/*
  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, -1, -7, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([9, 9, w])

primes_array = [
[2, 2, -w^2 + w + 4],\
[3, 3, -w^2 + w + 3],\
[5, 5, w^2 - 4],\
[8, 2, w^3 - w^2 - 4*w + 5],\
[17, 17, -w^2 + w + 5],\
[27, 3, -w^3 + 4*w - 2],\
[29, 29, -w^2 + w + 1],\
[29, 29, -w^3 + 4*w + 2],\
[37, 37, -2*w^3 + 3*w^2 + 9*w - 13],\
[41, 41, w^3 - w^2 - 5*w + 2],\
[43, 43, -w^3 + w^2 + 3*w - 4],\
[61, 61, w^2 - 2*w - 2],\
[61, 61, 2*w^2 - w - 8],\
[67, 67, -4*w^3 + 5*w^2 + 18*w - 22],\
[73, 73, -w^2 - 2*w + 2],\
[79, 79, w^2 - 2*w - 4],\
[89, 89, w^2 + w - 5],\
[89, 89, 2*w^3 - 2*w^2 - 9*w + 8],\
[89, 89, w^3 - 5*w - 5],\
[89, 89, -w^3 + 2*w^2 + 4*w - 4],\
[97, 97, w^3 + w^2 - 5*w - 4],\
[101, 101, -2*w^3 + 2*w^2 + 8*w - 11],\
[103, 103, w^3 - 5*w + 1],\
[109, 109, 2*w - 1],\
[109, 109, -w^3 + 2*w^2 + 3*w - 7],\
[113, 113, 2*w^2 - 7],\
[125, 5, -w^3 + 3*w^2 + 3*w - 8],\
[127, 127, 2*w^3 - 4*w^2 - 11*w + 16],\
[131, 131, 3*w^2 + 2*w - 8],\
[131, 131, -2*w^3 + 4*w^2 + 8*w - 13],\
[137, 137, w^3 - 3*w^2 - 7*w + 10],\
[149, 149, -w^3 + 4*w - 4],\
[149, 149, w^2 - 2*w - 8],\
[157, 157, -w - 4],\
[163, 163, -w^3 + 2*w^2 + 6*w - 10],\
[167, 167, -w^3 + 2*w^2 + 5*w - 11],\
[167, 167, -3*w^2 - 2*w + 10],\
[167, 167, -w^2 - w + 7],\
[167, 167, 2*w^2 - 2*w - 5],\
[173, 173, -3*w^3 + 2*w^2 + 13*w - 11],\
[179, 179, w^3 - 3*w^2 - 5*w + 10],\
[191, 191, -2*w^3 - 3*w^2 + 10*w + 14],\
[191, 191, 3*w^2 + w - 7],\
[193, 193, w^2 - w + 1],\
[193, 193, -w^2 - w - 1],\
[227, 227, 4*w^3 - 4*w^2 - 17*w + 20],\
[227, 227, -3*w^3 + 4*w^2 + 13*w - 17],\
[229, 229, -w^3 + 6*w - 2],\
[233, 233, -3*w^3 + 3*w^2 + 13*w - 16],\
[241, 241, -w^2 + 2*w - 2],\
[263, 263, 2*w^3 - 5*w^2 - 12*w + 16],\
[263, 263, -3*w^3 + 4*w^2 + 13*w - 19],\
[269, 269, w^3 + 2*w^2 - 5*w - 7],\
[269, 269, -w^3 - 2*w^2 + w + 5],\
[281, 281, 2*w^3 - 2*w^2 - 8*w + 1],\
[281, 281, w^2 - 8],\
[289, 17, 2*w^3 - 10*w - 7],\
