/*
  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([27,3,-w^3 + 5*w + 4])

primes_array = [
[3, 3, -w^3 + w^2 + 5*w],\
[3, 3, w - 1],\
[9, 3, w^3 - w^2 - 4*w],\
[13, 13, -2*w^3 + w^2 + 11*w + 3],\
[13, 13, w^3 - 2*w^2 - 4*w + 3],\
[16, 2, 2],\
[23, 23, -w^2 + 2],\
[23, 23, -w^3 + 2*w^2 + 4*w - 4],\
[49, 7, -w^3 + 2*w^2 + 5*w - 2],\
[49, 7, 2*w^3 - 2*w^2 - 9*w - 1],\
[53, 53, w^3 - 3*w^2 + 3],\
[53, 53, 2*w^3 - w^2 - 8*w - 3],\
[53, 53, -w^3 + w^2 + 6*w - 1],\
[61, 61, -w - 3],\
[61, 61, -w^3 + w^2 + 5*w - 4],\
[79, 79, w^3 - 2*w^2 - 4*w + 1],\
[79, 79, w^2 - 5],\
[101, 101, w^3 - 6*w],\
[101, 101, w^2 - 2*w - 4],\
[103, 103, 4*w^3 - 3*w^2 - 20*w - 3],\
[103, 103, w^3 - 2*w^2 - 1],\
[113, 113, 5*w^3 - 3*w^2 - 25*w - 6],\
[113, 113, 2*w - 3],\
[113, 113, w^3 - 3*w^2 - w + 4],\
[113, 113, 2*w^3 - 2*w^2 - 10*w - 1],\
[121, 11, -w^2 + 2*w - 2],\
[121, 11, -w^3 + 6*w + 6],\
[127, 127, 2*w^3 - 3*w^2 - 6*w + 3],\
[127, 127, -3*w^3 + 2*w^2 + 14*w + 1],\
[139, 139, 3*w^3 - 4*w^2 - 12*w + 6],\
[139, 139, 2*w^3 - w^2 - 8*w],\
[157, 157, -w^3 + 7*w + 5],\
[157, 157, w^3 - 7*w + 2],\
[173, 173, w^2 - 3*w - 3],\
[173, 173, -2*w^3 + w^2 + 11*w],\
[179, 179, -2*w^3 + w^2 + 8*w + 1],\
[179, 179, -3*w^3 + 4*w^2 + 12*w - 5],\
[181, 181, -2*w^3 + w^2 + 11*w + 6],\
[181, 181, w^2 - 3*w + 3],\
[191, 191, w^2 + w - 4],\
[191, 191, 2*w^3 - 3*w^2 - 9*w + 3],\
[199, 199, -w^3 - w^2 + 7*w + 6],\
[199, 199, -2*w^3 + 3*w^2 + 8*w - 2],\
[199, 199, -3*w^3 + 3*w^2 + 13*w],\
[199, 199, w^3 - 4*w - 4],\
[211, 211, -2*w^3 + 3*w^2 + 7*w - 6],\
[211, 211, w^3 - w^2 - 7*w + 2],\
[211, 211, 2*w^3 - w^2 - 9*w + 1],\
[211, 211, 2*w^3 - 2*w^2 - 11*w + 1],\
[233, 233, w^2 - 8],\
[233, 233, w^3 - 2*w^2 - 4*w - 2],\
[251, 251, 2*w^2 - 3*w - 4],\
[251, 251, 2*w^3 - 3*w^2 - 11*w + 4],\
[257, 257, -4*w^3 + 2*w^2 + 22*w + 5],\
[257, 257, 4*w^3 - 5*w^2 - 17*w + 8],\
[263, 263, 3*w^3 - 2*w^2 - 15*w],\
[263, 263, -w^3 + 2*w^2 + w - 3],\
[269, 269, -w^3 + w^2 + 2*w - 4],\
[269, 269, -w^3 - w^2 + 5*w + 4],\
[269, 269, 2*w^3 - 4*w^2 - 7*w + 7],\
[269, 269, -2*w^3 + 4*w^2 + 5*w],\
[277, 277, -4*w^3 + 3*w^2 + 19*w - 1],\
[277, 277, 2*w^3 - 3*w^2 - 5*w + 4],\
[283, 283, 3*w^3 - 3*w^2 - 13*w + 3],\
[283, 283, 2*w^3 - 2*w^2 - 7*w + 2],\
[289, 17, -w^3 + w^2 + 4*w - 5],\
