/*
  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, 6, -8, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([79, 79, 2/3*w^3 + w^2 - 10/3*w - 19/3])

primes_array = [
[9, 3, 1/3*w^3 - 8/3*w - 2/3],\
[9, 3, -w + 1],\
[11, 11, w],\
[11, 11, -1/3*w^3 + 2/3*w + 5/3],\
[11, 11, 1/3*w^3 - 8/3*w + 1/3],\
[11, 11, -2/3*w^3 + 13/3*w + 4/3],\
[16, 2, 2],\
[25, 5, 2/3*w^3 - 10/3*w - 1/3],\
[29, 29, -w - 3],\
[29, 29, -1/3*w^3 + 8/3*w - 10/3],\
[31, 31, w^3 - 6*w + 1],\
[31, 31, w^3 - 6*w - 2],\
[41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3],\
[41, 41, 2/3*w^3 - 13/3*w + 5/3],\
[59, 59, 2/3*w^3 - 13/3*w + 8/3],\
[59, 59, w^3 + w^2 - 6*w - 4],\
[79, 79, 2/3*w^3 + w^2 - 10/3*w - 19/3],\
[79, 79, 1/3*w^3 + w^2 - 2/3*w - 17/3],\
[89, 89, 4/3*w^3 - 23/3*w - 5/3],\
[89, 89, 1/3*w^3 + 2*w^2 - 5/3*w - 32/3],\
[101, 101, 5/3*w^3 + w^2 - 31/3*w - 16/3],\
[101, 101, 2/3*w^3 - w^2 - 4/3*w + 8/3],\
[109, 109, -w^3 - 2*w^2 + 7*w + 7],\
[109, 109, -1/3*w^3 + 2*w^2 + 2/3*w - 16/3],\
[131, 131, w^3 - w^2 - 7*w + 5],\
[131, 131, -2/3*w^3 - w^2 + 4/3*w + 13/3],\
[139, 139, -2/3*w^3 - w^2 + 13/3*w + 4/3],\
[139, 139, w^2 - w - 7],\
[151, 151, -5/3*w^3 + 28/3*w - 5/3],\
[151, 151, -1/3*w^3 + w^2 + 2/3*w - 16/3],\
[151, 151, w^3 + w^2 - 6*w - 3],\
[151, 151, -w^3 + w^2 + 7*w - 6],\
[179, 179, 5/3*w^3 + w^2 - 28/3*w - 19/3],\
[179, 179, -1/3*w^3 + w^2 + 11/3*w - 4/3],\
[181, 181, -4/3*w^3 - 2*w^2 + 20/3*w + 29/3],\
[181, 181, 1/3*w^3 + w^2 - 2/3*w - 20/3],\
[181, 181, w^3 + w^2 - 5*w - 7],\
[181, 181, 5/3*w^3 + w^2 - 28/3*w - 16/3],\
[191, 191, 2*w - 1],\
[191, 191, 2/3*w^3 - 16/3*w - 1/3],\
[199, 199, 1/3*w^3 - 11/3*w + 4/3],\
[199, 199, 1/3*w^3 - 11/3*w - 2/3],\
[211, 211, -2/3*w^3 - w^2 + 19/3*w + 16/3],\
[211, 211, 2/3*w^3 + w^2 - 19/3*w - 7/3],\
[229, 229, 1/3*w^3 - 11/3*w + 1/3],\
[239, 239, -5/3*w^3 - w^2 + 25/3*w + 19/3],\
[239, 239, -4/3*w^3 + w^2 + 20/3*w - 7/3],\
[251, 251, -2/3*w^3 + w^2 + 16/3*w - 14/3],\
