/*
  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, 7, -6, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([45, 15, -w^3 + 3*w^2 + 2*w - 6])

primes_array = [
[5, 5, -w^2 + 2*w + 3],\
[9, 3, w^3 - 3*w^2 - 2*w + 9],\
[9, 3, -w^3 + 5*w + 5],\
[11, 11, w],\
[11, 11, w - 1],\
[16, 2, 2],\
[19, 19, -w^3 + 2*w^2 + 3*w - 2],\
[19, 19, w^3 - w^2 - 4*w + 2],\
[29, 29, w^3 - 4*w^2 - w + 10],\
[29, 29, -w^3 + 3*w^2 + w - 7],\
[41, 41, 3*w^2 - 2*w - 10],\
[49, 7, -2*w^2 + 3*w + 8],\
[49, 7, w^3 - 2*w^2 - 2*w + 5],\
[71, 71, -w - 3],\
[71, 71, w - 4],\
[79, 79, -w^3 + w^2 + 3*w + 3],\
[79, 79, -w^3 + 2*w^2 + 2*w - 6],\
[89, 89, w^3 - 3*w^2 - 3*w + 7],\
[89, 89, w^3 - 6*w - 2],\
[101, 101, 2*w^3 - 5*w^2 - 3*w + 9],\
[101, 101, -w^3 + 3*w^2 + 3*w - 12],\
[109, 109, -w^3 + 4*w^2 + w - 9],\
[109, 109, -w^3 + 2*w^2 + 4*w - 2],\
[121, 11, -3*w^2 + 3*w + 10],\
[131, 131, -2*w^3 + 5*w^2 + 4*w - 9],\
[131, 131, -w^3 + w^2 + 3*w + 5],\
[131, 131, -w^3 + 2*w^2 + 2*w - 8],\
[131, 131, -w^3 + 6*w^2 - 18],\
[149, 149, -w^3 - 2*w^2 + 7*w + 8],\
[149, 149, -2*w^3 + 6*w^2 + 5*w - 16],\
[151, 151, w^2 + w - 5],\
[151, 151, w^2 - 3*w - 3],\
[169, 13, w^3 - 4*w^2 - w + 6],\
[169, 13, 2*w^3 - 5*w^2 - 4*w + 6],\
[179, 179, w^3 - 4*w - 6],\
[179, 179, -w^3 + 3*w^2 + w - 9],\
[181, 181, w^3 + 3*w^2 - 7*w - 15],\
[181, 181, w^3 - 6*w^2 + 2*w + 18],\
[191, 191, 5*w^2 - 7*w - 15],\
[191, 191, -4*w^2 + 5*w + 12],\
[199, 199, -w^3 + 3*w^2 + 3*w - 6],\
[199, 199, w^3 - 6*w - 1],\
[211, 211, w^3 - 7*w - 4],\
[211, 211, -w^3 + 3*w^2 + 4*w - 10],\
[229, 229, -w^3 - w^2 + 5*w + 9],\
[229, 229, -w^3 + 4*w^2 - 12],\
[239, 239, 4*w^2 - 5*w - 17],\
[239, 239, 4*w^2 - 3*w - 18],\
[241, 241, -w^3 + 3*w^2 + w - 2],\
[241, 241, -w^3 + 3*w^2 + 4*w - 9],\
[241, 241, w^3 - 7*w - 3],\
[241, 241, -w^3 + 4*w - 1],\
[271, 271, -w - 4],\
[271, 271, w^3 - 3*w^2 - 2*w + 2],\
[271, 271, -2*w^3 + 6*w^2 + 2*w - 7],\
[271, 271, w - 5],\
[281, 281, w^3 - 2*w^2 - 3*w - 1],\
[281, 281, -w^3 + w^2 + 4*w - 5],\
[289, 17, -2*w^3 + w^2 + 9*w + 1],\
[289, 17, -2*w^3 + 5*w^2 + 5*w - 9],\
