/*
  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, 1, -7, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([16, 8, -1/2*w^3 + w^2 + 5/2*w - 5])

primes_array = [
[2, 2, w],\
[2, 2, -1/2*w^3 + w^2 + 3/2*w],\
[11, 11, 1/2*w^3 - w^2 - 5/2*w + 4],\
[11, 11, -w^3 + w^2 + 6*w + 1],\
[13, 13, -w^3 + w^2 + 6*w - 3],\
[19, 19, -1/2*w^3 + w^2 + 5/2*w],\
[23, 23, -1/2*w^3 + 2*w^2 - 1/2*w - 2],\
[31, 31, -w^3 + w^2 + 6*w - 1],\
[31, 31, 1/2*w^3 - 7/2*w],\
[43, 43, 1/2*w^3 - w^2 - 9/2*w + 2],\
[67, 67, w^2 - w - 5],\
[67, 67, -1/2*w^3 + 11/2*w + 2],\
[73, 73, w^2 + w + 1],\
[79, 79, 1/2*w^3 + w^2 - 5/2*w - 2],\
[81, 3, -3],\
[83, 83, 2*w^3 - 2*w^2 - 12*w + 5],\
[83, 83, -1/2*w^3 - w^2 + 1/2*w + 2],\
[89, 89, -3/2*w^3 + 2*w^2 + 17/2*w - 2],\
[89, 89, 3/2*w^3 - 2*w^2 - 21/2*w + 2],\
[97, 97, 1/2*w^3 - w^2 - 1/2*w - 2],\
[97, 97, -w^3 + w^2 + 8*w + 1],\
[103, 103, 2*w^3 - 2*w^2 - 14*w + 1],\
[103, 103, w^2 - w - 3],\
[107, 107, -3/2*w^3 + 2*w^2 + 21/2*w - 4],\
[109, 109, 5/2*w^3 - 3*w^2 - 33/2*w + 4],\
[109, 109, -1/2*w^3 + w^2 + 9/2*w - 4],\
[113, 113, -w^3 + w^2 + 8*w - 3],\
[113, 113, 1/2*w^3 - w^2 - 1/2*w + 2],\
[121, 11, 5/2*w^3 - 2*w^2 - 31/2*w + 2],\
[127, 127, -3/2*w^3 + w^2 + 23/2*w],\
[127, 127, -1/2*w^3 + 11/2*w],\
[131, 131, w^2 - 3*w - 3],\
[137, 137, 1/2*w^3 - w^2 - 5/2*w + 6],\
[139, 139, -3/2*w^3 + 2*w^2 + 13/2*w + 2],\
[151, 151, -1/2*w^3 + 2*w^2 + 3/2*w - 10],\
[157, 157, 7/2*w^3 - 5*w^2 - 47/2*w + 10],\
[157, 157, 7/2*w^3 - 4*w^2 - 45/2*w + 4],\
[167, 167, 1/2*w^3 + w^2 - 5/2*w - 4],\
[167, 167, 2*w + 5],\
[173, 173, -w^3 + w^2 + 4*w - 7],\
[181, 181, w^3 - 2*w^2 - 9*w + 1],\
[193, 193, -1/2*w^3 + 2*w^2 + 3/2*w - 8],\
[197, 197, -1/2*w^3 - w^2 + 13/2*w + 2],\
[197, 197, -w^3 + w^2 + 4*w - 1],\
[197, 197, w^3 - 2*w^2 - 7*w + 7],\
[197, 197, -2*w^3 + 3*w^2 + 13*w - 11],\
[199, 199, -2*w^3 + 3*w^2 + 11*w - 5],\
[239, 239, -w^3 + 2*w^2 + 5*w - 1],\
[239, 239, 7/2*w^3 - 4*w^2 - 45/2*w + 6],\
[241, 241, 1/2*w^3 - 2*w^2 - 7/2*w + 8],\
[257, 257, 3/2*w^3 - 2*w^2 - 17/2*w - 4],\
[257, 257, 1/2*w^3 - 7/2*w - 6],\