[307, 307, w^3 + 2*w^2 - 6*w - 8],\
[307, 307, -2*w^3 + 2*w^2 + 8*w - 7],\
[313, 313, 5*w^3 - 6*w^2 - 24*w + 28],\
[313, 313, 2*w^3 - 2*w^2 - 10*w + 5],\
[337, 337, w^3 - 2*w - 2],\
[347, 347, -w^3 + 4*w^2 + w - 13],\
[349, 349, -2*w^3 + w^2 + 9*w - 7],\
[353, 353, 2*w^3 + w^2 - 7*w + 1],\
[359, 359, -3*w^2 - 3*w + 5],\
[359, 359, 3*w^3 - 5*w^2 - 13*w + 16],\
[367, 367, 3*w^3 - 6*w^2 - 15*w + 23],\
[367, 367, -w^3 + 2*w^2 + 6*w - 4],\
[367, 367, -2*w^3 + 2*w^2 + 8*w - 5],\
[367, 367, 2*w^3 - 3*w^2 - 12*w + 16],\
[373, 373, -2*w^3 + 4*w^2 + 12*w - 17],\
[373, 373, 3*w^2 + 4*w - 8],\
[383, 383, -w^3 + 2*w^2 + 3*w - 13],\
[383, 383, 2*w^2 - 2*w - 11],\
[389, 389, 5*w^3 - 6*w^2 - 24*w + 26],\
[389, 389, -w^3 + 3*w - 5],\
[401, 401, w^3 + 2*w^2 - 4*w - 10],\
[401, 401, 2*w^3 - w^2 - 7*w + 5],\
[409, 409, -2*w^2 + 3*w + 10],\
[419, 419, w^2 - 3*w - 1],\
[419, 419, -w^3 + 5*w^2 + 9*w - 14],\
[421, 421, 2*w^3 - 8*w - 7],\
[431, 431, -3*w^2 + 16],\
[431, 431, -3*w^3 + 11*w - 7],\
[433, 433, 4*w^3 - 7*w^2 - 18*w + 26],\
[439, 439, 3*w - 2],\
[439, 439, -w^3 + w^2 + 5*w - 8],\
[443, 443, 2*w^3 - w^2 - 6*w + 4],\
[443, 443, w^2 + 3*w - 1],\
[449, 449, -2*w^3 + 5*w^2 + 7*w - 17],\
[457, 457, 4*w^3 - 6*w^2 - 19*w + 22],\
[479, 479, 3*w^2 - 3*w - 13],\
[479, 479, -4*w^3 + 8*w^2 + 22*w - 31],\
[479, 479, 3*w^3 - 7*w^2 - 17*w + 26],\
[479, 479, -6*w^3 + 7*w^2 + 28*w - 32],\
[487, 487, -2*w^3 + 4*w^2 + 7*w - 8],\
[491, 491, 3*w^2 - w - 11],\
[491, 491, 4*w^3 - 5*w^2 - 18*w + 20],\
[521, 521, -3*w^3 + 11*w - 5],\
[521, 521, 3*w^3 - 3*w^2 - 15*w + 14],\
[523, 523, -2*w^3 + 5*w^2 + 8*w - 14],\
[529, 23, -2*w^3 + 3*w^2 + 8*w - 14],\
[529, 23, 3*w^3 - 2*w^2 - 14*w + 4],\
[541, 541, -w^3 + 2*w^2 + 7*w - 11],\
[541, 541, -w^3 - w^2 + 5*w - 2],\
[547, 547, -3*w - 2],\
[547, 547, w^3 - 2*w^2 - 7*w + 7],\
[547, 547, w^3 - 4*w^2 - 8*w + 10],\
[547, 547, 2*w^3 - 3*w^2 - 7*w + 11],\
[563, 563, 2*w^3 - 4*w^2 - 12*w + 13],\
