/*
  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, -6, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

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

primes_array = [
[2, 2, w],\
[2, 2, w - 3],\
[3, 3, w^2 - w - 5],\
[9, 3, w^2 + w - 1],\
[13, 13, 2*w^2 - 2*w - 11],\
[29, 29, -w^2 - 3*w - 1],\
[29, 29, -w^2 + w + 3],\
[29, 29, 2*w - 5],\
[31, 31, -w^2 + w + 1],\
[37, 37, -2*w - 1],\
[43, 43, -2*w - 3],\
[47, 47, 2*w - 3],\
[59, 59, 4*w^2 + 8*w - 1],\
[61, 61, -w^2 - w - 1],\
[67, 67, w^2 + w - 5],\
[71, 71, w^2 - w - 9],\
[73, 73, -2*w^2 + 2*w + 9],\
[73, 73, 6*w^2 + 10*w - 7],\
[73, 73, 3*w^2 - 3*w - 17],\
[79, 79, -2*w^2 + 2*w + 15],\
[83, 83, -2*w^2 - 4*w + 3],\
[89, 89, -3*w^2 + 7*w + 5],\
[97, 97, w^2 + 3*w + 3],\
[103, 103, -3*w^2 - 3*w + 7],\
[107, 107, 3*w^2 - 3*w - 19],\
[107, 107, 2*w^2 - 15],\
[107, 107, -4*w^2 - 6*w + 5],\
[109, 109, w^2 + 5*w - 1],\
[109, 109, w^2 - 3*w - 3],\
[109, 109, 6*w^2 + 12*w - 5],\
[125, 5, -5],\
[127, 127, w^2 + 3*w - 3],\
[127, 127, 2*w^2 + 4*w + 1],\
[137, 137, -4*w^2 + 6*w + 15],\
[139, 139, -3*w^2 + w + 15],\
[149, 149, 2*w^2 - 13],\
[157, 157, -2*w^2 + 7],\
[169, 13, 2*w^2 + 6*w - 1],\
[173, 173, 2*w^2 + 2*w - 7],\
[173, 173, -2*w^2 - 6*w + 3],\
[173, 173, -3*w^2 - 5*w + 5],\
[191, 191, 2*w^2 - 4*w - 7],\
[193, 193, 6*w^2 - 4*w - 35],\
[197, 197, -3*w^2 + 5*w + 9],\
[199, 199, w^2 - 3*w - 5],\
[211, 211, w^2 + 5*w + 5],\
[211, 211, 3*w^2 - w - 23],\
[211, 211, w^2 - 3*w - 9],\
[229, 229, -w^2 + w - 3],\
[229, 229, w^2 - 7*w + 11],\
[229, 229, w^2 + 5*w - 3],\
[233, 233, 7*w^2 - 5*w - 41],\
[241, 241, 2*w^2 - 3],\
[257, 257, -2*w - 7],\
[263, 263, 4*w^2 - 4*w - 25],\
[269, 269, 2*w^2 - 2*w - 5],\
[271, 271, 5*w^2 + 11*w - 1],\
[281, 281, 3*w^2 + 3*w - 5],\
[283, 283, 2*w^2 - 2*w - 3],\
[307, 307, 3*w^2 - 3*w - 13],\
[313, 313, w^2 + 3*w - 5],\
[331, 331, 4*w - 9],\
[337, 337, 2*w^2 + 4*w - 5],\
[343, 7, -7],\
[347, 347, 4*w^2 - 2*w - 29],\
[353, 353, -4*w - 3],\
[353, 353, 5*w^2 - 5*w - 33],\
[353, 353, 2*w^2 - 2*w - 17],\
[367, 367, 2*w - 9],\
[379, 379, -4*w^2 + 4*w + 19],\
[383, 383, -4*w - 5],\
[389, 389, 2*w^2 + 6*w + 5],\
[397, 397, 4*w - 7],\
[401, 401, 4*w - 13],\
[409, 409, -4*w^2 + 8*w + 11],\
[421, 421, 9*w^2 - 5*w - 57],\
[421, 421, 4*w^2 - 6*w - 17],\
[421, 421, 5*w^2 - 5*w - 29],\
[431, 431, -6*w^2 + 10*w + 21],\
[463, 463, 5*w^2 - 3*w - 27],\
[467, 467, 3*w^2 + w - 9],\
[479, 479, 3*w^2 - 5*w - 7],\
[487, 487, 3*w^2 - 5*w - 13],\
[487, 487, 4*w^2 - 21],\
[487, 487, 3*w^2 - 3*w - 1],\
[491, 491, -5*w^2 - 9*w + 1],\
[499, 499, -7*w^2 - 11*w + 7],\
[503, 503, 3*w^2 + 5*w + 1],\
[509, 509, 4*w^2 - 17],\
[521, 521, 3*w^2 + w - 3],\
[521, 521, 2*w^2 + 4*w + 3],\
[521, 521, 4*w^2 + 10*w + 1],\
[523, 523, -3*w^2 - w + 15],\
[523, 523, -10*w^2 - 20*w + 3],\
[523, 523, 3*w^2 + 7*w + 3],\
[547, 547, 8*w^2 - 4*w - 55],\
[557, 557, 3*w^2 - 3*w - 11],\
[563, 563, 8*w^2 - 4*w - 49],\
[571, 571, 6*w^2 + 14*w + 1],\
[587, 587, -2*w^2 + 2*w - 3],\
[587, 587, 7*w^2 - 5*w - 47],\
[587, 587, -w^2 + w - 5],\