[257, 257, 7/2*w^3 - 5*w^2 - 43/2*w + 10],\
[257, 257, 3*w^3 - 4*w^2 - 19*w + 7],\
[269, 269, 1/2*w^3 + w^2 + 3/2*w + 2],\
[269, 269, -2*w^3 + 3*w^2 + 11*w - 11],\
[277, 277, 5/2*w^3 - 4*w^2 - 35/2*w + 8],\
[281, 281, -3/2*w^3 + w^2 + 23/2*w - 2],\
[283, 283, -w^3 - w^2 + 8*w + 11],\
[283, 283, -3*w^2 + 3*w + 19],\
[311, 311, 3/2*w^3 - 2*w^2 - 17/2*w - 2],\
[331, 331, -4*w^3 + 7*w^2 + 25*w - 21],\
[337, 337, -5/2*w^3 + 2*w^2 + 35/2*w + 4],\
[337, 337, -2*w^3 + 3*w^2 + 11*w - 9],\
[347, 347, 2*w^3 - 4*w^2 - 12*w + 13],\
[347, 347, 5/2*w^3 - 5*w^2 - 29/2*w + 20],\
[347, 347, -1/2*w^3 - 1/2*w - 6],\
[347, 347, -1/2*w^3 + 3*w^2 + 1/2*w - 16],\
[349, 349, 1/2*w^3 + w^2 - 9/2*w - 4],\
[353, 353, -1/2*w^3 + 2*w^2 + 7/2*w],\
[359, 359, 3/2*w^3 - w^2 - 19/2*w + 2],\
[373, 373, -2*w^3 + 2*w^2 + 12*w + 1],\
[373, 373, -1/2*w^3 + w^2 + 9/2*w - 6],\
[379, 379, w^2 - w - 9],\
[379, 379, -2*w^3 + 3*w^2 + 13*w - 5],\
[383, 383, w^2 + w - 5],\
[389, 389, 9/2*w^3 - 6*w^2 - 59/2*w + 14],\
[397, 397, 2*w^2 - 2*w - 15],\
[401, 401, -1/2*w^3 + 2*w^2 + 3/2*w - 4],\
[419, 419, -3/2*w^3 + w^2 + 15/2*w - 2],\
[419, 419, w^3 + w^2 - 8*w - 15],\
[431, 431, -5/2*w^3 + 4*w^2 + 27/2*w - 16],\
[431, 431, 2*w^3 - 4*w^2 - 10*w + 11],\
[443, 443, 1/2*w^3 - 3*w^2 - 5/2*w + 14],\
[449, 449, 1/2*w^3 + w^2 - 9/2*w - 6],\
[457, 457, -w^3 + 2*w^2 + 7*w - 9],\
[461, 461, 4*w^3 - 5*w^2 - 27*w + 7],\
[487, 487, 2*w^2 - 9],\
[499, 499, 3/2*w^3 - w^2 - 15/2*w - 2],\
[499, 499, 3/2*w^3 - 4*w^2 - 17/2*w + 16],\
[503, 503, -5/2*w^3 + 3*w^2 + 37/2*w],\
[509, 509, -3*w^3 + 3*w^2 + 18*w + 7],\
[521, 521, -1/2*w^3 - 2*w^2 + 11/2*w + 18],\
[521, 521, -5/2*w^3 + 4*w^2 + 31/2*w - 8],\
[523, 523, -1/2*w^3 + 2*w^2 + 11/2*w],\
[523, 523, -1/2*w^3 - w^2 + 9/2*w],\
[547, 547, -7/2*w^3 + 6*w^2 + 45/2*w - 18],\
[547, 547, -w^3 - w^2 + 12*w + 5],\
[547, 547, 3/2*w^3 - w^2 - 19/2*w + 4],\
[547, 547, w^3 - w^2 - 6*w - 5],\
[557, 557, 1/2*w^3 + w^2 - 9/2*w - 12],\
[557, 557, 3/2*w^3 - 2*w^2 - 21/2*w],\
[557, 557, 2*w^3 - 2*w^2 - 12*w + 1],\
[557, 557, w^3 - 3*w^2 - 6*w + 11],\