[251, 251, -1/3*w^3 - w^2 - 1/3*w + 14/3],\
[269, 269, 2/3*w^3 + w^2 - 7/3*w - 19/3],\
[269, 269, 1/3*w^3 + 2*w^2 - 2/3*w - 32/3],\
[281, 281, w^3 - w^2 - 5*w + 1],\
[281, 281, 4/3*w^3 + w^2 - 20/3*w - 23/3],\
[289, 17, 2/3*w^3 + w^2 - 16/3*w - 4/3],\
[289, 17, 1/3*w^3 + w^2 - 11/3*w - 20/3],\
[331, 331, 1/3*w^3 + 2*w^2 - 8/3*w - 23/3],\
[331, 331, 2/3*w^3 + 2*w^2 - 13/3*w - 28/3],\
[349, 349, 4/3*w^3 - 17/3*w + 1/3],\
[349, 349, -5/3*w^3 + 28/3*w - 2/3],\
[361, 19, -1/3*w^3 + 5/3*w + 14/3],\
[361, 19, 4/3*w^3 - 20/3*w - 5/3],\
[379, 379, 4/3*w^3 + 2*w^2 - 23/3*w - 41/3],\
[379, 379, -5/3*w^3 + 31/3*w - 5/3],\
[389, 389, 5/3*w^3 + w^2 - 25/3*w - 16/3],\
[389, 389, -4/3*w^3 + w^2 + 20/3*w - 10/3],\
[401, 401, -1/3*w^3 - 3*w^2 + 5/3*w + 26/3],\
[401, 401, 1/3*w^3 + w^2 - 11/3*w - 23/3],\
[401, 401, -2/3*w^3 - 3*w^2 + 10/3*w + 52/3],\
[401, 401, 2/3*w^3 + w^2 - 16/3*w - 1/3],\
[409, 409, -2*w^3 - w^2 + 11*w + 2],\
[409, 409, -5/3*w^3 - 2*w^2 + 28/3*w + 37/3],\
[409, 409, 1/3*w^3 - 11/3*w - 17/3],\
[409, 409, 5/3*w^3 + 3*w^2 - 25/3*w - 37/3],\
[419, 419, -1/3*w^3 + 3*w^2 - 1/3*w - 37/3],\
[419, 419, 2*w^3 + 3*w^2 - 12*w - 13],\
[421, 421, -1/3*w^3 + 3*w^2 + 2/3*w - 31/3],\
[421, 421, -5/3*w^3 - w^2 + 25/3*w + 13/3],\
[421, 421, -w^3 + 7*w - 4],\
[421, 421, -4/3*w^3 + w^2 + 20/3*w - 13/3],\
[439, 439, -4/3*w^3 + w^2 + 14/3*w - 7/3],\
[439, 439, 4/3*w^3 - 17/3*w - 5/3],\
[439, 439, -7/3*w^3 - w^2 + 41/3*w + 17/3],\
[439, 439, 5/3*w^3 - 28/3*w - 4/3],\
[449, 449, w^3 + 3*w^2 - 6*w - 17],\
[449, 449, 1/3*w^3 + 3*w^2 - 8/3*w - 26/3],\
[449, 449, -4/3*w^3 + 26/3*w + 17/3],\
[449, 449, 7/3*w^3 + w^2 - 38/3*w - 8/3],\
[461, 461, -5/3*w^3 + w^2 + 28/3*w - 14/3],\
[461, 461, -2/3*w^3 - 2*w^2 + 10/3*w + 43/3],\
[461, 461, -2*w^3 - w^2 + 12*w + 4],\
[461, 461, -w^3 + 8*w - 5],\
[479, 479, w^3 + w^2 - 7*w - 2],\
[479, 479, -4/3*w^3 - w^2 + 23/3*w + 5/3],\
[499, 499, -4/3*w^3 - 2*w^2 + 23/3*w + 17/3],\
[499, 499, -4/3*w^3 - w^2 + 29/3*w + 8/3],\