[289, 17, w^3 - w^2 - 4*w - 4],\
[313, 313, -2*w^3 + 3*w^2 + 7*w - 9],\
[313, 313, -2*w^3 + w^2 + 9*w - 4],\
[337, 337, -2*w^3 + 3*w^2 + 8*w - 1],\
[337, 337, w^3 - 4*w - 5],\
[373, 373, -2*w^3 + w^2 + 10*w - 1],\
[373, 373, -w^3 + 2*w^2 + 2*w - 5],\
[389, 389, w^3 + w^2 - 7*w - 11],\
[389, 389, -2*w^2 + 3*w - 2],\
[419, 419, -w^3 + 3*w^2 + 4*w - 7],\
[419, 419, -w^3 + 3*w^2 + 4*w - 5],\
[433, 433, -2*w^3 + 2*w^2 + 6*w - 5],\
[433, 433, -4*w^3 + 4*w^2 + 18*w - 7],\
[433, 433, 2*w^3 - 3*w^2 - 11*w + 1],\
[433, 433, 7*w^3 - 4*w^2 - 35*w - 8],\
[443, 443, -3*w^3 + w^2 + 15*w + 3],\
[443, 443, -2*w^3 + 4*w^2 + 5*w - 6],\
[467, 467, 6*w^3 - 4*w^2 - 30*w - 3],\
[467, 467, -2*w^3 + 4*w^2 + 2*w - 3],\
[467, 467, 3*w - 4],\
[467, 467, 3*w^3 - 3*w^2 - 15*w - 1],\
[491, 491, -w^3 + 4*w^2 - 3*w + 2],\
[491, 491, -5*w^3 + 2*w^2 + 27*w + 13],\
[503, 503, 3*w^3 - 18*w - 11],\
[503, 503, -3*w^2 + 6*w + 1],\
[523, 523, w^3 - 3*w^2 - 3*w + 6],\
[523, 523, -3*w^3 + 16*w + 9],\
[523, 523, 2*w^2 - w - 5],\
[523, 523, 2*w^3 - 5*w^2 - 4*w + 5],\
[529, 23, 2*w^3 - 2*w^2 - 8*w - 5],\
[547, 547, -w^3 + 2*w^2 + w - 4],\
[547, 547, -2*w^3 + w^2 + 11*w - 2],\
[547, 547, -3*w^3 + 2*w^2 + 15*w - 1],\
[547, 547, w^2 - 3*w - 5],\
[563, 563, -3*w^3 + 4*w^2 + 10*w - 7],\
[563, 563, -3*w^3 + 3*w^2 + 15*w + 4],\
[569, 569, 2*w^3 - w^2 - 11*w - 7],\
[569, 569, w^2 - 3*w + 4],\
[571, 571, -3*w^3 + 2*w^2 + 12*w + 5],\
[571, 571, 4*w^3 - 5*w^2 - 16*w + 1],\
[601, 601, -5*w^3 + 4*w^2 + 26*w + 4],\
[601, 601, -w^2 + 6*w - 4],\
[607, 607, 2*w^3 - 4*w^2 - 6*w + 1],\
[607, 607, 2*w^3 - 2*w^2 - 11*w - 5],\
[607, 607, w^3 - w^2 - 7*w + 8],\
[607, 607, 2*w^3 - 10*w - 9],\
[625, 5, -5],\
[641, 641, -4*w^3 + 5*w^2 + 19*w - 13],\
[641, 641, -4*w^3 + 4*w^2 + 21*w - 1],\
[653, 653, 3*w^3 - 2*w^2 - 14*w + 2],\
[653, 653, 4*w^3 - 7*w^2 - 17*w + 10],\
[653, 653, -2*w^3 + w^2 + 14*w + 6],\
[653, 653, -2*w^3 + 3*w^2 + 6*w - 6],\
[659, 659, 3*w - 5],\
[659, 659, 3*w^3 - 3*w^2 - 15*w - 2],\
[673, 673, 2*w^3 - 4*w^2 - 9*w + 4],\
[673, 673, 3*w^3 - 3*w^2 - 16*w + 3],\
[673, 673, -w^3 + 2*w^2 + 4*w - 9],\
[673, 673, -w^3 + 3*w^2 + 5*w - 9],\
[677, 677, -3*w^3 + 4*w^2 + 15*w - 3],\
[677, 677, -w^3 + 2*w^2 + 7*w - 6],\
[701, 701, w^2 - 3*w - 6],\
[701, 701, 2*w^3 - w^2 - 11*w + 3],\