[569, 569, w^3 - 2*w^2 - 7*w + 1],\
[577, 577, w^3 + w^2 - 6*w - 5],\
[599, 599, -3/2*w^3 + 3*w^2 + 15/2*w - 10],\
[607, 607, -5/2*w^3 + 3*w^2 + 25/2*w + 4],\
[613, 613, w^3 - 9*w + 1],\
[619, 619, 3*w^2 - w - 17],\
[619, 619, w^3 - 3*w^2 - 6*w + 13],\
[625, 5, -5],\
[631, 631, 11/2*w^3 - 7*w^2 - 71/2*w + 16],\
[631, 631, -3/2*w^3 + 2*w^2 + 13/2*w],\
[643, 643, w^2 - 3*w - 5],\
[653, 653, 2*w^2 - 2*w - 11],\
[659, 659, 5*w^3 - 3*w^2 - 36*w - 9],\
[659, 659, -w^3 + 7*w - 1],\
[673, 673, 5/2*w^3 - 2*w^2 - 31/2*w - 4],\
[677, 677, 13/2*w^3 - 6*w^2 - 87/2*w + 2],\
[691, 691, -3/2*w^3 + 2*w^2 + 13/2*w - 2],\
[701, 701, -w^3 + 3*w^2 + 4*w - 13],\
[701, 701, w^3 - 2*w^2 - 9*w + 3],\
[709, 709, -w^2 - 5*w - 1],\
[739, 739, 3/2*w^3 - w^2 - 15/2*w],\
[743, 743, -2*w^3 + w^2 + 15*w + 1],\
[751, 751, -2*w^3 + 4*w^2 + 12*w - 11],\
[757, 757, -6*w^3 + 9*w^2 + 39*w - 23],\
[757, 757, -w^3 - w^2 + 8*w + 3],\
[757, 757, 13/2*w^3 - 8*w^2 - 83/2*w + 16],\
[757, 757, 3*w^3 - 4*w^2 - 17*w + 3],\
[761, 761, 7/2*w^3 - 5*w^2 - 47/2*w + 12],\
[769, 769, -3*w^2 + 7*w + 3],\
[773, 773, -3/2*w^3 - w^2 + 19/2*w + 4],\
[787, 787, -4*w^3 + 5*w^2 + 23*w - 17],\
[787, 787, 1/2*w^3 - w^2 - 9/2*w - 4],\
[809, 809, -2*w^3 + 2*w^2 + 16*w + 3],\
[809, 809, -4*w^3 + 5*w^2 + 25*w - 9],\
[811, 811, 3*w^3 - 3*w^2 - 20*w + 5],\
[811, 811, w^3 - 3*w^2 - 6*w + 17],\
[823, 823, 5/2*w^3 - w^2 - 33/2*w - 10],\
[829, 829, 5/2*w^3 - 3*w^2 - 33/2*w + 2],\
[829, 829, -3*w^3 + 2*w^2 + 23*w - 1],\
[839, 839, -11/2*w^3 + 7*w^2 + 67/2*w - 18],\
[853, 853, -w^3 + 2*w^2 + 3*w - 5],\
[853, 853, -5*w^3 + 7*w^2 + 32*w - 15],\
[857, 857, 3/2*w^3 - 5*w^2 - 15/2*w + 26],\
[859, 859, -w^3 + 3*w^2 + 4*w - 17],\
[863, 863, 4*w^3 - 4*w^2 - 26*w + 3],\
[877, 877, w^3 - 5*w - 5],\
[887, 887, 3/2*w^3 - w^2 - 11/2*w + 8],\
[907, 907, -9/2*w^3 + 7*w^2 + 53/2*w - 24],\
[919, 919, 7/2*w^3 - 6*w^2 - 41/2*w + 22],\
[929, 929, -2*w^3 + 3*w^2 + 11*w - 1],\
[937, 937, 2*w^3 - 4*w^2 - 12*w + 7],\
[947, 947, 9/2*w^3 - 5*w^2 - 61/2*w + 10],\
[953, 953, -1/2*w^3 + 2*w^2 + 3/2*w + 2],\
