/*
  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([9, 3, -7, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([45, 15, -w - 3])

primes_array = [
[5, 5, -1/3*w^3 - 2/3*w^2 + 7/3*w + 4],\
[5, 5, 1/3*w^3 - 4/3*w^2 - 1/3*w + 3],\
[9, 3, -w],\
[9, 3, 1/3*w^3 - 1/3*w^2 - 7/3*w + 1],\
[16, 2, 2],\
[19, 19, 2/3*w^3 - 2/3*w^2 - 11/3*w],\
[19, 19, -1/3*w^3 + 1/3*w^2 + 1/3*w + 1],\
[31, 31, 2/3*w^3 + 1/3*w^2 - 11/3*w - 2],\
[31, 31, 2/3*w^3 - 5/3*w^2 - 5/3*w + 5],\
[41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 6],\
[41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 3],\
[61, 61, 1/3*w^3 - 4/3*w^2 + 2/3*w + 4],\
[61, 61, 2/3*w^3 + 1/3*w^2 - 14/3*w - 2],\
[71, 71, -1/3*w^3 + 4/3*w^2 + 1/3*w - 7],\
[71, 71, 1/3*w^3 + 2/3*w^2 - 7/3*w],\
[89, 89, -1/3*w^3 + 1/3*w^2 + 10/3*w - 3],\
[89, 89, 1/3*w^3 - 1/3*w^2 - 10/3*w - 1],\
[101, 101, 1/3*w^3 - 1/3*w^2 - 7/3*w - 3],\
[101, 101, -2/3*w^3 + 5/3*w^2 + 11/3*w - 4],\
[101, 101, 2/3*w^3 - 5/3*w^2 - 8/3*w + 3],\
[101, 101, w - 4],\
[109, 109, 2/3*w^3 + 1/3*w^2 - 8/3*w - 4],\
[109, 109, -w^3 + 2*w^2 + 4*w - 4],\
[121, 11, w^3 - w^2 - 4*w + 1],\
[121, 11, -w^3 + w^2 + 4*w - 2],\
[131, 131, 4/3*w^3 - 1/3*w^2 - 22/3*w - 1],\
[131, 131, -w^3 + 7*w + 2],\
[139, 139, 2/3*w^3 - 2/3*w^2 - 14/3*w + 1],\
[139, 139, 2*w - 1],\
[139, 139, 1/3*w^3 + 5/3*w^2 - 7/3*w - 10],\
[139, 139, -2/3*w^3 + 8/3*w^2 + 5/3*w - 5],\
[149, 149, w^3 + w^2 - 7*w - 11],\
[149, 149, -w^3 - w^2 + 5*w + 4],\
[151, 151, -2/3*w^3 - 4/3*w^2 + 14/3*w + 5],\
[151, 151, 2/3*w^3 - 8/3*w^2 - 2/3*w + 9],\
[169, 13, -2/3*w^3 - 1/3*w^2 + 11/3*w],\
[169, 13, -2/3*w^3 + 5/3*w^2 + 5/3*w - 7],\
[179, 179, 4/3*w^3 - 7/3*w^2 - 13/3*w + 6],\
[179, 179, -4/3*w^3 + 1/3*w^2 + 19/3*w + 1],\
[181, 181, 5/3*w^3 - 8/3*w^2 - 20/3*w + 5],\
[181, 181, -1/3*w^3 + 1/3*w^2 + 10/3*w - 1],\
[181, 181, -4/3*w^3 + 1/3*w^2 + 16/3*w + 3],\
[191, 191, w^2 - 3*w - 1],\
[191, 191, -1/3*w^3 + 4/3*w^2 + 10/3*w - 3],\
[199, 199, -w^3 + 3*w^2 + w - 7],\
[199, 199, -4/3*w^3 + 1/3*w^2 + 19/3*w - 1],\
