/*
  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, -3, -6, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([9, 3, -w^2 + 3])

primes_array = [
[3, 3, w],\
[5, 5, w - 1],\
[7, 7, w^3 - w^2 - 4*w + 1],\
[11, 11, -w^2 + w + 2],\
[13, 13, -w^3 + w^2 + 3*w - 1],\
[16, 2, 2],\
[17, 17, -w^3 + 5*w + 2],\
[29, 29, -w^2 + w + 1],\
[37, 37, -w^3 + 2*w^2 + 4*w - 4],\
[41, 41, -w^3 + w^2 + 5*w + 1],\
[41, 41, w^2 - 5],\
[47, 47, w^3 - w^2 - 5*w - 2],\
[47, 47, -w^3 + 2*w^2 + 4*w - 1],\
[53, 53, w^3 - 4*w - 4],\
[53, 53, 2*w^3 - 2*w^2 - 9*w + 1],\
[61, 61, -w^3 + w^2 + 6*w - 2],\
[71, 71, w^3 - w^2 - 6*w - 2],\
[103, 103, 2*w^3 - 2*w^2 - 8*w + 1],\
[103, 103, -3*w^3 + 3*w^2 + 13*w - 5],\
[109, 109, 2*w^3 - w^2 - 8*w - 4],\
[109, 109, 2*w^3 - w^2 - 9*w - 1],\
[113, 113, -2*w^3 + w^2 + 11*w - 1],\
[113, 113, 2*w^3 - w^2 - 10*w - 4],\
[125, 5, 2*w^3 - 3*w^2 - 10*w + 4],\
[131, 131, -2*w^3 + w^2 + 9*w - 1],\
[139, 139, 2*w^3 - 4*w^2 - 7*w + 5],\
[139, 139, 3*w^3 - w^2 - 16*w - 5],\
[149, 149, w^2 + w - 4],\
[149, 149, 3*w^3 - 3*w^2 - 13*w + 1],\
[151, 151, w^2 - 2*w - 8],\
[157, 157, w^3 - w^2 - 7*w - 2],\
[157, 157, -2*w^2 + w + 7],\
[167, 167, 2*w^3 - 2*w^2 - 7*w + 5],\
[173, 173, -w^3 + 6*w - 2],\
[173, 173, -w^3 + 2*w^2 + 5*w - 7],\
[181, 181, -3*w^3 + 2*w^2 + 14*w + 1],\
[191, 191, -w^3 + 2*w^2 + 2*w - 5],\
[191, 191, -w^3 + 2*w^2 + 6*w - 5],\
[193, 193, -w^3 + 3*w^2 + 4*w - 4],\
[193, 193, 2*w^3 + 3*w^2 - 13*w - 20],\
[197, 197, w^3 - 2*w - 2],\
[199, 199, 2*w^3 - w^2 - 11*w - 2],\
[211, 211, -2*w^3 + 9*w + 5],\
[223, 223, -w^3 + w^2 + 4*w - 5],\
[223, 223, w^2 - 3*w - 2],\
[229, 229, 2*w^3 - 2*w^2 - 7*w + 1],\
[241, 241, 3*w^2 - 2*w - 13],\
[241, 241, -w^3 + w^2 + 5*w + 4],\
[257, 257, -3*w^3 + w^2 + 15*w + 8],\
[263, 263, -3*w^3 + 5*w^2 + 9*w - 7],\
[263, 263, 2*w^3 - w^2 - 9*w + 2],\
[263, 263, -w^3 + 4*w^2 + 2*w - 16],\
[263, 263, w^3 - w^2 - 2*w - 2],\
[271, 271, -3*w - 1],\
[281, 281, 2*w^3 - 3*w^2 - 8*w + 2],\
[281, 281, 3*w^3 - 4*w^2 - 12*w + 4],\
[283, 283, w^3 - 5*w - 7],\
[293, 293, -2*w^3 + w^2 + 10*w - 1],\
