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

NN = ZF.ideal([25, 25, -w^3 - w^2 + 3*w + 2])

primes_array = [
[5, 5, w^3 - 4*w],\
[7, 7, w + 1],\
[7, 7, -w^3 + 4*w - 1],\
[13, 13, -w^2 + 3],\
[16, 2, 2],\
[17, 17, w^3 + w^2 - 4*w - 2],\
[17, 17, -w^3 + 5*w - 2],\
[19, 19, -w^2 - w + 4],\
[19, 19, -w^2 - w + 1],\
[29, 29, 2*w^3 - w^2 - 9*w + 5],\
[41, 41, -w^3 + w^2 + 5*w - 3],\
[43, 43, w^3 - w^2 - 4*w + 2],\
[43, 43, w^3 - 6*w],\
[47, 47, -w^3 - w^2 + 5*w],\
[49, 7, w^2 + 2*w - 2],\
[59, 59, 2*w^3 - 8*w + 3],\
[67, 67, w^2 - w - 4],\
[79, 79, w^3 - w^2 - 4*w + 1],\
[81, 3, -3],\
[97, 97, w^3 + w^2 - 5*w + 1],\
[107, 107, -2*w^3 + 3*w^2 + 10*w - 12],\
[109, 109, 2*w^3 - w^2 - 9*w + 3],\
[125, 5, -3*w^3 - w^2 + 12*w + 1],\
[127, 127, -2*w^3 + w^2 + 10*w - 8],\
[137, 137, -w^3 + w^2 + 3*w - 5],\
[139, 139, w^3 - w^2 - 5*w + 1],\
[149, 149, 2*w^2 - w - 6],\
[149, 149, 2*w^3 + w^2 - 9*w],\
[163, 163, w^3 - 5*w - 3],\
[163, 163, 2*w^3 - 7*w],\
[173, 173, -2*w^3 + 11*w - 3],\
[173, 173, -w^3 + w^2 + 4*w - 8],\
[179, 179, 2*w^3 - w^2 - 8*w + 5],\
[181, 181, 2*w^3 - w^2 - 8*w],\
[181, 181, 2*w^3 - 9*w - 3],\
[191, 191, 2*w^2 - 5],\
[191, 191, -2*w^2 - w + 10],\
[193, 193, w^3 + w^2 - 3*w - 4],\
[193, 193, -w^3 + 2*w^2 + 3*w - 3],\
[197, 197, -2*w^3 + 9*w - 4],\
[197, 197, w^2 - 7],\
[199, 199, 3*w^3 - 13*w + 1],\
[199, 199, w^3 + 2*w^2 - 5*w - 7],\
[227, 227, -3*w^3 + w^2 + 13*w - 4],\
[227, 227, -w^3 + 2*w^2 + 7*w - 6],\
[241, 241, -2*w^3 + 7*w - 1],\
[257, 257, -2*w^3 + w^2 + 7*w - 5],\
[257, 257, 2*w^3 - w^2 - 7*w + 2],\
[263, 263, -w^3 + 7*w - 3],\
[269, 269, -2*w^3 + 2*w^2 + 11*w - 8],\
[271, 271, w^2 + 3*w - 2],\
[271, 271, w^2 + 2*w - 7],\
[277, 277, -2*w^3 + 11*w - 2],\
[281, 281, 2*w^3 - 11*w + 1],\
[283, 283, 2*w^3 - 9*w + 5],\
[283, 283, -2*w^3 + w^2 + 11*w - 7],\
[289, 17, -w^3 + 2*w^2 + 4*w - 4],\
[293, 293, -2*w^3 - w^2 + 7*w + 3],\
[307, 307, -3*w^3 + 2*w^2 + 13*w - 9],\
[311, 311, -w^3 + w^2 + 3*w - 7],\
[313, 313, -w^3 + 2*w^2 + 3*w - 2],\
[313, 313, w^2 - w - 8],\
