + 13],\
[163, 163, 3*w^2 - 7],\
[167, 167, 3*w^2 - 2*w - 7],\
[179, 179, 3*w^2 + 2*w - 10],\
[181, 181, -3*w^2 + 4],\
[191, 191, 5*w^2 - 3*w - 20],\
[193, 193, 2*w^2 - 3*w - 7],\
[193, 193, 3*w^2 - w - 7],\
[193, 193, 2*w^2 + 3*w - 7],\
[197, 197, w^2 - 3*w - 5],\
[197, 197, 2*w^2 + 2*w - 11],\
[197, 197, 3*w^2 - 5],\
[227, 227, -3*w^2 + 4*w + 8],\
[227, 227, 2*w^2 - 13],\
[227, 227, 2*w^2 - 3*w - 8],\
[229, 229, 3*w^2 - 3*w - 4],\
[233, 233, 3*w^2 - 17],\
[241, 241, w^2 - 4*w - 2],\
[241, 241, -w^2 + w - 4],\
[241, 241, 3*w^2 - w - 5],\
[251, 251, -4*w - 1],\
[257, 257, 3*w^2 - 2*w - 4],\
[257, 257, 2*w^2 + 3*w - 8],\
[263, 263, -2*w^2 - 4*w + 7],\
[269, 269, w - 7],\
[271, 271, 4*w^2 - 11],\
[277, 277, -w^2 + 2*w - 5],\
[281, 281, 2*w^2 - 4*w - 5],\
[311, 311, w^2 - 10],\
[313, 313, w^2 + 4*w - 8],\
[331, 331, -w^2 - w - 4],\
[337, 337, 5*w^2 - w - 25],\
[347, 347, 3*w^2 + 2*w - 13],\
[353, 353, 4*w^2 + w - 7],\
[359, 359, w^2 + w - 11],\
[361, 19, 2*w^2 + 3*w - 10],\
[367, 367, 2*w^2 + 3*w - 13],\
[383, 383, -2*w^2 - w - 2],\
[389, 389, 2*w^2 + 3*w - 11],\
[409, 409, w^2 - 4*w - 8],\
[419, 419, 3*w^2 - 4*w - 10],\
[419, 419, 6*w^2 - 4*w - 23],\
[419, 419, w - 8],\
[421, 421, 7*w^2 - 31],\
[431, 431, 2*w^2 - 4*w - 7],\
[439, 439, 5*w^2 + w - 13],\
[443, 443, -2*w^2 - 5*w + 8],\
[449, 449, 2*w^2 - 5*w - 4],\
[461, 461, 5*w^2 + w - 22],\
[463, 463, w^2 - 5*w - 2],\
[467, 467, 4*w^2 - 7],\
[487, 487, -5*w^2 + 3*w + 14],\
[499, 499, 5*w^2 - 14],\
[499, 499, 3*w^2 + 4*w - 10],\
[499, 499, 8*w^2 - 3*w - 29],\
[509, 509, w^2 - w - 11],\
[521, 521, 4*w^2 - w - 8],\
[541, 541, -w - 8],\
[547, 547, -5*w - 2],\
[557, 557, 4*w^2 - 4*w - 5],\
[569, 569, w^2 + 6*w - 1],\
[577, 577, 5*w^2 + 3*w - 16],\
[587, 587, 3*w^2 + 3*w - 13],\
[587, 587, 3*w^2 - 4*w - 14],\
[587, 587, 3*w^2 - 4*w - 13],\
[599, 599, 4*w^2 + 3*w - 14],\
[601, 601, 5*w^2 - w - 14],\
[607, 607, 4*w^2 - 3*w - 5],\
[617, 617, 5*w^2 - 13],\
[619, 619, 6*w^2 + w - 28],\
[631, 631, -5*w - 4],\
[631, 631, 6*w^2 - w - 19],\
[631, 631, 7*w^2 + 4*w - 20],\
[641, 641, 3*w^2 + w - 20],\
[643, 643, 3*w^2 + 3*w - 14],\
[643, 643, 4*w^2 - 23],\
[643, 643, 3*w^2 + 3*w - 17],\
[653, 653, 3*w - 11],\
[653, 653, 3*w^2 + 5*w - 10],\
[653, 653, 5*w^2 + 2*w - 19],\
[659, 659, 6*w^2 - 2*w - 19],\
[661, 661, w^2 - 5*w - 4],\
[677, 677, 3*w^2 + 3*w - 16],\
[691, 691, -w^2 + 3*w - 8],\
[733, 733, 5*w^2 - 2*w - 13],\
[739, 739, 4*w^2 + 4*w - 13],\