[521, 521, 2*w^3 - 11*w + 2],\
[521, 521, 4/3*w^3 - w^2 - 14/3*w + 1/3],\
[569, 569, w^3 + 3*w^2 - 6*w - 9],\
[569, 569, 4/3*w^3 + w^2 - 26/3*w - 32/3],\
[569, 569, -w^3 + 6*w - 6],\
[569, 569, -1/3*w^3 - 3*w^2 + 8/3*w + 50/3],\
[571, 571, 2/3*w^3 + 3*w^2 - 10/3*w - 46/3],\
[571, 571, 2*w^2 - 15],\
[599, 599, -w^3 + 2*w^2 + 4*w - 10],\
[599, 599, -w^2 - 3*w + 8],\
[601, 601, 5/3*w^3 + 3*w^2 - 28/3*w - 34/3],\
[601, 601, 1/3*w^3 - 3*w^2 - 2/3*w + 43/3],\
[619, 619, 2/3*w^3 - w^2 - 13/3*w + 2/3],\
[619, 619, 2/3*w^3 + w^2 - 7/3*w - 25/3],\
[619, 619, 2*w^3 + w^2 - 11*w - 6],\
[619, 619, -2/3*w^3 + 13/3*w - 17/3],\
[631, 631, 2/3*w^3 + 2*w^2 - 19/3*w - 31/3],\
[631, 631, w^3 + 2*w^2 - 8*w - 6],\
[631, 631, 2/3*w^3 + 2*w^2 - 4/3*w - 31/3],\
[631, 631, -2/3*w^3 + 2*w^2 + 16/3*w - 23/3],\
[659, 659, -2/3*w^3 + 1/3*w + 16/3],\
[659, 659, 5/3*w^3 - 34/3*w - 13/3],\
[661, 661, w^3 - 8*w + 2],\
[661, 661, 4/3*w^3 + w^2 - 23/3*w + 1/3],\
[661, 661, 5/3*w^3 + w^2 - 31/3*w - 10/3],\
[661, 661, -3*w - 1],\
[691, 691, -4/3*w^3 + 29/3*w + 2/3],\
[691, 691, 1/3*w^3 + 4/3*w - 5/3],\
[691, 691, -5/3*w^3 + w^2 + 25/3*w - 5/3],\
[691, 691, 2*w^3 + w^2 - 10*w - 7],\
[701, 701, -2/3*w^3 - 3*w^2 + 19/3*w + 46/3],\
[701, 701, w^3 - 2*w^2 - 6*w + 8],\
[701, 701, 4/3*w^3 + 2*w^2 - 17/3*w - 29/3],\
[701, 701, 5/3*w^3 + 3*w^2 - 28/3*w - 52/3],\
[709, 709, 1/3*w^3 - 14/3*w + 10/3],\
[709, 709, -2/3*w^3 + 19/3*w + 7/3],\
[719, 719, 3*w^2 - w - 10],\
[719, 719, -4/3*w^3 - 3*w^2 + 23/3*w + 47/3],\
[739, 739, 5/3*w^3 + w^2 - 34/3*w - 10/3],\
[739, 739, 4/3*w^3 + 2*w^2 - 23/3*w - 14/3],\
[761, 761, 7/3*w^3 + 2*w^2 - 41/3*w - 26/3],\
[761, 761, 2*w^3 + 3*w^2 - 12*w - 15],\
[769, 769, -2/3*w^3 + 19/3*w - 8/3],\
[769, 769, -4/3*w^3 + 2*w^2 + 17/3*w - 31/3],\
[769, 769, 1/3*w^3 - 14/3*w - 5/3],\
[769, 769, -1/3*w^3 + 3*w^2 + 2/3*w - 34/3],\
[809, 809, -4/3*w^3 + w^2 + 26/3*w - 7/3],\
[809, 809, w^3 + w^2 - 3*w - 7],\
