/*
  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([-1, -3, 6, 4, -5, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([53,53,-w^3 + 2*w + 2])

primes_array = [
[13, 13, w^5 - 5*w^3 + 4*w],\
[25, 5, w^5 - 5*w^3 + 6*w - 1],\
[25, 5, -w^3 + w^2 + 3*w - 1],\
[25, 5, w^5 - 4*w^3 - w^2 + 3*w + 2],\
[27, 3, w^4 - w^3 - 4*w^2 + 2*w + 2],\
[27, 3, w^4 - w^3 - 4*w^2 + 2*w + 3],\
[53, 53, -w^4 + w^3 + 3*w^2 - 2*w + 1],\
[53, 53, -w^4 + w^3 + 4*w^2 - 3*w - 4],\
[53, 53, -w^5 + w^4 + 4*w^3 - 4*w^2 - 3*w + 1],\
[53, 53, w^3 - 2*w - 2],\
[53, 53, w^5 - 5*w^3 - w^2 + 5*w],\
[53, 53, -w^4 + 4*w^2 + w - 4],\
[64, 2, -2],\
[79, 79, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 9*w + 2],\
[79, 79, -w^5 - w^4 + 5*w^3 + 4*w^2 - 6*w - 1],\
[79, 79, w^3 - w^2 - 4*w + 1],\
[79, 79, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 9*w + 3],\
[79, 79, -2*w^4 + w^3 + 7*w^2 - 3*w - 3],\
[79, 79, -w^5 + 6*w^3 - w^2 - 8*w + 1],\
[103, 103, 2*w^4 - 7*w^2 - w + 3],\
[103, 103, w^5 - w^4 - 4*w^3 + 5*w^2 + 2*w - 4],\
[103, 103, -w^5 - w^4 + 4*w^3 + 4*w^2 - 2*w - 3],\
[103, 103, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 4*w - 1],\
[103, 103, -w^4 + w^3 + 5*w^2 - 3*w - 4],\
[103, 103, 2*w^5 - w^4 - 10*w^3 + 3*w^2 + 10*w - 1],\
[131, 131, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w - 1],\
[131, 131, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 4*w + 2],\
[131, 131, w^5 - w^4 - 4*w^3 + 5*w^2 + 2*w - 5],\
[131, 131, 2*w^4 - 7*w^2 - w + 2],\
[131, 131, -2*w^5 + w^4 + 9*w^3 - 2*w^2 - 8*w],\
[131, 131, 2*w^5 - w^4 - 10*w^3 + 3*w^2 + 10*w],\
[157, 157, -w^5 - w^4 + 5*w^3 + 5*w^2 - 6*w - 3],\
[157, 157, -2*w^3 + w^2 + 5*w - 2],\
[157, 157, -w^5 - w^4 + 4*w^3 + 5*w^2 - 3*w - 3],\
[157, 157, -w^5 + 5*w^3 - 7*w],\
[157, 157, -2*w^5 + w^4 + 8*w^3 - 3*w^2 - 6*w + 2],\
[157, 157, -w^5 + w^4 + 5*w^3 - 4*w^2 - 7*w + 2],\
[181, 181, w^2 - 2*w - 2],\
[181, 181, 2*w^5 - 9*w^3 + 7*w],\
[181, 181, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 5*w + 4],\
[181, 181, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w + 1],\
[181, 181, 2*w^5 - 2*w^4 - 8*w^3 + 6*w^2 + 5*w - 2],\
[181, 181, w^4 - 2*w^2 - 2],\
[233, 233, 2*w^3 - w^2 - 5*w + 1],\
[233, 233, -w^5 - w^4 + 6*w^3 + 3*w^2 - 8*w],\
[233, 233, w^5 - w^4 - 5*w^3 + 4*w^2 + 7*w - 3],\
