/*
  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([-12, -11, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([8, 8, -w + 4])

primes_array = [
[2, 2, -w - 2],\
[2, 2, w + 1],\
[3, 3, w^2 - w - 9],\
[9, 3, -w^2 + 3*w + 5],\
[11, 11, -w^2 + w + 11],\
[13, 13, 2*w^2 - 4*w - 13],\
[23, 23, -w^2 + w + 7],\
[29, 29, -w^2 - w + 1],\
[41, 41, -2*w^2 + 2*w + 19],\
[41, 41, w^2 - 3*w - 7],\
[41, 41, w^2 - w - 5],\
[47, 47, 3*w^2 - 7*w - 17],\
[53, 53, -2*w - 1],\
[61, 61, -2*w + 7],\
[67, 67, 3*w^2 - 5*w - 23],\
[67, 67, 2*w^2 - 4*w - 11],\
[67, 67, 3*w^2 - 7*w - 13],\
[79, 79, w^2 + w - 5],\
[89, 89, 5*w^2 - 7*w - 47],\
[97, 97, 5*w^2 - 11*w - 29],\
[103, 103, -3*w^2 + 5*w + 25],\
[103, 103, 2*w^2 - 6*w - 5],\
[103, 103, -7*w^2 + 9*w + 67],\
[107, 107, w^2 - 3*w - 1],\
[109, 109, -5*w^2 + 7*w + 49],\
[113, 113, w^2 - w - 13],\
[121, 11, 9*w^2 - 13*w - 79],\
[125, 5, -5],\
[127, 127, -4*w^2 + 4*w + 41],\
[131, 131, 2*w + 7],\
[137, 137, -2*w^2 + 4*w + 7],\
[139, 139, -3*w^2 + 3*w + 29],\
[139, 139, w^2 - w - 1],\
[139, 139, 2*w - 1],\
[151, 151, -9*w^2 + 11*w + 85],\
[157, 157, 8*w^2 - 10*w - 73],\
[163, 163, -2*w^2 + 2*w + 7],\
[163, 163, 4*w^2 - 10*w - 17],\
[163, 163, -6*w - 7],\
[167, 167, 2*w^2 - 4*w - 17],\
[167, 167, -4*w^2 + 6*w + 37],\
[167, 167, -2*w^2 + 2*w + 5],\
[169, 13, -6*w^2 + 14*w + 29],\
[179, 179, -w^2 - 3*w + 1],\
[179, 179, 6*w^2 - 14*w - 31],\
[179, 179, -3*w^2 + 5*w + 29],\
[191, 191, -2*w + 5],\
[197, 197, 2*w^2 - 4*w - 19],\
[211, 211, -w^2 - w + 17],\
[211, 211, -2*w^2 + 2*w + 23],\
[211, 211, -5*w^2 + 7*w + 43],\
[241, 241, 2*w^2 - 2*w - 13],\
[241, 241, 3*w^2 - 7*w - 19],\
[241, 241, w^2 + w - 7],\
[257, 257, -4*w^2 + 8*w + 25],\
[269, 269, -2*w^2 + 29],\
[277, 277, 6*w^2 - 16*w - 29],\
[283, 283, 4*w^2 - 4*w - 37],\
[293, 293, -2*w^2 + 5],\
[307, 307, w^2 - 5*w - 17],\
[313, 313, 2*w^2 + 2*w - 5],\
[317, 317, 12*w^2 - 16*w - 109],\
[317, 317, w^2 + 3*w + 5],\
[317, 317, 3*w^2 - 5*w - 13],\
[331, 331, 6*w^2 - 8*w - 59],\
[337, 337, -5*w^2 + 5*w + 53],\
[343, 7, -7],\
[347, 347, 3*w^2 - 3*w - 23],\
[347, 347, -4*w^2 + 4*w + 43],\
[347, 347, -3*w^2 + w + 7],\
[349, 349, -w^2 + 5*w + 11],\
[353, 353, -5*w^2 + 11*w + 31],\
[359, 359, 6*w + 13],\
[359, 359, -13*w^2 + 17*w + 119],\
[367, 367, 2*w^2 - 8*w + 1],\
[373, 373, -4*w^2 + 10*w + 23],\
[379, 379, -11*w^2 + 15*w + 101],\
[383, 383, -3*w^2 + 9*w + 11],\
[397, 397, -5*w^2 + 11*w + 25],\