[317, 317, -2*w^3 - w^2 + 9*w - 1],\
[331, 331, w^3 + 2*w^2 - 5*w - 3],\
[337, 337, w^3 + 2*w^2 - 6*w - 4],\
[337, 337, w^3 - 2*w^2 - 6*w + 5],\
[347, 347, 2*w^3 + 2*w^2 - 9*w - 3],\
[347, 347, -2*w^3 + w^2 + 7*w - 4],\
[349, 349, w^2 - 2*w - 4],\
[353, 353, -w^3 + 2*w^2 + 2*w - 6],\
[359, 359, -w^3 + w^2 + 7*w - 6],\
[359, 359, -w^2 - w - 2],\
[361, 19, 2*w^3 - w^2 - 11*w],\
[367, 367, -2*w^3 + 2*w^2 + 8*w - 5],\
[373, 373, w^3 + w^2 - 2*w - 4],\
[379, 379, 2*w^3 - w^2 - 11*w + 4],\
[389, 389, 3*w^2 + w - 11],\
[397, 397, 3*w^3 - 13*w + 5],\
[401, 401, w - 5],\
[409, 409, 3*w^3 - 12*w + 4],\
[409, 409, 3*w^3 + 2*w^2 - 14*w - 4],\
[421, 421, 2*w^3 - w^2 - 10*w + 1],\
[421, 421, w^3 + 2*w^2 - 5*w - 4],\
[439, 439, -2*w^3 + 10*w - 5],\
[443, 443, -2*w^3 + 12*w - 5],\
[443, 443, -4*w^3 + w^2 + 18*w - 9],\
[443, 443, -2*w^3 + w^2 + 7*w - 9],\
[443, 443, 3*w^3 + 2*w^2 - 14*w - 5],\
[449, 449, w^3 + 2*w^2 - 2*w - 6],\
[449, 449, -2*w^3 + w^2 + 11*w - 5],\
[461, 461, 2*w^3 + 2*w^2 - 9*w - 4],\
[467, 467, -w^3 + w^2 + 7*w - 5],\
[487, 487, 2*w^2 + 4*w - 7],\
[491, 491, -w^3 + w^2 + 7*w - 4],\
[503, 503, 2*w^3 + w^2 - 9*w + 2],\
[521, 521, w^2 - 3*w - 3],\
[523, 523, -w^3 + 2*w^2 + 5*w - 4],\
[523, 523, -w^3 + 8*w - 5],\
[563, 563, w^3 + 3*w^2 - w - 8],\
[563, 563, w^3 - 2*w^2 - 8*w + 4],\
[563, 563, -w^3 + w^2 + 2*w - 5],\
[563, 563, -3*w^3 - w^2 + 10*w - 2],\
[577, 577, 3*w^3 - 16*w],\
[587, 587, w^3 + 2*w^2 - 6],\
[587, 587, -w^3 + 4*w - 6],\
[601, 601, -w - 5],\
[607, 607, w^3 - 2*w^2 - 4*w + 1],\
[613, 613, 4*w^3 - 17*w],\
[617, 617, -3*w^3 + w^2 + 14*w - 2],\
[617, 617, -4*w^3 + 17*w - 4],\
[619, 619, -3*w^3 + 12*w - 2],\
[643, 643, 3*w^3 - w^2 - 12*w + 3],\
[643, 643, -2*w^3 - w^2 + 11*w - 2],\
[647, 647, 2*w^3 + w^2 - 9*w + 3],\
[653, 653, w^3 + w^2 - 7*w - 4],\
[661, 661, -w^3 + 4*w^2 + 6*w - 15],\
[661, 661, 3*w^3 - 2*w^2 - 12*w + 10],\
[677, 677, -3*w^2 - 2*w + 12],\
[691, 691, -2*w^3 + 3*w^2 + 11*w - 10],\
[691, 691, -3*w^3 + 16*w - 3],\
[701, 701, 3*w^3 - 13*w + 6],\
[719, 719, -3*w^2 + w + 10],\
