/*
  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([11, 7, -6, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([55, 55, -3*w^2 + 2*w + 11])

primes_array = [
[5, 5, -w^2 + 2*w + 3],\
[9, 3, w^3 - 3*w^2 - 2*w + 9],\
[9, 3, -w^3 + 5*w + 5],\
[11, 11, w],\
[11, 11, w - 1],\
[16, 2, 2],\
[19, 19, -w^3 + 2*w^2 + 3*w - 2],\
[19, 19, w^3 - w^2 - 4*w + 2],\
[29, 29, w^3 - 4*w^2 - w + 10],\
[29, 29, -w^3 + 3*w^2 + w - 7],\
[41, 41, 3*w^2 - 2*w - 10],\
[49, 7, -2*w^2 + 3*w + 8],\
[49, 7, w^3 - 2*w^2 - 2*w + 5],\
[71, 71, -w - 3],\
[71, 71, w - 4],\
[79, 79, -w^3 + w^2 + 3*w + 3],\
[79, 79, -w^3 + 2*w^2 + 2*w - 6],\
[89, 89, w^3 - 3*w^2 - 3*w + 7],\
[89, 89, w^3 - 6*w - 2],\
[101, 101, 2*w^3 - 5*w^2 - 3*w + 9],\
[101, 101, -w^3 + 3*w^2 + 3*w - 12],\
[109, 109, -w^3 + 4*w^2 + w - 9],\
[109, 109, -w^3 + 2*w^2 + 4*w - 2],\
[121, 11, -3*w^2 + 3*w + 10],\
[131, 131, -2*w^3 + 5*w^2 + 4*w - 9],\
[131, 131, -w^3 + w^2 + 3*w + 5],\
[131, 131, -w^3 + 2*w^2 + 2*w - 8],\
[131, 131, -w^3 + 6*w^2 - 18],\
[149, 149, -w^3 - 2*w^2 + 7*w + 8],\
[149, 149, -2*w^3 + 6*w^2 + 5*w - 16],\
[151, 151, w^2 + w - 5],\
[151, 151, w^2 - 3*w - 3],\
[169, 13, w^3 - 4*w^2 - w + 6],\
[169, 13, 2*w^3 - 5*w^2 - 4*w + 6],\
[179, 179, w^3 - 4*w - 6],\
[179, 179, -w^3 + 3*w^2 + w - 9],\
[181, 181, w^3 + 3*w^2 - 7*w - 15],\
[181, 181, w^3 - 6*w^2 + 2*w + 18],\
[191, 191, 5*w^2 - 7*w - 15],\
[191, 191, -4*w^2 + 5*w + 12],\
[199, 199, -w^3 + 3*w^2 + 3*w - 6],\
[199, 199, w^3 - 6*w - 1],\
[211, 211, w^3 - 7*w - 4],\
[211, 211, -w^3 + 3*w^2 + 4*w - 10],\
[229, 229, -w^3 - w^2 + 5*w + 9],\
[229, 229, -w^3 + 4*w^2 - 12],\
[239, 239, 4*w^2 - 5*w - 17],\
[239, 239, 4*w^2 - 3*w - 18],\
[241, 241, -w^3 + 3*w^2 + w - 2],\
[241, 241, -w^3 + 3*w^2 + 4*w - 9],\
[241, 241, w^3 - 7*w - 3],\
[241, 241, -w^3 + 4*w - 1],\
[271, 271, -w - 4],\
[271, 271, w^3 - 3*w^2 - 2*w + 2],\
[271, 271, -2*w^3 + 6*w^2 + 2*w - 7],\
[271, 271, w - 5],\
[281, 281, w^3 - 2*w^2 - 3*w - 1],\
[281, 281, -w^3 + w^2 + 4*w - 5],\
[289, 17, -2*w^3 + w^2 + 9*w + 1],\
