/*
  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([4, 2, -5, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([1024, 16, 2*w^3 - 8*w])

primes_array = [
[4, 2, -w],\
[4, 2, 1/2*w^3 - 1/2*w^2 - 5/2*w + 1],\
[19, 19, 1/2*w^3 - 3/2*w^2 - 1/2*w + 4],\
[19, 19, 1/2*w^3 + 1/2*w^2 - 5/2*w - 1],\
[25, 5, w^3 - w^2 - 3*w + 1],\
[29, 29, -1/2*w^3 + 3/2*w^2 + 3/2*w - 1],\
[29, 29, -w^2 + 5],\
[31, 31, -w^3 + 2*w^2 + 3*w - 3],\
[31, 31, -1/2*w^3 - 1/2*w^2 + 3/2*w + 3],\
[41, 41, -1/2*w^3 - 1/2*w^2 + 5/2*w],\
[41, 41, -1/2*w^3 + 3/2*w^2 + 1/2*w - 5],\
[59, 59, w^3 - w^2 - 5*w + 1],\
[59, 59, 3/2*w^3 - 1/2*w^2 - 11/2*w - 1],\
[61, 61, 3/2*w^3 - 1/2*w^2 - 11/2*w - 2],\
[61, 61, -2*w^2 + w + 7],\
[71, 71, 1/2*w^3 - 3/2*w^2 - 5/2*w + 6],\
[71, 71, -1/2*w^3 + 5/2*w^2 - 1/2*w - 5],\
[71, 71, 1/2*w^3 + 3/2*w^2 - 7/2*w - 5],\
[71, 71, -w^3 + 3*w^2 + 2*w - 7],\
[81, 3, -3],\
[89, 89, -1/2*w^3 + 5/2*w^2 + 3/2*w - 6],\
[101, 101, w^2 - 2*w - 5],\
[101, 101, -1/2*w^3 - 1/2*w^2 + 7/2*w - 1],\
[109, 109, w^3 - 2*w^2 - 3*w + 1],\
[109, 109, -5/2*w^3 + 7/2*w^2 + 17/2*w - 8],\
[109, 109, 1/2*w^3 + 1/2*w^2 - 3/2*w - 5],\
[109, 109, -2*w^3 + 2*w^2 + 8*w - 1],\
[121, 11, 3/2*w^3 - 3/2*w^2 - 9/2*w + 1],\
[121, 11, -3/2*w^3 + 3/2*w^2 + 9/2*w - 2],\
[131, 131, -1/2*w^3 - 3/2*w^2 + 11/2*w + 6],\
[131, 131, -1/2*w^3 - 1/2*w^2 + 9/2*w + 2],\
[131, 131, -1/2*w^3 - 1/2*w^2 + 9/2*w + 1],\
[131, 131, -3/2*w^3 + 3/2*w^2 + 13/2*w - 5],\
[149, 149, 5/2*w^3 - 5/2*w^2 - 17/2*w],\
[149, 149, -1/2*w^3 + 3/2*w^2 + 3/2*w + 1],\
[151, 151, 2*w^3 - 2*w^2 - 7*w + 3],\
[151, 151, 5/2*w^3 - 5/2*w^2 - 19/2*w + 1],\
[181, 181, -w^3 + w^2 + w - 3],\
[181, 181, 1/2*w^3 - 5/2*w^2 + 1/2*w + 8],\
[191, 191, 1/2*w^3 + 3/2*w^2 - 9/2*w - 8],\
[191, 191, 3/2*w^3 - 3/2*w^2 - 13/2*w - 1],\
[211, 211, -1/2*w^3 + 3/2*w^2 - 1/2*w - 5],\
[211, 211, -w^3 + 5*w - 1],\
[229, 229, -2*w^3 + 4*w^2 + 6*w - 7],\
[229, 229, 3/2*w^3 - 3/2*w^2 - 15/2*w + 4],\
[239, 239, w^3 + w^2 - 4*w - 1],\
[239, 239, -3/2*w^3 + 1/2*w^2 + 15/2*w - 1],\
[241, 241, 1/2*w^3 + 1/2*w^2 - 1/2*w - 4],\
[241, 241, 3/2*w^3 - 5/2*w^2 - 11/2*w + 3],\
[269, 269, -w^3 + 2*w^2 + w - 5],\
[269, 269, -3/2*w^3 + 5/2*w^2 + 7/2*w - 8],\
[269, 269, 3/2*w^3 - 7/2*w^2 - 7/2*w + 4],\