[307, 307, w^3 - 3*w^2 - 3*w + 8],\
[307, 307, 3*w^3 - 15*w - 10],\
[311, 311, 2*w^3 - 3*w^2 - 7*w + 7],\
[313, 313, -w^3 + 7*w + 2],\
[337, 337, 3*w^3 - 17*w - 10],\
[337, 337, 2*w^3 - 2*w^2 - 9*w - 5],\
[343, 7, -2*w^3 + w^2 + 8*w + 2],\
[347, 347, -3*w^3 + 17*w + 14],\
[347, 347, -3*w^3 - w^2 + 17*w + 13],\
[349, 349, 2*w^3 - w^2 - 8*w - 1],\
[349, 349, -w^3 + 3*w^2 + 4*w - 8],\
[353, 353, -w^3 - w^2 + 7*w + 4],\
[359, 359, -2*w^3 + 9*w + 10],\
[367, 367, 3*w^3 - 4*w^2 - 9*w + 4],\
[367, 367, -w^3 + w^2 + 4*w + 4],\
[367, 367, -w^3 + 3*w^2 + 3*w - 7],\
[373, 373, -w^2 + w - 2],\
[373, 373, -2*w^3 + 2*w^2 + 11*w - 2],\
[379, 379, -2*w^3 + 4*w^2 + 7*w - 8],\
[379, 379, -w^2 - 2],\
[389, 389, -w^3 - 2*w^2 + 5*w + 11],\
[397, 397, -2*w^3 + 4*w^2 + 7*w - 7],\
[401, 401, w^3 - 3*w^2 - 2*w + 10],\
[401, 401, -2*w^3 + 11*w + 4],\
[401, 401, 3*w^3 - 2*w^2 - 14*w - 4],\
[401, 401, -3*w^3 + 4*w^2 + 15*w - 5],\
[421, 421, -w^3 + w^2 + 7*w - 1],\
[421, 421, -2*w^3 + 2*w^2 + 9*w - 7],\
[433, 433, w^2 - w - 8],\
[439, 439, -w^3 + 3*w^2 + 4*w - 7],\
[443, 443, -2*w^3 + w^2 + 10*w - 2],\
[449, 449, w^2 + 2*w - 4],\
[449, 449, 2*w^3 - 2*w^2 - 10*w - 1],\
[461, 461, 2*w^3 - 2*w^2 - 6*w + 5],\
[461, 461, -2*w^3 + 4*w^2 + 7*w - 13],\
[463, 463, -3*w^3 + 3*w^2 + 12*w - 4],\
[463, 463, w - 5],\
[467, 467, -3*w^3 + w^2 + 16*w + 8],\
[467, 467, 2*w^2 - 3*w - 10],\
[479, 479, 2*w^2 - 4*w - 1],\
[479, 479, -w^3 + 2*w^2 + 2*w - 7],\
[487, 487, -3*w^3 + 5*w^2 + 12*w - 10],\
[503, 503, w^3 + 2*w^2 - 8*w - 8],\
[509, 509, -3*w^3 + 5*w^2 + 11*w - 5],\
[521, 521, 2*w^2 - 3*w - 11],\
[523, 523, -2*w^3 - w^2 + 13*w + 11],\
[523, 523, -2*w^3 + 4*w^2 + 8*w - 7],\
[541, 541, 3*w^3 - 2*w^2 - 13*w + 1],\
[563, 563, w^3 - 3*w - 5],\
[563, 563, -w^3 + 3*w^2 + 2*w - 11],\
[569, 569, -2*w^3 + 4*w^2 + 9*w - 4],\
[569, 569, 3*w^2 - w - 14],\
[571, 571, -w^3 + 8*w - 1],\
[577, 577, 3*w^3 - 2*w^2 - 16*w + 1],\
[587, 587, -4*w^3 + 5*w^2 + 18*w - 7],\
[587, 587, 2*w^3 + w^2 - 10*w - 7],\
[599, 599, 2*w^3 + w^2 - 11*w - 8],\
[607, 607, w^3 - 3*w + 4],\
[613, 613, 3*w^3 - w^2 - 14*w - 5],\
