/*
  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, 0, -6, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([17,17,w^3 - 3*w - 1])

primes_array = [
[2, 2, w + 1],\
[7, 7, w],\
[9, 3, w^2 + w - 1],\
[9, 3, w^2 - w - 1],\
[17, 17, -w^3 + 3*w - 1],\
[17, 17, w^3 - 3*w - 1],\
[23, 23, -w^3 - w^2 + 3*w + 4],\
[23, 23, w^3 - w^2 - 3*w + 4],\
[41, 41, w^3 - w^2 - 2*w + 3],\
[41, 41, -w^3 - w^2 + 2*w + 3],\
[49, 7, w^2 - 6],\
[71, 71, w^3 + 3*w^2 - 3*w - 6],\
[71, 71, -2*w^2 - 2*w + 1],\
[73, 73, w^2 - 2*w - 2],\
[73, 73, w^2 + 2*w - 2],\
[79, 79, w^3 - w^2 - 5*w + 2],\
[79, 79, -w^3 - w^2 + 5*w + 2],\
[89, 89, 2*w - 1],\
[89, 89, -2*w - 1],\
[97, 97, -w^3 + w^2 + 4*w - 1],\
[97, 97, w^3 + w^2 - 4*w - 1],\
[103, 103, -2*w^3 + w^2 + 8*w - 2],\
[103, 103, w^3 - w^2 - 3*w + 6],\
[103, 103, -w^3 - w^2 + 3*w + 6],\
[103, 103, 2*w^3 + w^2 - 8*w - 2],\
[127, 127, w^3 + 3*w^2 - 4*w - 11],\
[127, 127, -w^3 + 3*w^2 + 4*w - 11],\
[137, 137, -w^3 - 3*w^2 + 2*w + 9],\
[137, 137, -2*w^2 + w + 10],\
[137, 137, -2*w^2 - w + 10],\
[137, 137, w^3 - 3*w^2 - 2*w + 9],\
[151, 151, 3*w^2 - 2*w - 10],\
[151, 151, w^3 + 3*w^2 - 4*w - 9],\
[167, 167, w^3 + 4*w^2 - 2*w - 8],\
[167, 167, 2*w^3 - 8*w - 3],\
[167, 167, 2*w^3 - 8*w + 3],\
[167, 167, 2*w^3 + 5*w^2 - 7*w - 13],\
[191, 191, -w^3 + 6*w - 2],\
[191, 191, w^3 - 6*w - 2],\
[193, 193, -2*w^3 + 2*w^2 + 6*w - 5],\
[193, 193, w^3 + 2*w^2 - 4*w - 4],\
[193, 193, -w^3 + 2*w^2 + 4*w - 4],\
[193, 193, -2*w^3 - 2*w^2 + 6*w + 5],\
[199, 199, 2*w^2 - 3*w - 6],\
[199, 199, w^3 + w^2 + w + 2],\
[199, 199, 2*w^3 + 2*w^2 - 4*w - 5],\
[199, 199, 2*w^2 + 3*w - 6],\
[223, 223, 2*w^3 + w^2 - 6*w - 6],\
[223, 223, w^3 - w^2 - 5*w - 2],\
[223, 223, 2*w^3 + 4*w^2 - 6*w - 9],\
[223, 223, -2*w^3 + w^2 + 6*w - 6],\
[239, 239, -2*w^3 - 5*w^2 + 5*w + 11],\
[239, 239, w^3 + 4*w^2 - 4*w - 10],\
[241, 241, -3*w^3 - 4*w^2 + 9*w + 11],\
[241, 241, -2*w^3 - w^2 + 6*w + 4],\
[257, 257, 2*w^3 + 5*w^2 - 6*w - 12],\
[257, 257, -w^3 - 4*w^2 + 3*w + 9],\
[263, 263, -w^3 - w^2 + 6*w + 3],\
[263, 263, w^3 - w^2 - 6*w + 3],\
[271, 271, -2*w^3 - w^2 + 7*w + 3],\
[271, 271, -w^3 + 2*w^2 + 3*w - 9],\