[269, 269, -w^3 - w^2 + 4*w + 7],\
[281, 281, -1/2*w^3 + 5/2*w^2 - 1/2*w - 9],\
[281, 281, 1/2*w^3 + 3/2*w^2 - 7/2*w - 1],\
[289, 17, -w^3 + 2*w^2 + 5*w - 1],\
[289, 17, -1/2*w^3 + 3/2*w^2 + 7/2*w - 7],\
[331, 331, 3/2*w^3 - 3/2*w^2 - 15/2*w + 2],\
[331, 331, -5/2*w^3 + 7/2*w^2 + 17/2*w - 4],\
[331, 331, 3*w - 1],\
[331, 331, -3/2*w^3 + 1/2*w^2 + 7/2*w + 3],\
[349, 349, -1/2*w^3 - 1/2*w^2 + 5/2*w - 3],\
[349, 349, -1/2*w^3 + 3/2*w^2 + 1/2*w - 8],\
[359, 359, -5/2*w^3 + 9/2*w^2 + 11/2*w - 6],\
[359, 359, -1/2*w^3 + 7/2*w^2 + 1/2*w - 6],\
[361, 19, -2*w^3 + 2*w^2 + 6*w - 3],\
[379, 379, 1/2*w^3 + 5/2*w^2 - 9/2*w - 7],\
[379, 379, w^3 - w^2 - 6*w + 3],\
[389, 389, 1/2*w^3 - 1/2*w^2 - 5/2*w - 4],\
[389, 389, w - 5],\
[401, 401, 2*w^2 + w - 7],\
[401, 401, -3/2*w^3 + 7/2*w^2 + 11/2*w - 6],\
[401, 401, w^3 - 3*w^2 - 3*w + 3],\
[401, 401, 2*w^2 - 9],\
[409, 409, -1/2*w^3 - 5/2*w^2 + 9/2*w + 8],\
[409, 409, 1/2*w^3 + 3/2*w^2 - 3/2*w - 7],\
[409, 409, -3/2*w^3 + 7/2*w^2 + 9/2*w - 5],\
[409, 409, -1/2*w^3 + 7/2*w^2 - 3/2*w - 7],\
[419, 419, 2*w^3 - 11*w - 5],\
[419, 419, 5/2*w^3 - 3/2*w^2 - 17/2*w + 2],\
[421, 421, -1/2*w^3 + 5/2*w^2 + 7/2*w - 2],\
[421, 421, -w^3 + w^2 + w + 5],\
[431, 431, 2*w^3 - w^2 - 6*w - 1],\
[431, 431, -5/2*w^3 + 7/2*w^2 + 15/2*w - 5],\
[439, 439, -w^3 + 7*w - 5],\
[439, 439, -w^3 + w + 3],\
[449, 449, -2*w^3 + 2*w^2 + 5*w - 3],\
[449, 449, -1/2*w^3 - 1/2*w^2 + 11/2*w - 1],\
[449, 449, 1/2*w^3 + 1/2*w^2 - 5/2*w - 7],\
[449, 449, 5/2*w^3 - 5/2*w^2 - 11/2*w + 5],\
[491, 491, -5/2*w^3 + 5/2*w^2 + 17/2*w - 2],\
[491, 491, 2*w^3 - 2*w^2 - 5*w + 1],\
[499, 499, w^3 + w^2 - 5*w - 1],\
[499, 499, -w^3 + 3*w^2 + w - 9],\
[521, 521, -w^3 + 4*w^2 + w - 9],\
[521, 521, 1/2*w^3 + 5/2*w^2 - 7/2*w - 7],\
[541, 541, -3/2*w^3 + 5/2*w^2 + 11/2*w - 1],\
[541, 541, 1/2*w^3 + 1/2*w^2 - 1/2*w - 6],\
[569, 569, -3/2*w^3 + 5/2*w^2 + 3/2*w - 5],\
[569, 569, -2*w^3 + 4*w^2 + 5*w - 13],\
[571, 571, -w^3 + 3*w^2 - 7],\
[571, 571, -w^3 + 2*w^2 + w - 7],\
[599, 599, 1/2*w^3 + 5/2*w^2 - 5/2*w - 11],\
[599, 599, 1/2*w^3 + 3/2*w^2 - 11/2*w - 8],\
[619, 619, -w - 5],\
[619, 619, -1/2*w^3 + 1/2*w^2 + 5/2*w - 6],\
[619, 619, -1/2*w^3 + 7/2*w^2 - 1/2*w - 9],\
