/*
  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([31, 31, 2/3*w^3 + 1/3*w^2 - 11/3*w - 2])

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 - 4*x^3 - 5*x^2 + 25*x - 18
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 3*e^3 - 8*e^2 - 26*e + 42, 2*e^3 - 5*e^2 - 18*e + 26, 2*e^3 - 5*e^2 - 18*e + 26, -e - 1, e^3 - 2*e^2 - 9*e + 10, -2*e^3 + 5*e^2 + 19*e - 26, -1, -3*e^3 + 7*e^2 + 26*e - 32, -7*e^3 + 19*e^2 + 61*e - 96, 6*e^3 - 15*e^2 - 52*e + 78, -6*e^3 + 16*e^2 + 49*e - 80, e^3 - 2*e^2 - 10*e + 10, -e^3 + e^2 + 10*e, -e^3 + 3*e^2 + 8*e - 12, 12*e^3 - 33*e^2 - 103*e + 168, 2*e^3 - 6*e^2 - 18*e + 30, 2*e^3 - 4*e^2 - 18*e + 18, 13*e^3 - 35*e^2 - 110*e + 174, -4*e^3 + 10*e^2 + 37*e - 48, -12*e^3 + 33*e^2 + 102*e - 162, -6*e^3 + 15*e^2 + 53*e - 76, e^3 - 2*e^2 - 7*e + 8, -3*e^3 + 7*e^2 + 25*e - 32, -9*e^3 + 23*e^2 + 80*e - 122, -4*e^3 + 11*e^2 + 37*e - 54, -12*e^3 + 33*e^2 + 99*e - 162, -e^3 + 3*e^2 + 4*e - 16, 10*e^3 - 28*e^2 - 84*e + 140, -5*e^3 + 11*e^2 + 43*e - 50, 5*e^3 - 11*e^2 - 47*e + 58, 9*e^3 - 23*e^2 - 75*e + 108, 3*e^3 - 8*e^2 - 28*e + 54, 13*e^3 - 33*e^2 - 112*e + 172, -13*e^3 + 35*e^2 + 111*e - 182, -e^3 + 2*e^2 + 9*e - 4, 11*e^3 - 30*e^2 - 91*e + 140, 10*e^3 - 29*e^2 - 86*e + 156, -17*e^3 + 47*e^2 + 139*e - 234, -11*e^3 + 29*e^2 + 90*e - 154, 7*e^3 - 20*e^2 - 59*e + 112, -5*e^3 + 12*e^2 + 43*e - 52, 9*e^3 - 25*e^2 - 75*e + 126, 2*e^3 - 5*e^2 - 21*e + 18, -13*e^3 + 32*e^2 + 114*e - 172, 13*e^3 - 35*e^2 - 107*e + 182, -e^3 + 4*e^2 + 7*e - 22, -4*e^3 + 13*e^2 + 33*e - 70, -8*e^3 + 25*e^2 + 66*e - 130, 9*e^3 - 25*e^2 - 73*e + 128, -3*e^3 + 8*e^2 + 22*e - 38, -2*e^3 + 5*e^2 + 21*e - 20, -3*e^3 + 5*e^2 + 26*e - 24, 4*e^3 - 11*e^2 - 32*e + 60, -9*e^3 + 22*e^2 + 84*e - 108, 2*e^3 - 7*e^2 - 11*e + 42, 5*e^3 - 13*e^2 - 46*e + 60, -10*e^3 + 25*e^2 + 85*e - 126, 5*e^3 - 12*e^2 - 42*e + 66, -13*e^3 + 34*e^2 + 108*e - 162, 13*e^3 - 34*e^2 - 108*e + 160, 2*e^3 - 5*e^2 - 15*e + 22, -9*e^3 + 24*e^2 + 75*e - 114, -3*e^3 + 10*e^2 + 19*e - 42, -12*e^3 + 29*e^2 + 107*e - 154, -29*e^3 + 78*e^2 + 250*e - 400, 6*e^3 - 18*e^2 - 51*e + 88, -8*e^3 + 20*e^2 + 75*e - 100, -21*e^3 + 58*e^2 + 177*e - 296, 15*e^3 - 42*e^2 - 122*e + 218, 7*e^3 - 15*e^2 - 63*e + 80, 7*e^3 - 20*e^2 - 53*e + 106, -8*e^3 + 19*e^2 + 66*e - 92, -3*e^3 + 8*e^2 + 16*e - 30, -16*e^3 + 43*e^2 + 137*e - 240, -20*e^3 + 54*e^2 + 176*e - 270, -6*e^3 + 14*e^2 + 52*e - 54, e^3 - 5*e^2 - 11*e + 40, -15*e^3 + 41*e^2 + 133*e - 224, -6*e^3 + 15*e^2 + 48*e - 72, 8*e^3 - 21*e^2 - 73*e + 102, 9*e^3 - 