/*
  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([5, -6, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([9, 9, w + 1])

primes_array = [
[3, 3, w - 2],\
[5, 5, w],\
[5, 5, -w + 3],\
[8, 2, 2],\
[9, 3, w^2 + w - 4],\
[13, 13, w + 3],\
[17, 17, -w^2 + w + 3],\
[23, 23, w^2 - 2],\
[23, 23, w^2 - 3],\
[23, 23, -w^2 + 8],\
[29, 29, w - 4],\
[37, 37, w^2 + w - 8],\
[41, 41, w^2 + 2*w - 4],\
[47, 47, 2*w^2 + w - 8],\
[59, 59, -2*w^2 - 3*w + 6],\
[61, 61, -2*w - 1],\
[67, 67, -2*w - 3],\
[79, 79, 2*w^2 - 9],\
[109, 109, w^2 + 2*w - 6],\
[109, 109, 2*w^2 + w - 14],\
[109, 109, -w^2 - w - 1],\
[113, 113, 2*w^2 + w - 12],\
[127, 127, w^2 - 3*w - 2],\
[131, 131, -4*w^2 - 5*w + 9],\
[137, 137, w^2 + 2*w - 7],\
[137, 137, 3*w + 8],\
[137, 137, 2*w^2 + w - 17],\
[139, 139, 3*w^2 + 2*w - 11],\
[149, 149, w - 6],\
[151, 151, w^2 + 2*w - 9],\
[157, 157, -w^2 - 2*w + 13],\
[157, 157, 2*w^2 - 7],\
[163, 163, w^2 + 3*w + 3],\
[163, 163, 2*w^2 - 3*w - 3],\
[163, 163, 2*w^2 - w - 7],\
[167, 167, 3*w^2 + w - 13],\
[169, 13, 2*w^2 + 5*w - 1],\
[173, 173, w^2 + 2*w + 2],\
[179, 179, 4*w^2 + 2*w - 19],\
[181, 181, w^2 + 3*w - 6],\
[191, 191, 3*w^2 + 2*w - 14],\
[193, 193, w^2 - 3*w - 3],\
[197, 197, 3*w^2 - 2*w - 13],\
[211, 211, -w - 6],\
[223, 223, -w^2 + w - 3],\
[223, 223, -3*w - 2],\
[223, 223, 2*w^2 - 2*w - 3],\
[229, 229, 2*w^2 - w - 4],\
[233, 233, 2*w + 7],\
[239, 239, -3*w - 4],\
[239, 239, w^2 - 11],\
[239, 239, 4*w^2 - w - 21],\
[241, 241, 3*w^2 + w - 19],\
[251, 251, -w^2 - 4*w - 1],\
[257, 257, w - 7],\
[263, 263, 3*w^2 + w - 12],\
[269, 269, 3*w^2 - 4*w - 11],\
[271, 271, 4*w^2 + 6*w - 11],\
[277, 277, -4*w^2 - 3*w + 17],\
[283, 283, 5*w^2 + 4*w - 18],\
[289, 17, 3*w^2 + 2*w - 9],\
[307, 307, -2*w^2 - w + 18],\
[307, 307, w^2 + 2*w + 3],\
[307, 307, -w^2 - 3],\
[313, 313, 3*w^2 - 13],\
[317, 317, 3*w^2 - 2*w - 12],\
[331, 331, w^2 + 3*w - 9],\
[331, 331, w^2 + 3*w - 11],\
[331, 331, w^2 + 3*w + 4],\
[343, 7, -7],\
[347, 347, 2*w^2 + 2*w - 13],\
[353, 353, 4*w^2 + 5*w - 13],\
[359, 359, 3*w^2 + 6*w - 1],\
[379, 379, w^2 - 3*w - 6],\
[383, 383, 3*w^2 - 23],\
[383, 383, w^2 - 12],\
