/*
  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([11, 0, -7, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([16, 2, 2])

primes_array = [
[4, 2, -w^2 + w + 3],\
[5, 5, w + 1],\
[5, 5, -w^3 + w^2 + 4*w - 4],\
[11, 11, w],\
[29, 29, w^3 - 2*w^2 - 3*w + 7],\
[29, 29, -w^3 - 2*w^2 + 3*w + 7],\
[31, 31, -w^3 + w^2 + 4*w - 2],\
[31, 31, -w^3 - w^2 + 4*w + 2],\
[41, 41, w^3 + 2*w^2 - 4*w - 6],\
[41, 41, w^3 - 5*w + 2],\
[49, 7, -w^3 + w^2 + 4*w - 1],\
[49, 7, w^3 + w^2 - 4*w - 1],\
[59, 59, -3*w^2 - w + 10],\
[59, 59, -3*w^2 + w + 10],\
[61, 61, -2*w^3 + 2*w^2 + 7*w - 5],\
[61, 61, 2*w^3 + w^2 - 8*w - 6],\
[71, 71, 2*w^2 + w - 9],\
[71, 71, 2*w^2 - w - 9],\
[81, 3, -3],\
[101, 101, 2*w^2 - w - 10],\
[101, 101, 2*w^2 + w - 10],\
[109, 109, 2*w^3 + w^2 - 8*w - 3],\
[109, 109, -2*w^3 + w^2 + 8*w - 3],\
[121, 11, -w^2 + 7],\
[131, 131, 4*w^2 - 2*w - 15],\
[131, 131, 2*w^2 + 2*w - 3],\
[131, 131, 2*w^2 - 2*w - 3],\
[131, 131, -w^3 - w^2 + 3*w + 7],\
[139, 139, w^3 + 4*w^2 - 5*w - 16],\
[139, 139, w^3 + 3*w^2 - 6*w - 10],\
[139, 139, 2*w^3 - w^2 - 8*w + 1],\
[139, 139, -4*w^2 - w + 12],\
[149, 149, 2*w^3 - w^2 - 8*w + 4],\
[149, 149, 2*w^3 + w^2 - 8*w - 4],\
[151, 151, 3*w^2 - 3*w - 8],\
[151, 151, 2*w^3 - 4*w^2 - 8*w + 17],\
[151, 151, w^2 - 2*w - 7],\
[151, 151, w^3 + 2*w^2 - 6*w - 8],\
[179, 179, w^3 - w^2 - 5*w + 1],\
[179, 179, -w^3 - w^2 + 5*w + 1],\
[191, 191, 2*w^3 - w^2 - 7*w + 2],\
[191, 191, 3*w^3 - 2*w^2 - 10*w + 2],\
[199, 199, w^3 - 4*w^2 - 3*w + 13],\
[199, 199, -w^3 - 4*w^2 + 3*w + 13],\
[211, 211, -w^3 + w^2 + 6*w - 5],\
[211, 211, w^3 - 5*w - 5],\
[211, 211, -w^3 + 5*w - 5],\
[211, 211, w^3 + w^2 - 6*w - 5],\
[229, 229, w^3 - 4*w^2 - 2*w + 14],\
[229, 229, 2*w^3 + 3*w^2 - 7*w - 12],\
[229, 229, 2*w^3 - 3*w^2 - 7*w + 12],\
[229, 229, 2*w^3 + w^2 - 7*w - 8],\
[241, 241, w^3 + w^2 - 6*w - 4],\
[241, 241, -w^3 + w^2 + 6*w - 4],\
[251, 251, 2*w^2 - w - 2],\
[251, 251, 2*w^2 + w - 2],\
[271, 271, 2*w^3 - 2*w^2 - 8*w + 5],\
[271, 271, -4*w^2 - w + 14],\
[271, 271, -4*w^2 + w + 14],\
[271, 271, 2*w^3 + 2*w^2 - 8*w - 5],\
[281, 281, -2*w^3 - w^2 + 9*w + 2],\
[281, 281, 2*w^3 - w^2 - 9*w + 2],\
[311, 311, -w^3 + 3*w - 5],\
[311, 311, w^3 - 3*w - 5],\
[331, 331, -3*w^3 + 3*w^2 + 12*w - 13],\
[331, 331, 3*w^3 + 3*w^2 - 12*w - 13],\
[349, 349, 2*w^3 + 2*w^2 - 7*w - 10],\
[349, 349, -2*w^3 + 2*w^2 + 7*w - 10],\
[361, 19, 4*w^2 - 15],\
[361, 19, -4*w^2 + 13],\
[379, 379, -w^3 + 3*w^2 + 3*w - 13],\
[379, 379, w^3 + 3*w^2 - 3*w - 13],\
[409, 409, 3*w^3 - 2*w^2 - 11*w + 2],\
