/*
  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([2, 0, -5, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([16, 4, -w^2 + 3])

primes_array = [
[2, 2, w],\
[2, 2, w + 1],\
[13, 13, -w^2 + w + 3],\
[13, 13, w^2 + w - 3],\
[19, 19, -w^3 + 3*w + 1],\
[19, 19, -w^3 + 3*w - 1],\
[43, 43, -w^2 + w - 1],\
[43, 43, w^2 + w + 1],\
[49, 7, w^3 + w^2 - 6*w - 3],\
[49, 7, w^3 - w^2 - 6*w + 3],\
[53, 53, 2*w^3 - w^2 - 9*w + 3],\
[53, 53, 2*w^3 + w^2 - 9*w - 3],\
[59, 59, w^3 - w^2 - 4*w + 1],\
[59, 59, -w^3 - w^2 + 4*w + 1],\
[67, 67, 3*w^3 - 13*w + 1],\
[67, 67, -w^3 + w^2 + 6*w - 5],\
[81, 3, -3],\
[83, 83, -2*w^3 - w^2 + 9*w + 7],\
[83, 83, 4*w^3 - 18*w - 1],\
[89, 89, -2*w^3 + 10*w + 1],\
[89, 89, -w^2 + w + 5],\
[89, 89, w^2 + w - 5],\
[89, 89, 2*w^3 - 10*w + 1],\
[101, 101, -2*w^3 + 8*w + 1],\
[101, 101, 2*w^3 - 8*w + 1],\
[103, 103, w^3 + 2*w^2 - 5*w - 7],\
[103, 103, w^3 + w^2 - 6*w - 7],\
[103, 103, -w^2 - 3*w + 1],\
[103, 103, -w^3 + 2*w^2 + 5*w - 7],\
[127, 127, -w^3 + 2*w^2 + 5*w - 5],\
[127, 127, w^3 + 3*w^2 - 4*w - 13],\
[127, 127, -w^3 + 3*w^2 + 4*w - 13],\
[127, 127, w^3 + 2*w^2 - 5*w - 5],\
[149, 149, -2*w^3 - w^2 + 7*w - 1],\
[149, 149, 2*w^3 - w^2 - 7*w - 1],\
[151, 151, -w^3 + w^2 + 2*w - 3],\
[151, 151, 2*w^3 - 10*w + 3],\
[151, 151, -2*w^3 + 10*w + 3],\
[151, 151, w^3 + w^2 - 2*w - 3],\
[157, 157, w^3 + 2*w^2 - 3*w - 1],\
[157, 157, -w^3 + 2*w^2 + 3*w - 1],\
[169, 13, 2*w^2 - 3],\
[179, 179, 2*w^3 + w^2 - 7*w - 3],\
[179, 179, -2*w^3 + w^2 + 7*w - 3],\
[229, 229, -w^3 + 2*w^2 + 3*w - 7],\
[229, 229, w^3 + 2*w^2 - 3*w - 7],\
[251, 251, -2*w^3 + 2*w^2 + 8*w - 9],\
[251, 251, -2*w^3 - 2*w^2 + 8*w + 9],\
[263, 263, 2*w^3 - w^2 - 3*w + 1],\
[263, 263, -2*w^3 + 2*w^2 + 10*w - 5],\
[263, 263, -2*w^3 - 2*w^2 + 10*w + 5],\
[263, 263, -2*w^3 - w^2 + 3*w + 1],\
[289, 17, 2*w^2 - 5],\
[293, 293, 4*w^3 - w^2 - 19*w + 7],\
[293, 293, -4*w^3 - w^2 + 19*w + 7],\
[307, 307, -w^3 + 3*w - 5],\
[307, 307, w^3 - 3*w - 5],\
[331, 331, 4*w^3 - 2*w^2 - 18*w + 7],\
[331, 331, 5*w^3 - 3*w^2 - 22*w + 11],\
[349, 349, -3*w^3 + 13*w + 3],\
[349, 349, -3*w^3 + 13*w - 3],\
[361, 19, 2*w^2 - 11],\
[373, 373, w^3 + 2*w^2 + w - 3],\
[373, 373, -w^3 + 2*w^2 - w - 3],\
[383, 383, -w^3 + 5*w + 5],\
[383, 383, -3*w^3 + 2*w^2 + 11*w - 5],\
[383, 383, 3*w^3 + 2*w^2 - 11*w - 5],\
[383, 383, w^3 - 5*w + 5],\
[389, 389, w^3 - 4*w^2 + w + 7],\
[389, 389, -2*w^3 - w^2 + 7*w + 7],\
[421, 421, -w^3 - w^2 - 2*w + 1],\
[421, 421, w^3 - w^2 + 2*w + 1],\
