/*
  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([-3, 0, 9, 0, -6, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([37, 37, w^4 - 5*w^2 - w + 5])

primes_array = [
[3, 3, w],\
[8, 2, w^3 - 3*w - 1],\
[37, 37, w^4 - 5*w^2 - w + 5],\
[37, 37, w^5 - 5*w^3 - w^2 + 4*w + 1],\
[37, 37, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w + 1],\
[37, 37, w^5 + w^4 - 5*w^3 - 4*w^2 + 5*w + 1],\
[37, 37, w^5 - 5*w^3 + w^2 + 4*w - 1],\
[37, 37, -w^4 + 5*w^2 - w - 5],\
[71, 71, w^5 - 5*w^3 - w^2 + 5*w + 4],\
[71, 71, w^5 - 4*w^3 - w^2 + w + 2],\
[71, 71, w^5 + w^4 - 5*w^3 - 5*w^2 + 4*w + 2],\
[71, 71, w^5 - w^4 - 5*w^3 + 5*w^2 + 4*w - 2],\
[71, 71, -w^5 + 4*w^3 - w^2 - w + 2],\
[71, 71, -w^5 + 5*w^3 - w^2 - 5*w + 4],\
[73, 73, w^5 - 5*w^3 - w^2 + 3*w + 2],\
[73, 73, 2*w^5 - w^4 - 10*w^3 + 4*w^2 + 9*w - 2],\
[73, 73, -w^5 + 4*w^3 - w^2 - 2*w + 2],\
[73, 73, w^5 - 4*w^3 - w^2 + 2*w + 2],\
[73, 73, -2*w^5 - w^4 + 10*w^3 + 4*w^2 - 9*w - 2],\
[73, 73, w^5 - 5*w^3 + w^2 + 3*w - 2],\
[107, 107, -2*w^5 - w^4 + 10*w^3 + 6*w^2 - 10*w - 5],\
[107, 107, -w^5 + 6*w^3 - w^2 - 6*w + 2],\
[107, 107, w^5 - w^4 - 4*w^3 + 6*w^2 + w - 7],\
[107, 107, -w^5 - w^4 + 4*w^3 + 6*w^2 - w - 7],\
[107, 107, w^5 - 6*w^3 - w^2 + 6*w + 2],\
[107, 107, w^5 - 6*w^3 + 8*w + 2],\
[109, 109, w^5 + w^4 - 4*w^3 - 5*w^2 + 2*w + 4],\
[109, 109, w^3 + w^2 - 4*w - 2],\
[109, 109, -w^5 - w^4 + 6*w^3 + 4*w^2 - 7*w - 2],\
[109, 109, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 2],\
[109, 109, -w^3 + w^2 + 4*w - 2],\
[109, 109, w^5 - w^4 - 4*w^3 + 5*w^2 + 2*w - 4],\
[179, 179, w^5 - w^4 - 5*w^3 + 4*w^2 + 4*w - 4],\
[179, 179, w^5 - w^4 - 5*w^3 + 5*w^2 + 5*w - 2],\
[179, 179, w^2 - w - 4],\
[179, 179, -w^2 - w + 4],\
[179, 179, -w^5 - w^4 + 5*w^3 + 5*w^2 - 5*w - 2],\
[179, 179, -w^5 - w^4 + 5*w^3 + 4*w^2 - 4*w - 4],\
[181, 181, 2*w^5 - 10*w^3 - w^2 + 9*w + 2],\
[181, 181, w^5 - w^4 - 5*w^3 + 4*w^2 + 6*w - 2],\
[181, 181, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 4],\
[181, 181, -w^5 - w^4 + 5*w^3 + 5*w^2 - 3*w - 4],\
[181, 181, w^5 + w^4 - 5*w^3 - 4*w^2 + 6*w + 2],\
[181, 181, -2*w^5 + 10*w^3 - w^2 - 9*w + 2],\
[251, 251, -w^5 - w^4 + 4*w^3 + 3*w^2 + 1],\
[251, 251, -w^5 - w^4 + 6*w^3 + 3*w^2 - 9*w + 1],\