[289, 17, -2*w^3 + 5*w^2 + 5*w - 9],\
[311, 311, w^3 - w^2 - 6*w + 1],\
[311, 311, -w^3 + 2*w^2 + 5*w - 5],\
[331, 331, -3*w^3 + 5*w^2 + 10*w - 5],\
[331, 331, 3*w^3 - 2*w^2 - 12*w - 6],\
[349, 349, -2*w^3 + 2*w^2 + 10*w + 3],\
[349, 349, -2*w^3 - w^2 + 12*w + 10],\
[349, 349, 2*w^3 + w^2 - 10*w - 12],\
[349, 349, -2*w^3 + 4*w^2 + 8*w - 13],\
[361, 19, -4*w^2 + 4*w + 13],\
[379, 379, w^3 + w^2 - 5*w - 10],\
[379, 379, -3*w^3 + 4*w^2 + 12*w - 7],\
[379, 379, w^3 + 4*w^2 - 8*w - 21],\
[379, 379, w^3 - 3*w - 6],\
[389, 389, 2*w^3 - 6*w^2 - 4*w + 13],\
[389, 389, -5*w^2 + 7*w + 13],\
[389, 389, 2*w^3 - 7*w^2 - 5*w + 20],\
[389, 389, 2*w^3 - 10*w - 5],\
[419, 419, -w^3 + 5*w^2 + w - 13],\
[419, 419, 4*w^2 - 7*w - 14],\
[419, 419, 2*w^3 - 3*w^2 - 5*w - 1],\
[419, 419, -w^3 - 2*w^2 + 8*w + 8],\
[421, 421, -2*w^2 + 5*w + 7],\
[421, 421, w^3 - 6*w^2 + 5*w + 13],\
[439, 439, 2*w^3 - 2*w^2 - 9*w + 3],\
[439, 439, w^3 + w^2 - 7*w - 4],\
[449, 449, 3*w^3 - 6*w^2 - 9*w + 8],\
[449, 449, 5*w^2 - 8*w - 17],\
[449, 449, 2*w^3 - w^2 - 8*w - 7],\
[449, 449, -3*w^3 + 3*w^2 + 12*w - 4],\
[479, 479, -2*w^3 + 3*w^2 + 5*w + 3],\
[479, 479, 3*w^3 - 7*w^2 - 9*w + 15],\
[499, 499, -w^3 + 4*w^2 + 4*w - 9],\
[499, 499, w^3 + w^2 - 9*w - 2],\
[509, 509, 2*w^3 + 2*w^2 - 11*w - 16],\
[509, 509, -2*w^3 + 8*w^2 + w - 23],\
[521, 521, -3*w^3 + w^2 + 13*w + 6],\
[521, 521, w^3 + 4*w^2 - 10*w - 16],\
[541, 541, 3*w^3 - 2*w^2 - 13*w - 3],\
[541, 541, -w^3 + 7*w^2 - w - 25],\
[541, 541, w^3 + 4*w^2 - 10*w - 20],\
[541, 541, -3*w^3 + 7*w^2 + 8*w - 15],\
[571, 571, w^3 + 4*w^2 - 9*w - 15],\
[571, 571, w^3 - 7*w^2 + 2*w + 19],\
[601, 601, -3*w^3 + 2*w^2 + 13*w + 5],\
[601, 601, 3*w^3 - 7*w^2 - 8*w + 17],\
[619, 619, w^3 - 3*w - 7],\
[619, 619, w^3 - 5*w^2 + 3*w + 13],\
[619, 619, 3*w^3 - 4*w^2 - 9*w - 2],\
[619, 619, -w^3 + 3*w^2 - 9],\
[641, 641, 6*w^2 - 4*w - 23],\
[641, 641, -3*w^3 + w^2 + 12*w + 5],\
[659, 659, -2*w^3 + 6*w^2 + 3*w - 17],\
[659, 659, -2*w^3 + 9*w + 10],\
[661, 661, 3*w^3 - 6*w^2 - 9*w + 14],\