[311, 311, w^3 - w^2 - 6*w + 1],\
[311, 311, -w^3 + 2*w^2 + 5*w - 5],\
[331, 331, -3*w^3 + 5*w^2 + 10*w - 5],\
[331, 331, 3*w^3 - 2*w^2 - 12*w - 6],\
[349, 349, -2*w^3 + 2*w^2 + 10*w + 3],\
[349, 349, -2*w^3 - w^2 + 12*w + 10],\
[349, 349, 2*w^3 + w^2 - 10*w - 12],\
[349, 349, -2*w^3 + 4*w^2 + 8*w - 13],\
[361, 19, -4*w^2 + 4*w + 13],\
[379, 379, w^3 + w^2 - 5*w - 10],\
[379, 379, -3*w^3 + 4*w^2 + 12*w - 7],\
[379, 379, w^3 + 4*w^2 - 8*w - 21],\
[379, 379, w^3 - 3*w - 6],\
[389, 389, 2*w^3 - 6*w^2 - 4*w + 13],\
[389, 389, -5*w^2 + 7*w + 13],\
[389, 389, 2*w^3 - 7*w^2 - 5*w + 20],\
[389, 389, 2*w^3 - 10*w - 5],\
[419, 419, -w^3 + 5*w^2 + w - 13],\
[419, 419, 4*w^2 - 7*w - 14],\
[419, 419, 2*w^3 - 3*w^2 - 5*w - 1],\
[419, 419, -w^3 - 2*w^2 + 8*w + 8],\
[421, 421, -2*w^2 + 5*w + 7],\
[421, 421, w^3 - 6*w^2 + 5*w + 13],\
[439, 439, 2*w^3 - 2*w^2 - 9*w + 3],\
[439, 439, w^3 + w^2 - 7*w - 4],\
[449, 449, 3*w^3 - 6*w^2 - 9*w + 8],\
[449, 449, 5*w^2 - 8*w - 17],\
[449, 449, 2*w^3 - w^2 - 8*w - 7],\
[449, 449, -3*w^3 + 3*w^2 + 12*w - 4],\
[479, 479, -2*w^3 + 3*w^2 + 5*w + 3],\
[479, 479, 3*w^3 - 7*w^2 - 9*w + 15],\
[499, 499, -w^3 + 4*w^2 + 4*w - 9],\
[499, 499, w^3 + w^2 - 9*w - 2],\
[509, 509, 2*w^3 + 2*w^2 - 11*w - 16],\
[509, 509, -2*w^3 + 8*w^2 + w - 23],\
[521, 521, -3*w^3 + w^2 + 13*w + 6],\
[521, 521, w^3 + 4*w^2 - 10*w - 16],\
[541, 541, 3*w^3 - 2*w^2 - 13*w - 3],\
[541, 541, -w^3 + 7*w^2 - w - 25],\
[541, 541, w^3 + 4*w^2 - 10*w - 20],\
[541, 541, -3*w^3 + 7*w^2 + 8*w - 15],\
[571, 571, w^3 + 4*w^2 - 9*w - 15],\
[571, 571, w^3 - 7*w^2 + 2*w + 19],\
[601, 601, -3*w^3 + 2*w^2 + 13*w + 5],\
[601, 601, 3*w^3 - 7*w^2 - 8*w + 17],\
[619, 619, w^3 - 3*w - 7],\
[619, 619, w^3 - 5*w^2 + 3*w + 13],\
[619, 619, 3*w^3 - 4*w^2 - 9*w - 2],\
[619, 619, -w^3 + 3*w^2 - 9],\
[641, 641, 6*w^2 - 4*w - 23],\
[641, 641, -3*w^3 + w^2 + 12*w + 5],\
[659, 659, -2*w^3 + 6*w^2 + 3*w - 17],\
[659, 659, -2*w^3 + 9*w + 10],\
[661, 661, 3*w^3 - 6*w^2 - 9*w + 14],\
[661, 661, 5*w^2 - 6*w - 18],\
[661, 661, -5*w^2 + 4*w + 19],\