[211, 211, -1/3*w^3 - 5/3*w^2 + 4/3*w + 8],\
[211, 211, w^3 - 3*w^2 - 4*w + 8],\
[229, 229, -4/3*w^3 + 4/3*w^2 + 13/3*w - 4],\
[229, 229, -5/3*w^3 + 5/3*w^2 + 23/3*w - 5],\
[229, 229, 1/3*w^3 - 7/3*w^2 + 2/3*w + 10],\
[229, 229, -2/3*w^3 + 2/3*w^2 + 2/3*w - 3],\
[239, 239, 4/3*w^3 - 1/3*w^2 - 19/3*w - 2],\
[239, 239, -2/3*w^3 + 2/3*w^2 + 11/3*w - 6],\
[251, 251, 2*w^2 - w - 8],\
[251, 251, -1/3*w^3 - 2/3*w^2 + 7/3*w - 2],\
[251, 251, 1/3*w^3 - 4/3*w^2 - 1/3*w + 9],\
[251, 251, -1/3*w^3 + 7/3*w^2 + 1/3*w - 7],\
[269, 269, -5/3*w^3 + 8/3*w^2 + 17/3*w - 8],\
[269, 269, -5/3*w^3 + 11/3*w^2 + 17/3*w - 12],\
[271, 271, 1/3*w^3 + 2/3*w^2 - 13/3*w - 3],\
[271, 271, 1/3*w^3 + 2/3*w^2 - 13/3*w - 2],\
[311, 311, -1/3*w^3 + 7/3*w^2 - 5/3*w - 8],\
[311, 311, -w^3 + 5*w - 1],\
[331, 331, -2/3*w^3 - 1/3*w^2 + 8/3*w + 6],\
[331, 331, -w^3 + 2*w^2 + 4*w - 2],\
[349, 349, -w^3 + 2*w^2 + w + 1],\
[349, 349, 1/3*w^3 - 1/3*w^2 - 7/3*w - 4],\
[349, 349, -4/3*w^3 + 13/3*w^2 + 10/3*w - 11],\
[349, 349, w - 5],\
[361, 19, -4/3*w^3 + 4/3*w^2 + 16/3*w - 3],\
[379, 379, 7/3*w^3 - 1/3*w^2 - 43/3*w - 9],\
[379, 379, 2*w^3 - 3*w^2 - 9*w + 8],\
[401, 401, 1/3*w^3 + 2/3*w^2 - 7/3*w - 8],\
[401, 401, -1/3*w^3 + 10/3*w^2 - 2/3*w - 15],\
[401, 401, -4/3*w^3 + 4/3*w^2 + 13/3*w - 3],\
[401, 401, -1/3*w^3 + 4/3*w^2 - 5/3*w + 3],\
[409, 409, -5/3*w^3 + 8/3*w^2 + 20/3*w - 7],\
[409, 409, 4/3*w^3 - 1/3*w^2 - 16/3*w - 1],\
[419, 419, 4/3*w^3 - 16/3*w^2 - 1/3*w + 10],\
[419, 419, 5/3*w^3 + 7/3*w^2 - 35/3*w - 17],\
[421, 421, -2*w^3 + 4*w^2 + 7*w - 10],\
[421, 421, -5/3*w^3 + 8/3*w^2 + 29/3*w - 7],\
[421, 421, 1/3*w^3 + 2/3*w^2 + 5/3*w - 4],\
[421, 421, -5/3*w^3 - 1/3*w^2 + 23/3*w + 5],\
[431, 431, -2/3*w^3 + 5/3*w^2 + 11/3*w - 1],\
[431, 431, -w^2 - w + 8],\
[439, 439, -2/3*w^3 + 11/3*w^2 - 1/3*w - 8],\
[439, 439, -2/3*w^3 + 11/3*w^2 + 8/3*w - 15],\
[479, 479, w^3 + 2*w^2 - 6*w - 10],\
[479, 479, -5/3*w^3 + 5/3*w^2 + 14/3*w],\
[479, 479, -5/3*w^3 + 5/3*w^2 + 23/3*w - 3],\