[661, 661, 5*w^2 - 6*w - 18],\
[661, 661, -5*w^2 + 4*w + 19],\
[661, 661, 3*w^3 - 3*w^2 - 12*w - 2],\
[691, 691, -3*w^3 + 5*w^2 + 10*w - 6],\
[691, 691, -3*w^3 + 4*w^2 + 11*w - 6],\
[701, 701, -w^3 - 4*w^2 + 9*w + 20],\
[701, 701, -w^3 + 4*w^2 + 3*w - 19],\
[701, 701, -w^3 + 4*w^2 + 4*w - 15],\
[701, 701, w^3 - 7*w^2 + 2*w + 24],\
[709, 709, 3*w^2 - 5*w - 13],\
[709, 709, 3*w^2 - w - 15],\
[719, 719, -w^3 + 3*w^2 + 4*w - 4],\
[719, 719, -w^3 + 7*w - 2],\
[739, 739, -w^3 + 4*w^2 + 3*w - 9],\
[739, 739, -2*w^3 - w^2 + 11*w + 8],\
[739, 739, 2*w^3 - 7*w^2 - 3*w + 16],\
[739, 739, -w^3 - w^2 + 8*w + 3],\
[751, 751, 3*w^3 - 5*w^2 - 10*w + 10],\
[751, 751, 2*w^2 - 5*w - 5],\
[761, 761, 3*w^3 - 3*w^2 - 11*w + 3],\
[761, 761, w^3 - 2*w^2 - 3*w - 2],\
[761, 761, -w^3 + w^2 + 4*w - 6],\
[761, 761, -3*w^3 + 6*w^2 + 8*w - 8],\
[769, 769, -w^3 - w^2 + 7*w + 1],\
[769, 769, -2*w^3 + 6*w^2 + 6*w - 15],\
[769, 769, -2*w^3 + 5*w^2 + 5*w - 7],\
[769, 769, 2*w^3 + w^2 - 12*w - 9],\
[809, 809, -2*w^3 + 8*w^2 + 4*w - 23],\
[809, 809, -3*w^3 + 11*w^2 + 2*w - 28],\
[829, 829, -3*w^3 + 7*w^2 + 7*w - 9],\
[829, 829, w^3 + 2*w^2 - 6*w - 15],\
[829, 829, -w^3 + 5*w^2 - w - 18],\
[829, 829, -3*w^3 + 2*w^2 + 12*w - 2],\
[839, 839, 2*w^3 - 9*w^2 + 20],\
[839, 839, -3*w^3 + 10*w^2 + 5*w - 30],\
[841, 29, 5*w^2 - 5*w - 16],\
[881, 881, -w^3 - 4*w^2 + 8*w + 18],\
[881, 881, 2*w^3 - 7*w^2 - w + 9],\
[881, 881, 2*w^3 - 6*w^2 - 3*w + 6],\
[881, 881, -w^3 + 7*w^2 - 3*w - 21],\
[911, 911, -w^3 - 4*w^2 + 8*w + 16],\
[911, 911, -w^3 - w^2 + 9*w + 5],\
[911, 911, -w^3 + 4*w^2 + 4*w - 12],\
[911, 911, w^3 - 7*w^2 + 3*w + 19],\
[919, 919, -4*w^3 + 7*w^2 + 14*w - 9],\
[919, 919, 4*w^3 - 5*w^2 - 16*w + 8],\
[929, 929, 3*w^2 - w - 16],\
[929, 929, 3*w^2 - 5*w - 14],\
[941, 941, -w^3 - w^2 + 9*w + 7],\
[941, 941, -3*w^3 + 7*w^2 + 7*w - 15],\
[941, 941, 3*w^3 - 2*w^2 - 12*w - 4],\
[941, 941, w^3 + 4*w^2 - 13*w - 16],\
[961, 31, -2*w^2 + 2*w + 13],\
[961, 31, 5*w^2 - 5*w - 18],\
[971, 971, w^3 + 4*w^2 - 9*w - 19],\