[563, 563, -w^3 + 4*w^2 + w - 7],\
[571, 571, -w^3 + 2*w^2 + 2*w - 8],\
[571, 571, 3*w^2 - 10],\
[577, 577, 2*w^3 + w^2 - 7*w - 5],\
[599, 599, 2*w^3 - 7*w + 2],\
[599, 599, -4*w^3 + 7*w^2 + 21*w - 29],\
[617, 617, -2*w^3 + w^2 + 11*w + 1],\
[631, 631, 3*w^2 + w - 13],\
[631, 631, 3*w^2 - 4*w - 10],\
[647, 647, 3*w^3 - 4*w^2 - 13*w + 11],\
[677, 677, -2*w^3 + 4*w^2 + 8*w - 17],\
[691, 691, -w^3 + 3*w^2 + 3*w - 14],\
[733, 733, -w^3 + 4*w^2 + 3*w - 11],\
[733, 733, 3*w^3 - w^2 - 15*w - 2],\
[733, 733, -w^3 + w^2 + 7*w - 2],\
[733, 733, 3*w^3 - 4*w^2 - 16*w + 16],\
[739, 739, -3*w^3 + w^2 + 11*w - 8],\
[743, 743, -3*w^3 + 2*w^2 + 12*w - 14],\
[751, 751, -2*w^3 + w^2 + 8*w - 2],\
[761, 761, -2*w^3 + 2*w^2 + 7*w - 10],\
[761, 761, 2*w^3 - 4*w^2 - 3*w + 8],\
[769, 769, -w^3 + w^2 + 7*w - 4],\
[769, 769, 2*w^3 + 2*w^2 - 8*w - 11],\
[787, 787, -2*w^3 + 3*w^2 + 6*w - 10],\
[787, 787, -2*w^2 - 3*w + 8],\
[797, 797, -2*w^3 + w^2 + 7*w - 1],\
[797, 797, -2*w^3 - 2*w^2 + 11*w + 10],\
[809, 809, -2*w^3 + 5*w^2 + 7*w - 19],\
[811, 811, 2*w^3 - 2*w^2 - 7*w - 2],\
[811, 811, 3*w^3 - 2*w^2 - 11*w - 1],\
[827, 827, 3*w^3 - 2*w^2 - 14*w + 10],\
[829, 829, 2*w^3 - 2*w^2 - 9*w + 2],\
[829, 829, 4*w^2 + 2*w - 11],\
[841, 29, 2*w^3 + w^2 - 6*w - 2],\
[863, 863, 2*w^3 - 8*w - 1],\
[883, 883, 3*w^3 - 4*w^2 - 15*w + 13],\
[887, 887, 3*w^3 - 2*w^2 - 12*w + 10],\
[907, 907, 2*w^3 - 3*w^2 - 9*w + 7],\
[911, 911, 2*w^3 + w^2 - 11*w - 5],\
[919, 919, -2*w^3 + 5*w - 8],\
[929, 929, -w^3 + 4*w^2 + 3*w - 13],\
[937, 937, w^3 - 5*w - 7],\
[937, 937, -2*w^3 + 3*w^2 + 10*w - 8],\
[971, 971, -2*w^3 + 4*w^2 + 11*w - 20],\
[971, 971, -4*w^3 + 4*w^2 + 19*w - 20],\
[977, 977, -4*w^3 + 6*w^2 + 19*w - 28],\
[983, 983, -4*w^2 - 4*w + 7],\
[991, 991, -5*w^3 + 8*w^2 + 26*w - 32],\
[991, 991, -6*w^3 + 6*w^2 + 26*w - 31],\
[997, 997, -3*w^3 + 8*w^2 + 18*w - 26]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^8 - 16*x^6 + 82*x^4 - 148*x^2 + 73
