/*
  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([3, -4, -8, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([17,17,w^3 - 3*w^2 - 2*w + 2])

primes_array = [
[3, 3, w],\
[3, 3, w^3 - 2*w^2 - 6*w + 1],\
[9, 3, w^3 - 2*w^2 - 5*w + 1],\
[16, 2, 2],\
[17, 17, w^3 - w^2 - 8*w - 5],\
[17, 17, -2*w^3 + 4*w^2 + 11*w - 2],\
[17, 17, -w^3 + 3*w^2 + 2*w - 2],\
[17, 17, w^3 - 2*w^2 - 4*w + 1],\
[25, 5, w^3 - 3*w^2 - 3*w + 1],\
[25, 5, w^2 - 2*w - 7],\
[29, 29, w^3 - 2*w^2 - 6*w - 1],\
[29, 29, w - 2],\
[53, 53, w^3 - 3*w^2 - 4*w + 4],\
[53, 53, -w^3 + 3*w^2 + 4*w - 5],\
[61, 61, -w^3 + 2*w^2 + 7*w - 4],\
[61, 61, -4*w^3 + 11*w^2 + 14*w - 8],\
[101, 101, -w^3 + 2*w^2 + 7*w - 1],\
[103, 103, -w^3 + 3*w^2 + 2*w - 5],\
[103, 103, -w^3 + w^2 + 8*w + 2],\
[113, 113, w^3 - 3*w^2 - 3*w + 7],\
[113, 113, w^2 - 2*w - 1],\
[127, 127, w^3 - 2*w^2 - 4*w - 2],\
[127, 127, -2*w^3 + 4*w^2 + 11*w + 1],\
[131, 131, -3*w^3 + 7*w^2 + 14*w - 8],\
[131, 131, -w^2 + w + 8],\
[131, 131, 2*w^3 - 5*w^2 - 9*w + 1],\
[131, 131, -3*w^3 + 6*w^2 + 16*w + 1],\
[139, 139, -2*w^3 + 5*w^2 + 9*w - 2],\
[139, 139, -w^2 + w + 7],\
[157, 157, -2*w^3 + 5*w^2 + 7*w - 5],\
[157, 157, -2*w^3 + 3*w^2 + 13*w + 2],\
[157, 157, -2*w^3 + 5*w^2 + 6*w - 2],\
[157, 157, -3*w^3 + 5*w^2 + 19*w + 4],\
[169, 13, -2*w^3 + 4*w^2 + 10*w - 1],\
[173, 173, 3*w^3 - 8*w^2 - 10*w + 7],\
[173, 173, 2*w^3 - 2*w^2 - 15*w - 8],\
[179, 179, -3*w^2 + 6*w + 17],\
[179, 179, -2*w^3 + 3*w^2 + 12*w + 1],\
[179, 179, -3*w^3 + 7*w^2 + 13*w - 7],\
[179, 179, -2*w^3 + 3*w^2 + 14*w + 4],\
[191, 191, w^3 - 3*w^2 - w + 4],\
[191, 191, -2*w^3 + 3*w^2 + 14*w + 2],\
[199, 199, -w^3 + 8*w + 10],\
[199, 199, 5*w^3 - 15*w^2 - 14*w + 14],\
[211, 211, -w - 4],\
[211, 211, -w^3 + 2*w^2 + 6*w - 5],\
[211, 211, 2*w^3 - 3*w^2 - 12*w - 4],\
[211, 211, -3*w^3 + 7*w^2 + 13*w - 4],\
[257, 257, -3*w^3 + 8*w^2 + 10*w - 2],\
[257, 257, 2*w^3 - 2*w^2 - 16*w - 7],\
[263, 263, 3*w^3 - 6*w^2 - 17*w + 1],\
[263, 263, w^3 - 2*w^2 - 3*w - 1],\
[269, 269, 5*w^3 - 13*w^2 - 19*w + 7],\
[269, 269, -3*w^3 + 6*w^2 + 18*w - 1],\
[277, 277, w^3 - 4*w^2 - w + 5],\
[277, 277, -w^3 + 4*w^2 + w - 11],\
[283, 283, -w^3 + 5*w^2 - w - 14],\
[283, 283, 2*w^3 - 7*w^2 - 4*w + 10],\
[283, 283, -w^3 + w^2 + 9*w + 1],\
[283, 283, w^2 - 4*w - 5],\
[311, 311, -w^3 + w^2 + 9*w + 5],\
[311, 311, w^2 - 4*w - 1],\
[313, 313, -4*w^3 + 9*w^2 + 19*w - 10],\