[383, 383, 3*w^2 + 3*w - 13],\
[401, 401, -6*w^2 - 7*w + 16],\
[409, 409, -4*w^2 + 3*w + 21],\
[421, 421, w^2 - 3*w - 9],\
[433, 433, w^2 - 3*w - 8],\
[433, 433, w^2 + 4*w - 8],\
[433, 433, 3*w^2 - w - 12],\
[439, 439, -4*w - 11],\
[449, 449, 2*w^2 + 3*w - 19],\
[457, 457, w^2 - 4*w - 3],\
[461, 461, 5*w^2 + 6*w - 16],\
[461, 461, -2*w^2 + 3*w + 16],\
[461, 461, 2*w^2 + 4*w - 9],\
[467, 467, 3*w^2 - 4*w - 12],\
[479, 479, 3*w^2 + w - 9],\
[487, 487, 5*w^2 + 2*w - 22],\
[491, 491, 3*w^2 - 11],\
[499, 499, 5*w^2 + w - 24],\
[503, 503, 3*w^2 + 2*w - 17],\
[503, 503, 3*w^2 + w - 7],\
[503, 503, 2*w^2 - 4*w - 7],\
[509, 509, 3*w - 11],\
[521, 521, 3*w^2 - 3*w - 16],\
[523, 523, -w - 8],\
[541, 541, 6*w^2 + 7*w - 19],\
[547, 547, 3*w^2 + 4*w - 12],\
[557, 557, 2*w^2 - 3*w - 12],\
[563, 563, 4*w^2 + 3*w - 12],\
[563, 563, 4*w^2 + w - 17],\
[563, 563, -w^2 - 5*w - 3],\
[569, 569, w^2 - 4*w - 4],\
[577, 577, w^2 - w - 13],\
[587, 587, 4*w^2 + 3*w - 8],\
[587, 587, 2*w^2 - 3*w - 13],\
[587, 587, 2*w^2 + 3*w - 18],\
[593, 593, 4*w - 13],\
[599, 599, w - 9],\
[613, 613, -w^2 + 3*w - 7],\
[613, 613, w^2 + 5*w + 2],\
[613, 613, 3*w^2 - 3*w - 17],\
[617, 617, w^2 + 5*w + 8],\
[641, 641, 4*w^2 + 3*w - 11],\
[641, 641, w^2 + w - 14],\
[641, 641, 3*w^2 - 2*w - 9],\
[647, 647, 2*w^2 + 3*w - 13],\
[653, 653, 3*w^2 - 8],\
[659, 659, 5*w^2 + 4*w - 21],\
[659, 659, 2*w^2 - 4*w - 19],\
[659, 659, -2*w^2 - 6*w + 9],\
[677, 677, -6*w - 13],\
[691, 691, 6*w - 11],\
[701, 701, 3*w^2 - 3*w - 19],\
[719, 719, 2*w^2 + 3*w - 16],\
[727, 727, 3*w^2 + 4*w - 13],\
[733, 733, 3*w^2 - w - 8],\
[739, 739, 2*w^2 + 4*w - 11],\
[739, 739, 4*w^2 - 3*w - 24],\
[739, 739, 3*w^2 - w - 6],\
[743, 743, 4*w^2 - 2*w - 17],\
[751, 751, -w - 9],\
[761, 761, 5*w^2 + 5*w - 19],\
[769, 769, w^2 - 4*w - 6],\
[769, 769, -5*w - 1],\
[769, 769, 2*w^2 - 19],\
[773, 773, 2*w^2 - 6*w - 3],\
[797, 797, 4*w^2 - 17],\
[809, 809, -7*w^2 - 4*w + 29],\
[811, 811, 6*w^2 + 3*w - 29],\
[823, 823, 4*w^2 + 2*w - 13],\
[823, 823, -w^2 - 4*w - 7],\
[823, 823, 4*w^2 + 4*w - 17],\
[827, 827, -2*w^2 - 3*w - 2],\
[829, 829, -w^2 + w - 6],\
[829, 829, -w^2 + 4*w - 9],\