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 0, -1/4*e^6 + 13/4*e^4 - 47/4*e^2 + 43/4, 1/2*e^4 - 4*e^2 + 13/2, -1/4*e^6 + 15/4*e^4 - 59/4*e^2 + 45/4, -e^3 + 5*e, -1/4*e^7 + 13/4*e^5 - 47/4*e^3 + 35/4*e, 1/4*e^7 - 17/4*e^5 + 83/4*e^3 - 99/4*e, 1/4*e^7 - 15/4*e^5 + 67/4*e^3 - 85/4*e, -1/4*e^7 + 11/4*e^5 - 23/4*e^3 - 19/4*e, 2*e, -3/4*e^7 + 41/4*e^5 - 161/4*e^3 + 163/4*e, 1/4*e^7 - 13/4*e^5 + 51/4*e^3 - 71/4*e, e^3 - 3*e, -1/4*e^6 + 11/4*e^4 - 35/4*e^2 + 57/4, -1/2*e^6 + 13/2*e^4 - 53/2*e^2 + 61/2, 1/4*e^6 - 15/4*e^4 + 55/4*e^2 - 1/4, -3/4*e^6 + 41/4*e^4 - 157/4*e^2 + 127/4, 1/4*e^7 - 11/4*e^5 + 19/4*e^3 + 47/4*e, -1/4*e^6 + 9/4*e^4 - 15/4*e^2 - 1/4, -1/4*e^7 + 13/4*e^5 - 55/4*e^3 + 91/4*e, -1/2*e^6 + 15/2*e^4 - 61/2*e^2 + 47/2, -2*e^4 + 19*e^2 - 27, 3/4*e^7 - 39/4*e^5 + 145/4*e^3 - 149/4*e, -1/2*e^7 + 15/2*e^5 - 69/2*e^3 + 87/2*e, 1/4*e^6 - 7/4*e^4 - 9/4*e^2 + 55/4, -1/4*e^7 + 19/4*e^5 - 103/4*e^3 + 125/4*e, e^5 - 10*e^3 + 25*e, -e^3 + 7*e, 1/2*e^6 - 13/2*e^4 + 39/2*e^2 + 5/2, 3/4*e^7 - 41/4*e^5 + 157/4*e^3 - 151/4*e, 1/2*e^6 - 19/2*e^4 + 97/2*e^2 - 95/2, -1/4*e^6 + 5/4*e^4 + 25/4*e^2 - 85/4, -3/4*e^6 + 29/4*e^4 - 57/4*e^2 + 15/4, -e^7 + 14*e^5 - 57*e^3 + 58*e, 1/2*e^7 - 11/2*e^5 + 27/2*e^3 + 3/2*e, 1/2*e^7 - 13/2*e^5 + 45/2*e^3 - 33/2*e, e^7 - 14*e^5 + 55*e^3 - 54*e, 3/2*e^6 - 35/2*e^4 + 103/2*e^2 - 51/2, e^5 - 10*e^3 + 17*e, -3/2*e^6 + 37/2*e^4 - 127/2*e^2 + 97/2, e^6 - 13*e^4 + 47*e^2 - 39, -e^4 + 9*e^2 - 14, -1/4*e^6 + 7/4*e^4 + 21/4*e^2 - 59/4, e^6 - 14*e^4 + 59*e^2 - 60, 1/2*e^6 - 13/2*e^4 + 41/2*e^2 - 33/2, -1/2*e^6 + 9/2*e^4 - 15/2*e^2 + 7/2, -1/4*e^7 + 19/4*e^5 - 91/4*e^3 + 65/4*e, 3/2*e^6 - 41/2*e^4 + 159/2*e^2 - 145/2, 3/2*e^7 - 43/2*e^5 + 173/2*e^3 - 169/2*e, 1/2*e^6 - 15/2*e^4 + 61/2*e^2 - 51/2, 1/2*e^7 - 13/2*e^5 + 41/2*e^3 - 5/2*e, -1/2*e^7 + 13/2*e^5 - 43/2*e^3 + 19/2*e, -1/4*e^6 + 21/4*e^4 - 99/4*e^2 + 79/4, -3/4*e^6 + 31/4*e^4 - 61/4*e^2 - 55/4, -1/2*e^7 + 15/2*e^5 - 57/2*e^3 + 