[661, 661, 3*w^3 - 3*w^2 - 12*w - 2],\
[691, 691, -3*w^3 + 5*w^2 + 10*w - 6],\
[691, 691, -3*w^3 + 4*w^2 + 11*w - 6],\
[701, 701, -w^3 - 4*w^2 + 9*w + 20],\
[701, 701, -w^3 + 4*w^2 + 3*w - 19],\
[701, 701, -w^3 + 4*w^2 + 4*w - 15],\
[701, 701, w^3 - 7*w^2 + 2*w + 24],\
[709, 709, 3*w^2 - 5*w - 13],\
[709, 709, 3*w^2 - w - 15],\
[719, 719, -w^3 + 3*w^2 + 4*w - 4],\
[719, 719, -w^3 + 7*w - 2],\
[739, 739, -w^3 + 4*w^2 + 3*w - 9],\
[739, 739, -2*w^3 - w^2 + 11*w + 8],\
[739, 739, 2*w^3 - 7*w^2 - 3*w + 16],\
[739, 739, -w^3 - w^2 + 8*w + 3],\
[751, 751, 3*w^3 - 5*w^2 - 10*w + 10],\
[751, 751, 2*w^2 - 5*w - 5],\
[761, 761, 3*w^3 - 3*w^2 - 11*w + 3],\
[761, 761, w^3 - 2*w^2 - 3*w - 2],\
[761, 761, -w^3 + w^2 + 4*w - 6],\
[761, 761, -3*w^3 + 6*w^2 + 8*w - 8],\
[769, 769, -w^3 - w^2 + 7*w + 1],\
[769, 769, -2*w^3 + 6*w^2 + 6*w - 15],\
[769, 769, -2*w^3 + 5*w^2 + 5*w - 7],\
[769, 769, 2*w^3 + w^2 - 12*w - 9],\
[809, 809, -2*w^3 + 8*w^2 + 4*w - 23],\
[809, 809, -3*w^3 + 11*w^2 + 2*w - 28],\
[829, 829, -3*w^3 + 7*w^2 + 7*w - 9],\
[829, 829, w^3 + 2*w^2 - 6*w - 15],\
[829, 829, -w^3 + 5*w^2 - w - 18],\
[829, 829, -3*w^3 + 2*w^2 + 12*w - 2],\
[839, 839, 2*w^3 - 9*w^2 + 20],\
[839, 839, -3*w^3 + 10*w^2 + 5*w - 30],\
[841, 29, 5*w^2 - 5*w - 16],\
[881, 881, -w^3 - 4*w^2 + 8*w + 18],\
[881, 881, 2*w^3 - 7*w^2 - w + 9],\
[881, 881, 2*w^3 - 6*w^2 - 3*w + 6],\
[881, 881, -w^3 + 7*w^2 - 3*w - 21],\
[911, 911, -w^3 - 4*w^2 + 8*w + 16],\
[911, 911, -w^3 - w^2 + 9*w + 5],\
[911, 911, -w^3 + 4*w^2 + 4*w - 12],\
[911, 911, w^3 - 7*w^2 + 3*w + 19],\
[919, 919, -4*w^3 + 7*w^2 + 14*w - 9],\
[919, 919, 4*w^3 - 5*w^2 - 16*w + 8],\
[929, 929, 3*w^2 - w - 16],\
[929, 929, 3*w^2 - 5*w - 14],\
[941, 941, -w^3 - w^2 + 9*w + 7],\
[941, 941, -3*w^3 + 7*w^2 + 7*w - 15],\
[941, 941, 3*w^3 - 2*w^2 - 12*w - 4],\
[941, 941, w^3 + 4*w^2 - 13*w - 16],\
[961, 31, -2*w^2 + 2*w + 13],\
[961, 31, 5*w^2 - 5*w - 18],\
[971, 971, w^3 + 4*w^2 - 9*w - 19],\
[971, 971, w^3 - 7*w^2 + 2*w + 23],\
[991, 991, 3*w^3 - 7*w^2 - 7*w + 14],\