[433, 433, 4*w^3 - 2*w^2 - 16*w + 5],\
[433, 433, -4*w + 1],\
[433, 433, 4*w + 1],\
[433, 433, -5*w^3 + 3*w^2 + 24*w - 13],\
[443, 443, -w^3 + w^2 + 3],\
[443, 443, -4*w^2 + 6*w + 7],\
[457, 457, 3*w^3 + w^2 - 12*w - 3],\
[457, 457, -2*w^3 + 6*w - 1],\
[457, 457, 2*w^3 - 6*w - 1],\
[457, 457, 3*w^3 - w^2 - 12*w + 3],\
[461, 461, w^3 - 3*w^2 - 2*w + 7],\
[461, 461, w^3 + 3*w^2 - 2*w - 7],\
[463, 463, -2*w^3 + 3*w^2 + 9*w - 11],\
[463, 463, 2*w^3 - w^2 - 5*w + 1],\
[463, 463, -2*w^3 - w^2 + 5*w + 1],\
[463, 463, -2*w^3 - 3*w^2 + 9*w + 11],\
[467, 467, w^3 - 7*w + 1],\
[467, 467, -w^3 + 7*w + 1],\
[491, 491, 2*w^2 - 2*w - 7],\
[491, 491, 2*w^2 + 2*w - 7],\
[509, 509, w^3 + 3*w^2 - 4*w - 11],\
[509, 509, -w^3 + 3*w^2 + 4*w - 11],\
[523, 523, 2*w^3 - w^2 - 11*w + 3],\
[523, 523, 2*w^3 + w^2 - 11*w - 3],\
[529, 23, 3*w^2 - w - 15],\
[529, 23, -3*w^2 - w + 15],\
[557, 557, 3*w^2 + w - 13],\
[557, 557, -3*w^2 + w + 13],\
[563, 563, -3*w^3 + 3*w^2 + 12*w - 7],\
[563, 563, 2*w^3 - 2*w^2 - 8*w + 3],\
[569, 569, w^3 + 3*w^2 - 6*w - 9],\
[569, 569, 4*w^3 - w^2 - 17*w + 1],\
[569, 569, -4*w^3 - w^2 + 17*w + 1],\
[569, 569, -w^3 + 3*w^2 + 6*w - 9],\
[587, 587, -w^3 + w^2 + 2*w - 5],\
[587, 587, w^3 + w^2 - 2*w - 5],\
[593, 593, 3*w^3 - 2*w^2 - 15*w + 5],\
[593, 593, -w^3 - 2*w^2 + 5*w + 1],\
[593, 593, w^3 - 2*w^2 - 5*w + 1],\
[593, 593, -3*w^3 - 2*w^2 + 15*w + 5],\
[599, 599, w^3 + w^2 - 3],\
[599, 599, -w^2 - w - 3],\
[599, 599, -w^2 + w - 3],\
[599, 599, -w^3 + w^2 - 3],\
[613, 613, -w^3 + w^2 + 6*w - 9],\
[613, 613, w^3 + w^2 - 6*w - 9],\
[625, 5, -5],\
[659, 659, -2*w^3 + 4*w^2 + 4*w - 1],\
[659, 659, 2*w^3 + 4*w^2 - 4*w - 1],\
[661, 661, w^3 + 3*w^2 - 6*w - 7],\
[661, 661, -2*w^3 + 6*w^2 - 7],\
[701, 701, -w^3 - 2*w^2 + 3*w + 9],\
[701, 701, -w^3 + 2*w^2 + 3*w - 9],\
[733, 733, w^3 + 2*w^2 - 7*w - 1],\
[733, 733, -4*w^3 + w^2 + 19*w - 11],\
[739, 739, -w^3 - w^2 + 2*w + 9],\
[739, 739, -w^3 + w^2 + 2*w - 9],\
[757, 757, 3*w^3 - w^2 - 10*w + 5],\
[757, 757, -3*w^3 - w^2 + 10*w + 5],\
[773, 773, -4*w^3 + 5*w^2 + 19*w - 23],\
[773, 773, -7*w^3 + 6*w^2 + 33*w - 29],\
[797, 797, 3*w^3 - 13*w + 5],\
[797, 797, 3*w^3 - 13*w - 5],\
[829, 829, -2*w^3 - w^2 + 9*w - 1],\
[829, 829, 2*w^3 - w^2 - 9*w - 1],\
[859, 859, 3*w^3 - w^2 - 14*w + 1],\
[859, 859, -3*w^3 - w^2 + 14*w + 1],\
[863, 863, -5*w^3 + 21*w + 3],\
[863, 863, -3*w^3 - w^2 + 12*w - 1],\
[863, 863, 3*w^3 - w^2 - 12*w - 1],\
[863, 863, -5*w^3 + 21*w - 3],\
[883, 883, -2*w^3 + 4*w^2 + 8*w - 17],\