[271, 271, w^3 + 2*w^2 - 3*w - 9],\
[271, 271, 2*w^3 - w^2 - 7*w + 3],\
[289, 17, 3*w^2 - 8],\
[313, 313, -w^3 + 2*w^2 - 6],\
[313, 313, w^3 + 2*w^2 - 6],\
[353, 353, -w^3 + 3*w - 5],\
[353, 353, w^3 - 3*w - 5],\
[359, 359, w^3 - w + 3],\
[359, 359, 4*w^3 + 5*w^2 - 14*w - 18],\
[361, 19, -w^3 + 2*w^2 + 4*w - 2],\
[361, 19, w^3 + 2*w^2 - 4*w - 2],\
[367, 367, -2*w^3 - 3*w^2 + 6*w + 10],\
[367, 367, -3*w - 2],\
[367, 367, 3*w - 2],\
[367, 367, w^3 + 2*w^2 - 7*w - 11],\
[401, 401, -2*w^3 + 6*w - 1],\
[401, 401, -3*w^3 + 3*w^2 + 11*w - 10],\
[401, 401, 3*w^3 + 3*w^2 - 11*w - 10],\
[401, 401, 2*w^3 - 6*w - 1],\
[409, 409, w^3 - 3*w^2 - 6*w + 13],\
[409, 409, -w^3 - 3*w^2 + 6*w + 13],\
[431, 431, w^3 - 4*w^2 - 2*w + 10],\
[431, 431, -w^3 - 4*w^2 + 2*w + 10],\
[433, 433, w^3 - w^2 - 6*w - 3],\
[433, 433, 2*w^3 + 4*w^2 - 7*w - 10],\
[463, 463, w^2 + 3*w - 3],\
[463, 463, w^2 - 3*w - 3],\
[487, 487, 2*w^2 + 2*w - 9],\
[487, 487, 2*w^2 - 2*w - 9],\
[521, 521, -w^3 - 4*w^2 + 5*w + 13],\
[521, 521, w^3 - 4*w^2 - 5*w + 13],\
[529, 23, -w^2 - 2],\
[577, 577, w^3 + 2*w^2 - 2*w - 8],\
[577, 577, -w^3 + 2*w^2 + 2*w - 8],\
[593, 593, -2*w^3 - 3*w^2 + 5*w + 9],\
[593, 593, w^3 + 2*w^2 - 8*w - 12],\
[599, 599, w^3 + 2*w^2 - 5*w - 3],\
[599, 599, -w^3 + 2*w^2 + 5*w - 3],\
[601, 601, -w^3 + 4*w^2 + 4*w - 8],\
[601, 601, w^3 + 4*w^2 - 4*w - 8],\
[607, 607, -w^3 + 8*w + 6],\
[607, 607, -3*w^3 - 5*w^2 + 7*w + 12],\
[607, 607, 2*w^2 - 4*w - 9],\
[607, 607, -2*w^3 - 3*w^2 + 11*w + 13],\
[617, 617, 3*w^3 + 2*w^2 - 11*w - 9],\
[617, 617, -2*w^3 + w^2 + 9*w - 1],\
[617, 617, 2*w^3 + w^2 - 9*w - 1],\
[617, 617, 3*w^3 - 12*w - 4],\
[625, 5, -5],\
[631, 631, w^3 - w^2 - 7*w - 4],\
[631, 631, 2*w^3 + 4*w^2 - 8*w - 11],\
[641, 641, -2*w^3 + 3*w^2 + 9*w - 9],\
[641, 641, w^3 - 2*w^2 - 7*w + 5],\
[641, 641, w^3 + 2*w^2 - 7*w - 5],\
[641, 641, 2*w^3 + 3*w^2 - 9*w - 9],\
[647, 647, -w^3 + 3*w^2 - 1],\
[647, 647, 2*w^3 - 3*w^2 - 8*w + 8],\
[647, 647, -2*w^3 - 3*w^2 + 8*w + 8],\
[647, 647, w^3 + 3*w^2 - 1],\
[727, 727, -3*w^3 - 2*w^2 + 11*w + 3],\
[727, 727, -2*w^3 + 5*w^2 + 9*w - 17],\