[811, 811, 4/3*w^3 + w^2 - 17/3*w - 23/3],\
[811, 811, -1/3*w^3 - 2*w^2 + 2/3*w + 38/3],\
[821, 821, 8/3*w^3 + 3*w^2 - 43/3*w - 34/3],\
[821, 821, 2/3*w^3 + 2*w^2 - 7/3*w - 34/3],\
[821, 821, -w^3 - w^2 + 4*w + 8],\
[821, 821, 2/3*w^3 + 4*w^2 - 7/3*w - 58/3],\
[829, 829, 4/3*w^3 + 3*w^2 - 26/3*w - 50/3],\
[829, 829, 7/3*w^3 - 41/3*w - 5/3],\
[839, 839, 2*w^3 + w^2 - 10*w - 3],\
[839, 839, 5/3*w^3 - w^2 - 25/3*w + 17/3],\
[841, 29, 5/3*w^3 - 25/3*w - 7/3],\
[859, 859, -7/3*w^3 - w^2 + 32/3*w + 29/3],\
[859, 859, 2*w^3 + w^2 - 14*w - 6],\
[859, 859, 1/3*w^3 + 2*w^2 - 2/3*w - 47/3],\
[859, 859, -5/3*w^3 - 2*w^2 + 31/3*w + 49/3],\
[881, 881, w^3 + 4*w^2 - 4*w - 20],\
[881, 881, 8/3*w^3 + 2*w^2 - 43/3*w - 34/3],\
[919, 919, -w^3 + 3*w - 7],\
[919, 919, 1/3*w^3 + 3*w^2 - 5/3*w - 35/3],\
[919, 919, -5/3*w^3 + 31/3*w - 23/3],\
[919, 919, 2/3*w^3 + 3*w^2 - 10/3*w - 43/3],\
[929, 929, w^2 - 2*w - 9],\
[929, 929, w^3 + w^2 - 7*w + 1],\
[961, 31, 2/3*w^3 - 10/3*w - 19/3],\
[971, 971, -w^3 - 3*w^2 + 7*w + 16],\
[971, 971, -5/3*w^3 + w^2 + 25/3*w - 8/3],\
[971, 971, 2*w^3 + w^2 - 10*w - 6],\
[971, 971, 1/3*w^3 + 2*w^2 - 8/3*w - 47/3]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 + 5*x^3 - 3*x^2 - 3*x + 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, e^3 + 5*e^2 - 3*e - 2, 2*e^3 + 11*e^2 - e - 5, e^2 + 6*e, -2*e^3 - 11*e^2 + 4, 2*e^3 + 10*e^2 - 5*e - 4, -e^3 - 5*e^2 + 2*e + 1, -4*e^3 - 20*e^2 + 11*e + 7, -2*e^2 - 10*e + 1, 2*e^3 + 11*e^2 - 7, -4*e^3 - 22*e^2 + e + 8, -e^2 - 7*e, -e - 10, -2*e^3 - 10*e^2 + 3*e + 2, 2*e^3 + 11*e^2 - 2*e - 9, -2*e^3 - 10*e^2 + 8*e + 2, 1, -4*e^3 - 21*e^2 + 6*e - 2, -e^3 - 7*e^2 - 7*e + 4, e^2 + 7*e + 4, -3*e^3 - 14*e^2 + 12*e - 1, 2*e^3 + 8*e^2 - 16*e - 9, e^3 + 3*e^2 - 9*e + 10, 3*e^3 + 14*e^2 - 13*e - 10, -9*e^3 - 48*e^2 + 7*e + 20, 5*e^3 + 26*e^2 - 7*e - 3, 3*e^3 + 18*e^2 + 8*e - 12, -e - 4, 4*e^3 + 20*e^2 - 12*e - 20, e^2 + 2*e, 6*e^3 + 36*e^2 + 11*e - 21, 3*e^2 + 17*e - 8, -2*e^3 - 