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

NN = ZF.ideal([13,13,w^2 - w - 2])

primes_array = [
[5, 5, -w^3 + w^2 + 5*w],\
[7, 7, -w^3 + 2*w^2 + 3*w - 3],\
[7, 7, w^3 - 2*w^2 - 3*w],\
[13, 13, -w^2 + w + 3],\
[13, 13, -w^2 + w + 2],\
[16, 2, 2],\
[23, 23, -w^2 + 3*w + 1],\
[23, 23, -2*w^3 + 3*w^2 + 9*w - 2],\
[25, 5, w^3 - 2*w^2 - 2*w + 2],\
[49, 7, w^3 - w^2 - 6*w - 1],\
[53, 53, 2*w^3 - 2*w^2 - 8*w - 3],\
[53, 53, 2*w^3 - 2*w^2 - 8*w - 1],\
[67, 67, 2*w^3 - 4*w^2 - 7*w + 2],\
[67, 67, -2*w^3 + 4*w^2 + 6*w - 1],\
[81, 3, -3],\
[83, 83, 2*w^3 - 3*w^2 - 6*w - 1],\
[83, 83, 3*w^3 - 4*w^2 - 12*w + 1],\
[103, 103, 3*w^3 - 4*w^2 - 12*w - 1],\
[103, 103, 2*w^3 - 3*w^2 - 6*w + 1],\
[107, 107, w^3 - w^2 - 3*w - 3],\
[107, 107, 2*w^3 - 2*w^2 - 9*w],\
[109, 109, -3*w^3 + 5*w^2 + 10*w - 5],\
[109, 109, -2*w^3 + 3*w^2 + 9*w + 1],\
[109, 109, -2*w^3 + 2*w^2 + 7*w],\
[109, 109, w^2 - 3*w - 4],\
[121, 11, 3*w^3 - 4*w^2 - 12*w],\
[121, 11, 2*w^2 - 3*w - 6],\
[139, 139, w^3 - 7*w - 1],\
[139, 139, -w^3 + 2*w^2 + 5*w - 4],\
[139, 139, 3*w^3 - 3*w^2 - 14*w - 2],\
[139, 139, -3*w^3 + 5*w^2 + 13*w - 2],\
[149, 149, -w^2 + 4*w + 1],\
[149, 149, -3*w^3 + 4*w^2 + 14*w - 1],\
[149, 149, 2*w^2 - 3*w - 5],\
[149, 149, -w^3 + 3*w^2 + 3*w - 4],\
[167, 167, -w^3 - w^2 + 7*w + 4],\
[167, 167, -4*w^3 + 6*w^2 + 15*w - 4],\
[169, 13, w^3 - w^2 - 6*w + 4],\
[173, 173, w^2 - 3*w - 5],\
[173, 173, 2*w^3 - 2*w^2 - 11*w],\
[179, 179, 2*w^3 - 2*w^2 - 9*w + 1],\
[179, 179, w^3 - w^2 - 3*w - 4],\
[179, 179, w^2 - 5],\
[179, 179, w^3 - 6*w - 1],\
[197, 197, w^3 + w^2 - 7*w - 6],\
[197, 197, 2*w^3 - 2*w^2 - 9*w + 2],\
[223, 223, -w^2 + 7],\
[223, 223, 2*w^2 - w - 4],\
[227, 227, 2*w^3 - 2*w^2 - 11*w - 2],\
[227, 227, w^3 - w^2 - 7*w - 1],\
[233, 233, w^3 - w^2 - 7*w],\
[233, 233, -2*w^3 + 2*w^2 + 11*w + 1],\
[257, 257, -2*w^3 + 4*w^2 + 8*w - 1],\
[257, 257, 2*w^2 - 4*w - 7],\
[277, 277, -4*w^3 + 6*w^2 + 15*w - 3],\
[277, 277, -3*w^3 + 5*w^2 + 9*w],\
[281, 281, -2*w^3 + 5*w^2 + 3*w - 5],\
[281, 281, -2*w^3 + 5*w^2 + 5*w - 7],\
[281, 281, 2*w^3 - 5*w^2 - 5*w + 4],\
[281, 281, -4*w^3 + 7*w^2 + 15*w - 4],\
[283, 283, 3*w^3 - 4*w^2 - 13*w - 3],\
[283, 283, -w^3 + 2*w^2 + w - 4],\
[313, 313, -2*w^3 + 4*w^2 + 5*w - 5],\
[313, 313, 3*w^3 - 5*w^2 - 11*w],\
