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

NN = ZF.ideal([47, 47, 2*w^2 - 3*w - 5])

primes_array = [
[2, 2, w],\
[8, 2, -w^3 + w^2 + 4*w - 1],\
[11, 11, w^3 - 2*w^2 - 2*w + 1],\
[23, 23, -w^3 + 4*w + 1],\
[23, 23, -w^2 + 2*w + 3],\
[31, 31, w^3 - 2*w^2 - w + 3],\
[37, 37, -w^3 + 3*w + 3],\
[37, 37, -2*w^3 + 3*w^2 + 6*w - 3],\
[41, 41, w^3 - 2*w^2 - 3*w + 1],\
[41, 41, -2*w^3 + 2*w^2 + 6*w - 1],\
[43, 43, -w^2 + w + 5],\
[47, 47, 2*w^2 - 3*w - 5],\
[53, 53, -w^3 + 3*w^2 + w - 7],\
[53, 53, -2*w^2 + 2*w + 5],\
[59, 59, 2*w^2 - w - 7],\
[61, 61, 2*w^2 - w - 3],\
[61, 61, 2*w^2 - w - 5],\
[67, 67, 2*w^3 - 2*w^2 - 7*w + 3],\
[67, 67, 2*w^3 - 2*w^2 - 7*w - 1],\
[71, 71, 2*w^3 - 4*w^2 - 4*w + 7],\
[81, 3, -3],\
[97, 97, -w^3 + 2*w^2 - 3],\
[101, 101, -w^3 + w^2 + 3*w + 3],\
[101, 101, 3*w^3 - 4*w^2 - 9*w + 3],\
[103, 103, -w^3 + 2*w^2 + w - 5],\
[103, 103, -2*w^3 + 4*w^2 + 4*w - 3],\
[107, 107, -w^3 + w^2 + 5*w - 1],\
[127, 127, 2*w^3 - 2*w^2 - 8*w + 1],\
[127, 127, -w^3 + 5*w - 1],\
[137, 137, w^3 - 6*w - 1],\
[149, 149, w^3 - 2*w - 3],\
[151, 151, 2*w^3 - 8*w - 3],\
[157, 157, 2*w^3 - w^2 - 6*w - 5],\
[163, 163, -w^3 + 6*w - 3],\
[173, 173, -4*w^3 + 5*w^2 + 12*w - 5],\
[181, 181, -2*w^3 + 5*w^2 + 5*w - 9],\
[191, 191, w^3 + w^2 - 3*w - 5],\
[193, 193, 2*w^3 - 3*w^2 - 6*w + 1],\
[197, 197, 3*w^3 - 4*w^2 - 10*w + 3],\
[199, 199, -3*w^2 + 4*w + 3],\
[199, 199, 2*w^3 - 4*w^2 - 6*w + 11],\
[211, 211, -2*w^2 + 5*w + 5],\
[223, 223, 2*w^2 + w - 5],\
[227, 227, w^2 + w - 5],\
[229, 229, w^3 - 4*w^2 - w + 9],\
[229, 229, 2*w^2 - 2*w - 9],\
[233, 233, -w^3 - 2*w^2 + 6*w + 7],\
[251, 251, -w^3 + 3*w + 7],\
[257, 257, -2*w^3 + w^2 + 7*w - 1],\
[269, 269, 3*w^3 - 4*w^2 - 11*w + 7],\
[277, 277, 3*w^3 - 4*w^2 - 7*w + 5],\
[277, 277, 3*w^3 - 2*w^2 - 12*w + 1],\
[277, 277, w^3 - 4*w^2 + 7],\
[277, 277, w^2 - 7],\
[281, 281, 3*w^2 - w - 7],\
[283, 283, -w^3 + 2*w^2 + 2*w - 7],\
[293, 293, -3*w^3 + 2*w^2 + 10*w - 1],\
[307, 307, -3*w^2 + 3*w + 7],\
[307, 307, 2*w^3 - 3*w^2 - 8*w + 5],\
[311, 311, -3*w^3 + 5*w^2 + 7*w - 3],\
[313, 313, w^2 - 2*w + 3],\
[313, 313, 2*w^3 - w^2 - 8*w + 1],\
[317, 317, -w^3 - w^2 + 5*w + 1],\
[317, 317, -w^3 + w^2 + 3*w - 5],\