[233, 233, w^5 + w^4 - 5*w^3 - 5*w^2 + 6*w + 4],\
[233, 233, -w^5 + 4*w^3 + 2*w^2 - 2*w - 3],\
[233, 233, 2*w^5 - w^4 - 8*w^3 + 3*w^2 + 6*w - 1],\
[311, 311, 2*w^5 - 2*w^4 - 9*w^3 + 6*w^2 + 8*w - 2],\
[311, 311, w^4 + w^3 - 3*w^2 - 4*w + 1],\
[311, 311, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 5*w + 2],\
[311, 311, w^5 - w^4 - 4*w^3 + 4*w^2 + w - 3],\
[311, 311, w^4 - 2*w^2 - w - 2],\
[311, 311, -2*w^5 + w^4 + 9*w^3 - 4*w^2 - 7*w + 2],\
[313, 313, -w^5 + 5*w^3 + w^2 - 4*w - 3],\
[313, 313, -w^5 + 6*w^3 - 7*w + 1],\
[313, 313, -w^4 + w^3 + 5*w^2 - 3*w - 3],\
[313, 313, -w^5 + w^4 + 3*w^3 - 4*w^2 + 4],\
[313, 313, -w^5 + 2*w^4 + 4*w^3 - 7*w^2 - 4*w + 2],\
[313, 313, -2*w^4 + w^3 + 7*w^2 - 2*w - 4],\
[337, 337, 2*w^5 - w^4 - 7*w^3 + 2*w^2 + 2*w + 2],\
[337, 337, w^5 - 2*w^4 - 3*w^3 + 7*w^2 + 2*w - 3],\
[337, 337, 2*w^5 - w^4 - 9*w^3 + 2*w^2 + 10*w - 1],\
[337, 337, -2*w^5 - w^4 + 9*w^3 + 4*w^2 - 8*w],\
[337, 337, 3*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 11*w - 5],\
[337, 337, w^4 - w^3 - 2*w^2 + 4*w],\
[389, 389, 2*w^5 - w^4 - 9*w^3 + 4*w^2 + 7*w],\
[389, 389, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 7*w + 3],\
[389, 389, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 5*w],\
[389, 389, -2*w^5 + 3*w^4 + 9*w^3 - 10*w^2 - 8*w + 2],\
[389, 389, -3*w^5 + w^4 + 14*w^3 - 2*w^2 - 12*w],\
[389, 389, -w^5 + w^4 + 5*w^3 - 3*w^2 - 3*w + 1],\
[443, 443, -2*w^5 + w^4 + 8*w^3 - w^2 - 7*w - 3],\
[443, 443, 2*w^5 - w^4 - 9*w^3 + 4*w^2 + 8*w],\
[443, 443, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 2*w - 4],\
[443, 443, 2*w^5 - w^4 - 9*w^3 + 2*w^2 + 8*w - 2],\
[443, 443, -2*w^4 + 2*w^3 + 7*w^2 - 4*w - 3],\
[443, 443, -3*w^5 + w^4 + 15*w^3 - 3*w^2 - 17*w + 2],\
[467, 467, 2*w^5 - 2*w^4 - 9*w^3 + 6*w^2 + 10*w - 1],\
[467, 467, w^5 + w^4 - 5*w^3 - 6*w^2 + 6*w + 5],\
[467, 467, -w^5 - 2*w^4 + 5*w^3 + 7*w^2 - 5*w - 3],\
[467, 467, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 13*w + 3],\
[467, 467, 3*w^5 - 2*w^4 - 13*w^3 + 6*w^2 + 11*w - 1],\
[467, 467, -2*w^5 + w^4 + 9*w^3 - w^2 - 8*w - 3],\
[521, 521, w^5 + 2*w^4 - 6*w^3 - 7*w^2 + 7*w + 3],\
[521, 521, 2*w^5 - 2*w^4 - 8*w^3 + 6*w^2 + 7*w - 1],\
[521, 521, -w^5 + w^4 + 6*w^3 - 4*w^2 - 10*w + 3],\
