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

NN = ZF.ideal([16, 2, 2])

primes_array = [
[4, 2, -1/3*w^3 + 2/3*w^2 + 4/3*w - 1],\
[7, 7, 1/3*w^3 - 2/3*w^2 - 7/3*w + 1],\
[7, 7, w - 2],\
[7, 7, w - 1],\
[9, 3, w],\
[9, 3, -1/3*w^3 + 2/3*w^2 + 7/3*w - 2],\
[23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*w - 2],\
[23, 23, -2/3*w^3 + 4/3*w^2 + 11/3*w - 4],\
[31, 31, -2/3*w^3 + 1/3*w^2 + 14/3*w + 2],\
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 7],\
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 4],\
[47, 47, 1/3*w^3 - 5/3*w^2 - 1/3*w + 5],\
[47, 47, w^2 - w - 4],\
[73, 73, -w^3 + 3*w^2 + 4*w - 8],\
[73, 73, -2/3*w^3 + 7/3*w^2 - 1/3*w - 4],\
[73, 73, -w^3 + w^2 + 7*w + 1],\
[73, 73, 2/3*w^3 - 4/3*w^2 - 14/3*w + 5],\
[79, 79, w^2 - 5],\
[79, 79, -2/3*w^3 + 7/3*w^2 + 8/3*w - 6],\
[97, 97, -2/3*w^3 + 7/3*w^2 + 5/3*w - 4],\
[97, 97, 1/3*w^3 + 1/3*w^2 - 7/3*w - 5],\
[103, 103, w^3 - 3*w^2 - 2*w + 5],\
[103, 103, -4/3*w^3 + 11/3*w^2 + 13/3*w - 10],\
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 3],\
[113, 113, w^3 - 2*w^2 - 5*w + 1],\
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 1],\
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 10],\
[127, 127, -2/3*w^3 + 1/3*w^2 + 14/3*w + 1],\
[127, 127, -2/3*w^3 + 7/3*w^2 + 2/3*w - 6],\
[137, 137, -1/3*w^3 + 5/3*w^2 - 2/3*w - 6],\
[137, 137, 1/3*w^3 + 1/3*w^2 - 10/3*w - 1],\
[151, 151, -w^3 + 2*w^2 + 3*w - 5],\
[151, 151, 1/3*w^3 + 1/3*w^2 - 10/3*w - 7],\
[169, 13, -2/3*w^3 + 7/3*w^2 + 5/3*w - 2],\
[169, 13, 1/3*w^3 + 1/3*w^2 - 7/3*w - 7],\
[191, 191, 1/3*w^3 - 2/3*w^2 + 2/3*w - 2],\
[191, 191, 2*w - 5],\
[191, 191, -w^3 + 2*w^2 + 6*w - 2],\
[191, 191, 2/3*w^3 - 4/3*w^2 - 14/3*w - 1],\
[239, 239, 1/3*w^3 - 2/3*w^2 - 7/3*w - 3],\
[239, 239, w - 5],\
[241, 241, 1/3*w^3 + 1/3*w^2 - 10/3*w],\
[241, 241, 4/3*w^3 - 17/3*w^2 + 2/3*w + 9],\
[241, 241, -1/3*w^3 + 5/3*w^2 - 2/3*w - 7],\
[241, 241, -4/3*w^3 + 14/3*w^2 + 13/3*w - 14],\
[257, 257, -w - 4],\
[257, 257, -1/3*w^3 + 2/3*w^2 + 7/3*w - 6],\