[409, 409, w^3 + 3*w^2 - 6*w - 9],\
[419, 419, -w^3 + w^2 + 6*w + 2],\
[419, 419, -3*w^3 + 5*w^2 + 13*w - 19],\
[421, 421, -w^3 + 3*w^2 + 7*w - 13],\
[421, 421, 2*w^3 - 2*w^2 - 9*w + 6],\
[421, 421, -2*w^3 - 2*w^2 + 9*w + 6],\
[421, 421, w^3 + 3*w^2 - 7*w - 13],\
[439, 439, 3*w^2 + 2*w - 13],\
[439, 439, 3*w^3 + 2*w^2 - 12*w - 10],\
[439, 439, -3*w^3 + 2*w^2 + 12*w - 10],\
[439, 439, 3*w^2 - 2*w - 13],\
[449, 449, 2*w^3 - 4*w^2 - 9*w + 14],\
[449, 449, -3*w^3 + 3*w^2 + 11*w - 9],\
[449, 449, 3*w^3 + 3*w^2 - 11*w - 9],\
[449, 449, w^3 - 2*w^2 - 5*w + 2],\
[461, 461, -w - 5],\
[461, 461, w - 5],\
[499, 499, w^3 - 5*w^2 - w + 17],\
[499, 499, -w^3 - 5*w^2 + w + 17],\
[509, 509, w^3 + 5*w^2 - 4*w - 16],\
[509, 509, w^2 - 2*w - 10],\
[509, 509, w^2 + 2*w - 10],\
[509, 509, w^3 - w^2 - 5*w + 9],\
[541, 541, 3*w^3 - 10*w - 6],\
[541, 541, -2*w^3 + 5*w^2 + 5*w - 16],\
[569, 569, 3*w^3 + 2*w^2 - 13*w - 6],\
[569, 569, -2*w^3 + 4*w^2 + 7*w - 10],\
[599, 599, w^3 + 2*w^2 - 7*w - 10],\
[599, 599, -w^3 + 2*w^2 + 7*w - 10],\
[601, 601, -3*w^3 - w^2 + 12*w + 6],\
[601, 601, 3*w^3 - w^2 - 12*w + 6],\
[619, 619, 3*w^3 - 12*w + 2],\
[619, 619, -3*w^3 + 12*w + 2],\
[631, 631, -w^3 + w^2 + 2*w - 8],\
[631, 631, w^3 + w^2 - 2*w - 8],\
[641, 641, w^3 + w^2 - 7*w - 1],\
[641, 641, -3*w^3 - 2*w^2 + 12*w + 4],\
[641, 641, 3*w^3 - 2*w^2 - 12*w + 4],\
[641, 641, w^3 - w^2 - 7*w + 1],\
[659, 659, -2*w^3 + w^2 + 10*w - 1],\
[659, 659, -w^3 + 5*w^2 + 4*w - 19],\
[659, 659, w^3 + 5*w^2 - 4*w - 19],\
[659, 659, 2*w^3 + w^2 - 10*w - 1],\
[691, 691, 3*w^2 - 2*w - 14],\
[691, 691, 3*w^2 + 2*w - 14],\
[701, 701, w^3 + 4*w^2 - 7*w - 12],\
[701, 701, 3*w^3 - w^2 - 12*w - 1],\
[719, 719, -w^3 + 7*w - 3],\
[719, 719, w^3 - 7*w - 3],\
[739, 739, 3*w^3 + w^2 - 12*w - 3],\
[739, 739, w^3 + 2*w^2 - 3*w - 1],\
[739, 739, -w^3 + 2*w^2 + 3*w - 1],\
[739, 739, -3*w^3 + w^2 + 12*w - 3],\
[751, 751, -2*w^3 + 6*w^2 + 7*w - 20],\
[751, 751, 2*w^3 + 6*w^2 - 7*w - 20],\
[761, 761, 2*w^3 + 6*w^2 - 8*w - 21],\
[761, 761, -2*w^3 + 6*w^2 + 8*w - 21],\
[769, 769, -2*w^3 + 6*w^2 + 8*w - 23],\
[769, 769, w^2 + 2*w + 3],\
[809, 809, 2*w^3 + 3*w^2 - 7*w - 14],\
[809, 809, -2*w^3 + 3*w^2 + 7*w - 14],\
[821, 821, -3*w^3 - w^2 + 11*w + 1],\
[821, 821, 3*w^3 - w^2 - 11*w + 1],\
[829, 829, 3*w^3 - w^2 - 12*w + 4],\
[829, 829, w^3 - 4*w^2 - 6*w + 12],\
[829, 829, -w^3 - 4*w^2 + 6*w + 12],\
[829, 829, -3*w^3 - w^2 + 12*w + 4],\
[839, 839, -w^3 + 3*w - 6],\
[839, 839, w^3 - 3*w - 6],\
[841, 29, 5*w^2 - 19],\
[859, 859, w^3 + 2*w^2 - 7*w - 9],\
[859, 859, -w^3 + 2*w^2 + 7*w - 9],\