[347, 347, -w^3 + 2*w^2 + w - 5],\
[347, 347, 3*w^3 - 4*w^2 - 13*w - 4],\
[353, 353, 3*w^3 - 5*w^2 - 12*w + 1],\
[353, 353, -w^3 + 3*w^2 - 5],\
[361, 19, 3*w^3 - 3*w^2 - 13*w - 9],\
[361, 19, 2*w^3 - 2*w^2 - 7*w + 4],\
[373, 373, -3*w^3 + 4*w^2 + 11*w + 1],\
[373, 373, -3*w^3 + 4*w^2 + 11*w],\
[383, 383, 3*w^3 - 5*w^2 - 13*w + 1],\
[383, 383, 2*w^2 - 5*w - 6],\
[397, 397, 4*w^3 - 5*w^2 - 19*w - 1],\
[397, 397, 2*w^3 - 2*w^2 - 9*w - 7],\
[431, 431, 2*w^3 - 2*w^2 - 7*w - 5],\
[431, 431, 3*w^3 - 6*w^2 - 11*w + 3],\
[431, 431, 3*w^3 - 3*w^2 - 13*w],\
[431, 431, w^3 - 4*w^2 + w + 8],\
[457, 457, 4*w^3 - 7*w^2 - 13*w],\
[457, 457, -3*w^2 + 4*w + 5],\
[463, 463, -3*w^3 + 5*w^2 + 11*w + 1],\
[463, 463, 2*w^3 - 4*w^2 - 5*w + 6],\
[487, 487, -w^3 + 4*w^2 - 7],\
[487, 487, -2*w^3 + 5*w^2 + 6*w - 5],\
[499, 499, -6*w^3 + 8*w^2 + 25*w - 1],\
[499, 499, -w^3 + 4*w^2 - 5],\
[499, 499, -2*w^3 + 5*w^2 + 6*w - 7],\
[499, 499, 3*w^3 - 5*w^2 - 7*w],\
[509, 509, -4*w^3 + 8*w^2 + 15*w - 8],\
[509, 509, 3*w^3 - 4*w^2 - 10*w - 2],\
[509, 509, -3*w^3 + 3*w^2 + 11*w + 6],\
[509, 509, 4*w^3 - 5*w^2 - 16*w],\
[521, 521, -4*w^3 + 5*w^2 + 15*w - 2],\
[521, 521, w^3 - w^2 - 8*w + 1],\
[521, 521, -4*w^3 + 7*w^2 + 12*w - 6],\
[521, 521, 3*w^3 - 3*w^2 - 16*w - 1],\
[523, 523, 3*w^3 - w^2 - 17*w - 8],\
[523, 523, -5*w^3 + 5*w^2 + 23*w + 4],\
[529, 23, 3*w^3 - 6*w^2 - 10*w + 2],\
[547, 547, -2*w^3 + 5*w^2 + 6*w - 6],\
[547, 547, -w^3 + 4*w^2 - 6],\
[557, 557, -3*w^3 + 6*w^2 + 9*w - 10],\
[557, 557, 3*w^3 - 6*w^2 - 9*w - 1],\
[571, 571, 4*w^3 - 4*w^2 - 17*w - 6],\
[571, 571, -2*w^3 + 6*w^2 + 3*w - 10],\
[571, 571, 4*w^3 - 5*w^2 - 18*w + 3],\
[571, 571, -3*w^3 + 7*w^2 + 9*w - 5],\
[587, 587, 3*w^3 - 3*w^2 - 14*w],\
[587, 587, w^3 - w^2 - 2*w - 4],\
[593, 593, -w^3 + 3*w^2 + 2*w - 10],\
[593, 593, -5*w^3 + 7*w^2 + 19*w - 4],\
[613, 613, 3*w^3 - 5*w^2 - 12*w - 4],\
[613, 613, -w^3 + 3*w^2 - 10],\
[631, 631, 4*w^3 - 8*w^2 - 13*w + 4],\
[631, 631, 3*w^3 - 4*w^2 - 15*w + 2],\
[631, 631, w^3 - 9*w - 1],\
[631, 631, 3*w^3 - 7*w^2 - 7*w + 9],\
[643, 643, 3*w^3 - 3*w^2 - 16*w - 2],\
[643, 643, w^3 - w^2 - 8*w],\
[647, 647, -w^3 + 4*w^2 + 2*w - 4],\
[647, 647, -3*w^2 + 4*w + 10],\
[673, 673, w^3 + 2*w^2 - 9*w - 8],\
[673, 673, -w^3 + 4*w^2 + 3*w - 7],\