[883, 883, 2*w^3 + 4*w^2 - 8*w - 17],\
[953, 953, -w^3 - w^2 + 8*w + 3],\
[953, 953, 4*w^3 - 2*w^2 - 16*w + 7],\
[953, 953, -3*w^3 + w^2 + 16*w - 7],\
[953, 953, w^3 - w^2 - 8*w + 3],\
[961, 31, -4*w^3 + 2*w^2 + 12*w + 3],\
[961, 31, 3*w^3 - w^2 - 10*w - 1],\
[971, 971, 3*w^3 - 4*w^2 - 13*w + 15],\
[971, 971, -4*w^3 + 3*w^2 + 19*w - 17],\
[977, 977, -2*w^3 + w^2 + 9*w + 3],\
[977, 977, -4*w^3 + w^2 + 19*w - 9],\
[977, 977, 3*w^3 - 15*w + 5],\
[977, 977, 2*w^3 + w^2 - 9*w + 3]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 9*x^2 + 12
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 0, 2*e, 2*e, -e^3 + 5*e, -e^3 + 5*e, 2*e^2 - 14, 2*e^2 - 14, -2*e^3 + 12*e, -2*e^3 + 12*e, -4*e^2 + 18, -4*e^2 + 18, 2*e^2 - 6, 2*e^2 - 6, e^3 - 5*e, e^3 - 5*e, -4*e^2 + 22, -2*e^2 + 6, -2*e^2 + 6, 2*e^3 - 16*e, 2*e, 2*e, 2*e^3 - 16*e, 2*e, 2*e, 8, 2*e^3 - 18*e, 2*e^3 - 18*e, 8, -8, -2*e^3 + 10*e, -2*e^3 + 10*e, -8, -4*e^2 + 18, -4*e^2 + 18, 2*e^3 - 18*e, 4*e^2 - 20, 4*e^2 - 20, 2*e^3 - 18*e, -2, -2, 26, -3*e^3 + 23*e, -3*e^3 + 23*e, 4*e^3 - 26*e, 4*e^3 - 26*e, e^3 - 13*e, e^3 - 13*e, -4*e^3 + 20*e, -4*e^2 + 12, -4*e^2 + 12, -4*e^3 + 20*e, 4*e^2 - 2, 4*e^2 - 6, 4*e^2 - 6, -6*e^2 + 34, -6*e^2 + 34, 6*e^2 - 10, 6*e^2 - 10, -4*e^2 + 26, -4*e^2 + 26, 4*e^2 - 10, -4*e^3 + 22*e, -4*e^3 + 22*e, -4*e^2 + 36, -2*e^3 + 10*e, -2*e^3 + 10*e, -4*e^2 + 36, -4*e^3 + 30*e, -4*e^3 + 30*e, 2*e, 2*e, -6*e, 2*e^3, 2*e^3, -6*e, -e^3 + 21*e, -e^3 + 21*e, 4*e^3 - 34*e, 10*e, 10*e, 4*e^3 - 34*e, 8*e^2 - 42, 8*e^2 - 42, 12*e^2 - 52, 0, 0, 12*e^2 - 52, e^3 - 13*e, e^3 - 13*e, -e^3 + 5*e, -e^3 + 5*e, -18, -18, 5*e^3 - 25*e, 5*e^3 - 25*e, 4*e^3 - 34*e, 4*e^3 - 34*e, 4*e^3 - 26*e, 4*e^3 - 26*e, 2*e^2 + 18, 2*e^2 + 18, -8*e^2 + 18, -6, -6, -8*e^2 + 18, -3*e^3 + 23*e, -3*e^3 + 23*e, -4*e^2 + 6, 18, 18, -4*e^2 + 6, 2*e^3 - 26*e, 4*e^2 - 36, 4*e^2 - 36, 2*e^3 - 26*e, 22, 22, 4*e^2 + 14, 2*e^2 - 6, 2*e^2 - 6, 4*e^2 - 38, 4*e^2 - 38, 10*e, 10*e, -4*e^2 + 10, -4*e^2 + 10, e^3 - 21*e, e^3 - 21*e, 4*e^3 - 18*e, 4*e^3 - 18*e, -4*e^3 + 30*e, -4*e^3 + 30*e, 8*e^2 - 42, 8*e^2 - 42, 12*e^2 - 38, 12*e^2 - 38, -2*e^2 - 34, -2*e^2 - 34, -6*e^3 + 38*e, -4*e^2 + 36, -4*e^2 + 36, -6*e^3 + 38*e, 3*e^3 - 7*e, 3*e^3 - 7*e, -2*e^3 - 4*e, 2*e^3, 2*e^3, -2*e^3 - 4*e, 10*e, 10*e, -10*e^2 + 54, -10*e^2 + 54, -12*e^2 + 78, 4*e^2 - 18, 4*e^2 - 18, -12*e^2 + 78]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([2,2,w+1])] = 1

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