[479, 479, -4/3*w^3 + 4/3*w^2 + 13/3*w - 2],\
[521, 521, -w^3 + 2*w^2 + 5*w + 1],\
[521, 521, 1/3*w^3 + 2/3*w^2 - 1/3*w - 10],\
[541, 541, 1/3*w^3 + 5/3*w^2 + 2/3*w - 7],\
[541, 541, 5/3*w^3 - 5/3*w^2 - 26/3*w],\
[569, 569, -w - 5],\
[569, 569, 1/3*w^3 - 1/3*w^2 - 7/3*w + 6],\
[571, 571, w^3 - 4*w - 8],\
[571, 571, -1/3*w^3 + 7/3*w^2 + 7/3*w - 4],\
[599, 599, 5/3*w^3 - 5/3*w^2 - 14/3*w + 4],\
[599, 599, -4/3*w^3 + 13/3*w^2 + 13/3*w - 15],\
[599, 599, 2*w^3 - 3*w^2 - 8*w + 5],\
[599, 599, 7/3*w^3 - 7/3*w^2 - 34/3*w + 6],\
[601, 601, -4/3*w^3 + 13/3*w^2 + 13/3*w - 13],\
[601, 601, -w^3 + 8*w + 5],\
[619, 619, -5/3*w^3 + 11/3*w^2 + 17/3*w - 13],\
[619, 619, 4/3*w^3 + 2/3*w^2 - 19/3*w - 2],\
[631, 631, w^3 - w^2 - 7*w + 1],\
[631, 631, 3*w - 2],\
[641, 641, -2/3*w^3 + 5/3*w^2 + 2/3*w - 7],\
[641, 641, -1/3*w^3 + 7/3*w^2 - 5/3*w - 9],\
[661, 661, -5/3*w^3 + 11/3*w^2 + 20/3*w - 9],\
[661, 661, w^3 + w^2 - 4*w - 7],\
[691, 691, -5/3*w^3 + 8/3*w^2 + 17/3*w - 6],\
[691, 691, 2*w^2 - 3*w - 10],\
[691, 691, 5/3*w^3 - 2/3*w^2 - 23/3*w - 1],\
[691, 691, 1/3*w^3 + 5/3*w^2 - 13/3*w - 3],\
[701, 701, 2/3*w^3 + 4/3*w^2 - 20/3*w - 11],\
[701, 701, -2/3*w^3 - 1/3*w^2 + 17/3*w + 9],\
[709, 709, -2/3*w^3 - 7/3*w^2 + 17/3*w + 11],\
[709, 709, -1/3*w^3 + 10/3*w^2 - 2/3*w - 8],\
[709, 709, -2/3*w^3 + 11/3*w^2 - 1/3*w - 10],\
[709, 709, 3*w^2 - 2*w - 14],\
[719, 719, 7/3*w^3 + 8/3*w^2 - 46/3*w - 21],\
[719, 719, -2*w^3 + 7*w^2 + 2*w - 13],\
[719, 719, 2/3*w^3 + 7/3*w^2 - 14/3*w - 9],\
[719, 719, -w^3 + 4*w^2 + 2*w - 13],\
[761, 761, 1/3*w^3 + 5/3*w^2 - 1/3*w - 8],\
[761, 761, 5/3*w^3 - 2/3*w^2 - 14/3*w + 1],\
[761, 761, -8/3*w^3 + 11/3*w^2 + 38/3*w - 11],\
[761, 761, -4/3*w^3 + 10/3*w^2 + 19/3*w - 9],\
[809, 809, -2/3*w^3 + 11/3*w^2 + 5/3*w - 15],\
[809, 809, 3*w^2 - w - 8],\
[811, 811, 2/3*w^3 + 7/3*w^2 - 14/3*w - 11],\
[811, 811, 1/3*w^3 + 8/3*w^2 - 13/3*w - 10],\
[811, 811, 1/3*w^3 - 10/3*w^2 + 5/3*w + 11],\
[811, 811, -w^3 + 4*w^2 + 2*w - 11],\
[821, 821, 2*w^3 - w^2 - 12*w - 5],\