[401, 401, w^2 - 5*w - 5],\
[419, 419, -4*w^2 + 8*w + 19],\
[421, 421, 3*w^2 - 9*w - 13],\
[431, 431, 3*w^2 - 5*w - 19],\
[433, 433, 6*w + 11],\
[433, 433, -7*w^2 + 11*w + 59],\
[433, 433, -w^2 + w - 1],\
[439, 439, -6*w^2 + 12*w + 41],\
[467, 467, 12*w^2 - 16*w - 115],\
[479, 479, 4*w^2 - 8*w - 23],\
[487, 487, w^2 - 5*w - 13],\
[491, 491, 2*w^2 - 4*w - 5],\
[503, 503, -3*w^2 + w + 19],\
[523, 523, -6*w^2 + 10*w + 49],\
[529, 23, 3*w^2 - 5*w - 17],\
[541, 541, w^2 + 7*w + 13],\
[541, 541, -4*w^2 + 12*w + 13],\
[541, 541, w^2 + w - 11],\
[547, 547, 7*w^2 - 19*w - 37],\
[563, 563, -5*w^2 + 9*w + 35],\
[569, 569, -3*w^2 + 9*w + 19],\
[577, 577, -10*w^2 + 14*w + 91],\
[593, 593, -4*w - 1],\
[599, 599, -6*w^2 + 6*w + 61],\
[613, 613, 5*w^2 - 7*w - 41],\
[613, 613, -3*w^2 - w + 5],\
[613, 613, 2*w^2 - 2*w - 25],\
[617, 617, -5*w^2 + 5*w + 47],\
[631, 631, w^2 - 7*w - 13],\
[647, 647, w^2 - 5*w + 1],\
[659, 659, 3*w^2 - w - 5],\
[661, 661, 5*w^2 - 5*w - 41],\
[661, 661, -w^2 - 7*w - 5],\
[661, 661, 8*w^2 - 18*w - 47],\
[673, 673, w^2 + 5*w + 1],\
[677, 677, 4*w + 13],\
[683, 683, -2*w^2 - 4*w + 5],\
[691, 691, -8*w^2 + 10*w + 71],\
[691, 691, 2*w^2 - 6*w - 17],\
[691, 691, 2*w^2 - 8*w - 11],\
[701, 701, w^2 - 5*w - 1],\
[709, 709, w^2 - 7*w + 13],\
[719, 719, 4*w^2 - 6*w - 13],\
[727, 727, 2*w^2 - 17],\
[743, 743, -4*w^2 + 4*w + 31],\
[751, 751, 2*w - 11],\
[757, 757, -7*w^2 + 9*w + 61],\
[761, 761, -13*w^2 + 15*w + 127],\
[773, 773, w^2 - w - 17],\
[773, 773, 5*w^2 - 13*w - 19],\
[773, 773, -6*w - 17],\
[787, 787, 4*w^2 - 12*w - 19],\
[797, 797, -18*w^2 + 22*w + 167],\
[797, 797, -3*w^2 + 7*w + 23],\
[797, 797, -11*w^2 + 13*w + 103],\
[811, 811, 4*w^2 - 10*w - 25],\
[811, 811, 8*w + 13],\
[811, 811, -8*w^2 + 10*w + 79],\
[821, 821, 3*w^2 - 3*w - 7],\
[839, 839, -9*w^2 + 19*w + 55],\
[841, 29, 9*w^2 - 13*w - 77],\
[857, 857, 2*w^2 + 2*w - 35],\
[859, 859, -4*w + 13],\
[877, 877, 2*w^2 - 6*w - 23],\
[881, 881, w^2 - 7*w + 11],\
[883, 883, 3*w^2 - w - 37],\
[887, 887, 3*w^2 - 3*w - 35],\
[907, 907, w^2 - 5*w + 7],\
[907, 907, -w^2 + 11*w + 17],\
[907, 907, -8*w^2 + 12*w + 77],\
[911, 911, 8*w + 17],\
[929, 929, 2*w^2 + 2*w - 11],\
[937, 937, 3*w^2 - 11*w - 17],\
[937, 937, -8*w^2 + 12*w + 71],\
[937, 937, 3*w^2 - w - 17],\
[941, 941, 3*w^2 + w - 13],\
[947, 947, -7*w^2 + 19*w + 31],\
[971, 971, 2*w^2 + 8*w + 11],\
[977, 977, 13*w^2 - 17*w - 125],\
