/*
  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 - 4*w - 1])

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^3 + 7*x^2 + x - 46
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [-1, -2*e^2 - 5*e + 21, 2*e^2 + 5*e - 21, 1, e, -6*e^2 - 15*e + 58, 4*e^2 + 8*e - 44, 3*e^2 + 6*e - 36, 5*e^2 + 13*e - 49, 6*e^2 + 13*e - 64, -9*e^2 - 21*e + 93, -2*e^2 - 5*e + 12, 8*e^2 + 19*e - 82, -3*e^2 - 7*e + 31, e^2 + 5*e - 11, -4*e^2 - 9*e + 39, e^2 + 2*e - 16, -13*e^2 - 33*e + 125, -3*e^2 - 8*e + 32, 9*e^2 + 20*e - 99, 11*e^2 + 26*e - 111, -e^2 - 2*e + 20, -9*e^2 - 24*e + 82, -2*e^2 - 4*e + 30, 2*e^2 + 9*e - 17, -4*e^2 - 11*e + 30, -13*e^2 - 34*e + 115, 11*e^2 + 27*e - 107, 2*e + 8, 9*e^2 + 21*e - 93, 3*e^2 + 11*e - 29, -6*e^2 - 14*e + 52, -2*e^2 - 3*e + 6, -12*e^2 - 31*e + 122, -6*e^2 - 13*e + 59, -4*e^2 - 6*e + 40, -10*e^2 - 28*e + 94, -2*e^2 - 3*e + 18, 2*e^2 + 6*e - 12, 4*e^2 + 12*e - 36, -4*e^2 - 14*e + 32, -6*e^2 - 13*e + 56, 11*e^2 + 28*e - 114, 2*e^2 - 18, 4*e^2 + 14*e - 34, 9*e^2 + 25*e - 89, 10*e^2 + 24*e - 102, 8*e^2 + 22*e - 74, -11*e^2 - 23*e + 133, 5*e^2 + 17*e - 41, 10*e^2 + 22*e - 100, 10*e^2 + 24*e - 84, -14*e^2 - 31*e + 142, 16*e^2 + 39*e - 171, -9*e^2 - 24*e + 92, -5*e^2 - 10*e + 66, 4*e^2 + 12*e - 32, 7*e^2 + 19*e - 51, 11*e^2 + 28*e - 117, -6*e - 18, 9*e^2 + 20*e - 106, -e^2 - 2*e + 25, 23*e^2 + 58*e - 226, -27*e^2 - 67*e + 263, -4*e + 4, -13*e^2 - 27*e + 139, 5*e^2 + 13*e - 31, e^2 + 6*e - 8, 4*e^2 + 7*e - 28, 5*e^2 + 20*e - 46, 24*e^2 + 60*e - 234, 17*e^2 + 36*e - 192, 4*e^2 + 10*e - 40, -9*e^2 - 27*e + 65, -15*e^2 - 36*e + 141, -16*e^2 - 34*e + 168, -28*e^2 - 70*e + 266, 10*e^2 + 24*e - 116, -10*e^2 - 23*e + 108, 7*e^2 + 17*e - 65, 11*e^2 + 26*e - 131, 2*e^2 + e - 21, -18*e^2 - 50*e + 164, -10*e^2 - 23*e + 104, -3*e^2 - 10*e + 34, -6*e^2 - 20*e + 60, -14*e^2 - 30*e + 138, 32*e^2 + 74*e - 334, 14*e^2 + 35*e - 143, 18*e^2 + 40*e - 186, 13*e^2 + 27*e - 147, 4*e^2 + 6*e - 60, -14*e^2 - 39*e + 142, -9*e^2 - 17*e + 83, -6*e^2 - 17*e + 68, -2*e^2 - 2*e + 20, 10*e^2 + 27*e - 84, 17*e^2 + 33*e - 189, 8*e^2 + 20*e - 86, 19*e^2 + 45*e - 177, 18*e^2 + 42*e - 182, -6*e^2 - 11*e + 45, 40*e^2 + 92*e - 422, -25*e^2 - 55*e + 261, 19*e^2 + 39*e - 207, -26*e^2 - 66*e + 240, -e^2 + 2*e - 1, 19*e^2 + 53*e - 177, 15*e^2 + 37*e - 167, -9*e^2 - 23*e + 75, 5*e^2 + 14*e - 70, -11*e^2 - 20*e + 136, 4*e^2 + 6*e - 26, -16*e^2 - 40*e + 134, 14*e^2 + 36*e - 150, -30*e^2 - 68*e + 312, -6*e^2 - 12*e + 50, -2*e^2 - 3*e + 57, 24*e^2 + 55*e - 268, 2*e^2 + 12*e - 30, 19*e^2 + 47*e - 179, 13*e^2 + 32*e - 124, 6*e^2 + 4*e - 78, -4*e + 6, 11*e^2 + 23*e - 127, -19*e^2 - 52*e + 178, -9*e^2 - 30*e + 68, -31*e^2 - 68*e + 330, -11*e^2 - 25*e + 107, -14*e^2 - 33*e + 176, 4*e^2 + 13*e - 23, 4*e^2 + 9*e - 28, -15*e^2 - 28*e + 184, 20*e^2 + 46*e - 196, 8*e^2 + 13*e - 105, -16*e^2 - 44*e + 128, 28*e^2 + 75*e - 261, 2*e^2 + 10*e - 14, 8*e^2 + 20*e - 82, -21*e^2 - 51*e + 219, -4*e^2 - 3*e + 46, 42*e^2 + 108*e - 388, 34*e^2 + 83*e - 342, -33*e^2 - 70*e + 362, -5*e^2 - 13*e + 59, -13*e^2 - 29*e + 113, -11*e^2 - 33*e + 81, 21*e^2 + 54*e - 209, -11*e^2 - 31*e + 87, -5*e + 10, -22*e^2 - 57*e + 176, -32*e^2 - 76*e + 340, -38, 15*e^2 + 33*e - 165, -24*e^2 - 63*e + 206, -16*e^2 - 44*e + 126, 3*e^2 + 3*e - 65, -9*e^2 - 33*e + 67, 26*e^2 + 58*e - 302, -27*e^2 - 63*e + 293, -18*e^2 - 38*e + 190, -18*e^2 - 46*e + 178, -32*e^2 - 68*e + 348, -13*e^2 - 41*e + 115, 16*e^2 + 36*e - 162, -20*e^2 - 43*e + 207, -31*e^2 - 74*e + 309, -14*e^2 - 29*e + 154, 13*e^2 + 23*e - 157, 43*e^2 + 110*e - 414, -22*e^2 - 52*e + 208, -21*e^2 - 52*e + 180, -3*e^2 + e + 25, 15*e^2 + 30*e - 155, -17*e^2 - 46*e + 164, -4*e^2 - 12*e + 44, 2*e^2 - e - 26, 24*e^2 + 60*e - 238]
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])] = -1

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