/*
  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, 3, 4, -5, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([27, 3, -w^3 + w^2 + 3*w - 2])

primes_array = [
[3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3],\
[9, 3, -w^4 + 5*w^2 - 3],\
[9, 3, -w^4 + w^3 + 5*w^2 - 3*w - 2],\
[13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1],\
[17, 17, w^4 - w^3 - 5*w^2 + 3*w + 1],\
[19, 19, -w^3 + w^2 + 4*w - 2],\
[23, 23, -w^2 + 3],\
[31, 31, w^3 - 4*w + 2],\
[32, 2, 2],\
[37, 37, w^3 - 3*w - 1],\
[53, 53, -2*w^4 + w^3 + 9*w^2 - 3*w - 2],\
[59, 59, -w^4 + 5*w^2 + w - 4],\
[61, 61, -w^4 + w^3 + 5*w^2 - 4*w],\
[67, 67, -w^4 + 6*w^2 + 2*w - 4],\
[71, 71, 2*w^4 - w^3 - 9*w^2 + 4*w + 5],\
[79, 79, 2*w^4 - w^3 - 10*w^2 + 2*w + 7],\
[83, 83, -w^4 + 2*w^3 + 5*w^2 - 7*w - 2],\
[83, 83, -w^4 + w^3 + 4*w^2 - 3*w + 3],\
[83, 83, w^4 - w^3 - 5*w^2 + 4*w - 1],\
[83, 83, -w^4 + w^3 + 4*w^2 - 4*w - 2],\
[83, 83, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],\
[97, 97, -w^4 - w^3 + 6*w^2 + 4*w - 4],\
[101, 101, -w^4 + 4*w^2 + 2*w - 2],\
[107, 107, -2*w^4 + w^3 + 11*w^2 - 3*w - 7],\
[127, 127, w^3 - w^2 - 4*w],\
[131, 131, w^4 - w^3 - 4*w^2 + 2*w - 1],\
[137, 137, w^4 - w^3 - 6*w^2 + 3*w + 3],\
[139, 139, -2*w^4 + w^3 + 9*w^2 - 3*w + 1],\
[157, 157, 2*w^4 - w^3 - 9*w^2 + 4*w + 2],\
[163, 163, -w^2 - 2*w + 3],\
[167, 167, w^4 - 5*w^2 + 2*w + 1],\
[169, 13, -2*w^4 + 2*w^3 + 9*w^2 - 7*w - 5],\
[169, 13, w^4 - 2*w^3 - 5*w^2 + 8*w + 2],\
[191, 191, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],\
[193, 193, 2*w^4 - 2*w^3 - 8*w^2 + 7*w],\
[199, 199, -w^4 + 2*w^3 + 5*w^2 - 9*w - 3],\
[211, 211, -3*w^4 + w^3 + 13*w^2 - 2*w - 1],\
[227, 227, 2*w^3 - w^2 - 9*w + 1],\
[233, 233, -w^3 + w^2 + 4*w + 1],\
[251, 251, -3*w^4 + 14*w^2 + w - 2],\
[257, 257, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 5],\
[257, 257, -w^4 + w^3 + 5*w^2 - 6*w - 1],\
[257, 257, 2*w^4 - 2*w^3 - 11*w^2 + 9*w + 9],\
[257, 257, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 3],\
[257, 257, 2*w^4 - w^3 - 10*w^2 + 5*w + 5],\
[269, 269, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],\
[271, 271, -w^4 + 4*w^2 + w - 3],\
[277, 277, -2*w^4 + 9*w^2 + w - 4],\
[281, 281, w^4 - w^3 - 3*w^2 + 3*w - 4],\
[283, 283, 2*w^4 - 3*w^3 - 8*w^2 + 13*w],\
[289, 17, -w^4 + w^3 + 7*w^2 - 5*w - 5],\
[289, 17, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 6],\