[983, 983, 11*w^2 - 15*w - 107],\
[983, 983, -5*w^2 + 13*w + 29],\
[983, 983, 2*w^2 - 8*w - 17],\
[991, 991, -21*w^2 + 25*w + 191],\
[997, 997, 3*w^2 - 3*w - 13]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 5*x^2 + 2
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, e, e, -e^3 + 4*e, e^2, e^3 - 5*e, e^3 - 3*e, 4*e, -2*e^3 + 9*e, 3*e^2 - 4, 0, -2*e^2 + 4, 3*e^3 - 13*e, -e^3 + 9*e, -6*e^3 + 23*e, -6*e^3 + 24*e, e^2 - 2, 3*e^3 - 15*e, -7*e^2 + 16, 5*e^3 - 26*e, 6*e^2 - 14, 12, -4*e^2 + 6, -e^2 + 18, -6, -7*e^2 + 20, 3*e^2, -4*e^2 + 12, 4*e^3 - 10*e, e^2 + 2, -3*e^2 + 10, -9*e^3 + 33*e, -3*e^2 + 16, -e^3, 7*e^3 - 29*e, -7*e^3 + 25*e, -e^3 + 6*e, 5*e^2 - 2, 7*e^3 - 34*e, -8*e^2 + 24, 8*e^2 - 16, 8, -8*e^3 + 27*e, -e^3 + e, -8*e^3 + 42*e, -6*e^2 + 14, 8*e^3 - 26*e, -6*e^2 + 8, -8*e^3 + 27*e, e^2 - 6, 3*e^3 - 7*e, -5*e, 4*e^2 + 6, e^3 - 8*e, -6*e^3 + 17*e, -16, -e^3 + 15*e, 3*e^3 - 16*e, -7*e^3 + 29*e, -2*e^2 + 18, 7*e^3 - 25*e, -11*e^3 + 43*e, -4*e^2 - 2, -6*e, 12*e^2 - 40, 7*e^2 - 16, -2*e^2 + 28, 15*e, -11*e^2 + 18, 10*e^3 - 49*e, 6*e^2 - 18, e^2 + 16, 4*e^2 + 4, -3*e^3 + 5*e, 2*e^3 - 18*e, -4, 8*e^2 - 4, 3*e^3 - 21*e, 6*e^3 - 18*e, e^3 - 18*e, 9*e^3 - 54*e, 6*e^3 - 30*e, -3*e^3 + e, 3*e^2 - 38, 3*e^2, 4*e^2 + 10, -8*e^2 + 28, 4*e^2 - 28, 4*e, -6*e^2 + 20, -15*e^2 + 50, -7*e^3 + 35*e, -3*e^2 + 26, -11*e^3 + 52*e, 12*e^2 - 46, 15*e^3 - 69*e, 10*e^3 - 62*e, -13*e^2 + 32, 10*e^3 - 55*e, -8*e^2 + 34, -5*e^2 + 20, -11*e, -16*e^3 + 70*e, -2*e^3 + 10*e, -8*e^3 + 32*e, 16*e^2 - 26, 7*e^3 - 44*e, -12*e^2 + 40, -9*e^3 + 37*e, 10*e^2 - 26, 4*e^3 - 30*e, 4*e^3 - 36*e, 16, 12*e^3 - 32*e, 8*e^2 - 42, 15*e^3 - 70*e, -11*e^3 + 32*e, 3*e^2 + 6, -12*e^3 + 46*e, -16*e^3 + 58*e, -10*e^2 + 22, 8*e^2 + 4, 8*e^3 - 14*e, 2*e^3 - 6*e, -8*e^2 + 32, 8*e^3 - 44*e, -5*e^3 + 36*e, -6*e^2 + 14, -2*e^2 - 26, 2*e^2 - 4, 5*e^3 - 4*e, 15*e^3 - 53*e, -4*e^2 + 14, -6*e^3 + 20*e, -e^2 + 22, -7*e^2 + 42, -8*e^2 + 8, -14*e^2 + 22, 11*e^3 - 35*e, 7*e^3 - 38*e, -20*e^3 + 73*e, -7*e^3 + 58*e, -4*e^2 + 2, 7*e^3 - 54*e, 6*e^3 - 47*e, 18*e^2 - 52, e^2 + 14, -8*e^2 + 44, -13*e^2 + 50, -20*e^2 + 68, -10*e^3 + 67*e, 14*e^3 - 70*e, 6*e^2 - 8, -16*e^3 + 88*e, -22*e^3 + 96*e, -8*e^3 + 23*e, 9*e^2 - 18, 5*e^2 + 36, -8*e^2 + 62, -4*e^2 + 8, -2*e^2, e^3 + 13*e, 18*e^3 - 78*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([2, 2, -w - 2])] = -1

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