[683, 683, 3*w^3 - 3*w^2 - 11*w - 2],\
[683, 683, 4*w^3 - 4*w^2 - 17*w - 5],\
[709, 709, -5*w^3 + 7*w^2 + 21*w + 1],\
[709, 709, -3*w^3 + 2*w^2 + 16*w + 4],\
[709, 709, 4*w^3 - 4*w^2 - 19*w - 2],\
[709, 709, -2*w^3 + 4*w^2 + 3*w - 4],\
[787, 787, 2*w^2 - w - 9],\
[787, 787, w^3 + w^2 - 7*w - 2],\
[811, 811, 4*w^3 - 7*w^2 - 11*w],\
[811, 811, 3*w^3 - 4*w^2 - 9*w - 1],\
[811, 811, -6*w^3 + 9*w^2 + 23*w - 5],\
[811, 811, 5*w^3 - 6*w^2 - 21*w - 2],\
[821, 821, -3*w^3 + 6*w^2 + 11*w - 2],\
[821, 821, 2*w^3 - 13*w - 5],\
[821, 821, -w^3 + 4*w^2 - w - 9],\
[821, 821, -w^3 + 3*w^2 + 5*w - 6],\
[841, 29, 2*w^3 - 2*w^2 - 12*w - 1],\
[857, 857, w^3 - 4*w^2 + 3*w + 6],\
[857, 857, -5*w^3 + 8*w^2 + 21*w - 3],\
[863, 863, -w^3 + 4*w^2 + w - 7],\
[863, 863, -w^3 + 4*w^2 + w - 6],\
[877, 877, -w^3 - w^2 + 8*w + 2],\
[877, 877, -w^3 + 3*w^2 + 4*w - 8],\
[883, 883, 4*w^3 - 7*w^2 - 14*w + 1],\
[883, 883, 3*w^3 - 6*w^2 - 8*w + 7],\
[929, 929, 3*w^3 - 5*w^2 - 12*w - 3],\
[929, 929, -w^3 + 3*w^2 - 9],\
[929, 929, 3*w - 5],\
[929, 929, 3*w^3 - 3*w^2 - 15*w + 2],\
[937, 937, w^3 - 2*w^2 - 6*w + 6],\
[937, 937, 2*w^3 - w^2 - 12*w],\
[941, 941, -w^3 + 4*w^2 + 2*w - 7],\
[941, 941, -4*w^3 + 6*w^2 + 13*w - 3],\
[941, 941, -5*w^3 + 7*w^2 + 19*w + 2],\
[941, 941, 3*w^2 - 4*w - 7],\
[953, 953, 3*w^3 - 5*w^2 - 12*w - 2],\
[953, 953, -w^3 + 3*w^2 - 8],\
[961, 31, 4*w^3 - 6*w^2 - 13*w - 2],\
[961, 31, 5*w^3 - 7*w^2 - 19*w + 3],\
[977, 977, -4*w^3 + 7*w^2 + 19*w - 6],\
[977, 977, -3*w^3 + 7*w^2 + 10*w - 6]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 - 3*x^2 - 5*x + 13
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -e^2 + 2*e + 4, e^2 - 7, e^2 - 2*e - 3, -1, -e^2 + e + 5, -e^2 + e + 5, -e^2 - 2*e + 9, -e^2 + 2*e + 1, -e, -e^2 + 13, 3*e^2 - 4*e - 9, -e^2 + e + 7, e^2 + 2*e - 8, -2*e + 3, 7, e^2 - 3*e - 11, -e^2 + 4*e + 4, e^2 - 4*e + 1, 2*e^2 - 6*e - 5, -4*e^2 + 6*e + 20, e^2 - 17, 2*e^2 - 2*e - 6, 2*e^2 - 2*e - 6, e^2 - 3*e - 13, e^2 - 6*e + 3, e^2 - 5, e^2 - 7*e - 1, -e^2 + 4*e - 2, -3*e^2 + 17, 2*e^2 - 2, -5*e^2 - 2*e + 41, -2*e^2 + 2*e, -e^2 + 6*e + 3, 2*e^2 - 2*e - 22, 14, -5*e^2 + 6*e + 25, -5*e^2 + 4*e + 32, -e^2 - 4*e + 17, 2*e^2 + 3*e - 28, -2*e^2 + 4*e + 4, -2*e^2 - 2*e + 12, 3*e^2 - 2*e - 7, 3*e^2 - 2*e - 7, -5, e^2 - 