[263, 263, -4/3*w^3 + 11/3*w^2 + 16/3*w - 8],\
[263, 263, 2/3*w^3 - 1/3*w^2 - 8/3*w - 3],\
[281, 281, -1/3*w^3 - 1/3*w^2 + 10/3*w - 2],\
[281, 281, -1/3*w^3 + 5/3*w^2 - 5/3*w - 5],\
[281, 281, -1/3*w^3 + 5/3*w^2 - 2/3*w - 9],\
[281, 281, -2/3*w^3 + 1/3*w^2 + 17/3*w],\
[289, 17, -w^3 + 2*w^2 + 4*w - 4],\
[289, 17, w^3 - 2*w^2 - 4*w + 2],\
[311, 311, -2/3*w^3 + 10/3*w^2 - 4/3*w - 9],\
[311, 311, -2/3*w^3 - 2/3*w^2 + 20/3*w + 5],\
[313, 313, -1/3*w^3 + 5/3*w^2 - 2/3*w - 8],\
[313, 313, -1/3*w^3 - 1/3*w^2 + 10/3*w - 1],\
[313, 313, w^3 - 4*w^2 - 2*w + 10],\
[313, 313, 1/3*w^3 + 4/3*w^2 - 10/3*w - 8],\
[337, 337, -4/3*w^3 + 5/3*w^2 + 19/3*w + 4],\
[337, 337, 5/3*w^3 - 13/3*w^2 - 17/3*w + 5],\
[353, 353, -4/3*w^3 + 8/3*w^2 + 25/3*w - 3],\
[353, 353, -1/3*w^3 - 1/3*w^2 + 16/3*w + 6],\
[353, 353, -2/3*w^3 + 10/3*w^2 + 2/3*w - 13],\
[353, 353, -5/3*w^3 + 7/3*w^2 + 29/3*w - 3],\
[367, 367, -5/3*w^3 + 4/3*w^2 + 35/3*w + 4],\
[367, 367, 2/3*w^3 - 7/3*w^2 - 8/3*w + 4],\
[367, 367, -4/3*w^3 + 14/3*w^2 + 1/3*w - 8],\
[367, 367, w^2 - 7],\
[383, 383, 4/3*w^3 - 14/3*w^2 - 10/3*w + 15],\
[383, 383, -5/3*w^3 + 13/3*w^2 + 20/3*w - 12],\
[383, 383, 2*w^2 - 2*w - 13],\
[383, 383, w^3 - w^2 - 8*w - 1],\
[401, 401, -4/3*w^3 + 11/3*w^2 + 22/3*w - 11],\
[401, 401, 5/3*w^3 - 10/3*w^2 - 29/3*w + 5],\
[409, 409, -w^3 + 2*w^2 + 3*w - 2],\
[409, 409, -1/3*w^3 + 8/3*w^2 - 2/3*w - 6],\
[409, 409, 1/3*w^3 - 8/3*w^2 + 2/3*w + 12],\
[409, 409, 4/3*w^3 - 8/3*w^2 - 19/3*w + 4],\
[433, 433, -w^3 + 3*w^2 - 1],\
[433, 433, -1/3*w^3 + 5/3*w^2 - 8/3*w - 5],\
[433, 433, -w^3 + w^2 + 8*w - 2],\
[433, 433, 5/3*w^3 - 7/3*w^2 - 32/3*w - 2],\
[439, 439, w^2 - 10],\
[439, 439, -2/3*w^3 + 4/3*w^2 + 17/3*w - 7],\
[449, 449, -2/3*w^3 + 4/3*w^2 + 11/3*w - 8],\
[449, 449, -4/3*w^3 + 5/3*w^2 + 22/3*w + 2],\
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 6],\
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 9],\
[463, 463, w^3 - 3*w^2 - 6*w + 11],\
[463, 463, -1/3*w^3 + 5/3*w^2 + 10/3*w - 4],\