[313, 313, -2*w^2 + 5*w + 11],\
[337, 337, -3*w^3 + 6*w^2 + 14*w + 2],\
[337, 337, -4*w^3 + 8*w^2 + 21*w + 1],\
[347, 347, -2*w^3 + 6*w^2 + 7*w - 10],\
[347, 347, -w^3 + 4*w^2 + 2*w - 7],\
[361, 19, -4*w^3 + 11*w^2 + 13*w - 10],\
[361, 19, -2*w^3 + 5*w^2 + 8*w - 10],\
[373, 373, 2*w^3 - 5*w^2 - 10*w + 2],\
[373, 373, -w^3 + 2*w^2 + 8*w - 2],\
[373, 373, -2*w^3 + 4*w^2 + 13*w - 1],\
[373, 373, -w^3 + 3*w^2 + 5*w - 8],\
[389, 389, -w^3 + 4*w^2 + 2*w - 10],\
[389, 389, -2*w^3 + 6*w^2 + 7*w - 7],\
[419, 419, -2*w^3 + 5*w^2 + 6*w - 1],\
[419, 419, -3*w^3 + 5*w^2 + 19*w + 5],\
[433, 433, w^2 - 4*w - 2],\
[433, 433, w^3 - w^2 - 9*w - 4],\
[439, 439, -w^3 + 4*w^2 + w - 2],\
[439, 439, w^3 - 4*w^2 - w + 14],\
[443, 443, -2*w^3 + 5*w^2 + 9*w + 1],\
[443, 443, -w^2 + w + 10],\
[467, 467, w^3 - w^2 - 8*w + 4],\
[467, 467, w^3 - 4*w^2 + 2],\
[491, 491, -4*w^3 + 12*w^2 + 10*w - 11],\
[491, 491, 2*w^3 - w^2 - 18*w - 11],\
[491, 491, 3*w^3 - 9*w^2 - 7*w + 11],\
[491, 491, 2*w^3 - 20*w - 19],\
[503, 503, w^3 - 3*w^2 - 4*w + 10],\
[503, 503, -w^3 + w^2 + 10*w - 2],\
[523, 523, -4*w^3 + 9*w^2 + 22*w - 7],\
[523, 523, w^3 - 3*w^2 - 7*w + 5],\
[529, 23, -2*w^3 + 4*w^2 + 10*w + 5],\
[529, 23, -2*w^3 + 4*w^2 + 10*w - 7],\
[547, 547, 2*w^3 - w^2 - 18*w - 14],\
[547, 547, 3*w^3 - 8*w^2 - 14*w + 8],\
[571, 571, 5*w^3 - 11*w^2 - 25*w + 5],\
[571, 571, -4*w^3 + 9*w^2 + 19*w - 11],\
[599, 599, -5*w^3 + 12*w^2 + 20*w - 4],\
[599, 599, 4*w^3 - 6*w^2 - 25*w - 11],\
[601, 601, -4*w^3 + 9*w^2 + 20*w - 5],\
[601, 601, w^3 - 9*w - 14],\
[601, 601, 3*w^3 - 8*w^2 - 11*w + 2],\
[601, 601, -2*w^3 + 3*w^2 + 14*w - 1],\
[641, 641, -2*w^3 + 3*w^2 + 15*w - 5],\
[641, 641, -4*w^3 + 8*w^2 + 23*w - 2],\
[647, 647, -3*w^3 + 8*w^2 + 13*w - 11],\
[647, 647, -w^3 + 4*w^2 + 3*w - 7],\
[659, 659, 3*w^3 - 7*w^2 - 14*w - 2],\
[659, 659, w^3 - w^2 - 6*w - 11],\
[673, 673, -3*w^3 + 7*w^2 + 11*w - 1],\
[673, 673, -4*w^3 + 7*w^2 + 24*w + 5],\
[677, 677, 5*w^3 - 12*w^2 - 21*w + 11],\
[677, 677, -2*w^3 + 6*w^2 + 5*w - 1],\
[677, 677, -2*w^3 + 6*w^2 + 8*w - 7],\
[677, 677, 2*w^3 - 6*w^2 - 8*w + 11],\
[719, 719, -2*w^3 + 6*w^2 + 10*w - 7],\
[719, 719, -4*w^3 + 10*w^2 + 20*w - 13],\
[727, 727, 3*w^3 - 10*w^2 - 8*w + 10],\
[727, 727, 3*w^3 - 8*w^2 - 13*w + 10],\
[727, 727, 2*w^3 - 8*w^2 - 3*w + 23],\
[727, 727, -w^3 + 4*w^2 + 3*w - 8],\
[751, 751, -7*w^3 + 18*w^2 + 25*w - 7],\
[751, 751, w^3 - 7*w^2 + 5*w + 26],\