[613, 613, 3*w^2 - w - 11],\
[619, 619, 2*w^2 - 5*w - 8],\
[631, 631, -2*w^3 - 2*w^2 + 12*w + 19],\
[647, 647, 2*w^3 - 2*w^2 - 11*w - 2],\
[653, 653, 3*w^3 - 5*w^2 - 8*w + 7],\
[677, 677, -2*w^3 + 4*w^2 + 5*w - 8],\
[683, 683, 2*w^3 + w^2 - 12*w - 8],\
[683, 683, -w^3 + 3*w^2 - 8],\
[683, 683, -2*w^3 + 3*w^2 + 11*w - 8],\
[683, 683, -2*w^3 + 3*w^2 + 10*w - 2],\
[691, 691, w^2 - 4*w - 1],\
[709, 709, 3*w^3 - 2*w^2 - 13*w - 2],\
[709, 709, w^3 - 2*w^2 - 3*w - 2],\
[727, 727, -w^3 + 4*w^2 + 2*w - 7],\
[727, 727, -w^3 + 8*w + 5],\
[733, 733, 3*w^2 - 2*w - 10],\
[733, 733, -w^3 + 5*w + 8],\
[757, 757, 3*w^3 - 14*w - 8],\
[761, 761, -w^3 + 3*w^2 + 5*w - 10],\
[773, 773, -2*w^3 + 3*w^2 + 9*w - 1],\
[787, 787, -w^3 + 4*w - 4],\
[797, 797, w^2 + 2*w - 5],\
[797, 797, w^3 + w^2 - 5*w - 1],\
[809, 809, w^3 + 3*w^2 - 9*w - 17],\
[811, 811, -4*w^2 + 3*w + 17],\
[811, 811, w^3 - 2*w^2 - 8*w + 2],\
[823, 823, -4*w^3 + 4*w^2 + 17*w - 4],\
[857, 857, 3*w^2 - w - 16],\
[881, 881, -2*w^3 + 9*w + 1],\
[883, 883, w^3 - w^2 - 4*w - 5],\
[887, 887, 3*w^2 + w - 10],\
[911, 911, w^3 - 3*w - 7],\
[919, 919, -w^3 + w^2 + 3*w - 7],\
[919, 919, 3*w^3 - 2*w^2 - 12*w + 2],\
[929, 929, 3*w^3 - w^2 - 16*w - 2],\
[929, 929, w^3 + w^2 - 6*w - 2],\
[941, 941, -3*w^3 + 5*w^2 + 14*w - 14],\
[941, 941, -w^3 + 2*w^2 + 6*w - 8],\
[947, 947, 3*w^3 - 3*w^2 - 13*w - 5],\
[947, 947, 3*w^3 - 4*w^2 - 12*w + 2],\
[953, 953, -3*w^3 + 5*w^2 + 10*w - 10],\
[961, 31, -w^3 + 4*w^2 + 3*w - 13],\
[961, 31, -4*w^3 + 3*w^2 + 17*w - 4],\
[967, 967, -w^3 - 4*w^2 + 9*w + 19],\
[967, 967, w^3 + 5*w^2 - 9*w - 26],\
[971, 971, -3*w^3 + 6*w^2 + 13*w - 7],\
[971, 971, w^3 + 2*w^2 - 7*w - 7],\
[977, 977, 2*w^3 - 3*w^2 - 4*w - 2],\
[983, 983, 3*w^3 - 5*w^2 - 7*w + 7],\
[991, 991, 2*w^2 - w - 13],\
[991, 991, w^3 - 8*w - 4]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 18*x^2 + 54