[331, 331, w^3 - w - 3],\
[331, 331, -w^3 + 2*w^2 - 5],\
[337, 337, -2*w^3 + 4*w^2 + 3*w - 7],\
[347, 347, 3*w^2 - w - 5],\
[349, 349, 2*w^2 - 7],\
[353, 353, 3*w^2 - 2*w - 5],\
[367, 367, 3*w^3 - 2*w^2 - 9*w - 1],\
[389, 389, 2*w^3 - 5*w^2 - 6*w + 11],\
[401, 401, -w^3 + 4*w^2 + w - 7],\
[401, 401, 3*w^2 - 2*w - 7],\
[409, 409, 3*w^3 - 4*w^2 - 7*w + 3],\
[409, 409, -2*w^3 + 6*w^2 + w - 7],\
[421, 421, 2*w^3 - 7*w - 1],\
[421, 421, 4*w^3 - 7*w^2 - 11*w + 9],\
[433, 433, w^3 - 3*w^2 - 5*w + 7],\
[439, 439, -w^3 + 7*w - 1],\
[457, 457, -2*w^3 + 4*w^2 + 5*w - 1],\
[457, 457, 2*w^3 - 4*w^2 - 6*w + 3],\
[467, 467, -2*w^3 - w^2 + 12*w + 5],\
[467, 467, -3*w^3 + 5*w^2 + 9*w - 5],\
[479, 479, w^2 + 2*w - 5],\
[479, 479, 3*w^3 - 2*w^2 - 11*w + 1],\
[499, 499, 2*w^3 - w^2 - 7*w + 3],\
[499, 499, w^3 - 4*w^2 + 2*w + 7],\
[499, 499, 4*w^3 - 3*w^2 - 13*w - 1],\
[499, 499, -w^2 + w - 3],\
[503, 503, 4*w^3 - 6*w^2 - 11*w + 5],\
[509, 509, 3*w^3 - 12*w - 5],\
[509, 509, w^2 - w - 7],\
[521, 521, -4*w^3 + 8*w^2 + 11*w - 11],\
[521, 521, w^3 - 2*w - 7],\
[523, 523, 4*w^3 - 7*w^2 - 7*w + 5],\
[523, 523, 2*w^3 - 3*w^2 - 3*w + 7],\
[529, 23, -2*w^3 + 3*w^2 + 2*w + 3],\
[557, 557, -4*w^3 + 4*w^2 + 13*w - 5],\
[557, 557, -2*w^3 + 2*w^2 + 9*w - 3],\
[563, 563, 4*w^3 - 4*w^2 - 15*w + 3],\
[569, 569, 2*w^2 - 4*w - 7],\
[571, 571, 3*w^3 - 4*w^2 - 6*w + 3],\
[571, 571, w^3 - 7*w - 1],\
[577, 577, -w^2 - 3],\
[593, 593, 2*w^3 - 2*w^2 - 9*w + 1],\
[593, 593, 2*w^2 - 9],\
[599, 599, 5*w^3 - 6*w^2 - 15*w + 5],\
[601, 601, -2*w^3 + 6*w^2 + 4*w - 5],\
[601, 601, 3*w^3 - 5*w^2 - 11*w + 7],\
[613, 613, -5*w^3 + 7*w^2 + 13*w - 3],\
[613, 613, 4*w^3 - 5*w^2 - 10*w + 7],\
[617, 617, 4*w^3 - 2*w^2 - 13*w - 3],\
[625, 5, -5],\
[631, 631, 4*w^2 - 2*w - 5],\
[631, 631, 4*w^3 - 5*w^2 - 11*w + 5],\
[641, 641, w^3 + 2*w^2 - 4*w - 9],\
[641, 641, 2*w^3 - 5*w^2 - 2*w + 9],\
[643, 643, w^3 - 4*w - 7],\
[647, 647, -w - 5],\
[653, 653, -w^3 + w^2 + 5*w - 7],\
[673, 673, -2*w^3 + w^2 + 8*w - 3],\
[673, 673, -w^3 + 5*w - 5],\
[673, 673, -3*w^3 + 7*w^2 + 5*w - 13],\
[673, 673, -w^2 + 5*w - 1],\
[677, 677, -2*w^3 + 6*w^2 + 5*w - 9],\
[701, 701, 3*w^3 - 2*w^2 - 7*w - 1],\
[701, 701, 4*w^3 - 6*w^2 - 9*w + 7],\