[479, 479, -1/3*w^3 + 5/3*w^2 - 2/3*w + 1],\
[479, 479, 5/3*w^3 - 10/3*w^2 - 23/3*w + 9],\
[487, 487, -2/3*w^3 + 7/3*w^2 + 2/3*w - 8],\
[487, 487, -2/3*w^3 + 1/3*w^2 + 14/3*w - 1],\
[503, 503, w^3 - 4*w^2 - 2*w + 16],\
[503, 503, -1/3*w^3 + 5/3*w^2 - 2/3*w - 11],\
[521, 521, -4/3*w^3 + 11/3*w^2 + 10/3*w - 7],\
[521, 521, 4/3*w^3 - 5/3*w^2 - 22/3*w],\
[529, 23, -1/3*w^3 + 2/3*w^2 + 4/3*w - 6],\
[569, 569, -1/3*w^3 + 8/3*w^2 + 1/3*w - 10],\
[569, 569, -2/3*w^3 + 10/3*w^2 + 5/3*w - 10],\
[577, 577, -2/3*w^3 + 10/3*w^2 + 5/3*w - 12],\
[577, 577, -1/3*w^3 + 8/3*w^2 + 1/3*w - 8],\
[593, 593, w^3 - w^2 - 5*w - 5],\
[593, 593, 4/3*w^3 - 11/3*w^2 - 13/3*w + 4],\
[599, 599, 2*w^2 - 3*w - 7],\
[599, 599, 2/3*w^3 - 1/3*w^2 - 8/3*w - 4],\
[599, 599, -1/3*w^3 + 8/3*w^2 - 5/3*w - 9],\
[599, 599, -4/3*w^3 + 11/3*w^2 + 16/3*w - 7],\
[607, 607, 4/3*w^3 - 8/3*w^2 - 22/3*w + 9],\
[607, 607, -2/3*w^3 + 4/3*w^2 + 2/3*w - 5],\
[625, 5, -5],\
[631, 631, -1/3*w^3 + 2/3*w^2 + 13/3*w - 5],\
[631, 631, -2/3*w^3 + 4/3*w^2 + 17/3*w - 1],\
[641, 641, -2/3*w^3 + 7/3*w^2 + 14/3*w - 7],\
[641, 641, 2/3*w^3 - 7/3*w^2 - 14/3*w + 8],\
[647, 647, -2/3*w^3 - 5/3*w^2 + 26/3*w + 8],\
[647, 647, -1/3*w^3 + 8/3*w^2 + 1/3*w - 9],\
[647, 647, -2/3*w^3 + 10/3*w^2 + 5/3*w - 11],\
[647, 647, -2/3*w^3 + 13/3*w^2 - 10/3*w - 13],\
[673, 673, -4/3*w^3 + 5/3*w^2 + 25/3*w - 2],\
[673, 673, -w^3 + 3*w^2 + w - 7],\
[727, 727, -2/3*w^3 + 7/3*w^2 + 2/3*w - 11],\
[727, 727, -2/3*w^3 + 1/3*w^2 + 14/3*w - 4],\
[743, 743, w^2 - w - 10],\
[743, 743, 1/3*w^3 - 5/3*w^2 - 1/3*w - 1],\
[769, 769, -w^3 + 2*w^2 + 7*w - 4],\
[769, 769, 3*w - 2],\
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 8],\
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 10],\
[823, 823, -w^3 + 2*w^2 + 7*w - 5],\
[823, 823, 3*w - 1],\
[839, 839, 4/3*w^3 - 2/3*w^2 - 25/3*w - 8],\
[839, 839, 2/3*w^3 - 7/3*w^2 - 14/3*w + 12],\
[857, 857, -1/3*w^3 + 5/3*w^2 - 5/3*w - 7],\
[857, 857, -2/3*w^3 + 1/3*w^2 + 17/3*w - 2],\
[863, 863, -2/3*w^3 + 7/3*w^2 - 7/3*w - 2],\
