/*
  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([13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1])

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^2 + 3*x + 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -2*e - 4, 3, -1, e, e, -2*e - 9, 4*e + 5, e + 2, 2*e + 3, -9*e - 16, -3*e + 6, -2*e - 7, -7, 4*e + 5, -4*e - 6, -4*e - 15, -e + 2, -4*e - 14, -2*e - 9, 14*e + 21, 6*e + 12, 13, 1, 3*e + 15, 16*e + 23, -e - 5, 6*e + 17, -2*e + 15, 8*e + 10, 4*e - 5, -e + 12, -10*e - 14, e - 11, -e - 9, 9*e + 6, 15, -16*e - 28, -17*e - 25, -5*e + 10, -18*e - 24, 7*e - 2, 3*e + 1, 7*e - 7, -6*e - 17, -12*e - 8, 10*e + 35, -6*e - 13, -6*e - 20, 12*e + 22, 25*e + 36, -15*e - 16, -8*e - 33, e - 4, 10*e + 17, 17*e + 9, -12*e - 35, 27*e + 43, 5*e + 29, -7*e - 22, 27*e + 36, 11*e + 19, -7*e + 2, -e + 5, 13*e + 11, -9*e + 13, 13*e + 39, -18*e - 13, -12*e - 4, -18*e - 34, 19*e + 24, -6*e - 10, 10*e - 11, -16*e - 23, 4*e - 15, 20*e + 15, 20*e + 35, -3*e - 9, -3*e - 24, 19*e + 37, 22*e + 31, -9*e - 22, 5*e - 13, -6*e - 15, 2*e - 17, -19*e - 39, -5*e - 7, -14*e - 28, -6*e - 28, -8*e - 12, -21*e - 23, -9*e - 32, -12*e - 25, -26*e - 47, -12*e - 27, 16*e + 24, 11*e + 3, -28*e - 49, 11*e + 15, -29*e - 51, 16*e + 17, 18*e + 17, e - 15, 9*e + 3, 5*e - 11, -25*e - 30, 16*e + 29, 11*e + 15, 7*e + 13, -28*e - 44, -25*e - 34, -6*e - 31, -23*e - 29, -16*e - 17, -10*e - 14, 39, 18*e + 39, 10*e + 14, 6*e + 1, 7*e + 45, -4*e + 3, -17*e - 24, 12*e + 19, -24*e - 52, -21*e - 20, 5*e - 5, -25*e - 57, 11, 10*e - 9, -3*e + 23, 2*e + 3, 9*e - 22, -10*e - 7, -19*e - 17, -33*e - 61, -20*e - 14, -26*e - 37, -20*e - 16, 16*e + 5, -9*e - 59, -11*e - 6, -30*e - 51, -15*e - 30, -5*e - 11, -14*e - 28, -12*e + 23, -25*e - 9, 31*e + 39, 19*e + 61, -13*e - 56, -42*e - 69, 11*e + 10, 12*e + 9, 7*e + 52, 42*e + 55, 9*e + 48, -18*e - 52, -4*e + 21, 4*e + 19, -5*e - 35, 24*e + 60, 4*e + 33, -12*e + 15, e + 29, 6*e - 42, 11*e - 14]
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^4 + w^3 + 4*w^2 - 3*w - 1])] = 1

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