[521, 521, 2*w^5 - 3*w^4 - 9*w^3 + 10*w^2 + 9*w - 3],\
[521, 521, -2*w^5 + 11*w^3 - 13*w + 1],\
[521, 521, 3*w^5 - w^4 - 13*w^3 + 3*w^2 + 11*w - 1],\
[547, 547, w^4 + w^3 - 5*w^2 - 3*w + 1],\
[547, 547, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 10*w + 6],\
[547, 547, w^4 - w^3 - w^2 + 3*w - 3],\
[547, 547, 2*w^4 + w^3 - 9*w^2 - 2*w + 6],\
[547, 547, 3*w^5 - w^4 - 13*w^3 + 3*w^2 + 11*w - 2],\
[547, 547, -w^5 + w^4 + 6*w^3 - 4*w^2 - 10*w + 2],\
[571, 571, -w^5 + 4*w^3 + 3*w^2 - 3*w - 4],\
[571, 571, -w^5 - w^4 + 3*w^3 + 6*w^2 - 5],\
[571, 571, w^5 - 6*w^3 + 2*w^2 + 9*w - 4],\
[571, 571, 2*w^5 - 9*w^3 - w^2 + 10*w + 1],\
[571, 571, 2*w^5 - 11*w^3 + w^2 + 13*w - 3],\
[571, 571, -w^5 + 7*w^3 - w^2 - 12*w + 2],\
[599, 599, -2*w^5 + w^4 + 8*w^3 - 5*w^2 - 5*w + 5],\
[599, 599, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w - 2],\
[599, 599, w^4 - 2*w^3 - 5*w^2 + 7*w + 4],\
[599, 599, -2*w^5 + 10*w^3 - w^2 - 11*w + 3],\
[599, 599, w^5 - 3*w^4 - 5*w^3 + 11*w^2 + 7*w - 5],\
[599, 599, -w^5 + w^4 + 6*w^3 - 3*w^2 - 8*w + 3],\
[677, 677, -w^5 - 2*w^4 + 6*w^3 + 8*w^2 - 9*w - 4],\
[677, 677, -3*w^5 + 2*w^4 + 13*w^3 - 7*w^2 - 12*w + 4],\
[677, 677, w^4 + w^3 - 5*w^2 - 5*w + 4],\
[677, 677, -2*w^5 + 9*w^3 - w^2 - 7*w + 1],\
[677, 677, w^5 - 5*w^3 + 2*w^2 + 5*w - 5],\
[677, 677, -w^5 - w^4 + 7*w^3 + 2*w^2 - 10*w + 1],\
[701, 701, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 2*w - 6],\
[701, 701, w^5 + w^4 - 6*w^3 - 5*w^2 + 6*w + 4],\
[701, 701, -w^5 - w^4 + 3*w^3 + 4*w^2 + 2*w - 2],\
[701, 701, 3*w^5 - 2*w^4 - 14*w^3 + 5*w^2 + 14*w],\
[701, 701, 3*w^5 - 2*w^4 - 14*w^3 + 8*w^2 + 13*w - 5],\
[701, 701, -w^5 + 2*w^4 + 3*w^3 - 7*w^2 + 2*w + 3],\
[727, 727, w^5 + 2*w^4 - 5*w^3 - 8*w^2 + 6*w + 3],\
[727, 727, -w^5 + 4*w^3 + 3*w^2 - 3*w - 6],\
[727, 727, -w^5 - w^4 + 5*w^3 + 2*w^2 - 4*w + 2],\
[727, 727, 3*w^5 - 2*w^4 - 13*w^3 + 6*w^2 + 12*w - 3],\
[727, 727, -2*w^5 + 11*w^3 - w^2 - 13*w + 1],\
[727, 727, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 5*w - 4],\
[857, 857, -w^5 + 3*w^4 + 5*w^3 - 11*w^2 - 5*w + 5],\
[857, 857, -w^5 - 2*w^4 + 4*w^3 + 9*w^2 - 3*w - 7],\
[857, 857, 2*w^5 + w^4 - 9*w^3 - 4*w^2 + 6*w + 3],\
[857, 857, 2*w^5 - 3*w^4 - 7*w^3 + 10*w^2 + 2*w - 5],\
[857, 857, -w^5 + 6*w^3 - 2*w^2 - 8*w + 2],\
[857, 857, -3*w^3 + w^2 + 9*w],\