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, e, 2, 1/3*e^3 - 4*e, -e^2 + 8, -e^2 + 11, 1/3*e^3 - 3*e, -e, e^2 - 4, -1/3*e^3 + 7*e, -2/3*e^3 + 7*e, -1/3*e^3 + 2*e, 1/3*e^3 - 4*e, 3*e, -1/3*e^3 + 3*e, e^2 - 10, -1/3*e^3 + 6*e, -10, 2*e^2 - 10, -2*e^2 + 14, -e^2 + 2, 1/3*e^3 - 5*e, e^3 - 11*e, -2/3*e^3 + 9*e, -2/3*e^3 + 4*e, -10, 14, 1/3*e^3 - 11*e, 2/3*e^3 - 11*e, -10, 8, -e^2 + 8, 2/3*e^3 - 14*e, -e^3 + 11*e, -e^3 + 11*e, -e^2 + 14, -1/3*e^3 + 2*e, e^3 - 16*e, 2*e^2 - 10, -e^2 + 8, 1/3*e^3 - 3*e, -2*e^2 + 32, 8, -16, -10, 3*e^2 - 28, 2*e^2 - 16, 3*e^2 - 16, -2/3*e^3 + 13*e, 1/3*e^3 - 4*e, 2*e^3 - 22*e, -2/3*e^3 + 2*e, e^3 - 10*e, 2*e^2 - 40, -4/3*e^3 + 17*e, -2/3*e^3 + 17*e, -4*e^2 + 44, -2/3*e^3 + 7*e, 2*e^2 - 4, -2*e^2 + 14, -e^3 + 16*e, 5*e^2 - 52, -4*e^2 + 44, -e^2 - 10, 2*e^2 - 40, -2/3*e^3 + 10*e, -1/3*e^3 + 4*e, 5*e^2 - 34, 2, -1/3*e^3 - e, 2/3*e^3 - 12*e, -4*e^2 + 38, 2*e^2 + 2, 2*e^2 - 34, -e^2 + 26, 3*e^2 - 40, 4*e^2 - 40, 2*e^2 + 2, -2*e^3 + 23*e, -4*e^2 + 32, -1/3*e^3 + 7*e, 2/3*e^3 - 5*e, -2*e^3 + 25*e, -e^3 + 7*e, 20, -3*e^2 + 26, 5*e^2 - 52, -4*e^2 + 20, 1/3*e^3 - 10*e, -3*e, 2*e^3 - 21*e, -e^3 + 5*e, 5/3*e^3 - 25*e, 4*e^2 - 22, 6*e^2 - 58, -1/3*e^3, 4/3*e^3 - 18*e, e^3 - 16*e, -2/3*e^3 + 2*e, -2*e^2 + 14, -e^3 + 18*e, -5/3*e^3 + 19*e, 7/3*e^3 - 27*e, -4*e^2 + 56, 4*e^2 - 22, -4*e^2 + 38, -8/3*e^3 + 28*e, 5/3*e^3 - 26*e, -2/3*e^3 + 11*e, -e, 2*e^2 - 10, e^2 - 22, -1/3*e^3 + 2*e, -5/3*e^3 + 26*e, -8/3*e^3 + 28*e, -4*e^2 + 44, e^2 - 22, 2*e^2 - 16, 2*e^2 - 34, 2*e^2 + 2, 2/3*e^3 - 12*e, 7*e, 2*e^3 - 31*e, 8/3*e^3 - 30*e, e^3 - 12*e, 1/3*e^3 + 6*e, -4/3*e^3 + 24*e, 6*e^2 - 46, -4*e^2 + 56, -10, 2*e^2 + 20, 44, -5*e^2 + 32, -e^2 + 8, 4*e^2 - 40, -1/3*e^3 + 7*e, -5/3*e^3 + 15*e, 2*e^2 - 28, 7/3*e^3 - 35*e, 2/3*e^3 - 5*e, -1/3*e^3 + 3*e, 4*e^2 - 40, 2*e^2 - 22, -4*e^2 + 44, -1/3*e^3 + 11*e, -4/3*e^3 + 27*e, -2*e^2 + 14, -e^3 + 22*e, -e^3 + 2*e, 2*e^2 + 2, -4*e^2 + 32, -e^3 + 11*e, e^3 - e, -1/3*e^3 + e, -7/3*e^3 + 25*e, -4/3*e^3 + 14*e, 8/3*e^3 - 28*e, -7/3*e^3 + 27*e, -e^2 + 2, -e^2 + 20, -6*e^2 + 32, -46, 7/3*e^3 - 30*e, 5/3*e^3 - 24*e, -e^3 + 13*e, 2/3*e^3 - 4*e, 10*e^2 - 94, -4*e^2 + 32]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([3, 3, w])] = 1

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