8*e^2 + 19*e + 9, -10*e^3 - 50*e^2 + 23*e + 15, e^2 + 7*e - 9, 4*e^3 + 20*e^2 - 10*e - 14, 3*e^3 + 11*e^2 - 25*e - 2, -e^3 - 6*e^2 - 3*e + 19, -e^3 - 7*e^2 - 10*e + 3, -2*e^3 - 11*e^2 + e + 7, -4*e^3 - 16*e^2 + 28*e + 7, 8*e^3 + 43*e^2 - e - 17, -7*e^3 - 36*e^2 + 10*e + 15, 3*e^3 + 18*e^2 + 12*e - 1, 4*e^3 + 17*e^2 - 24*e - 4, -e^3 - 6*e^2 - 6*e - 13, -5*e^3 - 22*e^2 + 29*e + 14, 7*e^3 + 43*e^2 + 17*e - 35, 5*e^3 + 26*e^2 - 5*e - 2, 2*e^3 + 11*e^2 - 3, e^3 - e^2 - 28*e + 6, -6*e^3 - 31*e^2 + 11*e - 10, 2*e^3 + 11*e^2 + e - 23, 5*e^3 + 26*e^2 - 6*e, 9*e^3 + 47*e^2 - 11*e - 2, -6*e^3 - 28*e^2 + 24*e + 4, 3*e^3 + 13*e^2 - 26*e - 19, 4*e^3 + 23*e^2 + 8*e - 3, -e^2 - 6*e + 20, -2*e^3 - 9*e^2 + 8*e + 1, -3*e^3 - 14*e^2 + 11*e + 5, -2*e^3 - 6*e^2 + 20*e - 18, -6*e^3 - 28*e^2 + 29*e + 16, 9*e^3 + 44*e^2 - 27*e - 9, -3*e^3 - 21*e^2 - 18*e + 9, 6*e^3 + 26*e^2 - 41*e - 21, 5*e^3 + 29*e^2 + 14*e - 8, 2*e^3 + 14*e^2 + 14*e - 23, 5*e^3 + 23*e^2 - 32*e - 18, -e^2 + 2*e + 9, -4*e^3 - 26*e^2 - 20*e + 4, -4*e^3 - 16*e^2 + 32*e + 4, 2*e^3 + 8*e^2 - 10*e + 1, e^3 + 9*e^2 + 21*e - 1, -15*e^3 - 77*e^2 + 27*e + 19, 15*e^3 + 80*e^2 - 17*e - 32, 10*e^3 + 58*e^2 + 13*e - 38, 21*e^3 + 111*e^2 - 31*e - 36, -5*e^3 - 29*e^2 - 11*e + 25, 3*e^3 + 18*e^2 + 6*e - 21, 6*e^3 + 33*e^2 + e + 6, e^3 + 2*e^2 - 13*e + 14, e^3 + 4*e^2 - 6*e + 11, -4*e^3 - 25*e^2 - 12*e + 28, -2*e^3 - 12*e^2 - 10*e - 8, 17*e^3 + 93*e^2 - 3*e - 38, 2*e^3 + 6*e^2 - 26*e - 1, 6*e^2 + 30*e - 1, -15*e^3 - 80*e^2 + 16*e + 27, -15*e^3 - 83*e^2 + 6*e + 44, 3*e^3 + 15*e^2 - 5*e - 6, -9*e^3 - 54*e^2 - 19*e + 33, 11*e^3 + 56*e^2 - 24*e - 6, 7*e^3 + 40*e^2 + 7*e - 16, -5*e^3 - 34*e^2 - 27*e + 29, -5*e^3 - 24*e^2 + 14*e + 15, -12*e^3 - 65*e^2 + 5*e + 34, 10*e^3 + 50*e^2 - 26*e + 3, -6*e^3 - 34*e^2 + 6*e + 36, -9*e^3 - 44*e^2 + 33*e + 37, e^3 - 36*e - 8, 8*e^3 + 38*e^2 - 40*e - 14, 12*e^3 + 65*e^2 - 7*e - 20, -8*e^3 - 41*e^2 + 24*e + 33, 4*e^3 + 24*e^2 + 5*e - 39, -3*e^2 - 14*e + 