[709, 709, 4*w^3 - 5*w^2 - 10*w + 1],\
[719, 719, w^3 - 2*w^2 - 3*w - 3],\
[727, 727, -3*w^3 + 4*w^2 + 11*w - 1],\
[727, 727, -2*w^3 + 6*w^2 + w - 11],\
[733, 733, -2*w^3 + 2*w^2 + 7*w + 5],\
[733, 733, -w^3 + 6*w^2 - 4*w - 9],\
[743, 743, -4*w^3 + 6*w^2 + 8*w - 7],\
[743, 743, -2*w^3 + 11*w + 3],\
[757, 757, -3*w^3 + 4*w^2 + 9*w + 1],\
[769, 769, -2*w^3 + 9*w + 1],\
[787, 787, -4*w^3 + 7*w^2 + 10*w - 7],\
[787, 787, -3*w^2 + w + 11],\
[787, 787, w^3 - w^2 - 3*w - 5],\
[787, 787, -2*w^3 + w^2 + 10*w + 1],\
[797, 797, 4*w^3 - 6*w^2 - 9*w + 5],\
[797, 797, -2*w^3 + 3*w^2 + 5*w - 9],\
[809, 809, -w^3 + 4*w^2 + 4*w - 9],\
[811, 811, -w^3 + 3*w^2 - w - 7],\
[811, 811, w^2 - 5*w - 1],\
[829, 829, -2*w^3 + 4*w^2 + 2*w - 7],\
[839, 839, 2*w^2 + w - 7],\
[839, 839, 4*w^3 - 4*w^2 - 12*w + 7],\
[841, 29, 2*w^2 - 6*w - 3],\
[841, 29, -4*w^3 + 6*w^2 + 14*w - 9],\
[857, 857, -2*w^3 + 6*w^2 + 3*w - 11],\
[881, 881, 4*w^2 - 4*w - 7],\
[883, 883, 4*w^3 - 4*w^2 - 11*w - 1],\
[883, 883, -3*w^3 + 6*w^2 + 7*w - 5],\
[919, 919, -w^3 + 5*w^2 - w - 7],\
[929, 929, -w^3 + 5*w^2 + w - 11],\
[941, 941, -3*w^3 + w^2 + 13*w + 1],\
[947, 947, 2*w^3 - 3*w^2 - 9*w + 3],\
[947, 947, 2*w^3 - 6*w^2 - 4*w + 11],\
[953, 953, -2*w^3 + 7*w - 1],\
[971, 971, 3*w^3 - w^2 - 11*w - 1],\
[977, 977, -w^3 + 5*w^2 - w - 9],\
[983, 983, w^3 - w - 5],\
[983, 983, 3*w^3 - 2*w^2 - 10*w + 3],\
[991, 991, -2*w^3 - 2*w^2 + 9*w + 5],\
[991, 991, -6*w^3 + 8*w^2 + 17*w - 11]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 - 6*x + 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -e, -e^2 - e + 6, e^2 + 2*e - 3, e^2 - 1, -2*e + 6, e + 5, -e^2 + e + 6, e + 5, -e^2 - 3*e + 6, -e^2 + e + 4, -1, e^2 - 3*e, e^2 - e - 6, 3*e + 5, 2*e^2 - 5*e - 11, e^2 - e - 6, -3*e^2 + 7, 2*e^2 + 3*e - 5, e^2 + 3*e - 12, -4*e^2 + e + 13, 2*e^2 + 4*e - 12, e^2 - e - 14, -e^2 - e, -e^2 + 5*e + 8, e - 9, 4*e^2 + e - 9, 6*e^2 + e - 19, 8, -4*e^2 + 2*e + 16, e^2 + e - 8, -5*e^2 - 4*e + 17, -4*e^2 - e + 11, -5*e^2 + 25, -2*e^2 - 2*e + 6, -4*e^2 + 3*e + 15, 2*e^2 + 3*e - 25, 3*e^2 + e - 18, 5*e^2 + 5*e - 20, -e - 11, -8*e - 8, e + 11, -e^2 + e, -2*e^2 - 6*e - 4, -2*e^2 + e - 5, -3*e^2 - 3*e + 8, e^2 + 6*e - 1, e^2 + e - 2, -6*e^2 - 2*e + 30, 2*e^2 + e - 9, -2*e^2 - 6*e + 26, -7*e^2 - e + 18, -4*e + 10, -5*e^2 - 8*e + 31, -e^2 - 11*e + 10, 5*e^2 - 6*e - 23, 4*e^2 + 4*e - 10, -3*e^2 + e + 10, 4*e^2 + 2*e - 18, 4*e^2 - 5*e - 27, -4*e^2 - 3*e + 9, -7*e^2 - e + 26, 2*e^2 - 6*e - 14, -3*e^2 + e, 5*e - 5, 3*e^2 - 9*e - 10, 3*e^2 - 6*e - 7, 5*e^2 + 7*e - 28, 3*e^2 + 12*e - 9, -e^2 - 2*e + 17, 2*e^2 + 2*e - 12, 3*e^2 + 14*e - 19, 3*e^2 - 9*e - 20, 2*e^2 - 3*e - 13, 7*e^2 + 9*e - 30, -3*e^2 - e - 6, -7*e - 3, -6*e + 8, -e^2 - e + 40, -9*e^2 + 2*e + 35, 4*e^2 - 5*e - 1, -4*e^2 - 3*e + 33, 2*e^2 - 5*e - 37, 3*e^2 - 5*e - 14, e^2 - 13*e - 8, 2*e^2 + 4*e - 34, -2*e^2 - 11*e + 5, 3*e^2 + 11*e - 22, -2*e^2 + 15*e + 19, -4*e^2 + 8*e + 24, 3*e^2 - 7*e - 4, -5*e^2 + 39, -7*e^2 - 10*e + 27, -4*e^2 - 4*e + 26, -3*e^2 + 3*e - 6, -e - 23, 3*e^2 + 4*e - 19, -9*e^2 + e + 22, e^2 + 2*e - 17, -4*e^2 + 17*e + 29, -8*e^2 + 5*e + 35, -3*e^2 + 11*e + 30, 13*e^2 + e - 46, 3*e^2 - 6*e - 17, 7*e^2 + 5*e - 10, 8*e^2 + e - 19, -2*e^2 + 3*e - 15, -e^2 + 6*e + 7, e^2 + 11*e + 10, 6*e^2 + 5*e - 25, -3*e^2 - 13*e + 26, -7*e^2 - 14*e + 27, 5*e^2 - 4*e - 31, 3*e^2 - 1, -4*e^2 + 2*e + 6, 16*e, 3*e^2 + 5*e - 10, -5*e^2 + 6*e + 13, e^2 + 3*e - 4, e^2 + 9*e + 6, -9*e^2 - e + 32, -3*e + 9, -6*e^2 + 20, -3*e^2 - 14*e + 23, -3*e^2 + 7*e + 26, 5*e^2 + e - 24, 3*e^2 + 4*e - 13, -4*e^2 + 3*e + 23, 8*e^2 - 8*e - 38, 2*e^2 - 7*e + 17, -6*e^2 + e + 21, -5*e^2 - 6*e - 5, 4*e^2 + 2*e - 48, -6*e^2 - 4*e + 16, 7*e^2 + e - 32, -6*e^2 + 14*e + 40, 6*e^2 + 6*e - 22, e^2 - 23, 13*e^2 + 5*e - 34, -2*e^2 - 4*e + 42, -4*e^2 + 2*e - 2, 4*e^2 + 7*e + 1, -e^2 + 10*e + 21, -7*e^2 + 13*e + 32, -2*e^2 - 3*e + 7, 3*e^2 + 3*e - 14, 2*e^2 - 12*e - 2, 12*e^2 + 3*e - 41, -2*e^2 - 4*e - 10, -13*e^2 - 4*e + 41, -4*e^2 - 4*e + 46, 2*e^2 - 6*e - 10, -e^2 + 3*e - 32, 10*e - 20, 3*e^2 + 9*e - 4, 9*e^2 + 15*e - 32, -e^2 - 11*e - 8, -12*e^2 - 9*e + 51, 3*e^2 + 4*e - 17, 2*e^2 + 9*e + 1, 10*e^2 + e - 43, 9*e^2 + 15*e - 34, 7*e^2 + 13*e - 12, 6*e^2 - 6*e - 18, -5*e^2 - 2*e + 7, 12*e^2 + 14*e - 58, 10*e^2 + 5*e - 31, -e^2 - 17*e + 18]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([47, 47, 2*w^2 - 3*w - 5])] = 1

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