/*
  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([7, 6, -5, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([28, 14, -w^3 + 3*w^2 + w - 4])

primes_array = [
[4, 2, -w^2 + w + 3],\
[7, 7, w],\
[7, 7, -w^2 + 2*w + 1],\
[7, 7, w^2 - 2],\
[7, 7, w - 1],\
[23, 23, -w^3 + w^2 + 3*w - 1],\
[23, 23, w^3 - 2*w^2 - 2*w + 2],\
[31, 31, w^2 - 5],\
[31, 31, -w^2 + 2*w + 4],\
[41, 41, -w^3 + 2*w^2 + 2*w - 6],\
[41, 41, w^3 - w^2 - 3*w - 3],\
[47, 47, w^2 - 2*w - 5],\
[47, 47, w^2 - 6],\
[71, 71, -w^3 + 2*w^2 + 3*w - 1],\
[71, 71, -w^3 + w^2 + 4*w - 3],\
[79, 79, -w - 3],\
[79, 79, w - 4],\
[81, 3, -3],\
[89, 89, w^2 - 3*w - 2],\
[89, 89, w^2 + w - 4],\
[97, 97, 2*w^3 - 5*w^2 - 4*w + 9],\
[97, 97, -2*w^3 + w^2 + 8*w + 2],\
[113, 113, w^3 - 2*w^2 - 4*w + 2],\
[113, 113, w^3 - w^2 - 5*w + 3],\
[121, 11, 2*w^2 - w - 9],\
[121, 11, -2*w^2 + 3*w + 8],\
[127, 127, w^3 - 3*w^2 - 2*w + 5],\
[127, 127, w^3 - 5*w - 1],\
[137, 137, 2*w - 1],\
[167, 167, 2*w^3 - 2*w^2 - 7*w - 2],\
[167, 167, w^3 - 7*w - 4],\
[167, 167, 2*w^3 - w^2 - 9*w - 2],\
[167, 167, 2*w^3 - 4*w^2 - 5*w + 9],\
[191, 191, 2*w^3 - 4*w^2 - 5*w + 5],\
[191, 191, -2*w^3 + 2*w^2 + 7*w - 2],\
[193, 193, -2*w^3 + 3*w^2 + 6*w - 6],\
[193, 193, -2*w^3 + 2*w^2 + 7*w - 4],\
[193, 193, -2*w^3 + 4*w^2 + 5*w - 3],\
[193, 193, 2*w^3 - 3*w^2 - 6*w + 1],\
[199, 199, -w^3 - 2*w^2 + 6*w + 10],\
[199, 199, -w^3 + 5*w^2 - w - 13],\
[223, 223, -w^3 + 2*w^2 + 3*w - 8],\
[223, 223, w^3 - w^2 - 4*w - 4],\
[233, 233, -w^2 - 1],\
[233, 233, w^2 - 2*w + 2],\
[239, 239, 3*w^2 - 2*w - 13],\
[239, 239, 3*w^2 - 4*w - 12],\
[241, 241, w^3 - w^2 - 6*w + 3],\
[241, 241, w^3 - 2*w^2 - 5*w + 3],\
[257, 257, -w^3 + 3*w^2 + 4*w - 9],\
[257, 257, -2*w^3 + 4*w^2 + 8*w - 11],\
[257, 257, -w^3 + 2*w^2 + 4],\
[257, 257, w^3 - 7*w - 3],\
[263, 263, -w^3 + w^2 + 3*w + 5],\
[263, 263, w^3 - 5*w^2 + 12],\
[263, 263, -w^3 - 2*w^2 + 7*w + 8],\
[263, 263, -w^3 + 7*w + 2],\
[271, 271, -w^3 + 3*w^2 + 2*w - 3],\
[271, 271, w^3 - 5*w + 1],\
[289, 17, 3*w^2 - 3*w - 8],\