[293, 293, -2*w^4 + 11*w^2 - 7],\
[317, 317, w^3 - 6*w],\
[337, 337, -w^4 + 2*w^3 + 5*w^2 - 9*w - 1],\
[337, 337, w^4 - 2*w^3 - 5*w^2 + 6*w + 3],\
[337, 337, -w^4 + w^3 + 5*w^2 - 4*w - 6],\
[337, 337, -2*w^3 + 8*w + 1],\
[337, 337, w^2 + w - 5],\
[347, 347, -2*w^4 + w^3 + 9*w^2 - 5*w - 3],\
[349, 349, 3*w^4 - 2*w^3 - 13*w^2 + 8*w + 2],\
[353, 353, w^4 - 3*w^3 - 4*w^2 + 12*w + 2],\
[359, 359, 2*w^3 - w^2 - 7*w + 3],\
[361, 19, w^4 - 2*w^3 - 4*w^2 + 7*w + 3],\
[361, 19, w^4 - w^3 - 3*w^2 + 3*w - 3],\
[367, 367, -2*w^4 + w^3 + 9*w^2 - w - 2],\
[379, 379, -2*w^3 + w^2 + 7*w + 2],\
[379, 379, -2*w^4 + w^3 + 10*w^2 - 4*w],\
[379, 379, 2*w^3 - w^2 - 6*w - 2],\
[379, 379, -2*w^3 + 2*w^2 + 6*w - 3],\
[379, 379, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6],\
[383, 383, 2*w^4 + w^3 - 10*w^2 - 7*w + 4],\
[383, 383, -2*w^3 + w^2 + 10*w],\
[383, 383, w^3 + w^2 - 5*w - 1],\
[383, 383, 3*w^4 + 3*w^3 - 15*w^2 - 15*w + 4],\
[383, 383, -2*w^4 + 3*w^3 + 10*w^2 - 10*w - 2],\
[389, 389, 3*w^4 - w^3 - 14*w^2 + 3*w + 3],\
[397, 397, w^4 - 6*w^2 - 2*w + 5],\
[397, 397, w^4 - 2*w^3 - 2*w^2 + 6*w - 5],\
[397, 397, 2*w^4 - 2*w^3 - 11*w^2 + 10*w + 6],\
[397, 397, w^4 - w^3 - 3*w^2 + 4*w - 6],\
[397, 397, w^3 + w^2 - 3*w - 4],\
[401, 401, w^3 - w^2 - 6*w + 3],\
[401, 401, -w^4 + 4*w^2 + 2*w - 3],\
[401, 401, -w^4 + 6*w^2 - w - 7],\
[421, 421, -w^4 - w^3 + 5*w^2 + 5*w - 4],\
[421, 421, -w^4 + w^3 + 6*w^2 - 3*w - 2],\
[421, 421, -w^4 - w^3 + 6*w^2 + 6*w - 5],\
[421, 421, 3*w^4 - w^3 - 15*w^2 + w + 10],\
[421, 421, 2*w^4 - 9*w^2 + 2*w + 4],\
[431, 431, 2*w^4 - 11*w^2 - 2*w + 11],\
[439, 439, 2*w^4 - 3*w^3 - 9*w^2 + 10*w + 6],\
[443, 443, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 2],\
[449, 449, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],\
[461, 461, 3*w^4 - 4*w^3 - 14*w^2 + 14*w + 5],\
[463, 463, w^3 - 6*w - 1],\
[467, 467, 3*w^4 - 3*w^3 - 13*w^2 + 10*w],\
[487, 487, 3*w^4 - w^3 - 14*w^2 + w + 3],\
[487, 487, 2*w^2 - w - 4],\
[487, 487, -2*w^4 + 2*w^3 + 9*w^2 - 8*w - 5],\
[487, 487, 2*w^4 - 10*w^2 - 3*w + 4],\
[487, 487, -2*w^4 + w^3 + 8*w^2 - 3*w - 1],\
[499, 499, -w^4 + 2*w^3 + 4*w^2 - 8*w - 2],\
[499, 499, -2*w^4 + w^3 + 10*w^2 - w - 9],\
[499, 499, -w^3 - 2*w^2 + 2*w + 5],\
[499, 499, -w^4 + 4*w^2 + 4*w],\
[499, 499, -2*w^4 + w^3 + 12*w^2 - 4*w - 9],\
[509, 509, 3*w^4 - 2*w^3 - 16*w^2 + 6*w + 9],\