[727, 727, w^3 - 3*w^2 - 4*w + 6],\
[743, 743, -w^3 + 2*w^2 + 5*w - 3],\
[761, 761, -w^3 + 3*w^2 + 3*w - 12],\
[761, 761, 3*w^3 - 2*w^2 - 14*w + 7],\
[769, 769, -3*w^3 + 3*w^2 + 16*w - 13],\
[769, 769, w^3 + 2*w^2 - 7*w - 6],\
[773, 773, w^3 - 8*w + 4],\
[773, 773, 4*w^2 + w - 16],\
[787, 787, w^2 + 4*w - 2],\
[797, 797, -2*w^3 + 3*w^2 + 10*w - 10],\
[797, 797, -w^3 - w^2 + 4*w - 4],\
[797, 797, w^3 + 2*w^2 - 3*w - 9],\
[797, 797, -3*w^3 + 3*w^2 + 11*w - 10],\
[821, 821, w^2 + w - 9],\
[827, 827, w^3 - 8*w + 1],\
[827, 827, -2*w^3 + w^2 + 9*w - 11],\
[827, 827, 3*w^3 - 14*w + 5],\
[827, 827, 2*w^3 + w^2 - 7*w - 5],\
[829, 829, -2*w^3 - 2*w^2 + 9*w - 1],\
[839, 839, -3*w^3 + 15*w - 5],\
[853, 853, -4*w^3 + 15*w - 8],\
[853, 853, -w^3 + 2*w^2 + 6*w - 3],\
[859, 859, -w^3 + w^2 + w - 4],\
[859, 859, w^3 + w^2 - w - 5],\
[881, 881, -w^3 + w^2 + 2*w - 6],\
[881, 881, w^3 + w^2 - 2*w - 6],\
[883, 883, w^3 + 3*w^2 - 3*w - 6],\
[887, 887, w^3 + 3*w^2 - 3*w - 8],\
[919, 919, 2*w^3 - w^2 - 13*w],\
[929, 929, -w^3 - 3*w^2 + 2*w + 10],\
[929, 929, w^3 + w^2 - 5*w - 8],\
[941, 941, w^3 + 2*w^2 - 3*w - 10],\
[953, 953, -w^3 + 5*w - 7],\
[953, 953, -2*w^3 + 3*w^2 + 9*w - 9],\
[953, 953, -2*w^3 - w^2 + 10*w - 3],\
[953, 953, -3*w^3 + 11*w - 1],\
[961, 31, -w^3 + 3*w^2 + 2*w - 10],\
[961, 31, -3*w^2 + 2*w + 7],\
[967, 967, w^3 + 3*w^2 - 3*w - 7],\
[967, 967, 4*w^3 + w^2 - 20*w + 1],\
[983, 983, 2*w^3 - w^2 - 12*w + 8],\
[983, 983, 2*w^3 + w^2 - 6*w - 4],\
[991, 991, 2*w^3 - 12*w + 3]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 19*x^2 + 75
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, e, e, -1/5*e^3 + 9/5*e, -e^2 + 11, 1/5*e^3 - 14/5*e, e^2 - 6, -e^2 + 14, -3/5*e^3 + 32/5*e, -2/5*e^3 + 13/5*e, 1/5*e^3 - 24/5*e, 2/5*e^3 - 23/5*e, 2*e^2 - 14, 2*e^2 - 15, -5, 3/5*e^3 - 32/5*e, -2/5*e^3 + 18/5*e, 0, 3*e^2 - 25, -3*e^2 + 25, 2/5*e^3 - 8/5*e, 4/5*e^3 - 51/5*e, 3/5*e^3 - 22/5*e, 3*e^2 - 29, -3/5*e^3 + 12/5*e, 8/5*e^3 - 77/5*e, 8/5*e^3 - 77/5*e, -3*e^2 + 27, -4*e^2 + 35, 7/5*e^3 - 93/5*e, -2*e^2 + 27, 2*e^2 - 24, -7/5*e^3 + 83/5*e, 4/5*e^3 - 31/5*e, -8, -4/5*e^3 + 71/5*e, 3*e^2 - 15, -2*e^2 + 37, -2/5*e^3 + 28/5*e, -e^2 + 27, -2*e^2 + 21, -e^3 + 14*e, 6/5*e^3 - 89/5*e, 3/5*e^3 - 27/5*e, 18, -2/5*e^3 + 8/5*e, -3/5*e^3 + 37/5*e, e^3 - 8*e, 2*e^2 - 9, -3*e^2 + 27, -3*e^2 + 19, -e^2 + 26, -17, 2*e^2 - 21, 2*e^2 - 19, e^2 - 10, -2/5*e^3 + 13/5*e, -3*e^2 + 21, -9/5*e^3 + 101/5*e, 7/5*e^3 - 88/5*e, -7/5*e^3 + 98/5*e, -3*e^2 + 11, 3*e^2 - 39, 1/5*e^3 - 24/5*e, 1/5*e^3 - 39/5*e, 2/5*e^3 - 58/5*e, 3*e^3 - 31*e, -6/5*e^3 + 44/5*e, -3/5*e^3 + 7/5*e, -1/5*e^3 - 16/5*e, -4*e^2 + 51, e^2 - 9, 1/5*e^3 + 26/5*e, -7/5*e^3 + 88/5*e, 5*e^2 - 31, e^2 - 29, e^3 - 4*e, 4*e^2 - 38, -4*e^2 + 48, -11/5*e^3 + 109/5*e, -7/5*e^3 + 108/5*e, -14/5*e^3 + 161/5*e, 1/5*e^3 + 1/5*e, -2*e^2 + 13, 3*e^2 - 48, -2*e^2 + 42, 3*e, 1/5*e^3 - 4/5*e, -e^2 + 24, 4*e^2 - 21, -6/5*e^3 + 59/5*e, -7*e^2 + 51, -3*e^2 + 25, 5*e^2 - 48, 2*e^2 - 9, 4/5*e^3 - 31/5*e, -e^3 + 7*e, -4*e^2 + 25, 2*e^2 - 24, 7*e^2 - 69, 4/5*e^3 - 36/5*e, -8/5*e^3 + 142/5*e, -8*e^2 + 85, -e^2 - 18, -9*e^2 + 84, 4*e^2 - 29, 9/5*e^3 - 116/5*e, 1/5*e^3 - 54/5*e, 3/5*e^3 - 102/5*e, 3/5*e^3 - 102/5*e, 9/5*e^3 - 96/5*e, 13/5*e^3 - 132/5*e, 4, 8*e^2 - 69, 11/5*e^3 - 94/5*e, 6/5*e^3 - 44/5*e, -2*e^3 + 22*e, -2*e^2 + 51, 2*e^3 - 29*e, -7*e^2 + 85, 4*e^2 - 9, 8/5*e^3 - 152/5*e, -11/5*e^3 + 114/5*e, 9/5*e^3 - 81/5*e, -4/5*e^3 + 21/5*e, 7/5*e^3 - 88/5*e, 5, -e^3 + 14*e, -2*e^2 + 27, -1/5*e^3 - 41/5*e, -9*e^2 + 89, -3*e^3 + 33*e, 3*e^2 - 54, -42, -14*e, e^2 - 12, 4*e^2 - 63, 6*e^2 - 66, 9*e^2 - 78, -6*e^2 + 87, -7*e^2 + 68, -6*e^2 + 39, 17/5*e^3 - 158/5*e, -13/5*e^3 + 162/5*e, -6/5*e^3 + 14/5*e, e^2 + 16, 3/5*e^3 - 37/5*e, -8*e^2 + 84, 12/5*e^3 - 163/5*e, 4/5*e^3 - 96/5*e, e^2 - 14, -6*e^2 + 69, -11*e^2 + 99, 27, -10*e^2 + 99, 11/5*e^3 - 119/5*e, -21, -9/5*e^3 + 136/5*e, -2*e^3 + 22*e, -2/5*e^3 - 42/5*e, -4/5*e^3 + 106/5*e, 5*e^2 - 82, -3*e^2 - 9, -3*e^2 + 21, -e^2 - 4]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([5, 5, w^3 - 4*w])] = 1

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