8, 6*e^3 + 27*e^2 - 33*e - 38, 8*e^3 + 40*e^2 - 13*e + 15, 9*e^3 + 44*e^2 - 37*e - 32, 9*e^3 + 42*e^2 - 38*e - 6, -11*e^3 - 58*e^2 + 19*e + 12, 3*e^2 + 21*e - 15, -14*e^3 - 76*e^2 + e + 29, -7*e^3 - 33*e^2 + 24*e - 15, 7*e^3 + 45*e^2 + 38*e - 25, 2*e^2 + 13*e - 13, 12*e^3 + 62*e^2 - 24*e - 45, -3*e^3 - 27*e^2 - 47*e + 25, -12*e^3 - 65*e^2 + 25, -2*e^3 - 8*e^2 + 13*e + 24, -14*e^3 - 79*e^2 - 11*e + 40, 6*e^3 + 34*e^2 + 5*e - 12, -16*e^3 - 82*e^2 + 37*e + 23, -13*e^3 - 68*e^2 + 23*e + 30, -4*e^3 - 22*e^2 - 3*e + 23, -4*e^3 - 24*e^2 - 4*e + 42, -21*e^3 - 111*e^2 + 23*e + 49, 13*e^3 + 73*e^2 - e - 49, 7*e^3 + 33*e^2 - 30*e + 13, -3*e^3 - 25*e^2 - 49*e + 20, -11*e^3 - 61*e^2 - 3*e + 15, 13*e^3 + 59*e^2 - 74*e - 32, 5*e^3 + 36*e^2 + 46*e - 36, 13*e^3 + 66*e^2 - 39*e - 40, -3*e^3 - 25*e^2 - 43*e + 11, -13*e^3 - 75*e^2 - 13*e + 48, 7*e^3 + 42*e^2 + 16*e - 55, 25*e^3 + 131*e^2 - 40*e - 45, 16*e^3 + 88*e^2 - 34, 6*e^3 + 29*e^2 - 16*e - 19, -7*e^3 - 32*e^2 + 34*e + 10, e^3 + 3*e^2 - 16*e - 8, -16*e^3 - 87*e^2 + 3*e + 50, -21*e^3 - 109*e^2 + 42*e + 51, 3*e^2 + 26*e + 18, -9*e^3 - 45*e^2 + 28*e + 23, -2*e^3 - 6*e^2 + 18*e - 20, 9*e^3 + 49*e^2 - 10*e - 24, 19*e^3 + 101*e^2 - 30*e - 52, 23*e^3 + 117*e^2 - 52*e - 49, -3*e^3 - 15*e^2 - 3*e - 15, 15*e^3 + 78*e^2 - 27*e - 41, -5*e^3 - 31*e^2 - 11*e + 16, 5*e^3 + 29*e^2 + 6*e + 2, -e^3 - 2*e^2 + 9*e - 34, 4*e^3 + 18*e^2 - 18*e - 5, 7*e^3 + 41*e^2 + 15*e - 33, -4*e^3 - 27*e^2 - 31*e + 11, -e^3 - 14*e^2 - 39*e - 10, 2*e^3 + 20*e^2 + 34*e - 35, -e^2 - 4*e + 32, -5*e^3 - 21*e^2 + 39*e + 22, 14*e^3 + 74*e^2 - 9*e - 34, 6*e^3 + 29*e^2 - 29*e - 17, 5*e^2 + 29*e - 6, 3*e^3 + 24*e^2 + 38*e - 8, -18*e^3 - 99*e^2 + 9*e + 34, -9*e^3 - 47*e^2 + 17*e + 31, 6*e^3 + 39*e^2 + 39*e - 27, -11*e^3 - 59*e^2 + 24*e + 56, 3*e^3 + 12*e^2 - 17*e + 1]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([79, 79, 2/3*w^3 + w^2 - 10/3*w - 19/3])] = -1

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