[971, 971, w^3 - 7*w^2 + 2*w + 23],\
[991, 991, 3*w^3 - 7*w^2 - 7*w + 14],\
[991, 991, -3*w^3 + 2*w^2 + 12*w + 3]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - 22*x^4 + 128*x^2 - 160
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1, e, 1/8*e^5 - 9/4*e^3 + 8*e, 1, e^2 - 8, -1/4*e^4 + 9/2*e^2 - 13, -1/2*e^3 + 6*e, 1/8*e^5 - 9/4*e^3 + 9*e, -1/8*e^5 + 9/4*e^3 - 8*e, -e, 2, -3*e, -1/2*e^3 + 5*e, -e^2 + 12, 1/2*e^4 - 7*e^2 + 12, -1/2*e^3 + 4*e, -1/8*e^5 + 9/4*e^3 - 7*e, 1/8*e^5 - 7/4*e^3 + 4*e, 1/2*e^3 - 7*e, -2*e^2 + 22, -e^4 + 15*e^2 - 38, -3/8*e^5 + 27/4*e^3 - 26*e, -1/8*e^5 + 7/4*e^3 - 4*e, 2, -1/2*e^4 + 7*e^2 - 8, -1/2*e^4 + 8*e^2 - 28, 1/2*e^4 - 9*e^2 + 32, 1/2*e^4 - 6*e^2 + 12, -3/2*e^3 + 15*e, 1/8*e^5 - 7/4*e^3 + 6*e, -1/2*e^4 + 9*e^2 - 28, -1/2*e^4 + 7*e^2 - 28, 1/8*e^5 - 11/4*e^3 + 8*e, 3/2*e^3 - 15*e, 1/2*e^5 - 17/2*e^3 + 26*e, 1/4*e^5 - 5*e^3 + 20*e, -1/2*e^4 + 5*e^2 + 2, 2*e^2 - 18, -1/2*e^4 + 5*e^2 + 12, e^4 - 17*e^2 + 52, -1/2*e^3 + 12*e, 3/2*e^3 - 16*e, 1/2*e^4 - 9*e^2 + 32, -e^4 + 14*e^2 - 28, 3/2*e^3 - 17*e, 1/4*e^5 - 7/2*e^3 + 5*e, 3/8*e^5 - 25/4*e^3 + 25*e, -1/4*e^5 + 9/2*e^3 - 14*e, -1/2*e^4 + 7*e^2 - 18, -2*e^2 + 2, 1/2*e^4 - 10*e^2 + 42, 2*e^2 + 2, 1/2*e^4 - 7*e^2 + 12, -1/2*e^4 + 11*e^2 - 28, e^4 - 17*e^2 + 52, -1/2*e^4 + 10*e^2 - 48, -e^4 + 16*e^2 - 38, 2, 1/8*e^5 - 13/4*e^3 + 12*e, -1/4*e^5 + 7/2*e^3 - 9*e, -8, -e^4 + 15*e^2 - 28, e^4 - 12*e^2 + 12, 1/2*e^4 - 2*e^2 - 28, -1/4*e^5 + 5*e^3 - 19*e, -1/4*e^5 + 4*e^3 - 11*e, -3/8*e^5 + 17/4*e^3 + 2*e, -3*e^3 + 31*e, 2, 7/8*e^5 - 59/4*e^3 + 51*e, -1/8*e^5 + 5/4*e^3 - 5*e, -1/8*e^5 + 13/4*e^3 - 25*e, -1/4*e^5 + 4*e^3 - 16*e, 1/2*e^3 - e, 1/4*e^5 - 5*e^3 + 21*e, 1/8*e^5 - 15/4*e^3 + 22*e, -1/8*e^5 + 7/4*e^3 - 12*e, -3/8*e^5 + 31/4*e^3 - 31*e, -3/8*e^5 + 25/4*e^3 - 23*e, 1/8*e^5 - 7/4*e^3 + e, -1/2*e^5 + 19/2*e^3 - 38*e, -1/2*e^4 + 9*e^2 - 38, 3/2*e^4 - 21*e^2 + 42, -3/8*e^5 + 31/4*e^3 - 41*e, 3/8*e^5 - 31/4*e^3 + 37*e, 1/4*e^5 - 9/2*e^3 + 27*e, 1/8*e^5 + 5/4*e^3 - 28*e, 3/8*e^5 - 31/4*e^3 + 