[719, 719, w^3 - w^2 - 8*w + 7],\
[719, 719, -4*w^3 + 5*w^2 + 21*w - 11],\
[797, 797, -2*w^3 + w^2 + 13*w + 2],\
[797, 797, -2*w^3 + w^2 + 13*w - 1],\
[809, 809, -w^3 + w^2 + 6*w - 7],\
[809, 809, w^3 - w^2 - 6*w - 5],\
[823, 823, -w^3 + 2*w^2 + 6*w - 8],\
[823, 823, 2*w^3 - 3*w^2 - 10*w],\
[823, 823, -7*w^3 + 5*w^2 + 34*w + 6],\
[823, 823, 6*w^3 - 6*w^2 - 27*w + 1],\
[829, 829, -w^3 + 4*w^2 + w - 11],\
[829, 829, 4*w^3 - 3*w^2 - 18*w - 3],\
[841, 29, -3*w^3 + 3*w^2 + 12*w - 1],\
[841, 29, -3*w^3 + 3*w^2 + 12*w - 2],\
[857, 857, -3*w^3 + 2*w^2 + 12*w - 1],\
[857, 857, 2*w^2 - 9],\
[857, 857, -4*w^3 + 5*w^2 + 16*w - 7],\
[857, 857, 2*w^3 - 4*w^2 - 8*w + 3],\
[881, 881, -3*w^3 + 2*w^2 + 12*w + 2],\
[881, 881, 4*w^3 - 5*w^2 - 16*w + 4],\
[883, 883, -4*w^3 + 4*w^2 + 22*w + 1],\
[883, 883, -2*w^3 + 2*w^2 + 14*w - 7],\
[887, 887, 8*w^3 - 6*w^2 - 41*w - 4],\
[887, 887, -w^3 + 3*w^2 - 5*w + 1],\
[907, 907, -3*w^3 + 4*w^2 + 10*w - 4],\
[907, 907, 4*w^3 - 3*w^2 - 18*w],\
[907, 907, w^3 - w^2 - 5*w - 5],\
[907, 907, w - 6],\
[911, 911, -3*w^3 + 6*w^2 + 7*w],\
[911, 911, 5*w^3 - 2*w^2 - 25*w - 13],\
[911, 911, 2*w^3 - w^2 - 8*w - 6],\
[911, 911, -3*w^3 + 4*w^2 + 12*w],\
[919, 919, -2*w^3 + 3*w^2 + 11*w - 10],\
[919, 919, -w^3 - w^2 + 8*w + 10]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 + 2*x^3 - 31*x^2 + 28*x + 16
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1, -1/8*e^3 - 3/8*e^2 + 3*e - 1, 1, -1/16*e^3 - 7/16*e^2 + 3/4*e + 5, e, -1/16*e^3 + 1/16*e^2 + 13/4*e - 4, 1/16*e^3 + 7/16*e^2 - 3/4*e, 1/4*e^3 + 3/4*e^2 - 7*e + 4, -5/16*e^3 - 19/16*e^2 + 35/4*e + 3, 1/2*e^2 + 3/2*e - 3, 1/8*e^3 + 7/8*e^2 - 1/2*e - 8, 1/2*e^3 + 3/2*e^2 - 15*e + 6, 1/16*e^3 + 7/16*e^2 - 3/4*e - 3, -3/8*e^3 - 5/8*e^2 + 23/2*e - 9, 1/4*e^3 + 3/4*e^2 - 7*e - 2, -1/8*e^3 + 1/8*e^2 + 7/2*e - 8, -1/2*e^3 - 3/2*e^2 + 14*e - 1, 5/16*e^3 + 19/16*e^2 - 35/4*e - 5, -1/2*e^2 - 5/2*e + 15, -1/16*e^3 - 15/16*e^2 - 3/4*e + 16, 3/16*e^3 + 21/16*e^2 - 13/4*e - 9, -9/16*e^3 - 31/16*e^2 + 55/4*e + 5, 3/16*e^3 + 5/16*e^2 - 21/4*e + 5, 1/8*e^3 - 1/8*e^2 - 11/2*e + 10, 1/8*e^3 + 7/8*e^2 - 5/2*e - 8, 1/4*e^3 + 3/4*e^2 - 7*e - 8, -1/16*e^3 - 7/16*e^2 - 1/4*e - 5, -7/16*e^3 - 17/16*e^2 + 53/4*e + 1, -1/16*e^3 - 7/16*e^2 + 3/4*e + 9, -3/16*e^3 - 5/16*e^2 + 17/4*e - 1, 3/8*e^3 + 5/8*e^2 - 