6*e - 19, 7*e^2 - 10*e - 31, e^2 - 4*e + 2, 4*e^2 - 2*e - 30, 5*e^2 - 4*e - 19, e^2 - 5, -3*e^2 - 2*e + 25, e^2 - 8*e - 10, -4*e^2 + 2*e + 26, e^2 - e + 3, e^2 + 2*e - 1, 4*e^2 - 4*e - 6, -4*e^2 + 4*e + 38, 7*e^2 - 9*e - 35, -2*e^2 + 4*e - 6, -4*e^2 + 8*e + 9, 6*e^2 + e - 50, -2*e^2 + 3*e + 4, 7*e^2 - 10*e - 25, -2*e^2 + 10*e + 10, -2*e^2 + 10*e + 10, 3*e^2 - 6*e - 12, 2*e + 13, -4*e^2 + 4*e + 34, -2*e^2 + 4*e - 10, -2*e^2 + 3*e + 22, 2*e^2 + 2*e - 4, -2*e^2 - 2*e - 8, 3*e^2 - 11*e - 15, 7*e^2 - 57, 6*e^2 - 6*e - 27, e^2 + 4*e + 8, 4*e^2 - 4*e - 17, 6*e^2 - 6*e - 28, -2*e^2 - e + 20, 7*e^2 - 8*e - 33, -6*e^2 + 30, -6*e^2 + 4*e + 40, 7*e^2 - 2*e - 56, 8*e^2 + 2*e - 64, 3*e^2 - 4*e - 37, 2*e - 30, -6*e^2 - 3*e + 48, 2*e^2 + 10*e - 24, 2*e^2 + 4*e - 16, 6*e^2 - 2*e - 58, -e^2 - 4*e + 38, -9*e^2 + 6*e + 49, -6*e^2 + 10*e + 8, 8*e^2 - 6*e - 52, -2*e^2 + 8*e + 28, 4*e^2 + 4*e - 38, -8*e^2 - 2*e + 52, -2*e - 10, 10*e^2 - 8*e - 40, e^2 + 2*e + 13, -4*e^2 + 6*e + 40, -9*e^2 - 6*e + 75, -7*e^2 + 8*e + 27, -13*e^2 + 14*e + 60, e^2 + 6*e - 11, -2*e^2 + 8*e + 22, 3*e^2 + 3*e - 51, 9*e^2 - 8*e - 47, -2*e^2 - 6*e + 10, 3*e^2 - 6*e - 9, 5*e^2 - 8*e - 21, -8*e^2 + 42, -4*e^2 - 6*e + 52, 3*e^2 - 3*e - 15, 4*e^2 + 3*e - 34, -4*e^2 + 2*e + 22, -3*e^2 + 6*e + 25, -e^2 - 8*e + 30, -e^2 + 2*e - 11, -7*e^2 + 12*e + 47, -4*e^2 - 4*e + 32, -2*e^2 + 10*e + 30, -e^2 - 8*e + 9, -7*e^2 + 8*e + 59, -6*e^2 + 10*e + 35, -3*e^2 + 43, 2*e^2 - 6*e - 40, e^2 + e + 3, 8*e^2 - 5*e - 52, e^2 - 8*e + 15, -2*e^2 + e + 14, 2*e^2 - 6*e - 4, -7*e^2 + 10*e + 45, -3*e^2 + 9*e + 19, 9*e^2 - 12*e - 35, 13*e^2 - 16*e - 57, -4*e^2 - 4*e + 17, 11*e, -2*e^2 + 16*e + 7, -5*e^2 + 14*e + 15, 3*e^2 + 10*e - 19, -11*e^2 - 6*e + 86, 3*e^2 + 6*e - 56, -e^2 + 9*e + 13, e^2 - 12*e - 3, 6*e^2 + 5*e - 26, -e^2 - 9*e - 5, -7*e^2 + 10*e + 33, 9*e^2 - 12*e - 47, 7*e^2 + 2*e - 27, 13*e^2 - 14*e - 77, -3*e - 18, 13*e^2 - 14*e - 77, 4*e^2 - 32, 7*e^2 - 8*e - 57, 4*e^2 - 14*e - 42, -7*e^2 + 8*e + 19, 7*e^2 + 7*e - 45, 6*e^2 - 3*e - 40, -14*e^2 + 10*e + 82, 12*e^2 - 4*e - 77, -6*e^2 + 18*e + 34, 8*e^2 - 2*e - 35, -2*e^2 + 6*e - 22, 3*e^2 + 2*e - 66]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([13,13,w^2 - w - 2])] = 1

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