[863, 863, 5/3*w^3 - 7/3*w^2 - 35/3*w + 1],\
[881, 881, -2/3*w^3 + 1/3*w^2 + 14/3*w + 8],\
[881, 881, -2/3*w^3 + 13/3*w^2 - 4/3*w - 18],\
[881, 881, -5/3*w^3 + 7/3*w^2 + 35/3*w + 3],\
[881, 881, -2/3*w^3 + 7/3*w^2 + 2/3*w + 1],\
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 7],\
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 18],\
[911, 911, 5/3*w^3 - 4/3*w^2 - 29/3*w - 3],\
[911, 911, 2*w^3 - 6*w^2 - 5*w + 13],\
[929, 929, 2*w^3 - 4*w^2 - 9*w + 2],\
[929, 929, 5/3*w^3 - 10/3*w^2 - 17/3*w],\
[929, 929, -4/3*w^3 + 14/3*w^2 + 13/3*w - 11],\
[929, 929, 1/3*w^3 + 4/3*w^2 - 7/3*w - 9],\
[937, 937, -w^3 + 4*w^2 + w - 13],\
[937, 937, 2/3*w^3 + 2/3*w^2 - 17/3*w - 3],\
[953, 953, -2*w^3 + w^2 + 15*w + 8],\
[953, 953, -5/3*w^3 + 19/3*w^2 - 1/3*w - 11],\
[961, 31, 4/3*w^3 - 8/3*w^2 - 16/3*w + 5],\
[967, 967, -4/3*w^3 + 2/3*w^2 + 31/3*w + 11],\
[967, 967, 7/3*w^3 - 5/3*w^2 - 46/3*w - 9],\
[977, 977, 2*w^3 - 5*w^2 - 8*w + 10],\
[977, 977, -w^3 + 3*w^2 + 2*w - 14],\
[977, 977, -4/3*w^3 + 8/3*w^2 + 25/3*w - 11],\
[977, 977, -w^3 + w^2 + 6*w - 7],\
[983, 983, -5/3*w^3 + 16/3*w^2 + 8/3*w - 12],\
[983, 983, -1/3*w^3 + 11/3*w^2 - 11/3*w - 15],\
[983, 983, 3*w^2 - 2*w - 16],\
[983, 983, -5/3*w^3 + 16/3*w^2 + 11/3*w - 14],\
[991, 991, 4/3*w^3 - 8/3*w^2 - 22/3*w + 1],\
[991, 991, 2/3*w^3 - 4/3*w^2 - 2/3*w - 3]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - 44*x^4 + 552*x^2 - 2048
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, 1/32*e^5 - 9/8*e^3 + 33/4*e, e, 1/32*e^5 - 9/8*e^3 + 33/4*e, 1/64*e^5 - 7/16*e^3 + 13/8*e, 1/64*e^5 - 7/16*e^3 + 13/8*e, -1/2*e^2 + 8, -1/2*e^2 + 8, 1/4*e^4 - 8*e^2 + 48, -1/64*e^5 + 11/16*e^3 - 53/8*e, -1/64*e^5 + 11/16*e^3 - 53/8*e, -1/32*e^5 + 9/8*e^3 - 37/4*e, -1/32*e^5 + 9/8*e^3 - 37/4*e, 5/64*e^5 - 43/16*e^3 + 137/8*e, 1/4*e^4 - 8*e^2 + 50, 1/4*e^4 - 8*e^2 + 50, 5/64*e^5 - 43/16*e^3 + 137/8*e, -1/32*e^5 + 9/8*e^3 - 33/4*e, -1/32*e^5 + 9/8*e^3 - 33/4*e, 1/64*e^5 - 7/16*e^3 + 29/8*e, 1/64*e^5 - 7/16*e^3 + 29/8*e, -1/4*e^4 + 8*e^2 - 48, -1/4*e^4 + 8*e^2 - 48, -5/64*e^5 + 47/16*e^3 - 177/8*e, -5/64*e^5 + 47/16*e^3 - 177/8*e, 3/64*e^5 - 25/16*e^3 + 71/8*e, 3/64*e^5 - 25/16*e^3 + 71/8*e, -1/4*e^4 + 19/2*e^2 - 72, -1/4*e^4 + 19/2*e^2 - 72, -1/4*e^4 + 15/2*e^2 - 38, -1/4*e^4 + 15/2*e^2 - 38, -1/4*e^4 + 9*e^2 - 64, -1/4*e^4 + 9*e^2 - 64, 3/64*e^5 - 25/16*e^3 + 111/8*e, 3/64*e^5 - 25/16*e^3 + 111/8*e, -5/32*e^5 + 41/8*e^3 - 129/4*e, -1/8*e^5 + 9/2*e^3 - 34*e, -5/32*e^5 + 41/8*e^3 - 129/4*e, -1/8*e^5 + 9/2*e^3 - 34*e, -1/2*e^4 + 16*e^2 - 104, -1/2*e^4 + 16*e^2 - 104, 1/4*e^4 - 10*e^2 + 74, 5/64*e^5 - 51/16*e^3 + 225/8*e, 1/4*e^4 - 10*e^2 + 74, 5/64*e^5 - 51/16*e^3 + 225/8*e, 1/2*e^4 - 16*e^2 + 98, 1/2*e^4 - 16*e^2 + 98, 1/16*e^5 - 9/4*e^3 + 37/2*e, 1/16*e^5 - 9/4*e^3 + 37/2*e, -1/4*e^4 + 10*e^2 - 78, -1/2*e^4 + 17*e^2 - 102, -1/4*e^4 + 10*e^2 - 78, -1/2*e^4 + 17*e^2 - 102, -1/4*e^4 + 7*e^2 - 14, 5/2*e^2 - 30, 1/4*e^4 - 11*e^2 + 88, 1/4*e^4 - 11*e^2 + 88, -1/4*e^4 + 10*e^2 - 86, -1/4*e^4 + 10*e^2 - 86, -1/64*e^5 + 3/16*e^3 + 43/8*e, -1/64*e^5 + 3/16*e^3 + 43/8*e, -1/64*e^5 + 11/16*e^3 - 85/8*e, -1/64*e^5 + 11/16*e^3 - 85/8*e, -9/64*e^5 + 75/16*e^3 - 261/8*e, 1/4*e^4 - 15/2*e^2 + 50, -9/64*e^5 + 75/16*e^3 - 261/8*e, 1/4*e^4 - 15/2*e^2 + 50, -2*e^2 + 24, -1/8*e^5 + 4*e^3 - 22*e, -2*e^2 + 24, -1/8*e^5 + 4*e^3 - 22*e, 3*e^2 - 48, 3*e^2 - 48, -5/32*e^5 + 41/8*e^3 - 117/4*e, -5/32*e^5 + 41/8*e^3 - 117/4*e, -3/64*e^5 + 29/16*e^3 - 127/8*e, -3/64*e^5 + 29/16*e^3 - 127/8*e, -1/2*e^4 + 29/2*e^2 - 70, 3/64*e^5 - 25/16*e^3 + 39/8*e, 3/64*e^5 - 25/16*e^3 + 39/8*e, -1/2*e^4 + 29/2*e^2 - 70, 9/64*e^5 - 63/16*e^3 + 125/8*e, e^4 - 33*e^2 + 210, e^4 - 33*e^2 + 210, 9/64*e^5 - 63/16*e^3 + 125/8*e, -9/32*e^5 + 81/8*e^3 - 281/4*e, -9/32*e^5 + 81/8*e^3 - 281/4*e, 3/4*e^4 - 25*e^2 + 154, 3/4*e^4 - 25*e^2 + 154, 3/64*e^5 - 33/16*e^3 + 167/8*e, 3/64*e^5 - 33/16*e^3 + 167/8*e, 1/8*e^5 - 5*e^3 + 38*e, 1/8*e^5 - 5*e^3 + 38*e, e^4 - 69/2*e^2 + 224, e^4 - 69/2*e^2 + 224, 1/4*e^4 - 5*e^2, 1/4*e^4 - 5*e^2, -1/4*e^4 + 