[727, 727, 2*w^3 + 5*w^2 - 9*w - 17],\
[727, 727, 3*w^3 - 2*w^2 - 11*w + 3],\
[743, 743, 3*w^2 - 2*w - 12],\
[743, 743, 3*w^2 + 2*w - 12],\
[751, 751, 2*w^3 - 2*w^2 - 6*w + 9],\
[751, 751, -2*w^3 - 2*w^2 + 6*w + 9],\
[761, 761, -3*w^3 + 5*w^2 + 6*w - 11],\
[761, 761, 3*w^3 + 5*w^2 - 6*w - 11],\
[769, 769, w^3 - 7*w - 3],\
[769, 769, -w^3 + 7*w - 3],\
[823, 823, w^3 + 2*w^2 - 5*w - 1],\
[823, 823, w^3 - 2*w^2 - 5*w + 1],\
[839, 839, 4*w^2 + w - 12],\
[839, 839, -3*w^3 + w^2 + 13*w - 8],\
[839, 839, 3*w^3 + w^2 - 13*w - 8],\
[839, 839, 4*w^2 - w - 12],\
[841, 29, 2*w^3 + 4*w^2 - 7*w - 16],\
[841, 29, -2*w^3 + 4*w^2 + 7*w - 16],\
[857, 857, 3*w^2 - 3*w - 11],\
[857, 857, 3*w^2 + 3*w - 11],\
[863, 863, w^3 + 3*w^2 - 5*w - 6],\
[863, 863, -w^3 + 3*w^2 + 5*w - 6],\
[881, 881, w^3 - w^2 - 5*w - 4],\
[881, 881, 2*w^2 + 4*w + 3],\
[887, 887, w^2 - w - 9],\
[887, 887, 3*w^3 + 4*w^2 - 11*w - 11],\
[887, 887, -3*w^3 + 4*w^2 + 11*w - 11],\
[887, 887, w^2 + w - 9],\
[911, 911, w^2 - 4*w - 8],\
[911, 911, -3*w^3 - 4*w^2 + 9*w + 13],\
[919, 919, 2*w^3 - w^2 - 8*w - 2],\
[919, 919, -2*w^3 - w^2 + 8*w - 2],\
[929, 929, 4*w^3 + w^2 - 16*w - 2],\
[929, 929, 2*w^3 - 11*w - 6],\
[937, 937, 3*w^3 - 3*w^2 - 10*w + 5],\
[937, 937, -3*w^3 - 3*w^2 + 10*w + 5],\
[953, 953, w^3 - 9*w - 11],\
[953, 953, 5*w^2 + 3*w - 17],\
[953, 953, 5*w^2 - 3*w - 17],\
[953, 953, -2*w^3 - w^2 + 8*w + 10],\
[961, 31, 4*w^2 - 11],\
[961, 31, 4*w^2 - 13],\
[967, 967, w^3 - w^2 - 6*w - 5],\
[967, 967, 2*w^2 + 3*w + 2],\
[983, 983, -w^2 + w - 3],\
[983, 983, 2*w^3 - 2*w^2 - 11*w + 10],\
[983, 983, 2*w^3 + 2*w^2 - 11*w - 10],\
[983, 983, -w^2 - w - 3],\
[991, 991, w^3 + 3*w^2 - 10*w - 15],\
[991, 991, -2*w^3 - 4*w^2 + w + 6]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 6*x^2 - 2*x + 5
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -e^3 + 2*e^2 + 4*e - 3, e^3 - 2*e^2 - 3*e + 8, -2*e^3 + 3*e^2 + 7*e - 6, 2*e^3 - 3*e^2 - 9*e + 6, -1, -3*e^3 + 3*e^2 + 10*e - 6, -e^3 + 3*e^2 + 5*e - 9, e^3 - 3*e^2 - 5*e + 7, 2*e^2 - 10, -2*e^3 - 2*e^2 + 9*e + 7, e^3 - e^2 - 9*e + 1, -e^3 + 2*e^2 + e, -4*e^3 + 2*e^2 + 18*e + 2, 3*e^3 - 6*e^2 - 7*e + 14, 3*e^3 - 8*e^2 - 11*e + 18, -3*e^3 + 6*e^2 + 5*e - 16, -2*e^3 + 4*e^2 + 2*e - 8, -e^3 + 2*e^2 + 2*e + 5, 5*e^3 - 8*e^2 - 15*e + 20, -2*e^3 + 4*e^2 + 6*e + 2, -e^3 + 7*e + 4, -3*e^3 + 3*e^2 + 15*e - 1, 6*e^3 - 4*e^2 - 24*e + 6, 5*e^3 - 2*e^2 - 22*e - 3, -3*e^2 - 4*e + 19, 3*e^3 - 5*e^2 - 10*e + 6, -4*e^2 + 4*e + 8, -2*e^3 + 4*e^2 + 10*e - 12, -3*e^3 - e^2 + 10*e + 8, -2*e^3 - e^2 + 14*e + 7, 6*e^3 - 12*e^2 - 20*e + 28, -4*e^3 + 4*e^2 + 20*e - 2, 5*e^3 - 3*e^2 - 20*e - 2, e^3 - 4*e^2 + 3*e + 10, -6*e^3 + 8*e^2 + 24*e - 8, 8*e^3 - 8*e^2 - 30*e + 12, 2*e^3 - 11*e^2 - 5*e + 36, -e^2 + 5*e + 2, 4*e^3 - 20*e + 2, e^3 - 7*e^2 - 3*e + 21, -5*e^3 + 10*e^2 + 18*e - 17, -e^3 + 8*e - 1, -6*e^3 + 10*e^2 + 20*e - 18, -8*e^3 + 8*e^2 + 34*e - 14, 2*e^3 - 14*e + 4, -6*e^3 + 2*e^2 + 28*e - 6, -5*e^3 + 9*e^2 + 15*e - 27, -3*e^3 - 2*e^2 + 20*e + 13, e^3 - 8*e + 3, 3*e^3 + 7*e^2 - 18*e - 14, 4*e + 6, -4*e^3 + 3*e^2 + 18*e - 15, 5*e^3 - 6*e^2 - 27*e + 10, -3*e^3 + 2*e^2 + 17*e + 12, 4*e^3 - 4*e^2 - 14*e + 20, -4*e^3 + e^2 + 24*e + 3, 9*e^3 - 8*e^2 - 33*e + 22, 5*e^3 - 8*e^2 - 24*e + 5, e^3 + 2*e^2 - 5*e - 16, e^3 - e^2 - 10*e - 12, e^3 - 13*e^2 + 3*e + 29, -4*e^3 + 2*e^2 + 24*e + 4, 9*e^3 - 7*e^2 - 40*e + 2, -10*e^3 + 10*e^2 + 39*e - 17, e^3 - 3*e^2 - 14*e + 18, 2*e^3 + 10*e^2 - 17*e - 23, -8*e^3 + 4*e^2 + 30*e + 2, 6*e^3 - 2*e^2 - 20*e + 2, -6*e^3 + 18*e^2 + 20*e - 54, 16*e^3 - 12*e^2 - 58*e + 10, -11*e^3 + 18*e^2 + 42*e - 25, 8*e^3 - 14*e^2 - 30*e + 28, -3*e^3 + 6*e^2 + 12*e - 19, 5*e^3 - 14*e^2 - 18*e + 43, 4*e^3 - 8*e^2 - 10*e + 26, -2*e^3 + 10*e^2 - 2*e - 20, -e^3 - 6*e^2 - 3*e + 42, 8*e^3 - 8*e^2 - 24*e + 20, -e^3 + 8*e^2 - 2*e - 7, e^3 - 5*e^2 - 6*e - 4, -2*e^3 + 14*e^2 + 6*e - 38, -2*e^3 - 8*e^2 + 12*e + 16, 11*e^3 - 7*e^2 - 45*e + 5, 9*e^3 - 8*e^2 - 23*e + 8, -6*e^3 + 4*e^2 + 18*e + 2, 5*e^3 + 2*e^2 - 19*e - 22, 5*e^3 - 2*e^2 - 14*e + 7, 8*e^3 - 15*e^2 - 22*e + 33, -12*e^3 + 20*e^2 + 30*e - 56, 10*e^2 - 4*e - 22, -18*e^3 + 18*e^2 + 64*e - 28, 10*e^3 - 12*e^2 - 42*e + 2, 3*e^3 - 17*e^2 - 11*e + 39, 18*e^3 - 13*e^2 - 68*e + 21, 10*e^3 - 14*e^2 - 41*e + 5, -e^3 + 7*e^2 + 4*e - 42, -e^3 - 2*e^2 - 2*e + 15, -5*e^3 + 21*e - 4, 3*e^3 + 5*e^2 - 17*e + 7, -7*e^3 + 14*e^2 + 21*e - 36, 2*e^3 + 5*e^2 - 21*e - 8, -6*e^3 + 10*e^2 + 26*e - 16, 2*e^3 - e^2 - 22*e + 9, -3*e^3 - 10*e^2 + 15*e + 34, -2*e^3 + 24, 3*e^3 - e^2 - 15*e - 9, -7*e^3 + 4*e^2 + 29*e, -9*e^3 + 10*e^2 + 37*e - 36, 14*e^3 - 12*e^2 - 60*e + 16, 7*e^3 - 10*e^2 - 27*e - 14, 3*e^3 + 4*e^2 - 15*e - 28, 3*e^3 - 11*e^2 - 9*e + 41, e^3 - 3*e^2 + 13*e - 3, -3*e^3 - 6*e^2 + 19*e + 30, 9*e^2 - 7*e - 18, -2*e^3 - 10*e^2 + 18*e + 28, 9*e^3 - 14*e^2 - 25*e + 24, 7*e^3 + 4*e^2 - 23*e - 22, 8*e^3 - 3*e^2 - 38*e + 1, 9*e^3 - 6*e^2 - 30*e - 11, -15*e^3 + 15*e^2 + 55*e - 41, -4*e^3 + 14*e^2 - 11*e - 47, -4*e^3 + 9*e^2 + 11*e - 24, 10*e^2 - 3*e - 7, 14*e^3 - 6*e^2 - 54*e - 6, -8*e^3 + 13*e^2 + 22*e - 33, 14*e^3 - 20*e^2 - 53*e + 49, -2*e^3 + 12*e^2 + 16*e - 30, 13*e^3 - 19*e^2 - 44*e + 48, -9*e^3 + 5*e^2 + 35*e + 5, 11*e^3 - 6*e^2 - 43*e + 12, -6*e^3 - 2*e^2 + 36*e + 6, 7*e^3 - 14*e^2 - 31*e + 36, 2*e^3 + 6*e^2 - 4*e - 44, 3*e^3 - 9*e^2 + 5*e + 21, 8*e^3 - 6*e^2 - 50*e + 18, 6*e^3 - 18*e^2 - 20*e + 30, -12*e^3 + 18*e^2 + 46*e - 30, 8*e^3 - 8*e^2 - 25*e - 1, 6*e^3 - 8*e^2 - 20*e + 18, 8*e^3 - 4*e^2 - 28*e + 32, 10*e^3 - 10*e^2 - 42*e - 12, 3*e^3 - 3*e^2 - 7*e + 1, -4*e^3 + 8*e^2 - 4*e - 24, -4*e^3 + 2*e^2 + 20*e + 32, 14*e^3 - 22*e^2 - 50*e + 48, -4*e^3 + 11*e^2 + 11*e - 58, -3*e^3 + 4*e^2 + 25*e - 14, -9*e^3 + 17*e^2 + 22*e - 30, 2*e^3 - 16*e^2 + 16*e + 42, -6*e^3 - e^2 + 28*e + 9, 23*e^3 - 26*e^2 - 78*e + 59, 13*e^3 - 18*e^2 - 30*e + 41, -6*e^3 + 8*e^2 + 18*e - 30, 18*e^3 - 11*e^2 - 61*e + 6, -2*e^3 + 16*e + 30, 11*e^3 - 12*e^2 - 54*e - 9, 7*e^3 + 11*e^2 - 37*e - 37, 2*e^3 - 6*e^2 - 5*e + 19, -2*e^3 + 14*e^2 + 15*e - 49, 5*e^3 - 15*e^2 - 21*e + 27, -4*e^3 - 4*e^2 + 26*e, 8*e^3 - 10*e^2 - 46*e + 14, -7*e^3 + 31*e^2 + 22*e - 94, -6*e^3 + 10*e^2 + 2*e - 32, -4*e^2 - 16*e + 26, 4*e^3 + 14*e^2 - 18*e - 42, -7*e^3 + 3*e^2 + 29*e + 3, 22*e^3 - 39*e^2 - 74*e + 89, -7*e^3 + 2*e^2 + 25*e + 8, e^3 + 6*e^2 - e - 56]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([17,17,w^3 - 3*w - 1])] = 1

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