[251, 251, -w^5 - 2*w^4 + 6*w^3 + 10*w^2 - 8*w - 7],\
[251, 251, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 8*w - 7],\
[251, 251, 2*w^5 - 2*w^4 - 11*w^3 + 9*w^2 + 12*w - 7],\
[251, 251, 2*w^5 + w^4 - 9*w^3 - 6*w^2 + 6*w + 5],\
[289, 17, -2*w^4 + 9*w^2 - 7],\
[289, 17, w^4 - 3*w^2 - 1],\
[289, 17, w^4 - 6*w^2 + 5],\
[359, 359, 2*w^5 - 9*w^3 + 5*w - 2],\
[359, 359, 2*w^4 - 10*w^2 + w + 8],\
[359, 359, -w^5 + w^4 + 5*w^3 - 3*w^2 - 3*w - 2],\
[359, 359, 2*w^5 - 11*w^3 + 13*w + 2],\
[359, 359, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 9*w + 8],\
[359, 359, -2*w^5 + 9*w^3 - 5*w - 2],\
[361, 19, 3*w^4 - 12*w^2 + 4],\
[361, 19, -3*w^4 + 14*w^2 - 7],\
[361, 19, w^4 - 4*w^2 - 1],\
[397, 397, -w^5 + w^4 + 4*w^3 - 5*w^2 - 2*w + 5],\
[397, 397, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 1],\
[397, 397, w^3 + w^2 - 4*w - 1],\
[397, 397, -w^3 + w^2 + 4*w - 1],\
[397, 397, -w^5 - w^4 + 6*w^3 + 4*w^2 - 7*w - 1],\
[397, 397, -w^5 - w^4 + 4*w^3 + 5*w^2 - 2*w - 5],\
[431, 431, -2*w^5 + 9*w^3 + w^2 - 7*w - 1],\
[431, 431, -2*w^5 - w^4 + 9*w^3 + 4*w^2 - 6*w - 4],\
[431, 431, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 2],\
[431, 431, -w^5 - w^4 + 6*w^3 + 5*w^2 - 9*w - 2],\
[431, 431, 2*w^5 - w^4 - 9*w^3 + 4*w^2 + 6*w - 4],\
[431, 431, 2*w^5 - 9*w^3 + w^2 + 7*w - 1],\
[433, 433, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 2],\
[433, 433, -w^5 + w^4 + 3*w^3 - 5*w^2 + w + 4],\
[433, 433, -2*w^3 - w^2 + 7*w + 2],\
[433, 433, -2*w^3 + w^2 + 7*w - 2],\
[433, 433, w^5 + w^4 - 3*w^3 - 5*w^2 - w + 4],\
[433, 433, -w^5 - w^4 + 7*w^3 + 4*w^2 - 10*w - 2],\
[467, 467, w^5 + w^4 - 5*w^3 - 3*w^2 + 6*w - 2],\
[467, 467, -w^5 + 2*w^4 + 5*w^3 - 9*w^2 - 3*w + 8],\
[467, 467, -w^5 + w^4 + 4*w^3 - 6*w^2 - w + 5],\
[467, 467, 2*w^5 - w^4 - 10*w^3 + 6*w^2 + 9*w - 4],\
[467, 467, -w^5 - 2*w^4 + 5*w^3 + 9*w^2 - 3*w - 8],\
[467, 467, 2*w^5 + w^4 - 9*w^3 - 4*w^2 + 7*w + 2],\
[503, 503, w^5 + w^4 - 5*w^3 - 4*w^2 + 4*w + 5],\
[503, 503, w^4 - 2*w^3 - 5*w^2 + 6*w + 5],\
[503, 503, -w^4 + 2*w^3 + 4*w^2 - 6*w - 1],\
[503, 503, w^4 + 2*w^3 - 4*w^2 - 6*w + 1],\
[503, 503, -w^4 - 2*w^3 + 5*w^2 + 6*w - 5],\
[503, 503, w^5 + 2*w^4 - 6*w^3 - 8*w^2 + 7*w + 5],\