[289, 17, 3*w^2 - 3*w - 10],\
[311, 311, -2*w^3 + 2*w^2 + 7*w - 1],\
[311, 311, -w^3 + 5*w^2 - w - 11],\
[311, 311, w^3 + 2*w^2 - 6*w - 8],\
[311, 311, -2*w^3 + 4*w^2 + 5*w - 6],\
[337, 337, w^3 - w^2 - 2*w - 4],\
[337, 337, 2*w^3 - 9*w - 4],\
[337, 337, 2*w^3 - 6*w^2 - 3*w + 11],\
[337, 337, -w^3 + 2*w^2 + w - 6],\
[353, 353, -w^3 + 4*w^2 - 12],\
[353, 353, w^3 + w^2 - 5*w - 9],\
[359, 359, 2*w^3 - 4*w^2 - 6*w + 5],\
[359, 359, 2*w^3 - 2*w^2 - 8*w + 3],\
[361, 19, -w^3 + 5*w^2 - 13],\
[361, 19, w^3 + 2*w^2 - 7*w - 9],\
[367, 367, 2*w^3 - 10*w - 5],\
[367, 367, w^3 - w^2 - 6*w + 2],\
[367, 367, -w^3 + 2*w^2 + 5*w - 4],\
[367, 367, 2*w^3 - 6*w^2 - 4*w + 13],\
[401, 401, -w^3 + 2*w^2 + 5*w - 5],\
[401, 401, -w^3 + w^2 + 6*w - 1],\
[409, 409, 2*w^3 - 3*w^2 - 6*w + 4],\
[409, 409, 3*w^3 - 3*w^2 - 12*w + 1],\
[409, 409, -3*w^3 + 6*w^2 + 9*w - 11],\
[409, 409, -2*w^3 + 3*w^2 + 6*w - 3],\
[431, 431, 3*w^2 - 5*w - 10],\
[431, 431, w^3 + w^2 - 8*w - 3],\
[457, 457, -2*w^3 + 4*w^2 + 4*w - 5],\
[457, 457, -2*w^3 + 2*w^2 + 6*w - 1],\
[463, 463, -2*w^3 + 7*w^2 + w - 16],\
[463, 463, 3*w^3 - 6*w^2 - 8*w + 8],\
[503, 503, -w^3 + 3*w^2 + 2*w - 12],\
[503, 503, -w^3 + 4*w^2 + 4*w - 10],\
[521, 521, w^3 + w^2 - 6*w - 2],\
[521, 521, -w^3 + 4*w^2 + w - 6],\
[529, 23, w^2 - w - 8],\
[569, 569, w^3 - 3*w - 6],\
[569, 569, -w^3 + 3*w^2 - 8],\
[577, 577, -w^3 + 4*w^2 - w - 10],\
[577, 577, w^3 + w^2 - 4*w - 8],\
[593, 593, -w^3 + 3*w^2 - 12],\
[593, 593, w^3 - 3*w - 10],\
[599, 599, w^3 + 3*w^2 - 7*w - 9],\
[599, 599, -w^3 + 3*w^2 + 3*w - 13],\
[601, 601, w^2 + 2*w - 4],\
[601, 601, w^2 - 4*w - 1],\
[607, 607, 2*w^2 - 13],\
[607, 607, w^3 + 2*w^2 - 7*w - 5],\
[607, 607, 2*w^3 - 3*w^2 - 8*w + 8],\
[607, 607, 2*w^2 - 4*w - 11],\
[617, 617, w^3 + w^2 - 9*w - 1],\
[617, 617, w^3 - w^2 - 4*w - 5],\
[617, 617, -w^3 + 2*w^2 + 3*w - 9],\
[617, 617, -w^3 + 4*w^2 + 4*w - 8],\
[625, 5, -5],\
[631, 631, -w^3 + 5*w^2 - 2*w - 13],\
[631, 631, w^3 + 2*w^2 - 5*w - 11],\
[641, 641, w^2 - 2*w + 3],\