[859, 859, 2*w^4 - 2*w^3 - 5*w^2 + 5*w - 2],\
[859, 859, -2*w^5 + 9*w^3 - 6*w - 2],\
[859, 859, -w^4 - w^3 + 2*w^2 + 3*w + 4],\
[859, 859, -w^5 - 2*w^4 + 6*w^3 + 9*w^2 - 7*w - 7],\
[859, 859, -2*w^5 - w^4 + 10*w^3 + 4*w^2 - 9*w - 4],\
[859, 859, -w^5 - w^4 + 4*w^3 + 4*w^2 - 3*w + 1],\
[883, 883, 2*w^5 + w^4 - 10*w^3 - 4*w^2 + 11*w],\
[883, 883, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 10*w + 3],\
[883, 883, 3*w^5 - w^4 - 13*w^3 + 2*w^2 + 12*w + 1],\
[883, 883, 3*w^5 - 2*w^4 - 12*w^3 + 6*w^2 + 8*w - 3],\
[883, 883, w^5 - 6*w^3 + w^2 + 10*w - 3],\
[883, 883, -2*w^5 + 11*w^3 - w^2 - 14*w + 3],\
[911, 911, -3*w^5 + 15*w^3 - 15*w + 1],\
[911, 911, -2*w^5 + 2*w^4 + 8*w^3 - 8*w^2 - 6*w + 3],\
[911, 911, -2*w^5 + 11*w^3 - 13*w + 2],\
[911, 911, -2*w^5 + 10*w^3 + 2*w^2 - 10*w - 1],\
[911, 911, 2*w^4 - 2*w^3 - 8*w^2 + 6*w + 7],\
[911, 911, -3*w^4 + 12*w^2 - 7],\
[937, 937, 3*w^4 - w^3 - 10*w^2 + 2*w + 4],\
[937, 937, -w^5 + 2*w^4 + 5*w^3 - 9*w^2 - 6*w + 5],\
[937, 937, -w^5 + 2*w^4 + 2*w^3 - 6*w^2 + 3*w + 3],\
[937, 937, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 12*w - 3],\
[937, 937, -2*w^5 + 11*w^3 + w^2 - 14*w],\
[937, 937, 2*w^5 + w^4 - 9*w^3 - 5*w^2 + 7*w + 5],\
[961, 31, -3*w^5 + w^4 + 12*w^3 - 3*w^2 - 8*w + 2],\
[961, 31, -2*w^5 + 2*w^4 + 10*w^3 - 7*w^2 - 13*w + 3],\
[961, 31, -w^5 + w^4 + 7*w^3 - 5*w^2 - 10*w + 4]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x
K = QQ
e = 1

hecke_eigenvalues_array = [-2, -8, 1, -5, 2, -4, -6, -9, 6, 1, -9, 0, -11, -17, 1, -8, 4, 1, 1, 5, -13, 17, -16, -4, 14, -12, 0, -6, -9, 12, -15, -4, 8, -7, -16, 8, -4, -25, 17, 8, -7, 2, -4, -6, -9, -12, -21, 21, 30, 33, -6, -6, -18, 15, -15, 1, -20, 1, 13, -17, 13, 10, 4, -14, -8, 25, 28, -36, 33, -6, 9, 6, 27, 12, -3, 9, 15, -30, 6, -3, -30, 24, -24, -18, 30, 18, 3, 12, -18, 0, -30, -28, -28, -22, 8, -13, -4, -5, -5, 40, 4, 22, 16, -18, 36, -3, 48, -30, 33, 45, -3, -6, 36, 24, 21, 6, -6, 30, -15, -45, -48, -35, 34, 34, 28, 28, 10, -15, 18, 54, 6, -42, -12, 28, 19, 7, -32, -17, 40, 25, 16, -26, -11, -44, -20, 48, -45, 36, -9, 24, 36, -38, -11, 13, -2, 7, -14, -34, 17, 8]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([53,53,-w^3 + 2*w + 2])] = -1

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