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

hecke_eigenvalues_array = [1, e, -1, -e - 3, -e - 1, -2*e - 2, 3*e + 6, -2*e - 10, 5*e + 5, -6*e - 14, 7*e + 10, -3*e - 4, -5*e - 8, 8*e + 14, 2*e, -3*e - 13, -3*e + 5, 10*e + 14, 3*e + 2, -e + 1, 3*e + 1, -3*e - 2, 8*e + 18, -7*e - 21, 5*e + 8, 9*e + 13, 10*e + 22, 5*e + 8, -6*e - 18, -13*e - 24, -4*e - 24, 6*e + 2, 8*e + 18, -7*e - 17, -9*e - 6, -8*e - 22, 5*e + 2, -4*e - 26, 6*e + 18, -8, 4, -19*e - 25, -5*e + 8, -16*e - 16, 12*e + 14, 16*e + 22, -16*e - 24, -8*e - 22, -16*e - 20, -3*e - 18, -20*e - 32, -22*e - 34, 18*e + 22, -8*e - 24, -e - 15, 11*e + 11, 3*e + 5, -e - 22, 4*e + 12, -6*e - 4, -13*e - 35, 18*e + 30, 8*e + 28, 27*e + 41, 23*e + 34, 9*e + 34, -7*e + 15, -2*e - 26, 2*e + 2, 12*e + 18, -15*e - 23, -10*e - 36, -9*e - 15, -5*e - 24, 6*e + 4, -7*e - 20, 28*e + 36, 6, 2*e - 12, -7*e - 14, 2*e - 6, -15*e - 1, 7*e + 26, 12, -13*e - 27, -16*e - 24, 16*e + 26, 9*e - 17, 32*e + 50, 18*e + 36, 2*e + 18, -25*e - 32, 8*e - 2, -35*e - 54, -30, -22*e - 46, -20*e - 24, 5*e + 12, -20*e - 20, -6*e - 22, -27*e - 27, 22*e + 34, -24*e - 52, -24*e - 28, -12*e - 38, -18*e - 24, -30*e - 50, 4*e + 46, -18*e - 24, 9*e + 12, 6*e - 28, -4*e - 14, -5*e + 1, -9*e - 13, 8*e + 36, 2*e + 40, 22*e + 12, 10*e - 4, 26*e + 40, -11*e + 2, -17*e - 31, 6*e - 8, 14*e + 36, 2*e + 10, -17*e - 7, -3*e - 27, 11*e + 5, 29*e + 47, -5*e - 8, -10*e - 12, 3*e - 3, 5*e - 7, 21*e + 54, -5*e + 15, 6*e + 10, -15*e - 19, 6*e + 10, 6, 30*e + 36, -3*e - 36, 16*e + 30, 2*e + 30, -9*e + 25, -23*e - 48, -6*e - 34, -30*e - 30, 13*e + 38, 12*e + 38, -22*e - 18, e + 6, -e - 42, 8*e + 50, -2*e + 48, 23*e + 42, -24*e - 56, 34*e + 58, -26*e - 34, -4*e + 8, 24*e + 42, -14*e + 18, 10*e - 34, 8*e + 30, -6*e - 24, -12*e - 48, 10*e + 14, -27*e - 52, 29*e + 63, -21*e - 42, 19*e + 19, 8*e + 36, -33*e - 35, -20*e - 44, -18*e - 14, -3*e + 9, 5*e + 24, -44*e - 76, -12*e - 40, -23*e - 40]
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]