[829, 829, -2*w^2 + 3*w + 21],\
[839, 839, 6*w^2 - 2*w - 31],\
[841, 29, w^2 + 3*w + 6],\
[853, 853, -7*w^2 - 6*w + 27],\
[859, 859, 6*w - 1],\
[881, 881, w^2 + 5*w - 16],\
[907, 907, 2*w^2 + 7*w + 7],\
[919, 919, w^2 - 4*w - 9],\
[953, 953, -4*w^2 - w + 32],\
[967, 967, 5*w^2 - 5*w - 18],\
[971, 971, 3*w^2 - 4*w - 24],\
[977, 977, 2*w^2 + 4*w + 3],\
[997, 997, w^2 + 5*w - 12],\
[997, 997, 5*w^2 - 3*w - 22],\
[997, 997, 3*w^2 + 8*w - 2]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [0, e, e - 3, 2*e - 3, -e + 1, -e + 2, -e + 3, 6, 2*e - 6, -6, -e + 6, e - 4, -3*e + 6, 2*e, 2*e - 12, -3*e + 5, -4*e + 2, 2*e + 8, -4*e + 14, -3*e - 7, -3*e - 7, -e + 12, -6*e + 8, 2*e + 6, -5*e + 3, -e + 9, -3*e + 6, -10, -3*e + 3, -2*e - 16, 3*e - 10, e + 11, 14, -4*e + 8, -4, 18, e + 14, 4*e - 18, -4*e - 6, -5*e + 2, -6*e + 18, 3*e + 11, -5*e + 18, -6*e + 14, -2*e + 8, -4, -2*e + 8, -3*e + 8, 7*e - 3, 4*e + 6, -2*e + 12, 2*e + 18, 11*e - 13, 2*e + 6, e + 6, -12*e + 18, 5*e, 4*e + 14, e - 4, -4*e - 10, -5*e + 5, -22, 8, 6*e - 4, 14, -7*e + 15, 2*e - 4, 8*e - 10, -8*e + 20, 6*e - 10, 4*e - 6, 9*e - 30, 6*e - 6, -2*e - 28, 8*e - 12, -10*e + 24, -10*e + 6, 11*e - 3, -10, 9*e - 16, 3*e + 8, -e + 20, -12*e + 26, 20, 11*e - 21, 3*e + 5, 13*e - 21, 13*e - 21, -7*e, 18, -10*e, 4*e + 14, -2*e, -6*e + 26, -8*e + 12, 14*e - 30, -6*e, -3*e + 3, 4*e - 6, -10*e + 14, 5*e + 8, -10*e + 2, 12*e - 30, 4*e, -4*e - 12, -2*e + 6, 3*e - 3, 3*e - 25, 6*e - 24, -12*e + 24, -16*e + 24, 7*e - 21, 8*e + 12, 9*e - 25, 5*e - 1, -34, 11*e - 39, -3*e + 30, 11*e - 39, -9*e + 12, 4*e - 12, e + 15, -10*e + 30, 4*e + 6, -10*e + 30, 5*e + 3, 16*e - 34, 5*e - 3, -30, 2, 13*e - 19, 12*e - 22, 6*e + 14, -22*e + 32, -6*e, 6*e - 34, -5*e + 12, -e - 19, -5*e + 20, 3*e - 13, -17*e + 18, 9*e - 36, -5*e, -2*e - 10, -4*e + 20, 12*e - 16, -10*e - 4, -4*e + 12, -3*e - 37, -3*e + 23, -12*e + 2, -30, -9*e - 4, 3*e - 22, -4*e - 16, 3*e - 36, -8*e - 10, 4*e - 4, 12*e - 18, 8, -4*e + 42, -15*e + 42, 4*e + 38, -3*e + 20, e - 4]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}

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