21/2*e + 10, 1/8*e^3 + 7/8*e^2 - 5/2*e - 8, 1/8*e^3 + 7/8*e^2 - 5/2*e + 3, -1/2*e^2 - 9/2*e + 5, -1/16*e^3 - 15/16*e^2 - 7/4*e + 12, 13/16*e^3 + 27/16*e^2 - 99/4*e + 15, -5/16*e^3 - 35/16*e^2 + 23/4*e + 19, -1/16*e^3 - 7/16*e^2 - 9/4*e - 5, -3/4*e^3 - 9/4*e^2 + 22*e - 3, 3/4*e^3 + 9/4*e^2 - 18*e + 13, -1/2*e^3 - 3/2*e^2 + 11*e - 12, -1/2*e^3 - 3/2*e^2 + 15*e + 6, -5/16*e^3 - 11/16*e^2 + 45/4*e - 2, 11/16*e^3 + 29/16*e^2 - 77/4*e + 7, -1/16*e^3 + 9/16*e^2 + 23/4*e - 9, -3/8*e^3 - 17/8*e^2 + 6*e + 17, 3/4*e^3 + 9/4*e^2 - 18*e, -5/16*e^3 - 3/16*e^2 + 39/4*e - 1, -1/2*e^3 - 3/2*e^2 + 13*e - 20, -e^2 - 3*e + 10, 5/8*e^3 + 7/8*e^2 - 20*e + 19, 1/4*e^3 + 5/4*e^2 - 7/2*e - 3, 3/8*e^3 + 5/8*e^2 - 21/2*e + 12, -15/16*e^3 - 41/16*e^2 + 109/4*e - 5, 5/16*e^3 + 3/16*e^2 - 51/4*e + 18, 1/16*e^3 - 9/16*e^2 - 7/4*e + 10, -17/16*e^3 - 55/16*e^2 + 111/4*e - 14, -1/2*e^2 - 3/2*e + 15, 7/8*e^3 + 17/8*e^2 - 53/2*e + 7, 3/8*e^3 + 5/8*e^2 - 27/2*e + 6, 7/8*e^3 + 9/8*e^2 - 55/2*e + 19, 3/8*e^3 + 13/8*e^2 - 19/2*e + 4, -1/16*e^3 - 7/16*e^2 - 9/4*e + 3, -3/16*e^3 - 13/16*e^2 + 31/4*e + 16, 1/8*e^3 - 1/8*e^2 - 3/2*e + 14, e^3 + 3*e^2 - 26*e + 6, 11/16*e^3 + 37/16*e^2 - 83/4*e - 2, 3/4*e^3 + 11/4*e^2 - 39/2*e + 3, 3/4*e^3 + 5/4*e^2 - 27*e + 17, 3/4*e^3 + 13/4*e^2 - 18*e - 4, -15/16*e^3 - 57/16*e^2 + 77/4*e + 9, -11/8*e^3 - 37/8*e^2 + 73/2*e - 6, 9/16*e^3 + 47/16*e^2 - 31/4*e - 17, -e^3 - 5/2*e^2 + 49/2*e - 19, -9/16*e^3 - 15/16*e^2 + 91/4*e - 19, 1/16*e^3 - 9/16*e^2 - 39/4*e + 3, -1/8*e^3 - 7/8*e^2 + 1/2*e + 20, -5/16*e^3 - 35/16*e^2 + 11/4*e + 29, -9/8*e^3 - 23/8*e^2 + 65/2*e - 28, 5/8*e^3 + 11/8*e^2 - 35/2*e + 20, e^2 + 2*e - 32, 21/16*e^3 + 83/16*e^2 - 123/4*e - 1, 3/16*e^3 + 5/16*e^2 - 17/4*e - 9, 7/8*e^3 + 17/8*e^2 - 59/2*e + 16, -7/16*e^3 - 25/16*e^2 + 43/4*e - 10, 3/16*e^3 + 37/16*e^2 - 1/4*e - 15, -15/16*e^3 - 41/16*e^2 + 109/4*e - 26, 1/8*e^3 - 1/8*e^2 - 11/2*e + 14, 9/16*e^3 + 31/16*e^2 - 71/4*e - 3, 3/4*e^3 + 9/4*e^2 - 25*e - 6, -7/8*e^3 - 25/8*e^2 + 43/2*e + 10, 1/4*e^3 - 1/4*e^2 - 16*e + 12, 1/2*e^3 + 7/2*e^2 - 8*e - 36, 3/16*e^3 + 5/16*e^2 - 13/4*e + 13, 1/2*e^3 + e^2 - 27/2*e + 3, -13/8*e^3 - 43/8*e^2 + 77/2*e - 2, -1/16*e^3 - 7/16*e^2 - 1/4*e + 3, e^3 + 4*e^2 - 20*e - 16, 3/16*e^3 - 11/16*e^2 - 33/4*e + 17, 5/16*e^3 + 3/16*e^2 - 39/4*e + 