[641, 641, -w^3 + 6*w - 2],\
[641, 641, -w^3 + 3*w^2 + 3*w - 3],\
[641, 641, -w^2 - 2],\
[647, 647, -2*w^3 + 5*w^2 + 4*w - 6],\
[647, 647, 2*w^3 - 4*w^2 - 5*w - 1],\
[647, 647, 3*w^3 - 7*w^2 - 7*w + 13],\
[647, 647, -2*w^3 + w^2 + 8*w - 1],\
[673, 673, -w^3 + 7*w - 2],\
[673, 673, -w^3 + 3*w^2 + 4*w - 4],\
[719, 719, -2*w^3 + 5*w^2 + 3*w - 12],\
[719, 719, 4*w^2 - 5*w - 10],\
[719, 719, -4*w^2 + 3*w + 11],\
[719, 719, 2*w^3 - w^2 - 7*w - 6],\
[727, 727, -w - 5],\
[727, 727, w - 6],\
[743, 743, w^2 + 2*w - 11],\
[743, 743, w^3 + 4*w^2 - 11*w - 11],\
[751, 751, -2*w^2 + w + 13],\
[751, 751, 2*w^2 - 3*w - 12],\
[769, 769, 2*w^3 - 6*w^2 - 3*w + 17],\
[769, 769, 3*w^3 - 2*w^2 - 11*w + 2],\
[809, 809, w^3 + 3*w^2 - 6*w - 11],\
[809, 809, -3*w^3 + 5*w^2 + 8*w - 11],\
[823, 823, -w^3 + w^2 + 2*w - 6],\
[823, 823, 4*w^2 - 5*w - 11],\
[823, 823, -4*w^2 + 3*w + 12],\
[823, 823, -w^3 + 4*w^2 + 4*w - 12],\
[839, 839, -2*w^3 + w^2 + 6*w + 5],\
[839, 839, -3*w^3 + 8*w^2 + 7*w - 17],\
[839, 839, 3*w^3 - w^2 - 14*w - 5],\
[839, 839, -2*w^3 + 5*w^2 + 2*w - 10],\
[857, 857, w^3 + w^2 - 8*w - 2],\
[857, 857, -w^3 + 4*w^2 + 3*w - 8],\
[863, 863, -w^3 + w^2 - w - 1],\
[863, 863, w^3 - 2*w^2 + 2*w - 2],\
[911, 911, -w^3 + 3*w^2 - 11],\
[911, 911, w^3 - 3*w - 9],\
[919, 919, -2*w^3 + w^2 + 8*w - 2],\
[919, 919, -2*w^3 + 5*w^2 + 4*w - 5],\
[929, 929, -2*w^3 + 2*w^2 + 9*w - 4],\
[929, 929, 2*w^3 - 9*w - 12],\
[929, 929, 2*w^3 - 6*w^2 - 3*w + 19],\
[929, 929, -2*w^3 + 4*w^2 + 7*w - 5],\
[937, 937, -w^3 + 2*w^2 - 6],\
[937, 937, -w^3 + 3*w^2 - 10],\
[937, 937, w^3 - 3*w - 8],\
[937, 937, w^3 - w^2 - w - 5],\
[953, 953, 2*w^3 - 6*w^2 - 3*w + 18],\
[953, 953, -3*w^3 + 3*w^2 + 11*w - 5],\
[961, 31, 4*w^2 - 4*w - 11],\
[967, 967, -w^3 + 7*w - 3],\
[967, 967, -2*w^3 + 2*w^2 + 11*w + 2],\
[967, 967, -w^3 - 4*w^2 + 9*w + 9],\
[967, 967, -w^3 + 3*w^2 + 4*w - 3],\
[977, 977, 2*w^2 - 5*w - 8],\
[977, 977, -w^3 + 2*w^2 - w + 6],\
[977, 977, -w^3 + w^2 - 6],\
[977, 977, 2*w^2 + w - 11],\