27/2*e, 1/4*e^7 - 11/4*e^5 + 23/4*e^3 + 3/4*e, 1/2*e^6 - 5/2*e^4 - 23/2*e^2 + 63/2, -3/2*e^7 + 41/2*e^5 - 163/2*e^3 + 169/2*e, -3/2*e^6 + 35/2*e^4 - 107/2*e^2 + 91/2, -1/4*e^7 + 17/4*e^5 - 91/4*e^3 + 115/4*e, 5/4*e^6 - 63/4*e^4 + 187/4*e^2 - 57/4, -1/2*e^7 + 15/2*e^5 - 75/2*e^3 + 133/2*e, 1/2*e^6 - 11/2*e^4 + 41/2*e^2 - 63/2, 5/4*e^6 - 53/4*e^4 + 99/4*e^2 + 101/4, 1/2*e^7 - 17/2*e^5 + 81/2*e^3 - 85/2*e, -3/2*e^6 + 35/2*e^4 - 93/2*e^2 - 3/2, -2*e^3 + 20*e, -e^4 + 5*e^2 + 10, 3/2*e^6 - 39/2*e^4 + 143/2*e^2 - 103/2, 1/2*e^7 - 11/2*e^5 + 25/2*e^3 + 25/2*e, 5/4*e^6 - 69/4*e^4 + 275/4*e^2 - 219/4, 9/4*e^6 - 121/4*e^4 + 443/4*e^2 - 379/4, -3/2*e^6 + 39/2*e^4 - 141/2*e^2 + 137/2, 2*e^6 - 24*e^4 + 77*e^2 - 45, -5/4*e^7 + 71/4*e^5 - 279/4*e^3 + 229/4*e, 3/4*e^7 - 43/4*e^5 + 173/4*e^3 - 181/4*e, 5/4*e^7 - 73/4*e^5 + 307/4*e^3 - 311/4*e, 5/4*e^6 - 79/4*e^4 + 379/4*e^2 - 409/4, 1/4*e^7 - 23/4*e^5 + 131/4*e^3 - 133/4*e, -e^6 + 9*e^4 - 15*e^2 + 3, e^4 - 11*e^2, 3/4*e^7 - 39/4*e^5 + 141/4*e^3 - 129/4*e, e^7 - 13*e^5 + 46*e^3 - 40*e, -3/2*e^6 + 35/2*e^4 - 103/2*e^2 + 27/2, -1/2*e^7 + 17/2*e^5 - 77/2*e^3 + 53/2*e, -2*e^6 + 24*e^4 - 69*e^2 + 25, -3*e^5 + 30*e^3 - 57*e, e^7 - 14*e^5 + 60*e^3 - 75*e, -e^7 + 13*e^5 - 45*e^3 + 31*e, -e^7 + 12*e^5 - 34*e^3 + e, -1/2*e^6 + 11/2*e^4 - 13/2*e^2 - 53/2, -1/2*e^7 + 15/2*e^5 - 73/2*e^3 + 131/2*e, e^7 - 12*e^5 + 36*e^3 - 11*e, -3/2*e^6 + 35/2*e^4 - 95/2*e^2 + 27/2, -1/2*e^7 + 9/2*e^5 - 3/2*e^3 - 77/2*e, 10*e, e^7 - 14*e^5 + 55*e^3 - 60*e, 1/2*e^6 - 11/2*e^4 + 23/2*e^2 + 3/2, -1/4*e^6 + 23/4*e^4 - 147/4*e^2 + 269/4, 3/4*e^7 - 45/4*e^5 + 193/4*e^3 - 215/4*e, 1/2*e^7 - 15/2*e^5 + 73/2*e^3 - 99/2*e, -3/4*e^6 + 59/4*e^4 - 305/4*e^2 + 329/4, -1/4*e^7 + 17/4*e^5 - 95/4*e^3 + 231/4*e, 3/4*e^6 - 27/4*e^4 + 25/4*e^2 + 47/4, -e^5 + 12*e^3 - 43*e, -1/2*e^7 + 11/2*e^5 - 23/2*e^3 - 7/2*e, 1/2*e^7 - 9/2*e^5 + 11/2*e^3 + 13/2*e, -e^6 + 13*e^4 - 34*e^2 - 20, -2*e^6 + 29*e^4 - 115*e^2 + 106, 3*e^5 - 