[821, 821, 1/3*w^3 + 5/3*w^2 - 7/3*w - 14],\
[829, 829, -2/3*w^3 + 11/3*w^2 + 5/3*w - 10],\
[829, 829, 3*w^2 - w - 13],\
[839, 839, 4/3*w^3 - 1/3*w^2 - 25/3*w + 1],\
[839, 839, -2/3*w^3 + 5/3*w^2 - 1/3*w - 6],\
[841, 29, 5/3*w^3 - 5/3*w^2 - 20/3*w + 1],\
[841, 29, -5/3*w^3 + 5/3*w^2 + 20/3*w - 4],\
[859, 859, -1/3*w^3 - 8/3*w^2 + 13/3*w + 11],\
[859, 859, -1/3*w^3 + 10/3*w^2 - 5/3*w - 10],\
[859, 859, 2/3*w^3 - 8/3*w^2 + 1/3*w + 9],\
[859, 859, -2/3*w^3 + 5/3*w^2 + 2/3*w - 8],\
[881, 881, -4/3*w^3 + 16/3*w^2 + 4/3*w - 11],\
[881, 881, 1/3*w^3 - 13/3*w^2 - 4/3*w + 18],\
[911, 911, -1/3*w^3 + 10/3*w^2 + 1/3*w - 14],\
[911, 911, -1/3*w^3 + 10/3*w^2 + 1/3*w - 9],\
[929, 929, 4/3*w^3 - 7/3*w^2 - 22/3*w + 4],\
[929, 929, 1/3*w^3 + 2/3*w^2 + 2/3*w - 6],\
[961, 31, 5/3*w^3 - 5/3*w^2 - 20/3*w + 2],\
[971, 971, -2/3*w^3 + 11/3*w^2 + 2/3*w - 13],\
[971, 971, 1/3*w^3 + 8/3*w^2 - 10/3*w - 9],\
[991, 991, -4/3*w^3 + 10/3*w^2 + 16/3*w - 17],\
[991, 991, -2*w^3 + 4*w^2 + 11*w - 11]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - x^3 - 13*x^2 - 3*x + 18
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [-1, e, -1, 1/3*e^3 - 4/3*e^2 - 7/3*e + 6, -e^2 + e + 5, -2/3*e^3 + 5/3*e^2 + 20/3*e - 4, 1/3*e^3 - 4/3*e^2 - 1/3*e + 8, e^3 - 2*e^2 - 7*e + 4, -e^3 + 2*e^2 + 8*e - 2, -1/3*e^3 - 2/3*e^2 + 16/3*e + 2, 2/3*e^3 - 2/3*e^2 - 20/3*e - 4, -4/3*e^3 + 7/3*e^2 + 40/3*e, 2/3*e^3 - 5/3*e^2 - 20/3*e + 6, -2/3*e^3 + 8/3*e^2 + 14/3*e - 14, 1/3*e^3 - 4/3*e^2 - 19/3*e + 10, e + 6, 2/3*e^3 + 1/3*e^2 - 26/3*e - 10, e^3 - 2*e^2 - 9*e + 12, e^3 - e^2 - 9*e, 1/3*e^3 - 4/3*e^2 - 1/3*e + 10, e^3 - 2*e^2 - 11*e, -e^3 + 2*e^2 + 9*e - 4, e - 4, 4/3*e^3 - 4/3*e^2 - 46/3*e - 4, -2*e + 10, -1/3*e^3 + 4/3*e^2 + 7/3*e - 4, -1/3*e^3 + 1/3*e^2 + 7/3*e + 2, -e^3 + 4*e^2 + 5*e - 10, 5/3*e^3 - 11/3*e^2 - 47/3*e - 2, 2*e^2 - 6*e - 8, e^3 - 5*e^2 - 3*e + 28, -4/3*e^3 + 10/3*e^2 + 37/3*e - 4, -1/3*e^3 - 5/3*e^2 + 13/3*e + 14, 2/3*e^3 - 2/3*e^2 - 20/3*e, -e^3 + 4*e^2 + 9*e - 14, 2*e^3 - 6*e^2 - 14*e + 14, 