[619, 619, 3*w^2 - 2*w - 7],\
[641, 641, -w^3 - 2*w^2 + 5*w + 3],\
[641, 641, 2*w^2 - 4*w - 7],\
[641, 641, 3/2*w^3 - 9/2*w^2 - 5/2*w + 13],\
[641, 641, -3/2*w^3 + 7/2*w^2 + 5/2*w - 9],\
[659, 659, -w^3 + 7*w + 1],\
[659, 659, 5/2*w^3 - 5/2*w^2 - 19/2*w - 1],\
[659, 659, -1/2*w^3 + 7/2*w^2 + 1/2*w - 8],\
[659, 659, 2*w^2 + 2*w - 7],\
[661, 661, 2*w^3 - 2*w^2 - 10*w + 1],\
[661, 661, w^3 - 5*w - 7],\
[691, 691, -w^3 + 3*w^2 + 4*w - 3],\
[691, 691, -3/2*w^3 + 7/2*w^2 + 1/2*w - 6],\
[691, 691, 5/2*w^3 - 1/2*w^2 - 23/2*w - 2],\
[691, 691, -1/2*w^3 + 5/2*w^2 + 5/2*w - 10],\
[701, 701, 5/2*w^3 - 5/2*w^2 - 23/2*w + 4],\
[701, 701, -1/2*w^3 + 1/2*w^2 + 7/2*w - 7],\
[701, 701, -2*w^3 + w^2 + 10*w - 3],\
[701, 701, -1/2*w^3 + 7/2*w^2 - 3/2*w - 12],\
[709, 709, -1/2*w^3 + 5/2*w^2 - 3/2*w - 8],\
[709, 709, -3/2*w^3 + 5/2*w^2 + 5/2*w - 7],\
[719, 719, -w^3 + 4*w^2 - w - 9],\
[719, 719, 3/2*w^3 + 3/2*w^2 - 17/2*w - 5],\
[739, 739, 3/2*w^3 + 1/2*w^2 - 11/2*w - 8],\
[739, 739, 2*w^3 - 4*w^2 - 5*w + 3],\
[761, 761, 3*w^2 - 11],\
[761, 761, 5/2*w^3 - 7/2*w^2 - 13/2*w + 4],\
[761, 761, -3/2*w^3 + 9/2*w^2 + 9/2*w - 7],\
[761, 761, 5/2*w^3 - 3/2*w^2 - 17/2*w - 1],\
[821, 821, -w^3 + w^2 + 7*w + 1],\
[821, 821, -4*w - 1],\
[821, 821, -w^3 + w^2 + 7*w - 5],\
[821, 821, 2*w^3 - 2*w^2 - 10*w + 5],\
[829, 829, 3*w^3 - 2*w^2 - 11*w + 1],\
[829, 829, -5/2*w^3 + 7/2*w^2 + 11/2*w - 5],\
[839, 839, 3/2*w^3 + 1/2*w^2 - 9/2*w - 6],\
[839, 839, 5/2*w^3 - 9/2*w^2 - 15/2*w + 6],\
[841, 29, 5/2*w^3 - 5/2*w^2 - 15/2*w + 4],\
[859, 859, -3*w^3 + 3*w^2 + 10*w - 1],\
[859, 859, 5/2*w^3 - 5/2*w^2 - 13/2*w],\
[881, 881, -3/2*w^3 + 7/2*w^2 + 9/2*w - 3],\
[881, 881, -w^3 + 3*w^2 + 2*w - 13],\
[881, 881, -5/2*w^3 + 9/2*w^2 + 19/2*w - 8],\
[881, 881, -7/2*w^3 + 11/2*w^2 + 25/2*w - 10],\
[911, 911, w^3 - w^2 + 2*w - 1],\
[911, 911, 7/2*w^3 - 11/2*w^2 - 17/2*w + 4],\
[911, 911, -3*w^3 + 2*w^2 + 11*w + 1],\
[911, 911, 5/2*w^3 - 7/2*w^2 - 11/2*w + 3],\
[919, 919, 1/2*w^3 + 1/2*w^2 + 3/2*w - 5],\
[919, 919, -5/2*w^3 + 7/2*w^2 + 21/2*w - 4],\
[929, 929, -3/2*w^3 + 9/2*w^2 + 3/2*w - 11],\
[929, 929, -1/2*w^3 + 1/2*w^2 + 11/2*w + 1],\
[929, 929, -3/2*w^3 + 3/2*w^2 + 17/2*w - 5],\
[929, 929, 3/2*w^3 + 3/2*w^2 - 15/2*w - 4],\