24*e^2 - 68*e + 118, 22*e^3 - 58*e^2 - 183*e + 296, -4*e^3 + 9*e^2 + 38*e - 58, 9*e^3 - 23*e^2 - 80*e + 130, -5*e^3 + 14*e^2 + 47*e - 66, -21*e^3 + 56*e^2 + 170*e - 276, -21*e^3 + 54*e^2 + 180*e - 280, -6*e^3 + 18*e^2 + 44*e - 88, 11*e^3 - 30*e^2 - 87*e + 138, -9*e^3 + 23*e^2 + 78*e - 144, 6*e^3 - 16*e^2 - 48*e + 96, -6*e^3 + 11*e^2 + 56*e - 60, -3*e^3 + 10*e^2 + 22*e - 42, 3*e^3 - 5*e^2 - 30*e + 42, 9*e^3 - 25*e^2 - 70*e + 142, -25*e^3 + 67*e^2 + 218*e - 362, 2*e^2 - 7*e - 12, -16*e^3 + 43*e^2 + 131*e - 228, -19*e^3 + 48*e^2 + 160*e - 236, -19*e^3 + 51*e^2 + 171*e - 266, 19*e^3 - 51*e^2 - 166*e + 264, -30*e^3 + 79*e^2 + 252*e - 396, -5*e^3 + 17*e^2 + 38*e - 96, -28*e^3 + 73*e^2 + 246*e - 384, -23*e^3 + 65*e^2 + 200*e - 346, -6*e^3 + 15*e^2 + 47*e - 76, 24*e^3 - 63*e^2 - 203*e + 314, 2*e^3 - 2*e^2 - 23*e + 14, 3*e^3 - 11*e^2 - 19*e + 62, 3*e^2 + 2*e - 16, 28*e^3 - 75*e^2 - 239*e + 384, 26*e^3 - 72*e^2 - 217*e + 360, -4*e^3 + 9*e^2 + 37*e - 44, -7*e^3 + 19*e^2 + 59*e - 104, 22*e^3 - 61*e^2 - 198*e + 332, 21*e^3 - 53*e^2 - 171*e + 242, 3*e^3 - 8*e^2 - 31*e + 62, -11*e^3 + 31*e^2 + 93*e - 142, -6*e^3 + 14*e^2 + 43*e - 60, 4*e^3 - 7*e^2 - 51*e + 36, e^3 + 2*e^2 - 10*e - 14, -11*e^3 + 29*e^2 + 102*e - 170, 21*e^3 - 61*e^2 - 174*e + 310, 3*e^3 - 6*e^2 - 36*e + 46, 5*e^3 - 17*e^2 - 25*e + 90, -e^3 + 3*e^2 - e - 30, 6*e^3 - 20*e^2 - 41*e + 102, 4*e^3 - 11*e^2 - 39*e + 54, -23*e^3 + 59*e^2 + 206*e - 318, -e^2 - e - 12, e^3 - 6*e^2 - 8*e + 42, -6*e^3 + 16*e^2 + 52*e - 78, -19*e^3 + 52*e^2 + 171*e - 288, -10*e^3 + 32*e^2 + 75*e - 168, -6*e^3 + 19*e^2 + 50*e - 92, -3*e^3 + 11*e^2 + 27*e - 74, -8*e^3 + 21*e^2 + 55*e - 98, -17*e^3 + 45*e^2 + 143*e - 218, -28*e^3 + 76*e^2 + 235*e - 396, e^3 + 2*e^2 - 6*e - 30, 31*e^3 - 79*e^2 - 262*e + 406, 16*e^3 - 45*e^2 - 127*e + 220, 5*e^2 - 4*e - 60, -14*e^3 + 36*e^2 + 114*e - 180, -23*e^3 + 66*e^2 + 193*e - 332, 22*e^3 - 61*e^2 - 186*e + 298, -30*e^3 + 83*e^2 + 252*e - 424, -19*e^3 + 50*e^2 + 152*e - 244, 30*e^3 - 81*e^2 - 249*e + 406, 7*e^3 - 23*e^2 - 48*e + 112, e^3 - 2*e^2 - 8*e - 30, -17*e^3 + 45*e^2 + 147*e - 252, 9*e^3 - 24*e^2 - 83*e + 114, -14*e^3 + 42*e^2 + 113*e - 222, -3*e^3 + 10*e^2 + 33*e - 84, -13*e^3 + 35*e^2 + 103*e - 204, 12*e^3 - 33*e^2 - 98*e + 146, 9*e^3 - 19*e^2 - 83*e + 114, 6*e^3 - 14*e^2 - 51*e + 54, -34*e^3 + 92*e^2 + 298*e - 460, 29*e^3 - 81*e^2 - 240*e + 380]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([31, 31, 2/3*w^3 + 1/3*w^2 - 11/3*w - 2])] = 1

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