/*
  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([31, 31, w^2 - 5])

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 + 5*x^3 + 3*x^2 - 5*x + 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -e^3 - 6*e^2 - 7*e + 2, 2*e^3 + 10*e^2 + 7*e - 7, e^3 + 5*e^2 + 5*e - 1, 2*e^3 + 11*e^2 + 11*e - 7, -2*e^3 - 11*e^2 - 9*e + 9, -4*e^3 - 22*e^2 - 24*e + 13, 1, -e^3 - 6*e^2 - 5*e + 4, -8*e^3 - 40*e^2 - 32*e + 20, 2*e^3 + 12*e^2 + 13*e - 14, -2*e^3 - 8*e^2 - 4*e - 3, -5*e^3 - 25*e^2 - 19*e + 16, -3*e^3 - 16*e^2 - 20*e + 1, 3*e^3 + 14*e^2 + 8*e - 8, -2*e^3 - 12*e^2 - 14*e + 5, 10*e^3 + 50*e^2 + 41*e - 21, -6*e^3 - 31*e^2 - 22*e + 16, 12*e^3 + 62*e^2 + 56*e - 34, -3*e^3 - 18*e^2 - 22*e + 13, 2*e^3 + 9*e^2 - 2*e - 13, e^3 + 12*e^2 + 25*e - 8, 4*e^3 + 18*e^2 + 9*e - 16, -7*e^3 - 32*e^2 - 17*e + 25, e^3 + 4*e^2 - 5*e - 10, -4*e^3 - 24*e^2 - 25*e + 20, -13*e^3 - 69*e^2 - 60*e + 38, e^3 + 3*e^2 - 4*e - 10, 3*e^3 + 20*e^2 + 30*e - 2, 8*e^3 + 41*e^2 + 29*e - 35, 16*e^3 + 81*e^2 + 67*e - 43, -8*e^3 - 41*e^2 - 28*e + 28, -8*e^3 - 44*e^2 - 48*e + 28, -3*e^3 - 11*e^2 - e - 5, -4*e^3 - 22*e^2 - 20*e + 11, -6*e^3 - 35*e^2 - 47*e + 12, 13*e^3 + 69*e^2 + 63*e - 36, 7*e^3 + 40*e^2 + 40*e - 39, 4*e^3 + 17*e^2 + 14*e + 5, e^3 - 2*e^2 - 26*e - 13, -15*e^3 - 84*e^2 - 86*e + 47, 2*e^2 + 4*e + 7, -14*e^3 - 76*e^2 - 68*e + 42, -2*e^3 - 6*e^2 + 4*e + 9, -6*e^3 - 26*e^2 - 7*e + 23, 6*e^3 + 26*e^2 + 3*e - 25, e^3 + 5*e^2 + 8*e - 7, 9*e^3 + 49*e^2 + 48*e - 19, -6*e^3 - 23*e^2 + 6*e + 24, 14*e^3 + 77*e^2 + 78*e - 37, -5*e^3 - 23*e^2 - 13*e + 26, -13*e^3 - 68*e^2 - 60*e + 35, 8*e^3 + 48*e^2 + 54*e - 17, 9*e^3 + 48*e^2 + 36*e - 43, e^3 + 4*e^2 + e - 3, 4*e^3 + 23*e^2 + 22*e - 33, -6*e^3 - 28*e^2 - 11*e + 19, 10*e^3 + 47*e^2 + 22*e - 39, 3*e^3 + 16*e^2 + 19*e + 4, -5*e^3 - 20*e^2 - 5*e + 6, -6*e^3 - 29*e^2 - 24*e + 19, 2*e^3 + 16*e^2 + 32*e - 7, -5*e^3 - 24*e^2 - 24*e + 1, 5*e^3 + 35*e^2 + 52*e - 25, -e^3 - 7*e^2 - 9*e - 8, -15*e^3 - 79*e^2 - 66*e + 43, 15*e^3 + 85*e^2 + 95*e - 41, -4*e^3 - 22*e^2 - 28*e + 2, 3*e^3 + 13*e^2 - 3*e - 6, 17*e^3 + 93*e^2 + 94*e - 53, -12*e^3 - 69*e^2 - 81*e + 34, -12*e^3 - 62*e^2 - 46*e + 39, -3*e^3 - 13*e^2 - 11*e + 6, 25*e^3 + 136*e^2 + 129*e - 74, 9*e^3 + 47*e^2 + 38*e - 22, -10*e^3 - 51*e^2 - 46*e + 33, -12*e^3 - 70*e^2 - 74*e + 43, 18*e^3 + 98*e^2 + 90*e - 43, 10*e^3 + 51*e^2 + 45*e - 27, -23*e^3 - 122*e^2 - 113*e + 71, 9*e^2 + 20*e - 25, 5*e^3 + 28*e^2 + 23*e - 30, -10*e^3 - 61*e^2 - 65*e + 60, 9*e^3 + 47*e^2 + 47*e - 22, -6*e^3 - 37*e^2 - 49*e - 2, 6*e^3 + 32*e^2 + 35*e - 35, -5*e^3 - 22*e^2 - 16*e + 11, -20*e^3 - 113*e^2 - 113*e + 78, -7*e^3 - 34*e^2 - 25*e + 16, -10*e^3 - 46*e^2 - 22*e + 37, 17*e^3 + 90*e^2 + 84*e - 33, 12*e^3 + 61*e^2 + 51*e - 36, -5*e^3 - 30*e^2 - 38*e + 2, -20*e^3 - 97*e^2 - 70*e + 50, -3*e^3 - 25*e^2 - 33*e + 41, -e^3 - 2*e^2 + 5*e - 21, e^3 + 10*e^2 + 15*e - 23, -21*e^3 - 109*e^2 - 78*e + 72, 5*e^3 + 25*e^2 + 18*e - 20, 6*e^3 + 39*e^2 + 47*e - 36, -3*e^3 - 10*e^2 + 6*e - 6, 20*e^3 + 102*e^2 + 76*e - 57, 5*e^3 + 19*e^2 - 5*e - 46, -e^3 - 16*e^2 - 37*e + 14, 20*e^3 + 114*e^2 + 119*e - 64, 3*e^3 + 11*e^2 + 6*e + 1, 26*e^3 + 136*e^2 + 116*e - 72, -8*e^2 - 27*e + 16, e^2 + 16*e + 33, -12*e^3 - 58*e^2 - 40*e + 4, 18*e^3 + 92*e^2 + 84*e - 55, e^3 + 19*e^2 + 51*e - 2, -22*e^3 - 115*e^2 - 101*e + 64, 18*e^3 + 90*e^2 + 65*e - 61, 2*e^3 + 21*e^2 + 52*e - 3, 9*e^3 + 43*e^2 + 24*e - 48, -8*e^3 - 32*e^2 + 2*e + 18, 9*e^3 + 58*e^2 + 91*e - 22, -e^3 - e^2 + 4*e + 7, 6*e^3 + 34*e^2 + 28*e - 28, -18*e^3 - 97*e^2 - 96*e + 33, 18*e^3 + 107*e^2 + 118*e - 67, -13*e^3 - 72*e^2 - 68*e + 56, 9*e^3 + 39*e^2 + 28*e + 16, 9*e^3 + 39*e^2 + 21*e - 12, -6*e^3 - 37*e^2 - 42*e + 20, 15*e^3 + 78*e^2 + 67*e - 64, -e^3 - 16*e^2 - 44*e + 7, 27*e^3 + 141*e^2 + 122*e - 62, -12*e^3 - 73*e^2 - 86*e + 20, 2*e^3 + 11*e^2 + 5*e - 1, -14*e^3 - 76*e^2 - 73*e + 15, -8*e^3 - 46*e^2 - 53*e + 32, 16*e^3 + 87*e^2 + 85*e - 59, 7*e^3 + 46*e^2 + 68*e - 11, 21*e^3 + 127*e^2 + 154*e - 75, -20*e^3 - 114*e^2 - 116*e + 76, 27*e^3 + 151*e^2 + 152*e - 99, -16*e^3 - 90*e^2 - 107*e + 38, 22*e^3 + 120*e^2 + 119*e - 73, -6*e^2 - 8*e + 4, 26*e^3 + 139*e^2 + 115*e - 82, -26*e^3 - 135*e^2 - 100*e + 78, -11*e^3 - 68*e^2 - 78*e + 59, -19*e^3 - 96*e^2 - 92*e + 32, -36*e^3 - 178*e^2 - 137*e + 86, 5*e^3 + 29*e^2 + 32*e - 61, -20*e^3 - 119*e^2 - 128*e + 92, 23*e^3 + 117*e^2 + 106*e - 62, -26*e^3 - 146*e^2 - 140*e + 93, 23*e^3 + 128*e^2 + 134*e - 77, 26*e^3 + 140*e^2 + 126*e - 61, -4*e^3 - 22*e^2 - 37*e - 12, 4*e^3 + 15*e^2 - 9*e - 38, 8*e^3 + 55*e^2 + 83*e - 13, -2*e^3 - 17*e^2 - 43*e - 8, -2*e^3 + 4*e^2 + 37*e - 1, 2*e^2 + 21*e + 18, -19*e^3 - 109*e^2 - 130*e + 45, 2*e^3 + 10*e^2 + e - 54, 5*e^3 + 22*e^2 + 12*e - 11, -27*e^3 - 150*e^2 - 153*e + 73, 4*e^3 + 30*e^2 + 54*e - 25, 3*e^3 + 8*e^2 - 14*e - 15, 23*e^3 + 134*e^2 + 151*e - 74, 8*e^3 + 56*e^2 + 84*e - 31, 17*e^3 + 86*e^2 + 70*e - 25, 17*e^3 + 95*e^2 + 88*e - 44, 14*e^3 + 65*e^2 + 42*e - 19, 24*e^3 + 114*e^2 + 79*e - 64, -21*e^3 - 124*e^2 - 149*e + 66, 31*e^3 + 163*e^2 + 150*e - 89, -11*e^3 - 53*e^2 - 32*e + 23, 16*e^3 + 91*e^2 + 97*e - 69, 27*e^3 + 133*e^2 + 86*e - 83, -5*e^3 - 24*e^2 - 21*e - 14, e^3 - 5*e^2 - 7*e + 47, 3*e^3 + 28*e^2 + 45*e - 43]
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, w^2 - 5])] = -1

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