[941, 941, -1/2*w^3 + 5/2*w^2 - 5/2*w - 7],\
[941, 941, -3/2*w^3 + 7/2*w^2 + 5/2*w - 10],\
[961, 31, 5/2*w^3 - 5/2*w^2 - 15/2*w + 2],\
[971, 971, 5/2*w^3 + 1/2*w^2 - 21/2*w - 4],\
[971, 971, 3*w^3 - 15*w - 5],\
[971, 971, -2*w^3 + 2*w^2 + 3*w - 5],\
[971, 971, -5/2*w^3 + 5/2*w^2 + 21/2*w - 10],\
[991, 991, -3/2*w^3 + 7/2*w^2 + 9/2*w - 2],\
[991, 991, 1/2*w^3 + 3/2*w^2 - 3/2*w - 10]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 - 48
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, 1, e, 1/2*e + 2, -e - 2, -1/2*e, 6, 1/2*e + 6, 1/2*e - 2, -e - 2, e - 2, -e, e + 8, -2, 6, -e - 4, -e + 4, 1/2*e - 2, -2*e, 3/2*e + 4, 3/2*e - 4, -e + 2, 14, e + 10, 1/2*e - 4, 1/2*e + 12, -10, 2, e - 10, -e, 12, -e, 1/2*e - 14, -e + 2, 2*e + 6, e + 4, 5/2*e - 2, -3/2*e - 4, -2*e - 2, 3/2*e + 2, 2*e + 8, -2*e + 4, -3/2*e + 2, -3*e + 2, 1/2*e - 12, e + 12, -e + 4, e + 14, -22, 6, 1/2*e - 20, -2*e + 6, 14, -3/2*e + 8, 2*e + 10, -4*e + 2, -2*e + 2, 2*e + 4, 4, 2*e - 4, -e - 8, -2, -10, 1/2*e - 26, 16, -4*e + 10, -4, e - 16, 7/2*e - 8, -2*e - 2, e - 26, -4*e - 6, -22, 10, -2*e - 6, -1/2*e - 12, -2*e + 26, -1/2*e - 28, -5/2*e - 10, -3*e, -e - 14, 3/2*e, e + 12, -e + 28, -4*e + 8, e + 20, -3*e - 18, -3/2*e, -2*e + 2, -4*e + 2, -3*e + 8, -e - 24, -e - 16, -2*e + 12, -5/2*e + 12, -3/2*e - 24, 7/2*e, -3*e - 14, -7/2*e + 16, 2*e + 2, 1/2*e + 2, -e - 24, 4*e, -e + 36, -28, 3*e - 8, -e - 32, -2*e - 4, 5/2*e + 16, e + 14, 2*e - 6, -6, 4*e + 12, 4*e - 4, 1/2*e - 22, 3*e + 24, 2*e + 30, 6, -3/2*e + 34, -2*e + 20, 9/2*e + 2, -2*e + 28, -7/2*e - 4, 6*e - 2, 2*e + 6, -3*e + 18, -2*e + 6, -e + 42, -3*e - 20, e + 12, -3*e - 24, -1/2*e - 2, -2*e + 18, -e - 18, -38, -e - 18, -2, -5*e + 10, 4*e - 2, 22, 1/2*e - 4, -3*e + 34, e - 12, e - 28, 3*e - 10, 4*e + 20, 11/2*e - 10, 2*e + 10, 2*e - 22, -5/2*e - 12, -7*e + 6, e - 28, -9/2*e - 6, e + 28, -4*e, 2*e + 16, -2*e + 40, 3/2*e + 36, 9/2*e + 8, 2*e - 6, -2*e - 30, -e + 10, 7/2*e + 16, -11/2*e + 16, 2*e - 36, -e - 24, 3*e + 16, -e - 24, -16, -3*e - 36]
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])] = -1
AL_eigenvalues[ZF.ideal([4, 2, 1/2*w^3 - 1/2*w^2 - 5/2*w + 1])] = -1

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