31*e^3 + 68*e, -3/2*e^7 + 41/2*e^5 - 151/2*e^3 + 105/2*e, -5/2*e^6 + 65/2*e^4 - 239/2*e^2 + 243/2, 1/2*e^6 - 7/2*e^4 - 5/2*e^2 + 11/2, -2*e^7 + 28*e^5 - 113*e^3 + 117*e, 3/2*e^6 - 35/2*e^4 + 115/2*e^2 - 135/2, 5/2*e^6 - 67/2*e^4 + 227/2*e^2 - 149/2, -3/2*e^7 + 41/2*e^5 - 155/2*e^3 + 141/2*e, e^5 - 15*e^3 + 50*e, -2*e^4 + 24*e^2 - 54, -1/2*e^7 + 17/2*e^5 - 89/2*e^3 + 141/2*e, 3/4*e^6 - 41/4*e^4 + 137/4*e^2 + 13/4, -e^6 + 12*e^4 - 36*e^2 + 27, 3/4*e^7 - 51/4*e^5 + 249/4*e^3 - 329/4*e, -3/4*e^6 + 23/4*e^4 + 31/4*e^2 - 211/4, 3/4*e^6 - 37/4*e^4 + 85/4*e^2 + 85/4, -1/4*e^7 + 11/4*e^5 - 27/4*e^3 - 7/4*e, -1/2*e^6 + 13/2*e^4 - 39/2*e^2 - 13/2, 3*e^5 - 27*e^3 + 48*e, -4*e^4 + 31*e^2 - 21, -3/2*e^7 + 45/2*e^5 - 193/2*e^3 + 195/2*e, 1/4*e^7 - 17/4*e^5 + 111/4*e^3 - 287/4*e, 2*e^7 - 29*e^5 + 118*e^3 - 115*e, 7/4*e^6 - 103/4*e^4 + 417/4*e^2 - 361/4, 1/2*e^7 - 15/2*e^5 + 79/2*e^3 - 137/2*e, 3/2*e^6 - 41/2*e^4 + 161/2*e^2 - 91/2, -9/4*e^7 + 121/4*e^5 - 451/4*e^3 + 371/4*e, -e^7 + 14*e^5 - 50*e^3 + 23*e, 1/4*e^7 - 9/4*e^5 + 23/4*e^3 - 95/4*e, 3*e^7 - 41*e^5 + 158*e^3 - 150*e, 7/2*e^6 - 89/2*e^4 + 297/2*e^2 - 191/2, 3*e^6 - 40*e^4 + 147*e^2 - 110, 9/4*e^7 - 129/4*e^5 + 531/4*e^3 - 523/4*e, -e^7 + 14*e^5 - 56*e^3 + 61*e, 1/2*e^7 - 11/2*e^5 + 31/2*e^3 - 1/2*e, -3/2*e^6 + 39/2*e^4 - 145/2*e^2 + 157/2, 3/2*e^6 - 37/2*e^4 + 127/2*e^2 - 177/2, 2*e^5 - 14*e^3 + 8*e, -5/2*e^6 + 65/2*e^4 - 239/2*e^2 + 259/2, -3*e^6 + 39*e^4 - 137*e^2 + 109, 3*e^6 - 39*e^4 + 128*e^2 - 86, 7/4*e^6 - 87/4*e^4 + 237/4*e^2 - 29/4, 7/4*e^6 - 81/4*e^4 + 241/4*e^2 - 207/4, e^7 - 15*e^5 + 69*e^3 - 91*e, e^6 - 6*e^4 - 15*e^2 + 52, -e^7 + 16*e^5 - 75*e^3 + 74*e, 9/4*e^6 - 93/4*e^4 + 203/4*e^2 + 9/4, -2*e^6 + 24*e^4 - 80*e^2 + 82, -e^4 + 12*e^2 - 15, -1/2*e^6 + 19/2*e^4 - 85/2*e^2 + 23/2, -11/4*e^7 + 157/4*e^5 - 653/4*e^3 + 659/4*e]
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^2 + w + 3])] = 1

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