13, -5/16*e^3 - 51/16*e^2 - 1/4*e + 23, 3/4*e^3 + 5/4*e^2 - 22*e + 10, 1/16*e^3 - 9/16*e^2 - 3/4*e + 5, -5/16*e^3 - 3/16*e^2 + 43/4*e - 9, 7/8*e^3 + 25/8*e^2 - 37/2*e - 10, -1/4*e^3 - 3/4*e^2 + 12*e - 6, -3/8*e^3 - 21/8*e^2 + 5/2*e + 15, 7/16*e^3 + 49/16*e^2 - 5/4*e - 29, -5/16*e^3 - 19/16*e^2 + 43/4*e + 5, 1/16*e^3 + 7/16*e^2 + 17/4*e - 14, -17/16*e^3 - 71/16*e^2 + 87/4*e + 5, -5/4*e^3 - 21/4*e^2 + 47/2*e + 19, 31/16*e^3 + 73/16*e^2 - 217/4*e + 23, -1/4*e^3 - 3/4*e^2 + 5*e - 14, 3/2*e^3 + 7/2*e^2 - 44*e + 31, e^2 - 26, -1/2*e^3 - 5/2*e^2 + 12*e - 6, -3/16*e^3 - 21/16*e^2 + 9/4*e + 11, -5/16*e^3 - 3/16*e^2 + 31/4*e - 37, -3/8*e^3 - 13/8*e^2 + 11/2*e - 22, -9/8*e^3 - 19/8*e^2 + 32*e - 3, -5/8*e^3 - 3/8*e^2 + 41/2*e - 30, 13/16*e^3 + 27/16*e^2 - 75/4*e + 29, 7/16*e^3 + 49/16*e^2 + 3/4*e - 40, 9/16*e^3 + 31/16*e^2 - 59/4*e + 5, -1/2*e^3 - 5/2*e^2 + 14*e + 34, 21/16*e^3 + 51/16*e^2 - 147/4*e + 27, 1/16*e^3 - 1/16*e^2 - 17/4*e - 22, 1/4*e^3 + 3/4*e^2 - 11*e + 12, -11/16*e^3 - 45/16*e^2 + 49/4*e + 8, 7/8*e^3 + 9/8*e^2 - 61/2*e + 16, 23/16*e^3 + 65/16*e^2 - 153/4*e + 22, -1/8*e^3 - 7/8*e^2 + 9/2*e + 12, 7/8*e^3 + 13/8*e^2 - 20*e + 37, 9/16*e^3 + 47/16*e^2 - 59/4*e - 22, -17/16*e^3 - 71/16*e^2 + 83/4*e + 15, -1/4*e^3 - 11/4*e^2 + e + 33, 17/16*e^3 + 39/16*e^2 - 115/4*e + 36, -e^2 - e + 11, -7/8*e^3 - 41/8*e^2 + 27/2*e + 26, -33/16*e^3 - 103/16*e^2 + 211/4*e - 8, -e - 24, -11/16*e^3 - 69/16*e^2 + 47/4*e + 24, 5/16*e^3 + 51/16*e^2 + 13/4*e - 51, -3/8*e^3 - 25/8*e^2 + 6*e + 45, -5/8*e^3 - 27/8*e^2 + 21/2*e + 13, -3/8*e^3 - 5/8*e^2 + 23/2*e - 16, -1/4*e^3 - 7/4*e^2 - 5*e + 12, 1/16*e^3 + 23/16*e^2 + 9/4*e - 51, -5/2*e^3 - 15/2*e^2 + 60*e - 8, -5/4*e^3 - 19/4*e^2 + 32*e + 1, 1/8*e^3 + 11/8*e^2 + 3*e - 3, -e^3 - 6*e^2 + 17*e + 38, -e^2 - 4*e + 16, -15/16*e^3 - 89/16*e^2 + 37/4*e + 55, 1/2*e^3 + 7/2*e^2 - 5*e - 46, -17/16*e^3 - 23/16*e^2 + 159/4*e - 29, -3/16*e^3 + 11/16*e^2 + 69/4*e - 25, 9/16*e^3 + 31/16*e^2 - 35/4*e - 6, -9/8*e^3 - 35/8*e^2 + 29*e + 23, 5/4*e^3 + 15/4*e^2 - 35*e - 2, -41/16*e^3 - 111/16*e^2 + 259/4*e - 24, 9/16*e^3 + 15/16*e^2 - 103/4*e + 7]
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^3 - w^2 - 5*w])] = -1
AL_eigenvalues[ZF.ideal([9,3,w^3 - w^2 - 4*w])] = -1

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