[521, 521, -2*w^4 + w^3 + 10*w^2 - 5*w - 4],\
[523, 523, w - 4],\
[529, 23, -w^3 + 2*w^2 + 4*w - 4],\
[529, 23, w^4 + 2*w^3 - 6*w^2 - 11*w + 4],\
[569, 569, -2*w^4 + 11*w^2 - 10],\
[571, 571, -2*w^4 + w^3 + 10*w^2 - 3*w - 2],\
[593, 593, -2*w^4 + 3*w^3 + 10*w^2 - 11*w - 8],\
[613, 613, -2*w^3 + 2*w^2 + 9*w - 5],\
[617, 617, -2*w^4 + 10*w^2 - w - 7],\
[631, 631, -2*w^4 + 2*w^3 + 7*w^2 - 8*w + 3],\
[641, 641, -2*w^4 + 2*w^3 + 8*w^2 - 9*w + 3],\
[643, 643, -3*w^4 + w^3 + 13*w^2 - w + 1],\
[643, 643, w^4 + w^3 - 3*w^2 - 6*w - 2],\
[643, 643, -w^4 - w^3 + 6*w^2 + 3*w - 4],\
[643, 643, 2*w^4 - w^3 - 10*w^2 + 4*w + 1],\
[643, 643, 2*w^4 - 2*w^3 - 8*w^2 + 6*w + 5],\
[647, 647, -2*w^3 + w^2 + 6*w - 4],\
[659, 659, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 10],\
[661, 661, 2*w^4 + 3*w^3 - 11*w^2 - 14*w + 6],\
[673, 673, w^4 - 3*w^3 - 3*w^2 + 10*w - 1],\
[683, 683, w^4 - 2*w^3 - 4*w^2 + 6*w - 4],\
[701, 701, w^3 - w^2 - 2*w + 4],\
[727, 727, -w^4 + 6*w^2 + 2*w - 7],\
[743, 743, w^4 - w^3 - 7*w^2 + 5*w + 9],\
[769, 769, w^4 + 2*w^3 - 5*w^2 - 9*w + 4],\
[787, 787, w^4 + w^3 - 7*w^2 - 3*w + 5],\
[797, 797, -2*w^4 - w^3 + 10*w^2 + 4*w - 4],\
[797, 797, w^4 - 3*w^3 - 5*w^2 + 12*w + 5],\
[797, 797, -3*w^4 + 2*w^3 + 13*w^2 - 8*w],\
[797, 797, -w^4 + w^3 + 3*w^2 - 5*w - 1],\
[797, 797, 2*w^4 - 11*w^2 + 2*w + 7],\
[809, 809, -4*w^4 + 2*w^3 + 18*w^2 - 6*w + 1],\
[809, 809, 3*w^4 - 3*w^3 - 14*w^2 + 11*w + 8],\
[809, 809, w^4 - 2*w^3 - 2*w^2 + 8*w - 4],\
[809, 809, 2*w^4 - 8*w^2 - 2*w + 3],\
[809, 809, 2*w^4 - 3*w^3 - 9*w^2 + 13*w + 4],\
[821, 821, -2*w^4 + 2*w^3 + 8*w^2 - 6*w + 3],\
[823, 823, -2*w^4 + 3*w^3 + 8*w^2 - 12*w - 1],\
[829, 829, 2*w^4 - 3*w^3 - 10*w^2 + 13*w + 4],\
[839, 839, -2*w^4 + 12*w^2 - w - 10],\
[863, 863, 2*w^4 - w^3 - 9*w^2 + 5],\
[877, 877, -w^4 + w^3 + 5*w^2 - 5*w - 7],\
[881, 881, 2*w^4 - w^3 - 9*w^2 + w - 1],\
[887, 887, 3*w^4 - 3*w^3 - 15*w^2 + 11*w + 6],\
[907, 907, -3*w^4 + w^3 + 13*w^2 - w - 5],\
[919, 919, w^4 + w^3 - 3*w^2 - 5*w - 3],\
[929, 929, 2*w^4 - 3*w^3 - 10*w^2 + 9*w + 4],\
[937, 937, -2*w^4 + 10*w^2 + 3*w - 7],\
[941, 941, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],\
[961, 31, 2*w^4 + 2*w^3 - 10*w^2 - 12*w + 5],\
[961, 31, -w^4 + 5*w^2 - w - 7],\
[967, 967, -2*w^4 + w^3 + 9*w^2 - 3*w + 2],\
[991, 991, -3*w^4 + w^3 + 15*w^2 - 4*w - 5],\
