/*
  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^4 + w^3 + 3*w^2 - 2*w + 1])

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^3 - 5*x^2 - 11*x + 42
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, e - 5, 1/3*e^2 - 1/3*e - 3, 1/3*e^2 - 4/3*e - 8, -e + 2, 2/3*e^2 - 8/3*e - 6, -1, -2/3*e^2 - 1/3*e + 12, -2*e + 4, -2/3*e^2 + 8/3*e + 10, -1/3*e^2 + 1/3*e + 7, e^2 - 3*e - 9, 2*e - 1, 4, 2/3*e^2 + 1/3*e - 11, -e^2 + e + 17, -1/3*e^2 + 7/3*e + 7, -4/3*e^2 + 7/3*e + 15, 4/3*e^2 - 10/3*e - 12, -1/3*e^2 + 10/3*e - 2, -e^2 + 3*e + 13, -2/3*e^2 - 1/3*e + 22, 4/3*e^2 - 13/3*e - 14, -e^2 + 3*e + 13, 2/3*e^2 - 11/3*e - 11, -1/3*e^2 - 2/3*e + 6, 1/3*e^2 + 2/3*e - 4, -1/3*e^2 - 2/3*e + 23, -e^2 + 3*e + 7, -4/3*e^2 + 4/3*e + 24, e^2 - 24, e^2 - e - 9, -4/3*e^2 + 1/3*e + 22, 4, -2*e^2 + 6*e + 16, -4/3*e^2 + 10/3*e + 20, 1/3*e^2 + 8/3*e - 8, 4/3*e^2 - 4/3*e - 12, -e^2 + 3*e + 17, -5/3*e^2 + 5/3*e + 27, 2*e^2 - 6*e - 18, 1/3*e^2 - 10/3*e + 3, 2*e + 4, 2/3*e^2 - 2/3*e - 14, 6, 1/3*e^2 - 4/3*e - 9, -e^2 + 3*e - 5, 4/3*e^2 - 13/3*e - 15, -e + 18, -4/3*e^2 + 4/3*e + 10, -7/3*e^2 + 4/3*e + 35, -e^2 + 5*e + 3, -e^2 + e + 17, 4/3*e^2 + 2/3*e - 26, 8/3*e^2 - 23/3*e - 16, e^2 - 3*e - 13, 1/3*e^2 + 2/3*e + 5, -1/3*e^2 - 2/3*e + 15, -11/3*e^2 + 29/3*e + 31, 5/3*e^2 - 11/3*e - 33, -2/3*e^2 + 2/3*e - 4, 6*e - 10, -2*e^2 + 8*e + 16, 2*e + 4, e^2 - e - 19, 6*e - 10, 4/3*e^2 - 7/3*e - 34, 3*e^2 - 9*e - 27, 2*e^2 + e - 37, -4/3*e^2 + 10/3*e + 24, -2/3*e^2 + 11/3*e + 9, 2/3*e^2 - 8/3*e - 22, 1/3*e^2 + 2/3*e - 14, -1/3*e^2 + 22/3*e - 1, -4/3*e^2 + 16/3*e + 14, -2/3*e^2 + 2/3*e + 8, 2*e^2 - 7*e - 12, -1/3*e^2 - 5/3*e + 5, 7*e - 11, -2/3*e^2 + 5/3*e + 20, 8/3*e^2 - 8/3*e - 34, -4/3*e^2 - 2/3*e + 8, -4/3*e^2 - 5/3*e + 20, -4*e - 8, -e^2 + 3*e + 35, 2/3*e^2 - 14/3*e - 14, 4/3*e^2 - 4/3*e - 14, 10/3*e^2 - 19/3*e - 38, -2/3*e^2 + 17/3*e + 3, 5/3*e^2 - 20/3*e + 2, e + 14, 2/3*e^2 + 7/3*e - 27, -1/3*e^2 - 2/3*e + 34, 5/3*e^2 - 14/3*e - 36, -4*e - 8, -2/3*e^2 + 2/3*e + 32, -4/3*e^2 - 2/3*e + 8, -28, -7/3*e^2 + 1/3*e + 49, -5/3*e^2 - 4/3*e + 41, -28, 2/3*e^2 - 23/3*e + 2, e^2 + 2*e - 26, -2/3*e^2 - 4/3*e + 6, -2/3*e^2 + 14/3*e + 2, 5/3*e^2 - 5/3*e - 37, 5/3*e^2 - 17/3*e - 23, -e^2 + 6*e, -1/3*e^2 - 14/3*e + 32, -4/3*e^2 + 7/3*e + 5, -2/3*e^2 - 4/3*e + 38, -2*e^2 + 3*e + 42, 7/3*e^2 - 22/3*e + 1, -1/3*e^2 + 1/3*e - 13, 4*e - 20, 1/3*e^2 - 4/3*e - 14, 1/3*e^2 - 1/3*e + 25, 7/3*e^2 - 25/3*e - 31, -2/3*e^2 - 7/3*e + 6, 2*e^2 - 2*e - 42, 5/3*e^2 - 32/3*e - 26, -11/3*e^2 + 8/3*e + 56, -7/3*e^2 - 5/3*e + 69, 2/3*e^2 - 11/3*e - 16, 2*e^2 - 4*e - 34, -2/3*e^2 - 7/3*e + 24, 2*e^2 - 5*e - 22, 3*e^2 - 8*e - 10, 6*e - 18, -1/3*e^2 + 7/3*e - 11, -4/3*e^2 + 4/3*e + 26, 2*e^2 - 4*e - 16, -2/3*e^2 - 16/3*e + 10, -11/3*e^2 + 5/3*e + 69, 4/3*e^2 - 25/3*e - 21, -8/3*e^2 + 20/3*e + 52, e^2 + 6*e - 9, -10*e + 4, -4*e, -1/3*e^2 - 5/3*e + 27, 1/3*e^2 + 26/3*e - 6, -4*e^2 + 8*e + 48, -8*e + 4, 4/3*e^2 - 13/3*e + 23, -2*e^2 + 7*e + 10, 5/3*e^2 - 11/3*e - 43, -11/3*e^2 - 4/3*e + 68, -2/3*e^2 + 11/3*e + 1, 2/3*e^2 - 14/3*e + 28, 4/3*e^2 + 17/3*e - 22, -4/3*e^2 + 4/3*e + 40, -2*e + 14, 3*e^2 - 10*e - 50, 4/3*e^2 - 7/3*e - 4, -1/3*e^2 + 7/3*e + 27, -11/3*e^2 + 14/3*e + 37, -e^2 + 3*e + 47, -e^2 - 3*e + 9, -14/3*e^2 + 44/3*e + 36, 4/3*e^2 - 28/3*e - 30]
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^4 + w^3 + 3*w^2 - 2*w + 1])] = 1

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