[991, 991, -3*w^3 + 2*w^2 + 12*w + 3]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 + 2*x - 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [-1, -1, e, 2*e + 1, -e - 3, e, e + 7, 3*e + 4, -3*e + 3, -2*e - 4, 2*e + 3, 4*e + 1, 2*e + 7, 8*e + 5, -3*e - 4, -3*e - 1, -7*e - 1, 6*e + 15, -4*e - 1, 6*e + 2, -7*e - 4, 2*e + 10, -4*e - 7, 3*e - 6, 13, 5*e + 5, -6*e - 4, 12*e + 12, 10*e + 9, -5*e + 9, -4*e + 5, -7*e - 16, 12*e + 6, -6*e + 2, -12*e - 10, e + 1, -e + 5, -13*e - 11, -3*e + 14, -12*e - 15, -5*e + 2, 14, -12*e - 12, 6*e + 12, -6*e - 12, 14*e + 17, -4*e - 17, 3*e + 10, -8*e - 23, -e - 22, 12*e + 18, 13*e + 5, -7*e - 24, e + 7, 8*e + 22, -7*e - 7, -18*e - 14, -13*e - 13, -14*e - 26, 13, -13*e - 5, 2*e + 15, -4*e + 2, -e - 11, -16*e - 20, -12*e - 20, 4, -5*e - 8, -18*e - 12, -16*e - 21, -11*e + 2, 10*e + 25, 4*e - 12, 2*e + 10, -2*e + 16, -12*e - 12, -4*e + 10, 16*e + 16, 6*e - 6, 9*e + 9, 2*e + 25, 12*e + 8, 12*e + 25, -8*e - 2, 5*e + 15, 27*e + 26, -6*e - 1, 10*e + 12, 12*e - 8, -11*e + 5, 5*e + 21, 24*e + 29, -7*e + 22, -4*e + 8, 18*e + 22, 6*e + 2, -13*e - 12, 12*e + 20, 19*e + 33, 2*e - 18, -2*e + 7, e + 27, 18*e + 27, -3*e - 8, 8*e + 18, -20*e - 18, -18*e - 21, -3*e + 27, 18*e + 38, 2, 7*e - 17, 21*e + 34, 20*e + 32, -2*e + 46, 2*e - 10, -9*e - 19, 4*e - 31, 10*e + 13, -7*e - 21, -5*e - 20, -3*e - 42, 15*e + 16, 19*e + 12, 15*e + 24, 11*e + 5, 15*e + 12, 7*e + 23, -11*e - 1, 6*e - 20, -3*e + 24, -8*e - 30, -20*e - 5, 4*e + 10, -2*e + 32, -11*e + 15, -12*e - 26, -6*e + 38, -30*e - 36, -7*e - 28, -15*e - 41, 4*e - 10, -22*e - 22, -10*e + 2, 22*e + 30, -20*e - 42, -15*e + 1, -14*e - 8, 18*e + 35, -27*e - 13, 4*e - 9, -e - 25, -24*e - 36, 7*e - 23, -3*e + 29, 11*e - 11, 10*e + 46, 18*e, -18*e - 10, -15*e - 29, 26*e + 38, 7*e + 5, -2*e - 38, -6*e + 10, -5*e - 4, 17*e - 3, -17*e - 16, 8*e + 18, 7*e + 28, 5*e + 32, -12*e - 19, -31*e - 21, 12*e - 26]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([5, 5, -w^2 + 2*w + 3])] = 1
AL_eigenvalues[ZF.ideal([9, 3, w^3 - 3*w^2 - 2*w + 9])] = 1

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