[997, 997, -3*w^4 + 15*w^2 - 5],\
[997, 997, 2*w^4 - w^3 - 9*w^2 + 6*w + 2],\
[997, 997, 2*w^4 - w^3 - 11*w^2 + w + 7],\
[997, 997, 2*w^3 - 5*w - 1],\
[997, 997, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 + x^3 - 10*x^2 - 19*x - 8
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1, e, 1, -e^3 + 11*e + 6, e^2 - 3*e - 8, 2*e^3 - e^2 - 18*e - 12, -e^3 - e^2 + 10*e + 12, -3*e^3 + 29*e + 26, 4*e^3 - e^2 - 38*e - 31, -2*e^3 + e^2 + 19*e + 8, 2*e^3 - 21*e - 22, 4*e^3 - e^2 - 37*e - 34, -2*e^3 + 2*e^2 + 15*e + 6, -2*e^3 + 17*e + 22, e^3 + 2*e^2 - 15*e - 22, -e^2, -2*e^3 + e^2 + 21*e + 14, 4*e^3 - 3*e^2 - 34*e - 18, -2*e^3 + 4*e^2 + 13*e - 6, 5*e^3 - 2*e^2 - 47*e - 40, 3*e^3 - 31*e - 28, 2*e^3 - 2*e^2 - 17*e - 10, -e^3 + 14*e + 2, -5*e^3 + 5*e^2 + 44*e + 20, -7*e^3 + e^2 + 66*e + 50, -2*e^3 - e^2 + 20*e + 26, -2*e^3 + 2*e^2 + 19*e - 2, -7*e^3 + e^2 + 63*e + 56, 2*e^3 - 3*e^2 - 12*e + 2, -2*e^3 + 6*e^2 + 13*e - 12, -e^3 + e^2 + 10*e + 4, e^3 - 4*e^2 - 5*e + 6, 2*e^3 - 2*e^2 - 21*e - 2, -2*e^3 - 2*e^2 + 20*e + 20, e^3 + 4*e^2 - 14*e - 32, 4*e^3 - 2*e^2 - 33*e - 20, 9*e^3 - 2*e^2 - 79*e - 68, 7*e^3 - 4*e^2 - 64*e - 44, 5*e^3 - 3*e^2 - 42*e - 26, e^3 - 14*e - 12, -2*e^2 - 2*e + 4, 2*e^3 - e^2 - 22*e, -8*e^3 + 3*e^2 + 73*e + 54, -5*e^3 - 4*e^2 + 53*e + 66, -4*e^3 + 7*e^2 + 24*e - 14, -9*e^3 + 2*e^2 + 83*e + 64, 5*e^3 - 2*e^2 - 46*e - 40, 5*e^3 - 4*e^2 - 39*e - 12, -3*e^3 + 2*e^2 + 23*e + 2, -3*e^3 + 3*e^2 + 28*e + 18, -4*e^3 + 2*e^2 + 34*e + 6, -3*e^3 + 27*e + 10, 2*e^2 - 3*e - 10, 5*e^3 - e^2 - 49*e - 54, -3*e^3 + 4*e^2 + 28*e + 14, -4*e^3 + 33*e + 26, -7*e^3 + 3*e^2 + 60*e + 32, 6*e^3 - 3*e^2 - 54*e - 50, 2*e^3 - 6*e^2 - 10*e + 22, -5*e^3 + 3*e^2 + 43*e + 30, -4*e^3 + e^2 + 34*e + 34, 3*e^3 + e^2 - 26*e - 18, -5*e^3 - 2*e^2 + 46*e + 50, -3*e^3 + e^2 + 35*e + 14, -4*e^3 + 5*e^2 + 37*e + 10, 2*e^3 + 3*e^2 - 23*e - 40, 6*e^3 - 4*e^2 - 55*e - 44, -2*e^2 - e + 18, 8*e^3 - 6*e^2 - 68*e - 42, 5*e^3 - 43*e - 36, -3*e^3 - 4*e^2 + 29*e + 44, -7*e^2 + 12*e + 38, 3*e^3 + 3*e^2 - 40*e - 44, -2*e^3 + 25*e + 24, -6*e^3 + 59*e + 58, 6*e^3 - 6*e^2 - 45*e - 16, 5*e^3 - 2*e^2 - 42*e - 30, -3*e^3 - 5*e^2 + 39*e + 54, 6*e^3 - 4*e^2 - 57*e - 32, 12*e^3 - 119*e - 102, -2*e^3 - 3*e^2 + 24*e + 36, -8*e^3 + e^2 + 81*e + 74, -13*e^3 + 4*e^2 + 120*e + 110, e^3 + 2*e^2 - 16*e - 26, -9*e^3 - 2*e^2 + 85*e + 84, -e^3 + 3*e^2 + 16*e - 18, -3*e^3 + 4*e^2 + 22*e + 2, 4*e^3 - 3*e^2 - 32*e - 16, 7*e^3 - 4*e^2 - 55*e - 46, -11*e^3 + 4*e^2 + 107*e + 82, -6*e^3 - 3*e^2 + 62*e + 52, -4*e^3 + 3*e^2 + 37*e + 16, 3*e^3 - 37*e - 28, 11*e^3 - 4*e^2 - 103*e - 86, -2*e^3 - e^2 + 23*e + 20, -8*e^3 + e^2 + 76*e + 70, 2*e^3 + 5*e^2 - 21*e - 44, 10*e^3 - 5*e^2 - 85*e - 56, -2*e^2 - 2*e + 24, -9*e^3 + 4*e^2 + 83*e + 54, 8*e^3 + 2*e^2 - 80*e - 60, -10*e^3 + 5*e^2 + 97*e + 60, -4*e^3 + 3*e^2 + 35*e + 20, -7*e^3 - e^2 + 75*e + 76, 6*e^3 + 5*e^2 - 69*e - 84, 7*e^3 + 3*e^2 - 70*e - 78, -7*e^3 + e^2 + 74*e + 54, -8*e^3 + e^2 + 70*e + 78, 7*e^3 + e^2 - 71*e - 70, -13*e^3 + 137*e + 124, e - 22, 6*e^3 - 5*e^2 - 48*e - 18, -2*e^3 - e^2 + 25*e + 8, 5*e^3 + 2*e^2 - 51*e - 72, -3*e^3 - e^2 + 34*e + 30, -16*e^3 + 2*e^2 + 149*e + 118, 6*e^3 + 3*e^2 - 65*e - 82, -2*e^3 + 5*e^2 + 12*e - 10, -e^3 + 5*e^2 + 10*e - 18, -4*e^3 + 9*e^2 + 29*e - 20, -10*e^3 + 4*e^2 + 85*e + 72, 12*e^3 - 126*e - 126, -3*e^3 - 4*e^2 + 27*e + 54, -12*e^3 + 2*e^2 + 118*e + 84, -13*e^3 + 9*e^2 + 111*e + 64, 11*e^3 - 3*e^2 - 96*e - 64, 16*e^3 - 165*e - 146, -8*e^3 - e^2 + 78*e + 94, -e^3 + 3*e^2 + 9*e - 12, -e^3 - 2*e^2 + 21*e + 2, 4*e^3 - 4*e^2 - 32*e + 12, 9*e^3 - e^2 - 80*e - 76, 6*e^3 - e^2 - 44*e - 50, -19*e^3 + 4*e^2 + 190*e + 168, 7*e^3 + 4*e^2 - 81*e - 84, -e^3 - 4*e^2 + 16*e + 22, 9*e^3 + 3*e^2 - 108*e - 100, 10*e^3 - 6*e^2 - 98*e - 62, -4*e^2 + 7*e + 24, 14*e^3 - e^2 - 139*e - 112, 13*e^3 - 4*e^2 - 118*e - 90, 18*e^3 - 5*e^2 - 163*e - 114, -19*e^3 + 4*e^2 + 174*e + 146, 4*e^3 + 3*e^2 - 35*e - 38, -4*e^3 + 8*e^2 + 18*e - 32, 14*e^3 - 6*e^2 - 133*e - 96, -7*e^3 + 2*e^2 + 75*e + 48, 13*e^3 - 4*e^2 - 127*e - 112, -2*e^3 + 5*e^2 + 2*e - 16, 2*e^3 + 6*e^2 - 30*e - 56, -4*e^3 - 2*e^2 + 49*e + 38, 5*e^3 - 3*e^2 - 36*e - 24, -2*e^3 - 6*e^2 + 31*e + 24, -19*e^3 + 6*e^2 + 180*e + 136, 9*e^3 + 3*e^2 - 95*e - 102, 6*e^3 - 2*e^2 - 54*e - 40, 5*e^3 - 52*e - 62, 18*e^3 - 2*e^2 - 181*e - 170, -8*e^3 + 8*e^2 + 68*e + 26, 5*e^3 - 5*e^2 - 51*e - 32, 4*e^3 - 50*e - 40, -10*e^3 + e^2 + 110*e + 74, 8*e^3 - 5*e^2 - 67*e - 62, 10*e^3 - 9*e^2 - 80*e - 44, -8*e^3 - 3*e^2 + 86*e + 86, -8*e^3 + 3*e^2 + 74*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([3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3])] = -1
AL_eigenvalues[ZF.ideal([9, 3, -w^4 + w^3 + 5*w^2 - 3*w - 2])] = -1

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