[541, 541, -2*w^5 + 2*w^4 + 11*w^3 - 9*w^2 - 12*w + 5],\
[541, 541, w^5 - w^4 - 4*w^3 + 6*w^2 + 3*w - 7],\
[541, 541, -w^5 - w^4 + 4*w^3 + 3*w^2 - 1],\
[541, 541, w^5 - w^4 - 4*w^3 + 3*w^2 - 1],\
[541, 541, -w^5 - w^4 + 4*w^3 + 6*w^2 - 3*w - 7],\
[541, 541, 2*w^5 + 2*w^4 - 11*w^3 - 9*w^2 + 12*w + 5],\
[577, 577, w^3 + w^2 - 5*w - 1],\
[577, 577, 2*w^5 + w^4 - 11*w^3 - 4*w^2 + 11*w + 1],\
[577, 577, -2*w^5 + w^4 + 9*w^3 - 5*w^2 - 7*w + 5],\
[577, 577, 2*w^5 + w^4 - 9*w^3 - 5*w^2 + 7*w + 5],\
[577, 577, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 11*w - 1],\
[577, 577, -w^3 + w^2 + 5*w - 1],\
[613, 613, 2*w^5 - 10*w^3 - w^2 + 8*w + 1],\
[613, 613, -2*w^5 + w^4 + 10*w^3 - 4*w^2 - 10*w + 1],\
[613, 613, w^4 - 5*w^2 + 2*w + 5],\
[613, 613, w^4 - 5*w^2 - 2*w + 5],\
[613, 613, 2*w^5 + w^4 - 10*w^3 - 4*w^2 + 10*w + 1],\
[613, 613, -2*w^5 + 10*w^3 - w^2 - 8*w + 1],\
[647, 647, -w^5 + 2*w^4 + 5*w^3 - 10*w^2 - 6*w + 8],\
[647, 647, 2*w^3 + w^2 - 5*w - 2],\
[647, 647, -w^5 - w^4 + 7*w^3 + 5*w^2 - 11*w - 4],\
[647, 647, w^5 - w^4 - 7*w^3 + 5*w^2 + 11*w - 4],\
[647, 647, 2*w^3 - w^2 - 5*w + 2],\
[647, 647, -w^5 - 2*w^4 + 5*w^3 + 10*w^2 - 6*w - 8],\
[683, 683, -2*w^5 + 10*w^3 - 7*w - 1],\
[683, 683, -3*w^5 + 15*w^3 - 14*w + 1],\
[683, 683, -2*w^5 - w^4 + 11*w^3 + 6*w^2 - 12*w - 5],\
[683, 683, 2*w^5 - w^4 - 11*w^3 + 6*w^2 + 12*w - 5],\
[683, 683, 3*w^5 - 15*w^3 + 14*w + 1],\
[683, 683, 2*w^5 - 10*w^3 + 7*w - 1],\
[719, 719, 3*w^5 + w^4 - 15*w^3 - 5*w^2 + 14*w + 4],\
[719, 719, -w^5 + 5*w^3 + w^2 - 7*w - 2],\
[719, 719, -w^5 + 7*w^3 - 11*w + 1],\
[719, 719, w^5 - 7*w^3 + 11*w + 1],\
[719, 719, w^5 - 5*w^3 + w^2 + 7*w - 2],\
[719, 719, -3*w^5 + w^4 + 15*w^3 - 5*w^2 - 14*w + 4],\
[757, 757, w^5 - w^4 - 3*w^3 + 5*w^2 - 2*w - 5],\
[757, 757, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w - 4],\
[757, 757, 2*w^5 - 9*w^3 - w^2 + 6*w + 4],\
[757, 757, 2*w^5 - 9*w^3 + w^2 + 6*w - 4],\
[757, 757, -w^5 - w^4 + 6*w^3 + 4*w^2 - 9*w - 4],\
[757, 757, w^5 + w^4 - 3*w^3 - 5*w^2 - 2*w + 5],\
[827, 827, -w^5 - w^4 + 4*w^3 + 4*w^2 - w - 4],\
[827, 827, 2*w^4 - 8*w^2 + w + 1],\
[827, 827, w^5 + 3*w^4 - 5*w^3 - 12*w^2 + 4*w + 5],\
[827, 827, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 4*w + 5],\