[961, 31, -1/2*w^3 + 2*w^2 - 1/2*w - 6],\
[977, 977, 5/2*w^3 - 2*w^2 - 31/2*w],\
[983, 983, -5/2*w^3 + 6*w^2 + 27/2*w - 24],\
[983, 983, -3/2*w^3 + 2*w^2 + 21/2*w - 10],\
[983, 983, -w^3 + 4*w^2 + 7*w - 15],\
[983, 983, -1/2*w^3 + 3*w^2 - 3/2*w - 4],\
[997, 997, 1/2*w^3 - w^2 - 5/2*w - 4],\
[997, 997, 1/2*w^3 + w^2 - 17/2*w - 2]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 - 5*x - 10
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, 1, e, -2, e - 2, e - 6, e - 2, -2*e + 6, 6, -e + 2, e + 2, 2*e - 8, -2*e + 6, 6, -e + 6, e - 8, e + 6, 2*e - 14, 8, -2*e + 10, -4*e + 8, 0, -6, -e - 10, 10, -2*e - 4, 2*e - 2, 10, -4, -e + 10, -8, -e + 18, -e, -e + 6, -e - 10, -2*e + 2, 2, -2*e + 22, 8, -6, -5*e + 8, -2*e + 12, e, -14, -5*e + 8, e, 4*e - 12, -4*e + 12, -4*e + 12, 4*e - 12, 14, e - 12, e + 6, -6*e + 14, e - 4, -2*e + 16, 4*e - 2, e + 8, -4*e + 4, e + 10, -e + 16, -2*e - 4, 6*e - 22, -2*e + 4, 2*e + 12, -6*e + 8, -2*e - 8, 2*e + 12, 5*e - 24, -2*e - 6, -e + 6, -2*e + 2, 0, e - 30, -2*e + 10, 2*e + 12, -34, 2*e - 30, -6*e + 10, 2*e + 16, -4*e + 14, -6*e + 26, -2*e - 14, 4*e - 4, 2*e + 6, e + 12, -3*e - 4, -e + 22, 4*e - 8, -8*e + 28, 0, -2*e + 32, -6*e + 34, 3*e + 16, -9*e + 24, -8*e + 12, 3*e + 26, -7*e + 22, -2*e + 8, -4*e + 12, -3*e + 10, 2*e - 14, 2*e - 4, 7*e - 4, 5*e - 16, 5*e - 8, -8*e + 8, -12, 4*e + 18, -28, 4*e - 4, 3*e + 8, -32, -7*e + 4, -2*e + 2, 4*e + 16, -5*e + 4, 4*e - 20, -4*e + 34, 6*e + 4, 9*e - 10, -8*e + 14, -26, 2*e - 46, 4*e - 24, -3*e + 16, -4*e + 8, 2*e + 14, 3*e - 12, 2*e - 12, 44, 5*e - 20, 4*e - 10, -7*e + 16, -4*e - 28, 4*e - 2, 8*e - 30, -4*e - 30, -5*e - 10, 7*e - 26, -4*e - 26, -3*e + 44, 5*e - 44, 3*e + 28, 6*e - 40, -2*e - 2, -7*e + 10, 4*e - 16, -6*e + 26, 4*e - 6, 4*e + 20, 11*e - 36, -11*e + 14, 50, -3*e + 24, 6*e - 30, -2*e + 42, -6*e - 6, -4*e + 2, e - 4, -9*e + 10, -e + 10, -8, 8*e - 38, 4*e - 8]
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])] = -1
AL_eigenvalues[ZF.ideal([2, 2, -1/2*w^3 + w^2 + 3/2*w])] = -1

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