[983, 983, -w^3 + 6*w^2 - 2*w - 16],\
[983, 983, w^3 + 3*w^2 - 7*w - 13]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 23*x^2 - 12*x + 37
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1, e, 1, -1/6*e^3 + 1/6*e^2 + 19/6*e + 1/3, -1/6*e^3 + 1/6*e^2 + 19/6*e + 1/3, -1/2*e^2 + 1/2*e + 9/2, 1/6*e^3 + 1/3*e^2 - 11/3*e - 35/6, -1/6*e^3 + 1/6*e^2 + 25/6*e - 8/3, 1/6*e^3 - 1/6*e^2 - 19/6*e + 5/3, 4, -1/3*e^3 + 1/3*e^2 + 16/3*e - 10/3, 1/6*e^3 + 1/3*e^2 - 17/3*e - 23/6, 1/3*e^3 + 1/6*e^2 - 35/6*e - 7/6, -1/3*e^3 + 1/3*e^2 + 13/3*e - 10/3, -1/3*e^3 + 1/3*e^2 + 25/3*e + 8/3, 1/6*e^3 - 2/3*e^2 - 14/3*e + 49/6, -1/3*e^3 + 1/3*e^2 + 22/3*e - 1/3, 1/2*e^2 - 1/2*e - 25/2, 1/3*e^3 + 1/6*e^2 - 35/6*e + 17/6, 1/3*e^3 - 1/3*e^2 - 28/3*e - 11/3, 1/6*e^3 + 5/6*e^2 - 25/6*e - 22/3, -1/6*e^3 - 1/3*e^2 + 17/3*e - 13/6, 1/3*e^3 - 4/3*e^2 - 16/3*e + 43/3, -1/3*e^3 - 2/3*e^2 + 19/3*e + 32/3, 1/3*e^3 - 1/3*e^2 - 10/3*e - 2/3, -1/6*e^3 + 1/6*e^2 + 19/6*e + 1/3, 1/6*e^3 - 1/6*e^2 - 25/6*e + 32/3, -1/3*e^3 + 4/3*e^2 + 19/3*e - 25/3, 1/2*e^2 - 5/2*e - 33/2, -1/2*e^2 - 5/2*e + 41/2, 2/3*e^3 - 2/3*e^2 - 38/3*e + 26/3, -2/3*e^3 + 5/3*e^2 + 35/3*e - 41/3, e^2 - e - 21, -1/3*e^3 + 1/3*e^2 + 10/3*e - 25/3, 1/3*e^3 - 5/6*e^2 - 17/6*e + 71/6, 1/3*e^3 - 1/3*e^2 - 13/3*e + 10/3, 1/2*e^2 + 5/2*e - 1/2, 1/3*e^3 - 4/3*e^2 - 16/3*e + 55/3, -1/3*e^3 + 4/3*e^2 + 16/3*e - 31/3, e^2 + e - 5, 1/2*e^3 - 1/2*e^2 - 25/2*e, 1/6*e^3 - 7/6*e^2 - 13/6*e + 8/3, 2/3*e^3 - 1/6*e^2 - 61/6*e - 95/6, 2/3*e^3 - 5/3*e^2 - 41/3*e + 41/3, -e^2 - e + 21, -1/3*e^3 + 1/3*e^2 + 31/3*e + 8/3, -1/6*e^3 + 7/6*e^2 - 5/6*e - 83/3, -1/2*e^2 + 1/2*e - 11/2, 1/2*e^3 + 1/2*e^2 - 23/2*e - 11, -1/2*e^2 - 9/2*e + 25/2, 2/3*e^3 - 2/3*e^2 - 44/3*e + 2/3, -1/3*e^3 + 1/3*e^2 + 13/3*e - 22/3, 1/6*e^3 - 7/6*e^2 - 43/6*e + 59/3, 1/3*e^3 + 7/6*e^2 - 35/6*e - 121/6, e^2 - 28, 1/6*e^3 - 2/3*e^2 - 8/3*e + 145/6, -1/6*e^3 + 7/6*e^2 + 13/6*e - 26/3, 1/3*e^3 - 4/3*e^2 - 13/3*e + 37/3, -3/2*e^2 - 5/2*e + 35/2, -1/2*e^3 - 1/2*e^2 + 23/2*e + 