[739, 739, 5*w^2 - 4*w - 11],\
[739, 739, w^2 - 5*w - 5],\
[743, 743, 8*w^2 - 3*w - 28],\
[751, 751, -w^2 - 6*w - 1],\
[757, 757, 4*w^2 - 5*w - 13],\
[761, 761, 5*w^2 + 2*w - 20],\
[809, 809, -w^2 - 6*w + 14],\
[811, 811, 5*w^2 - 11],\
[821, 821, w^2 + 6*w - 11],\
[821, 821, 3*w^2 + 5*w - 11],\
[821, 821, 2*w^2 + 5*w - 11],\
[823, 823, -w^2 + w - 7],\
[823, 823, 3*w^2 - 5*w - 10],\
[823, 823, w^2 - 5*w - 7],\
[827, 827, 6*w^2 - 17],\
[853, 853, 5*w^2 - 7*w - 4],\
[857, 857, 3*w^2 - 6*w - 7],\
[859, 859, 3*w^2 + 4*w - 13],\
[859, 859, -3*w - 10],\
[859, 859, 6*w^2 - w - 31],\
[863, 863, 2*w^2 - 7*w - 2],\
[863, 863, 2*w^2 - 5*w - 13],\
[863, 863, w - 10],\
[877, 877, 6*w^2 + 3*w - 20],\
[881, 881, 9*w^2 - 40],\
[883, 883, 5*w^2 - 4*w - 10],\
[907, 907, 5*w^2 - 8],\
[907, 907, 5*w^2 - 3*w - 11],\
[907, 907, 2*w^2 - w - 16],\
[919, 919, 4*w^2 + 3*w - 22],\
[929, 929, 7*w^2 + w - 20],\
[929, 929, 6*w^2 - 3*w - 17],\
[929, 929, 4*w^2 + 3*w - 17],\
[937, 937, 2*w - 11],\
[941, 941, 6*w^2 + 2*w - 23],\
[947, 947, 4*w^2 + w - 25],\
[947, 947, 5*w^2 - 28],\
[947, 947, -w^2 - 7],\
[953, 953, w^2 - 13],\
[971, 971, 4*w^2 - 6*w - 11],\
[971, 971, 2*w^2 - 6*w - 5],\
[971, 971, 2*w^2 + 5*w - 16],\
[977, 977, 2*w^2 - 5*w - 10],\
[983, 983, 2*w^2 + 5*w - 13],\
[991, 991, -w^2 - 2*w - 7],\
[997, 997, 5*w^2 - w - 10],\
[997, 997, -3*w^2 - 6*w + 11],\
[997, 997, 9*w^2 + 3*w - 25]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [e, -2*e, 2*e, -5, -e, -8, -4*e, -e, -1, 4*e, -8, 2*e, 10*e, -8, -9*e, 5*e, 3*e, 3*e, -2, 2, 0, -14, e, -8, 10, -2, 16, -10*e, -10*e, -12, 12*e, 13*e, 10*e, 4, 4, -10, -15*e, -7*e, -2, 0, -e, 10*e, 3*e, -16, -10*e, 2, -10*e, 15*e, 4, 8*e, -9*e, 10, -6*e, 18, -20*e, 14*e, -9*e, 6, -16, 18, -9*e, -2, 18*e, -24, 34, -4, 2, 12, -20*e, 24, -30, 32, -8, -30, -13*e, -7*e, 0, -28, -26, 17*e, -23*e, -18, 11*e, 14, 11*e, 5*e, 3*e, 6*e, 10, 6*e, 6, -6*e, -32, -24*e, 3*e, -8*e, 4, -28, 0, -28*e, -30, 0, 26*e, -10*e, -24, -3*e, -3*e, 44, 26, 14, 28, 42, -46, -18, -46, 18*e, 8*e, -22, 48, -9*e, 26, -18*e, 3*e, -10*e, -15*e, 18*e, 6, -30, -7*e, -4, 38, -18, -44, 24*e, -25*e, 25*e, 21*e, -13*e, -36, 6, -36, 13*e, -22*e, 10, -32, -14, -3*e, -25*e, -4*e, 12, -32, 12*e, -37*e, 34, -42, 18, -4, 52, 24, 16, -2*e, -16*e, 6, 31*e, 32, -2, 36*e, -52, -41*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([41, 41, -2*w^2 - w + 7])] = 1

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