11/2*e^2, -1/4*e^4 + 11/2*e^2, 1/4*e^4 - 7*e^2 + 42, 1/4*e^4 - 7*e^2 + 42, -1/2*e^4 + 33/2*e^2 - 78, -1/64*e^5 + 3/16*e^3 - 13/8*e, -1/64*e^5 + 3/16*e^3 - 13/8*e, -17/64*e^5 + 147/16*e^3 - 517/8*e, -17/64*e^5 + 147/16*e^3 - 517/8*e, -1/64*e^5 - 5/16*e^3 + 99/8*e, -1/64*e^5 - 5/16*e^3 + 99/8*e, -3/32*e^5 + 35/8*e^3 - 175/4*e, -3/16*e^5 + 23/4*e^3 - 67/2*e, -3/32*e^5 + 35/8*e^3 - 175/4*e, -3/16*e^5 + 23/4*e^3 - 67/2*e, -1/4*e^4 + 10*e^2 - 88, -1/4*e^4 + 10*e^2 - 88, 1/4*e^4 - 11*e^2 + 114, 3/8*e^5 - 25/2*e^3 + 84*e, 3/8*e^5 - 25/2*e^3 + 84*e, -15/64*e^5 + 129/16*e^3 - 387/8*e, -15/64*e^5 + 129/16*e^3 - 387/8*e, 1/2*e^4 - 19*e^2 + 128, 1/8*e^5 - 4*e^3 + 17*e, 1/8*e^5 - 4*e^3 + 17*e, 1/2*e^4 - 19*e^2 + 128, -1/2*e^4 + 14*e^2 - 70, -1/2*e^4 + 14*e^2 - 70, -3/4*e^4 + 26*e^2 - 168, -3/4*e^4 + 26*e^2 - 168, 1/4*e^4 - 7*e^2 + 24, 1/4*e^4 - 7*e^2 + 24, 17/64*e^5 - 143/16*e^3 + 421/8*e, 17/64*e^5 - 143/16*e^3 + 421/8*e, -5/64*e^5 + 31/16*e^3 - 1/8*e, -5/64*e^5 + 31/16*e^3 - 1/8*e, 1/16*e^5 - 11/4*e^3 + 39/2*e, 1/16*e^5 - 11/4*e^3 + 39/2*e, -5/16*e^5 + 43/4*e^3 - 139/2*e, -5/16*e^5 + 43/4*e^3 - 139/2*e, 3/4*e^4 - 23*e^2 + 122, 3/4*e^4 - 23*e^2 + 122, 1/32*e^5 - 9/8*e^3 + 25/4*e, 1/32*e^5 - 9/8*e^3 + 25/4*e, 3/2*e^2 - 30, -17/64*e^5 + 139/16*e^3 - 469/8*e, -17/64*e^5 + 139/16*e^3 - 469/8*e, 3/2*e^2 - 30, 1/4*e^5 - 17/2*e^3 + 51*e, 1/4*e^5 - 17/2*e^3 + 51*e, 3*e^2 - 72, 3*e^2 - 72, 18, 18, -3/64*e^5 + 29/16*e^3 - 31/8*e, -3/64*e^5 + 29/16*e^3 - 31/8*e, -2*e^2 + 34, -2*e^2 + 34, -1/2*e^4 + 27/2*e^2 - 70, -1/2*e^4 + 27/2*e^2 - 70, 1/4*e^4 - 7*e^2 + 50, -1/2*e^4 + 18*e^2 - 104, -1/2*e^4 + 18*e^2 - 104, e^2 - 22, 1/2*e^4 - 21*e^2 + 178, e^2 - 22, 1/2*e^4 - 21*e^2 + 178, 1/4*e^4 - 9*e^2 + 104, -3/16*e^5 + 23/4*e^3 - 61/2*e, -3/16*e^5 + 23/4*e^3 - 61/2*e, 1/4*e^4 - 9*e^2 + 104, -5/16*e^5 + 43/4*e^3 - 141/2*e, -5/16*e^5 + 43/4*e^3 - 141/2*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([4, 2, -1/3*w^3 + 2/3*w^2 + 4/3*w - 1])] = 1

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