11, -1/2*e^3 + 1/2*e^2 + 11/2*e - 5, -2/3*e^3 + 1/6*e^2 + 103/6*e + 11/6, e^3 - e^2 - 17*e, 1/2*e^3 - 1/2*e^2 - 23/2*e + 9, 1/2*e^2 + 5/2*e + 7/2, -3/2*e^2 + 1/2*e + 43/2, -1/2*e^3 + 3/2*e^2 + 21/2*e - 4, 1/3*e^3 + 7/6*e^2 - 35/6*e - 133/6, -e^2 - e + 11, 1/3*e^3 - 1/3*e^2 - 37/3*e - 2/3, -7/6*e^3 + 1/6*e^2 + 127/6*e + 34/3, 1/3*e^3 + 2/3*e^2 - 22/3*e - 11/3, -5/6*e^3 - 7/6*e^2 + 107/6*e + 47/3, 1/3*e^3 - 11/6*e^2 - 11/6*e + 173/6, -e^3 + 16*e + 5, -4/3*e^3 - 1/6*e^2 + 143/6*e + 37/6, 1/3*e^3 - 10/3*e^2 - 10/3*e + 115/3, 1/3*e^3 - 1/3*e^2 - 19/3*e + 10/3, -1/3*e^3 + 1/3*e^2 + 19/3*e + 32/3, 1/2*e^3 + e^2 - 13*e - 27/2, -1/3*e^3 + 5/6*e^2 + 35/6*e - 71/6, -1/2*e^3 + 1/2*e^2 + 23/2*e - 9, 2/3*e^3 - 13/6*e^2 - 73/6*e + 61/6, -1/2*e^3 + 1/2*e^2 + 9/2*e - 10, -1/6*e^3 + 1/6*e^2 + 31/6*e + 19/3, -1/3*e^3 - 2/3*e^2 + 4/3*e + 23/3, -e^3 + 5/2*e^2 + 35/2*e - 23/2, 5/6*e^3 + 7/6*e^2 - 119/6*e - 53/3, 2/3*e^3 - 5/3*e^2 - 41/3*e + 29/3, 5/3*e^3 - 1/6*e^2 - 199/6*e - 53/6, -2/3*e^3 - 1/3*e^2 + 47/3*e - 5/3, 2/3*e^3 + 4/3*e^2 - 35/3*e - 58/3, -2/3*e^3 + 5/3*e^2 + 29/3*e - 11/3, 7/6*e^3 - 5/3*e^2 - 65/3*e + 31/6, e^3 - 22*e - 23, -2/3*e^3 + 2/3*e^2 + 50/3*e + 34/3, e^3 - e^2 - 17*e - 8, -1/6*e^3 - 4/3*e^2 - 16/3*e + 197/6, 5/6*e^3 + 1/6*e^2 - 95/6*e - 53/3, -1/3*e^3 + 5/6*e^2 + 47/6*e - 23/6, -5/6*e^3 + 4/3*e^2 + 52/3*e - 65/6, 2/3*e^3 + 4/3*e^2 - 44/3*e - 58/3, 1/3*e^3 - 7/3*e^2 - 13/3*e + 130/3, -e^3 + 2*e^2 + 16*e + 1, -1/6*e^3 - 11/6*e^2 + 37/6*e + 100/3, 1/3*e^3 + 2/3*e^2 - 28/3*e - 35/3, -1/3*e^3 + 17/6*e^2 + 47/6*e - 155/6, 1/2*e^3 + 1/2*e^2 - 7/2*e - 13, -1/3*e^3 + 7/3*e^2 + 13/3*e - 40/3, 1/2*e^3 - 2*e^2 - 12*e + 31/2, 1/3*e^3 - 7/3*e^2 - 7/3*e + 136/3, 1/3*e^3 + 1/6*e^2 - 89/6*e - 31/6, -1/6*e^3 - 4/3*e^2 + 2/3*e + 83/6, 1/2*e^3 - 9/2*e^2 - 15/2*e + 45, 4/3*e^3 + 2/3*e^2 - 88/3*e - 50/3, 4/3*e^3 - 11/6*e^2 - 161/6*e + 35/6, 1/3*e^3 + 1/6*e^2 + 7/6*e - 31/6, 5/6*e^3 + 1/6*e^2 - 83/6*e - 23/3, e^3 - 2*e^2 - 22*e - 3, -1/3*e^3 - 1/6*e^2 + 23/6*e + 139/6, -e^2 + 22, -1/3*e^3 + 1/3*e^2 + 13/3*e + 8/3, -3/2*e^3 - 3/2*e^2 + 65/2*e + 17, -11/6*e^3 + 5/6*e^2 + 221/6*e + 11/3, -5/6*e^3 - 7/6*e^2 + 107/6*e - 1/3, -1/6*e^3 + 7/6*e^2 + 13/6*e + 64/3, -1/2*e^3 + 1/2*e^2 + 15/2*e + 21, 1/2*e^3 - 3/2*e^2 - 9/2*e + 26, -2/3*e^3 - 11/6*e^2 + 85/6*e + 179/6, 7/6*e^3 - 1/6*e^2 - 97/6*e - 67/3, 1/6*e^3 - 1/6*e^2 - 31/6*e - 49/3, -4/3*e^3 + 1/3*e^2 + 64/3*e + 8/3, 3/2*e^3 - 2*e^2 - 34*e + 17/2, -7/6*e^3 + 1/6*e^2 + 151/6*e - 32/3, -5/2*e^2 + 9/2*e + 45/2, 5/6*e^3 - 7/3*e^2 - 37/3*e + 191/6, -2/3*e^3 + 8/3*e^2 + 32/3*e - 110/3, 4*e - 22, -e^3 + 30*e - 3, 1/3*e^3 - 1/3*e^2 - 10/3*e + 10/3, 4/3*e^3 - 7/3*e^2 - 79/3*e + 37/3, -2/3*e^3 - 1/3*e^2 + 77/3*e + 37/3, 1/6*e^3 - 5/3*e^2 - 5/3*e + 79/6, -1/2*e^3 + 3*e^2 + 11*e - 53/2, -1/6*e^3 + 1/6*e^2 + 13/6*e + 22/3, 2*e^2 - 6*e - 38, -e^3 + 3/2*e^2 + 27/2*e - 41/2, 2/3*e^3 + 1/3*e^2 - 59/3*e + 23/3, -e^3 + 2*e^2 + 16*e - 25, e^3 - 1/2*e^2 - 29/2*e + 19/2, 7/6*e^3 - 2/3*e^2 - 62/3*e - 95/6, -1/3*e^3 + 1/3*e^2 + 37/3*e - 106/3, -3/2*e^3 + 7/2*e^2 + 47/2*e - 34, -1/2*e^3 + e^2 + 7*e - 13/2, 5/6*e^3 - 17/6*e^2 - 101/6*e + 124/3, -4/3*e^3 + 10/3*e^2 + 70/3*e - 40/3, 1/3*e^3 + 2/3*e^2 - 28/3*e + 7/3, -3*e^2 - e + 47, -5/6*e^3 - 13/6*e^2 + 95/6*e + 59/3, -5/6*e^3 + 11/6*e^2 + 71/6*e - 43/3, -1/6*e^3 + 13/6*e^2 - 11/6*e - 176/3, -4/3*e^3 - 1/6*e^2 + 209/6*e - 35/6, -2*e^2 + 2*e + 24, 5/6*e^3 + 1/6*e^2 - 107/6*e + 49/3, 1/6*e^3 + 5/6*e^2 - 79/6*e - 79/3, e^3 + 1/2*e^2 - 31/2*e - 15/2, -1/2*e^3 + 7/2*e^2 + 1/2*e - 44, 4/3*e^3 + 1/6*e^2 - 179/6*e + 23/6, 1/2*e^3 - 5/2*e^2 - 11/2*e + 23, e^3 - 1/2*e^2 - 23/2*e - 37/2, 5/6*e^3 - 4/3*e^2 - 22/3*e + 65/6, 5/3*e^3 - 5/3*e^2 - 83/3*e - 40/3, 4*e - 4, 1/6*e^3 + 17/6*e^2 - 1/6*e - 142/3, -e^2 - 3, 2/3*e^3 - 2/3*e^2 - 38/3*e + 62/3, -e^3 + 1/2*e^2 + 59/2*e + 5/2, 7/6*e^3 - 2/3*e^2 - 50/3*e + 13/6]
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, -w^2 + w + 3])] = -1
AL_eigenvalues[ZF.ideal([7, 7, -w^2 + 2*w + 1])] = -1

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