[911, 911, -2*w^3 + 4*w^2 + 9*w - 13],\
[911, 911, 2*w^3 + 4*w^2 - 9*w - 13],\
[919, 919, 2*w^3 + 4*w^2 - 7*w - 17],\
[919, 919, 3*w^3 + 2*w^2 - 11*w - 6],\
[919, 919, 3*w^3 - 2*w^2 - 11*w + 6],\
[919, 919, 2*w^3 - 4*w^2 - 7*w + 17],\
[941, 941, -2*w^3 + 2*w^2 + 7*w - 15],\
[941, 941, 2*w^3 + 2*w^2 - 7*w - 15],\
[961, 31, -2*w^2 + 13],\
[971, 971, w^3 + 2*w^2 - 7*w - 7],\
[971, 971, -w^3 + 2*w^2 + 7*w - 7],\
[991, 991, 3*w^3 + 2*w^2 - 11*w - 7],\
[991, 991, -3*w^3 + 2*w^2 + 11*w - 7]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 16*x^2 + 16
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, e, e, -1/2*e^3 + 6*e, -e^2 + 10, -e^2 + 10, 0, 0, 1/2*e^3 - 9*e, 1/2*e^3 - 9*e, -1/2*e^3 + 7*e, -1/2*e^3 + 7*e, 4, 4, e^3 - 15*e, e^3 - 15*e, -8, -8, 2, -e^2 + 2, -e^2 + 2, e, e, e^2 + 6, 2*e^2 - 20, -1/2*e^3 + 10*e, -1/2*e^3 + 10*e, 2*e^2 - 20, 4, 3/2*e^3 - 18*e, 3/2*e^3 - 18*e, 4, -3*e, -3*e, -4*e, 2*e^2 - 24, 2*e^2 - 24, -4*e, 1/2*e^3 - 14*e, 1/2*e^3 - 14*e, e^3 - 16*e, e^3 - 16*e, 2*e^2 - 8, 2*e^2 - 8, 3/2*e^3 - 22*e, -2*e^2 + 12, -2*e^2 + 12, 3/2*e^3 - 22*e, -10, -e^2 + 18, -e^2 + 18, -10, -1/2*e^3 + 11*e, -1/2*e^3 + 11*e, -2*e^2 + 20, -2*e^2 + 20, -e^3 + 8*e, -2*e^2 + 16, -2*e^2 + 16, -e^3 + 8*e, -3/2*e^3 + 19*e, -3/2*e^3 + 19*e, 2*e^2 - 8, 2*e^2 - 8, -3/2*e^3 + 22*e, -3/2*e^3 + 22*e, -2, -2, -4*e^2 + 42, e^2 + 22, 20, 20, -5/2*e^3 + 31*e, -5/2*e^3 + 31*e, 5/2*e^3 - 30*e, 5/2*e^3 - 30*e, -3*e, e^3 - 11*e, e^3 - 11*e, -3*e, 24, -2*e^3 + 28*e, -2*e^3 + 28*e, 24, 1/2*e^3 - e, -3/2*e^3 + 19*e, -3/2*e^3 + 19*e, 1/2*e^3 - e, -2, -2, -2*e^2 + 28, -2*e^2 + 28, 4*e^2 - 18, -2, -2, 4*e^2 - 18, 4*e^2 - 34, 4*e^2 - 34, 3/2*e^3 - 13*e, 3/2*e^3 - 13*e, -e^3 + 12*e, -e^3 + 12*e, -3/2*e^3 + 27*e, -3/2*e^3 + 27*e, -3/2*e^3 + 30*e, -3/2*e^3 + 30*e, 8, 8, -3/2*e^3 + 23*e, -5/2*e^3 + 27*e, -5/2*e^3 + 27*e, -3/2*e^3 + 23*e, 1/2*e^3 - 14*e, 2*e^2 - 4, 2*e^2 - 4, 1/2*e^3 - 14*e, 4*e^2 - 20, 4*e^2 - 20, -e^3 + 13*e, -e^3 + 13*e, 4*e^3 - 56*e, 4*e^3 - 56*e, -3/2*e^3 + 34*e, 2*e^2 - 36, 2*e^2 - 36, -3/2*e^3 + 34*e, -2*e^2 + 32, -2*e^2 + 32, e^2 - 26, e^2 - 26, -3*e^2 + 30, -3*e^2 + 30, e^2 - 10, e^2 - 10, e^3 - 7*e, e^3 - 7*e, -2*e^3 + 25*e, -e^3 + 21*e, -e^3 + 21*e, -2*e^3 + 25*e, -2*e^2 - 8, -2*e^2 - 8, 26, -3/2*e^3 + 30*e, -3/2*e^3 + 30*e, -e^3 + 24*e, -e^3 + 24*e, 4*e^2 - 8, 2*e^3 - 20*e, 2*e^3 - 20*e, 4*e^2 - 8, 3*e^2 - 6, 3*e^2 - 6, -3*e^2 + 62, 1/2*e^3 - 10*e, 1/2*e^3 - 10*e, -16*e, -16*e]
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

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