[757, 757, -4*w^3 + 9*w^2 + 18*w - 7],\
[757, 757, -3*w^3 + 5*w^2 + 17*w + 1],\
[797, 797, 2*w^2 - 4*w - 5],\
[797, 797, 2*w^3 - 6*w^2 - 6*w + 11],\
[823, 823, -2*w^3 + 2*w^2 + 15*w + 14],\
[823, 823, -3*w^3 + 8*w^2 + 10*w - 1],\
[829, 829, -6*w^3 + 16*w^2 + 22*w - 7],\
[829, 829, 5*w^3 - 13*w^2 - 20*w + 8],\
[829, 829, -w^3 + 7*w + 4],\
[829, 829, -5*w^3 + 12*w^2 + 23*w - 14],\
[841, 29, -3*w^3 + 6*w^2 + 15*w - 2],\
[859, 859, -4*w^3 + 7*w^2 + 24*w - 1],\
[859, 859, -5*w^3 + 15*w^2 + 14*w - 13],\
[881, 881, 3*w^3 - 7*w^2 - 16*w + 4],\
[881, 881, -w^3 + 3*w^2 + 6*w - 7],\
[883, 883, -2*w^3 + 3*w^2 + 12*w + 10],\
[883, 883, -3*w^3 + 7*w^2 + 13*w + 2],\
[887, 887, 4*w^3 - 11*w^2 - 11*w + 11],\
[887, 887, 2*w^3 - 7*w^2 - 3*w + 7],\
[887, 887, 4*w^3 - 5*w^2 - 29*w - 10],\
[887, 887, 3*w^2 - 7*w - 16],\
[907, 907, -w^3 + 3*w^2 + 8*w - 5],\
[907, 907, -5*w^3 + 11*w^2 + 28*w - 8],\
[911, 911, -4*w^3 + 10*w^2 + 18*w - 17],\
[911, 911, 2*w^2 - 2*w - 1],\
[919, 919, 4*w^3 - 7*w^2 - 25*w - 4],\
[919, 919, 2*w^3 - 5*w^2 - 5*w + 1],\
[937, 937, -w^3 + 3*w^2 + w - 8],\
[937, 937, -2*w^3 + 3*w^2 + 14*w - 2],\
[953, 953, -2*w^3 + w^2 + 17*w + 19],\
[953, 953, 4*w^3 - 11*w^2 - 13*w + 4],\
[961, 31, 2*w^3 - 3*w^2 - 11*w - 8],\
[961, 31, w^3 - 4*w^2 + 2*w + 11],\
[971, 971, 2*w^2 - 5*w - 4],\
[971, 971, -w^3 + 4*w^2 - 11],\
[991, 991, -2*w^3 + 6*w^2 + 9*w - 8],\
[991, 991, -2*w^3 + 3*w^2 + 13*w - 1],\
[991, 991, 3*w^3 - 8*w^2 - 14*w + 11],\
[991, 991, -2*w^3 + 5*w^2 + 7*w - 8],\
[997, 997, -2*w^3 + 5*w^2 + 6*w + 1],\
[997, 997, -3*w^3 + 5*w^2 + 19*w + 7],\
[997, 997, -6*w^3 + 12*w^2 + 33*w - 4],\
[997, 997, 4*w^3 - 11*w^2 - 15*w + 11]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [e, -e - 1, -e^2 - 2*e + 1, -e^2 - 2*e + 3, 3, -2*e^2 - e + 5, 1, e^2 + 2*e - 3, -2*e^2 - 5*e + 1, -3*e^2 - 3*e + 6, 3*e^2 + 2*e - 9, 2*e^2 + 2*e, 2*e^2 + 2*e - 11, -2*e^2 + e + 2, -2*e^2 - 6*e + 2, 5*e^2 + 7*e - 12, 6*e^2 + 7*e - 12, 4*e - 2, 6*e^2 + 11*e - 10, e^2 + 3*e - 13, 2*e + 3, -e^2 - 2*e - 5, -e^2 + 2*e + 12, -3*e^2 + e + 10, e^2 + 4*e - 16, -2*e^2 - 10*e + 9, 3*e^2 + 8*e - 8, 7*e^2 + 17*e - 9, -8*e^2 - 6*e + 17, 2*e^2 + 10*e - 2, 4*e^2 + 6*e + 8, 7*e^2 + 5*e - 12, e^2 + 10*e + 2, 9*e^2 + 13*e - 18, 7*e^2 + e - 27, 5*e^2 + 2*e - 16, 2*e^2 + 2*e + 4, 4*e^2 + 6*e - 8, -2*e^2 - 4*e - 14, 8*e^2 + 8*e - 22, e^2 - e - 6, -4*e^2 + 7*e + 23, 9*e^2 + 11*e - 11, 8*e^2 + 14*e - 18, -2*e^2 - e + 2, -8*e^2 - 18*e + 15, -5*e^2 - 12*e + 10, -3*e^2 - 6*e + 15, 2*e^2 + 2*e + 8, -8*e^2 - 16*e + 9, -2*e^2 - 6*e - 7, 6*e^2 - 16, 12*e^2 + 15*e - 18, -3*e^2 - 10*e - 8, 2*e^2 + 2*e - 20, -2*e^2 - 6*e + 2, 4*e^2 + 5*e - 18, -e^2 - 7*e + 6, e^2 + 7*e + 19, 7*e^2 + 3*e - 22, -9*e^2 - 11*e + 2, 4*e^2 + 8*e - 18, -3*e^2 - 3*e + 22, -10*e^2 - 8*e + 10, -4*e^2 - 9*e - 11, -8*e^2 - 20*e + 22, -e^2 - 14*e - 12, -8*e^2 - 12*e - 8, -6, e^2 + 3*e - 6, 2*e^2 + 4*e - 26, -e^2 - 9*e - 8, 8*e + 14, 25, 12*e^2 + 13*e - 16, -5*e^2 - 6*e, 10*e^2 + 21*e - 9, 7*e^2 + 17*e - 14, -6*e^2 + 33, 13*e^2 + 10*e - 33, 7*e^2 + 8*e - 11, 8*e^2 + 23*e - 20, -5*e^2 - 14*e + 31, 6*e^2 + 7*e - 20, 2*e^2 + 2*e + 30, -5*e^2 - 6*e - 16, 6*e^2 + 23*e - 17, e^2 + 19*e + 2, -15*e^2 - 20*e + 29, 13*e^2 + 13*e - 20, -5*e^2 - 10*e + 16, -e^2 - 25, -19*e^2 - 23*e + 32, -15*e^2 - 17*e + 19, 10*e^2 + 6*e - 36, -e^2 - 11*e + 5, -6*e^2 - 3*e + 34, 4*e^2 - 2*e - 38, -20*e^2 - 17*e + 40, -10*e^2 - 4*e + 14, -8*e^2 - 10*e - 8, -6*e^2 + 4*e + 6, -13*e^2 - 7*e + 32, 2*e^2 + 17*e + 8, 10*e^2 + 9*e - 13, 4*e^2 + 14*e + 12, -16*e^2 - 23*e + 8, 4*e^2 + e - 15, -e^2 - 7*e - 9, 3*e^2 + 6*e + 4, 18*e^2 + 20*e - 12, -2*e^2 - 27*e - 8, 3*e^2 + 25*e + 1, 16*e + 16, 6*e^2 + 18*e - 32, -19*e^2 - 31*e + 16, -13*e^2 - 26*e + 11, -4*e^2 + 12*e + 14, -8*e^2 - 15*e + 10, -11*e^2 - 13*e + 28, 7*e^2 - 9, -18*e^2 - 38*e + 27, -16*e^2 - 6*e + 50, -10*e^2 - 26*e + 6, -9*e - 10, -5*e^2 + 13*e + 31, -10*e^2 - 21*e + 20, -18*e^2 - 21*e + 22, 10*e^2 + 8*e, e^2 - 21*e - 26, -5*e^2 + 15*e + 20, 16*e^2 + 20*e - 44, 22*e^2 + 14*e - 57, -14*e^2 - 18*e + 46, 9*e^2 + 12*e + 15, 12*e^2 + 6*e - 40, 7*e^2 + 5*e - 23, 6*e^2 + 27*e - 4, -4*e^2 - 6*e + 12, -5*e^2 + 5*e + 44, -14*e^2 - 2*e + 54, 10*e^2 + 29*e - 6, -8*e^2 - 32*e - 7, 7*e^2 + 14*e - 13, -21*e^2 - 26*e + 44, 20*e^2 + 12*e - 51, 11*e^2 + 29*e - 34, 8*e^2 + 16*e, -4*e^2 - 20, -10*e^2 - 7*e + 24, 10*e^2 + 5*e - 20, -6*e^2 + 8*e + 33, -11*e - 8, 8*e^2 + 21*e - 6, -6*e^2 - 4*e - 10, 15*e^2 + 19*e - 26, 10*e^2 + 4*e - 16, 11*e^2 + 23*e - 10, 9*e^2 + 6*e + 7, 3*e^2 + 13*e - 20, 19*e^2 + 21*e - 40, -14*e^2 - 11*e + 40, -19*e^2 - 48*e + 37, -12*e - 19, 19*e^2 + 29*e - 19, -e^2 + 17*e + 12, 8*e^2 + 26*e + 18, -24*e^2 - 38*e + 28, -6*e^2 + e + 23]
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^2 - 2*w + 2])] = -1

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