[827, 827, -2*w^4 + 8*w^2 + w - 1],\
[827, 827, -w^5 + w^4 + 4*w^3 - 4*w^2 - w + 4],\
[829, 829, -w^5 - 3*w^4 + 5*w^3 + 14*w^2 - 4*w - 11],\
[829, 829, -w^5 - 2*w^4 + 5*w^3 + 11*w^2 - 5*w - 10],\
[829, 829, -w^5 + 3*w^4 + 5*w^3 - 13*w^2 - 4*w + 8],\
[829, 829, -w^5 - w^4 + 5*w^3 + 7*w^2 - 5*w - 7],\
[829, 829, 2*w^4 - 7*w^2 + w + 1],\
[829, 829, -w^5 + 3*w^4 + 5*w^3 - 14*w^2 - 4*w + 11],\
[863, 863, w^5 - w^4 - 6*w^3 + 6*w^2 + 7*w - 4],\
[863, 863, -w^5 - w^4 + 6*w^3 + 5*w^2 - 7*w - 1],\
[863, 863, -w^4 + w^3 + 4*w^2 - 4*w - 5],\
[863, 863, -w^4 - w^3 + 4*w^2 + 4*w - 5],\
[863, 863, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 1],\
[863, 863, -w^5 - w^4 + 6*w^3 + 6*w^2 - 7*w - 4],\
[937, 937, -w^5 + 4*w^4 + 5*w^3 - 16*w^2 - 4*w + 5],\
[937, 937, w^5 + w^4 - 4*w^3 - 3*w^2 - w - 2],\
[937, 937, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + w - 7],\
[937, 937, -2*w^5 + 10*w^3 - w^2 - 6*w + 2],\
[937, 937, w^5 - w^4 - 4*w^3 + 3*w^2 - w + 2],\
[937, 937, -w^5 - 4*w^4 + 5*w^3 + 16*w^2 - 4*w - 5],\
[971, 971, w^4 - 2*w^3 - 4*w^2 + 6*w + 2],\
[971, 971, 2*w^3 + w^2 - 6*w - 2],\
[971, 971, w^4 - 2*w^3 - 5*w^2 + 6*w + 4],\
[971, 971, w^4 + 2*w^3 - 5*w^2 - 6*w + 4],\
[971, 971, 2*w^3 - w^2 - 6*w + 2],\
[971, 971, w^4 + 2*w^3 - 4*w^2 - 6*w + 2]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [e, -e, 1, -2*e - 4, -2*e^2 - 6*e + 2, -5*e^2 - 4*e + 15, e^2 + 4*e - 3, 3*e^2 - 13, -e^2 + 2*e + 11, 4*e^2 + 2*e - 16, 5*e^2 - 21, 14, e^2 + 4*e - 5, 3*e^2 + 8*e - 3, 5*e^2 + 6*e - 18, 2*e^2 + 4*e - 9, -2*e^2 + 2*e + 5, -3*e^2 + 8, 3*e^2 + 10*e - 4, -5*e^2 - 6*e + 12, -e^2 - 7*e - 3, -6*e^2 - 11*e + 16, -4*e^2 - 7*e + 2, 8*e^2 + 9*e - 18, -e^2 - e + 7, -2*e^2 + 3*e + 10, 3*e^2 + 8*e + 1, -3*e^2 - 6*e - 3, 6*e + 10, -9*e^2 - 12*e + 17, -7*e^2 - 12*e + 7, 8*e^2 + 14*e - 14, -7*e^2 - 2*e + 31, e^2 - 6*e - 13, -5*e^2 - 6*e + 1, -e^2 - 10*e - 7, 7*e^2 + 12*e - 17, -5*e^2 - 6*e + 13, 4*e^2 - 4*e - 20, -6*e^2 - 12*e + 2, 8*e^2 + 12*e - 16, e^2 + 6*e - 11, 5*e^2 - 2*e - 23, 6*e^2 + 6*e - 28, -5*e^2 - 11*e - 7, e^2 + 7*e - 7, 6*e^2 + 9*e - 4, -4*e^2 - e - 4, -3*e^2 - e + 25, -6*e^2 - 17*e + 18, 4*e^2 + 2*e - 31, 2*e - 15, -7*e^2 - 10*e + 16, -14*e^2 - 6*e + 54, e^2 - 10*e - 7, -10*e^2 - 22*e + 6, -13*e^2 - 28*e + 21, 2*e^2 + 8*e + 4, 8*e^2 + 22*e - 8, -6*e - 11, 3*e^2 + 8*e - 8, 5*e^2 + 10*e - 20, -4*e^2 + 6*e + 8, -9*e^2 - 18*e + 9, 16*e^2 + 28*e - 26, -e^2 - 2*e - 23, 3*e^2 + 14*e + 17, -2*e^2 + 6*e + 14, -15*e^2 - 8*e + 63, -2*e^2 + 4*e, -9*e^2 - 10*e + 15, -8*e + 6, 17*e^2 + 30*e - 35, 12*e^2 + 10*e - 52, -8*e - 11, 17*e^2 + 6*e - 66, -8*e^2 - 12*e + 21, 18*e^2 + 14*e - 55, e^2 - 2*e + 6, -10*e^2 - 18*e + 25, -19*e^2 - 17*e + 57, 14*e^2 + 7*e - 58, 16*e^2 + 27*e - 32, -e^2 - e - 25, -12*e^2 - 19*e + 22, -e^2 + 9*e + 5, 4*e^2 + 8*e - 4, -7*e^2 - 6*e + 19, e^2 + 12*e - 11, -6*e^2 - 14*e + 16, -2*e - 6, 4*e^2 + 10*e - 10, -5*e^2 + 8*e + 15, 8*e^2 + 14*e - 8, 10*e^2 + 22*e - 10, -11*e^2 - 12*e + 25, 4*e^2 + 16*e - 10, -3*e^2 + 2*e + 7, 14*e^2 + 20*e - 17, -12*e^2 - 26*e + 27, 10*e^2 + 4*e - 61, 2*e^2 + 12*e - 13, -5*e^2 - 4*e + 26, -8*e^2 - 20*e + 13, e^2 - 10*e - 21, -9*e^2 - 2*e + 29, 21*e^2 + 18*e - 57, -2*e^2 - 10*e - 14, 9*e^2 + 6*e - 45, 14*e^2 + 10*e - 50, -11*e^2 - 6*e + 29, 3*e^2 - 10*e - 25, -5*e^2 + 2*e + 23, -6*e - 12, 14*e^2 + 16*e - 12, -16*e^2 - 8*e + 62, -25*e^2 - 35*e + 47, 2*e^2 + 15*e - 16, -14*e^2 - 21*e + 28, 25*e^2 + 33*e - 51, -9*e^2 - 7*e + 31, -2*e^2 + 3*e + 16, 9*e^2 + 2*e - 39, 13*e^2 + 18*e + 1, 9*e^2 + 22*e + 5, -21*e^2 - 10*e + 79, 7*e^2 - 55, 4*e^2 + 8*e - 34, -5*e^2 - 2*e - 3, -11*e^2 - 12*e + 33, -2*e^2 - 2*e - 2, -2*e^2 + 10*e + 6, -7*e^2 - 22*e + 3, -11*e^2 - 18*e + 23, -13*e^2 - 34*e + 13, 17*e^2 + 20*e - 19, 7*e^2 + 2*e - 51, 23*e^2 + 34*e - 31, 3*e^2 - 6*e - 31, 5*e^2 + 18*e - 1, 15*e^2 + 26*e - 35, 13*e^2 + 24*e - 35, 22*e^2 + 28*e - 52, -17*e^2 - 2*e + 65, 7*e^2 + 3, 19*e^2 + 36*e - 37, 2*e^2 + 2*e + 14, 6*e^2 + 14*e + 18, -14*e^2 - 24*e + 16, 4*e^2 + 2*e - 24, -8*e^2 - 18*e + 32, -19*e^2 - 16*e + 59, -14*e^2 - 38*e + 22, 12*e^2 + 20*e - 22, -18*e^2 - 32*e + 32, 2*e^2 + 8*e + 16, -10*e^2 - 14*e + 18, 20*e^2 + 24*e - 22, 3*e^2 + 20*e - 17, 3*e^2 - 14*e - 47, -5*e^2 - 18*e + 29, -15*e^2 - 18*e + 67, 27*e^2 + 34*e - 55, e^2 - 2*e - 25]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([37, 37, w^4 - 5*w^2 - w + 5])] = -1

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