[593, 593, 2*w^2 - 4*w - 13],\
[601, 601, -4*w^2 - 10*w + 5],\
[607, 607, 2*w^2 + 8*w - 1],\
[617, 617, -5*w^2 - 9*w + 7],\
[619, 619, w^2 - w - 13],\
[631, 631, 6*w^2 - 4*w - 33],\
[631, 631, -4*w - 11],\
[631, 631, 3*w^2 - w - 11],\
[641, 641, -6*w - 1],\
[653, 653, 3*w^2 + w - 7],\
[659, 659, -2*w - 9],\
[661, 661, w^2 + 3*w - 9],\
[673, 673, 3*w^2 + 3*w + 1],\
[673, 673, 2*w^2 - 12*w + 17],\
[673, 673, 6*w - 5],\
[677, 677, -4*w^2 - 4*w + 5],\
[683, 683, 3*w^2 - 5*w - 3],\
[691, 691, 3*w^2 + 9*w - 1],\
[709, 709, 3*w^2 + 3*w - 11],\
[719, 719, 6*w^2 + 14*w + 3],\
[727, 727, -7*w^2 + 15*w + 15],\
[727, 727, 14*w^2 + 24*w - 13],\
[727, 727, w^2 + 3*w - 11],\
[733, 733, 3*w^2 + w - 25],\
[739, 739, 5*w^2 - 7*w - 19],\
[739, 739, -6*w^2 - 8*w + 9],\
[739, 739, -w^2 + 3*w - 7],\
[751, 751, -w^2 - 5*w + 21],\
[751, 751, w^2 - 9*w + 1],\
[751, 751, 3*w^2 - w - 3],\
[757, 757, 6*w^2 - 6*w - 35],\
[761, 761, 2*w^2 + 2*w - 19],\
[769, 769, w^2 - 5*w - 5],\
[773, 773, -2*w^2 - 3],\
[797, 797, 4*w^2 + 2*w - 3],\
[823, 823, -6*w^2 - 10*w + 9],\
[827, 827, 7*w^2 + 17*w + 3],\
[857, 857, 9*w^2 + 19*w - 3],\
[859, 859, 8*w^2 - 6*w - 45],\
[863, 863, 14*w^2 + 24*w - 15],\
[863, 863, 2*w^2 + 2*w - 13],\
[863, 863, 8*w^2 + 12*w - 13],\
[877, 877, -4*w^2 + 23],\
[881, 881, -10*w^2 - 16*w + 11],\
[911, 911, w^2 - 3*w - 15],\
[919, 919, -7*w^2 + 13*w + 19],\
[929, 929, -w^2 - 7*w - 1],\
[937, 937, 3*w^2 - 3*w - 7],\
[941, 941, 3*w^2 + 9*w + 1],\
[947, 947, 12*w^2 - 10*w - 71],\
[961, 31, 5*w^2 - 3*w - 25],\
[967, 967, 6*w^2 - 2*w - 37],\
[967, 967, 18*w^2 - 12*w - 109],\
[967, 967, -8*w^2 + 18*w + 13],\
[971, 971, 3*w^2 - 3*w - 5],\
[971, 971, 11*w^2 - 9*w - 69],\
[971, 971, 8*w - 5],\
[977, 977, 6*w^2 - 2*w - 45],\
[977, 977, 11*w^2 - 9*w - 67],\
[977, 977, -5*w^2 + 5*w + 23],\
[983, 983, -2*w^2 - 4*w + 29],\
[991, 991, 3*w^2 - 3*w - 25],\
[991, 991, w^2 - 5*w - 11],\
[991, 991, -3*w^2 + 5*w + 21],\
[997, 997, 5*w^2 + 13*w + 3],\
[997, 997, 19*w^2 - 15*w - 113],\
[997, 997, -w^2 - 3*w - 7]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [e, -1, 0, -e, 3*e, -3*e, e, -8, -6*e, -5*e, 2*e, 0, 0, -2, -8, 12, 11*e, -10, 9*e, -4, -16, 11*e, -4, -10*e, 4*e, 8, -6*e, -9*e, -7*e, -12, -16, -16, 8, -7*e, -12*e, 12, 15*e, 15*e, -12, -14, 2, 2*e, -9*e, -5*e, 2*e, 6*e, 24, -16*e, -20, 14, 0, 5*e, 15*e, 10, 0, 17*e, 2*e, -17*e, -4*e, 2*e, -10, -4, 20, -20, -8, -25*e, -13*e, 4, -8, 2*e, -4*e, 2, -8, -34, 13*e, -20, -e, 21*e, 18*e, -28*e, 18*e, 6*e, 8*e, 20, 0, -12, -20*e, -12, -17*e, -7*e, -12, 29*e, 12, -12, 40, -12, 19*e, 36, -12*e, 36, -32, -16, -11*e, 14, -10*e, -40, -12, 6*e, -44, 30*e, -21*e, 25*e, 16, 14, 20, -34, 8, 15*e, 0, 22*e, 4, -48, -38*e, -18*e, 8, -30, 14*e, -4*e, 4, 8, 4*e, -6*e, 17*e, -20, 9*e, 34, -e, -52, -2*e, 7*e, -22*e, -28, -8, -36, 16, -27*e, -20, 2*e, 25*e, -35*e, -37*e, 10*e, e, 32, 10*e, 28*e, 22*e, 2*e, 12, -18, -35*e, -7*e, -12, 12, 16*e, -24*e, -9*e, -15*e, 28]
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 - w - 5])] = 1

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