34*e, 5/2*e^3 - 27*e, -1/8*e^5 + 17/4*e^3 - 27*e, -1/4*e^5 + 7*e^3 - 42*e, 1/8*e^5 - 11/4*e^3 + 21*e, 1/8*e^5 - 19/4*e^3 + 37*e, -1/8*e^5 + 5/4*e^3 + 12*e, -3/8*e^5 + 29/4*e^3 - 22*e, e^4 - 12*e^2 + 2, -e^4 + 16*e^2 - 38, 1/2*e^4 - 9*e^2 + 2, e^4 - 12*e^2 + 22, 1/2*e^4 - 6*e^2 + 22, 1/2*e^4 - 2*e^2 - 18, -1/2*e^4 + 11*e^2 - 48, e^4 - 18*e^2 + 52, -1/2*e^4 + 6*e^2 + 2, 2, e^5 - 16*e^3 + 46*e, 3/8*e^5 - 25/4*e^3 + 19*e, -7/8*e^5 + 61/4*e^3 - 59*e, -5/8*e^5 + 47/4*e^3 - 49*e, 2*e^2 + 2, -e^4 + 10*e^2 + 2, -3/8*e^5 + 23/4*e^3 - 23*e, -3/8*e^5 + 23/4*e^3 - 19*e, 2*e^2 - 18, -3*e^2 + 2, 4*e^2 - 18, e^4 - 20*e^2 + 62, e^4 - 12*e^2 + 12, 1/2*e^4 - 6*e^2 + 12, e^4 - 18*e^2 + 62, -3/2*e^4 + 24*e^2 - 58, -18, -18, 3/8*e^5 - 21/4*e^3 + 12*e, 1/4*e^5 - 6*e^3 + 37*e, -1/4*e^5 + 5*e^3 - 14*e, -5/8*e^5 + 41/4*e^3 - 35*e, 1/2*e^5 - 8*e^3 + 30*e, -1/4*e^5 + 3*e^3 + 4*e, -3/2*e^3 + 22*e, e^5 - 18*e^3 + 58*e, 1/2*e^4 - 13*e^2 + 52, -e^4 + 10*e^2 - 8, -e^4 + 14*e^2 - 38, 2*e^4 - 30*e^2 + 82, -3/2*e^4 + 22*e^2 - 38, -e^4 + 10*e^2 + 2, -1/8*e^5 + 13/4*e^3 - 22*e, -1/2*e^5 + 19/2*e^3 - 39*e, 3/8*e^5 - 29/4*e^3 + 18*e, -5/8*e^5 + 47/4*e^3 - 38*e, 1/4*e^5 - 9/2*e^3 + 11*e, 1/4*e^5 - e^3 - 25*e, 1/2*e^5 - 17/2*e^3 + 39*e, 5/8*e^5 - 49/4*e^3 + 38*e, -3/8*e^5 + 21/4*e^3 - 6*e, 2*e^3 - 13*e, -3/8*e^5 + 33/4*e^3 - 37*e, -7/8*e^5 + 57/4*e^3 - 41*e, -e^4 + 12*e^2 + 2, 1/2*e^4 - 11*e^2 + 62, -1/2*e^4 + 3*e^2 + 22, -e^4 + 14*e^2 + 2, 1/2*e^4 - 4*e^2 - 38, -3/2*e^4 + 26*e^2 - 88, 32, 3/2*e^4 - 22*e^2 + 72, -e^4 + 19*e^2 - 68, -5/2*e^3 + 20*e, 1/4*e^5 - 3*e^3 - 6*e, -1/2*e^5 + 9*e^3 - 39*e, -5/8*e^5 + 45/4*e^3 - 30*e, -1/2*e^4 + 5*e^2 - 38, 3/2*e^4 - 20*e^2 + 22, e^4 - 16*e^2 + 62, 2*e^4 - 31*e^2 + 82, 1/2*e^4 - 4*e^2 + 2, 3/2*e^4 - 27*e^2 + 62, 1/2*e^4 - 7*e^2 - 8, -1/2*e^4 + 5*e^2 - 8, -2*e^4 + 28*e^2 - 48, 1/2*e^4 - 13*e^2 + 52]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([5, 5, -w^2 + 2*w + 3])] = -1
AL_eigenvalues[ZF.ideal([11, 11, w])] = -1

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