4/3*e^3 - 7/3*e^2 - 58/3*e + 6, -4/3*e^3 + 7/3*e^2 + 34/3*e - 4, 2/3*e^3 - 8/3*e^2 - 17/3*e + 26, -2*e^3 + 4*e^2 + 18*e - 10, 4*e^2 - 5*e - 26, 5/3*e^3 - 14/3*e^2 - 35/3*e + 16, 5/3*e^3 - 20/3*e^2 - 17/3*e + 38, -2/3*e^3 - 4/3*e^2 + 32/3*e + 10, 7/3*e^3 - 16/3*e^2 - 49/3*e + 18, -1/3*e^3 + 4/3*e^2 + 19/3*e - 2, 3*e^3 - 6*e^2 - 24*e + 8, -2*e^3 + 6*e^2 + 13*e - 22, -7/3*e^3 + 22/3*e^2 + 49/3*e - 20, -5/3*e^3 + 17/3*e^2 + 35/3*e - 24, e^3 - 4*e^2 - 13*e + 22, 4*e^2 - 6*e - 20, 2/3*e^3 - 2/3*e^2 - 20/3*e + 8, -5/3*e^3 + 14/3*e^2 + 32/3*e - 14, -10/3*e^3 + 22/3*e^2 + 85/3*e - 16, 3*e^3 - 6*e^2 - 23*e + 18, 5/3*e^3 - 11/3*e^2 - 47/3*e + 20, -2/3*e^3 + 2/3*e^2 + 26/3*e - 8, 1/3*e^3 - 4/3*e^2 - 16/3*e - 2, e^3 - e^2 - 11*e + 6, 2*e^2 - 10*e - 20, 1/3*e^3 + 11/3*e^2 - 25/3*e - 28, e^3 - 4*e^2 - 3*e + 30, 1/3*e^3 - 4/3*e^2 - 1/3*e + 16, -e^3 + 15*e + 2, 2/3*e^3 - 2/3*e^2 - 11/3*e - 26, 11/3*e^3 - 26/3*e^2 - 98/3*e + 18, 2*e^3 - 6*e^2 - 12*e + 26, -e^3 + 2*e^2 + 5*e - 8, 8/3*e^3 - 20/3*e^2 - 74/3*e - 2, 2*e^3 - 4*e^2 - 20*e + 14, 10/3*e^3 - 16/3*e^2 - 94/3*e + 14, -2/3*e^3 - 4/3*e^2 + 23/3*e + 20, -7/3*e^3 + 16/3*e^2 + 37/3*e - 10, -3*e^3 + 7*e^2 + 23*e, -5/3*e^3 + 11/3*e^2 + 47/3*e + 4, -2/3*e^3 - 10/3*e^2 + 32/3*e + 22, e^3 - 3*e^2 - 7*e + 22, -3*e^3 + 6*e^2 + 31*e - 8, -3*e^3 + 8*e^2 + 21*e - 30, 5/3*e^3 - 8/3*e^2 - 77/3*e + 2, 8/3*e^3 - 14/3*e^2 - 68/3*e + 6, -3*e^3 + 6*e^2 + 21*e - 16, 2*e^3 - 4*e^2 - 16*e + 2, -2*e^3 + 4*e^2 + 12*e + 4, 7/3*e^3 - 16/3*e^2 - 76/3*e - 8, -8/3*e^3 + 14/3*e^2 + 86/3*e - 14, -4/3*e^3 + 13/3*e^2 + 22/3*e - 20, 2*e^3 - 6*e^2 - 20*e + 8, -3*e^3 + 10*e^2 + 15*e - 48, 4*e^2 - 10*e - 24, -5/3*e^3 + 20/3*e^2 + 14/3*e - 32, -7/3*e^3 + 1/3*e^2 + 85/3*e + 14, 7/3*e^3 - 10/3*e^2 - 85/3*e - 8, -4/3*e^3 + 4/3*e^2 + 43/3*e + 8, 4/3*e^3 - 4/3*e^2 - 67/3*e - 10, -e^3 + 4*e^2 + e - 26, -5/3*e^3 + 14/3*e^2 + 29/3*e - 20, 1/3*e^3 + 11/3*e^2 - 31/3*e - 32, -5*e - 2, -11/3*e^3 + 26/3*e^2 + 110/3*e - 16, 7/3*e^3 - 28/3*e^2 - 19/3*e + 52, -4/3*e^3 + 22/3*e^2 + 13/3*e - 46, -1/3*e^3 - 5/3*e^2 + 19/3*e + 26, -1/3*e^3 + 10/3*e^2 + 1/3*e - 10, -5/3*e^3 + 14/3*e^2 + 35/3*e - 36, -4/3*e^3 + 16/3*e^2 + 10/3*e - 38, -3*e^3 + 10*e^2 + 20*e - 28, 3*e^3 - 6*e^2 - 25*e + 14, -4/3*e^3 + 10/3*e^2 + 16/3*e - 8, 2*e^3 - 10*e^2 - 6*e + 44, 10/3*e^3 - 22/3*e^2 - 112/3*e + 22, -2/3*e^3 + 2/3*e^2 + 56/3*e - 14, -e^3 + 6*e^2 + 9*e - 50, -3*e^3 + 12*e^2 + 11*e - 56, -5/3*e^3 + 8/3*e^2 + 59/3*e - 30, -8/3*e^3 + 14/3*e^2 + 101/3*e + 6, -2*e^3 + 22*e + 8, -11/3*e^3 + 26/3*e^2 + 80/3*e - 36, -5/3*e^3 + 2/3*e^2 + 65/3*e + 4, 4*e^3 - 10*e^2 - 26*e + 36, -e^3 + 17*e + 16, 3*e^3 - 12*e^2 - 19*e + 46, -e^3 - 2*e^2 + 8*e + 34, 14/3*e^3 - 44/3*e^2 - 104/3*e + 48, -8/3*e^3 + 23/3*e^2 + 62/3*e - 8, 2*e^2 - 9*e - 6, 1/3*e^3 + 8/3*e^2 - 7/3*e - 38, -5/3*e^3 + 17/3*e^2 + 71/3*e - 32, 7/3*e^3 - 37/3*e^2 - 43/3*e + 64, 8/3*e^3 - 14/3*e^2 - 53/3*e - 4, 7/3*e^3 - 28/3*e^2 - 25/3*e + 28, -e^3 + 9*e + 30, -8/3*e^3 + 20/3*e^2 + 56/3*e - 14, -10/3*e^3 + 22/3*e^2 + 67/3*e - 10, -2*e^3 + 2*e^2 + 24*e + 22, 14/3*e^3 - 26/3*e^2 - 140/3*e + 18, -2*e + 16, 3*e^3 - 10*e^2 - 17*e + 28, -5*e^3 + 14*e^2 + 39*e - 48, -11/3*e^3 + 17/3*e^2 + 119/3*e + 4, 6*e^3 - 12*e^2 - 52*e + 16, 4/3*e^3 + 2/3*e^2 - 58/3*e - 22, -7/3*e^3 + 16/3*e^2 + 73/3*e - 10, -2/3*e^3 + 8/3*e^2 + 20/3*e - 20, -5/3*e^3 + 14/3*e^2 + 53/3*e - 4, -11/3*e^3 + 20/3*e^2 + 98/3*e + 2, e^3 + 4*e^2 - 21*e - 34, -8/3*e^3 + 17/3*e^2 + 44/3*e - 12, 2*e^3 - 7*e^2 - 16*e - 2, 1/3*e^3 - 22/3*e^2 + 41/3*e + 38, 8/3*e^3 - 29/3*e^2 - 50/3*e + 26, -7/3*e^3 + 16/3*e^2 + 25/3*e - 4, -1/3*e^3 + 13/3*e^2 + 19/3*e - 58, e^3 - 9*e - 6, 12*e - 18, 3*e^3 - 14*e^2 - 13*e + 84, -2/3*e^3 + 17/3*e^2 + 26/3*e - 54, -11/3*e^3 + 23/3*e^2 + 107/3*e - 2, -13/3*e^3 + 28/3*e^2 + 139/3*e - 22, e^3 + 4*e^2 - 19*e - 34, 4*e^3 - 16*e^2 - 30*e + 68]
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, -1/3*w^3 - 2/3*w^2 + 7/3*w + 4])] = 1
AL_eigenvalues[ZF.ideal([9, 3, -w])] = 1

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