/*
  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([25, 5, -11, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([25,5,1/5*w^3 - 1/5*w^2 - 11/5*w + 1])

primes_array = [
[5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7],\
[5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*w + 4],\
[11, 11, w + 1],\
[11, 11, 1/5*w^3 - 1/5*w^2 - 11/5*w + 2],\
[16, 2, 2],\
[19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1],\
[19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w],\
[19, 19, -2/5*w^3 + 2/5*w^2 + 17/5*w],\
[19, 19, w - 1],\
[29, 29, w^2 - w - 8],\
[29, 29, 2/5*w^3 - 2/5*w^2 - 7/5*w + 2],\
[31, 31, 1/5*w^3 - 1/5*w^2 - 1/5*w + 2],\
[31, 31, 2/5*w^3 - 2/5*w^2 - 17/5*w + 3],\
[61, 61, -2/5*w^3 - 3/5*w^2 + 12/5*w + 5],\
[61, 61, -4/5*w^3 - 6/5*w^2 + 39/5*w + 15],\
[61, 61, 3/5*w^3 + 2/5*w^2 - 18/5*w - 3],\
[61, 61, 7/5*w^3 + 3/5*w^2 - 62/5*w - 13],\
[71, 71, 1/5*w^3 - 6/5*w^2 - 6/5*w + 4],\
[71, 71, w^2 - 8],\
[79, 79, 3/5*w^3 + 12/5*w^2 - 33/5*w - 22],\
[79, 79, 1/5*w^3 + 4/5*w^2 - 11/5*w - 9],\
[81, 3, -3],\
[101, 101, 3/5*w^3 + 7/5*w^2 - 28/5*w - 15],\
[101, 101, 1/5*w^3 + 9/5*w^2 - 16/5*w - 16],\
[121, 11, 3/5*w^3 - 3/5*w^2 - 18/5*w + 1],\
[131, 131, 1/5*w^3 + 4/5*w^2 - 16/5*w - 8],\
[131, 131, w^2 - 2*w - 2],\
[149, 149, 4/5*w^3 + 1/5*w^2 - 29/5*w - 5],\
[149, 149, 4/5*w^3 - 9/5*w^2 - 19/5*w + 6],\
[151, 151, 2/5*w^3 + 3/5*w^2 - 17/5*w - 3],\
[151, 151, -2/5*w^3 + 7/5*w^2 + 7/5*w - 8],\
[151, 151, 4/5*w^3 + 1/5*w^2 - 29/5*w - 2],\
[151, 151, -4/5*w^3 + 9/5*w^2 + 19/5*w - 9],\
[179, 179, -3/5*w^3 + 8/5*w^2 + 18/5*w - 6],\
[179, 179, 2/5*w^3 + 3/5*w^2 - 12/5*w - 6],\
[211, 211, 1/5*w^3 + 4/5*w^2 - 6/5*w - 10],\
[211, 211, 2/5*w^3 - 7/5*w^2 - 12/5*w + 2],\
[229, 229, 6/5*w^3 - 11/5*w^2 - 41/5*w + 11],\
[229, 229, 2*w^2 - 2*w - 9],\
[229, 229, -4/5*w^3 - 1/5*w^2 + 19/5*w + 2],\
[229, 229, -3/5*w^3 + 3/5*w^2 + 28/5*w - 3],\
[239, 239, 11/5*w^3 + 4/5*w^2 - 96/5*w - 19],\
[239, 239, -6/5*w^3 + 1/5*w^2 + 51/5*w + 3],\
[241, 241, 1/5*w^3 + 4/5*w^2 - 1/5*w - 6],\
[241, 241, 4/5*w^3 - 4/5*w^2 - 29/5*w - 1],\
[251, 251, -1/5*w^3 + 6/5*w^2 + 11/5*w - 12],\
[251, 251, -1/5*w^3 + 11/5*w^2 + 1/5*w - 15],\
[269, 269, 2/5*w^3 + 3/5*w^2 - 17/5*w - 2],\
[269, 269, -2/5*w^3 + 7/5*w^2 + 7/5*w - 9],\
[271, 271, 1/5*w^3 - 1/5*w^2 + 4/5*w],\
[271, 271, 3/5*w^3 - 3/5*w^2 - 28/5*w + 2],\
[281, 281, w^3 - 3*w^2 - 4*w + 13],\
[281, 281, -3/5*w^3 + 3/5*w^2 + 28/5*w - 6],\
[281, 281, -6/5*w^3 + 26/5*w^2 + 11/5*w - 16],\
[281, 281, w^3 + w^2 - 8*w - 9],\
[289, 17, 2/5*w^3 + 3/5*w^2 - 7/5*w - 5],\
[289, 17, 4/5*w^3 - 9/5*w^2 - 29/5*w + 8],\
[311, 311, -w^2 + 2*w + 11],\
[311, 311, -1/5*w^3 + 6/5*w^2 + 1/5*w - 1],\
[311, 311, 1/5*w^3 + 4/5*w^2 - 16/5*w + 1],\
[311, 311, -1/5*w^3 - 4/5*w^2 + 11/5*w + 10],\
[349, 349, -1/5*w^3 - 9/5*w^2 + 11/5*w + 16],\
[349, 349, 2/5*w^3 - 12/5*w^2 - 7/5*w + 7],\
[359, 359, -4/5*w^3 + 4/5*w^2 + 19/5*w - 1],\
[359, 359, -w^3 + 3*w^2 + 5*w - 12],\
[359, 359, w^3 - w^2 - 7*w + 2],\
[359, 359, 4/5*w^3 + 6/5*w^2 - 29/5*w - 11],\
[379, 379, 4/5*w^3 + 6/5*w^2 - 34/5*w - 9],\
[379, 379, 3/5*w^3 - 3/5*w^2 - 28/5*w + 7],\
[389, 389, w^2 - 3*w - 8],\
[389, 389, -4/5*w^3 + 14/5*w^2 + 19/5*w - 16],\
[401, 401, -3*w^2 + 2*w + 23],\
[401, 401, 1/5*w^3 - 16/5*w^2 + 4/5*w + 11],\
[409, 409, -w^3 + 2*w^2 + 3*w - 7],\
[409, 409, 7/5*w^3 - 2/5*w^2 - 57/5*w - 2],\
[419, 419, -1/5*w^3 + 1/5*w^2 + 16/5*w - 2],\
[419, 419, 1/5*w^3 - 1/5*w^2 - 16/5*w],\
[421, 421, 2/5*w^3 + 3/5*w^2 - 27/5*w - 3],\
[421, 421, 1/5*w^3 + 4/5*w^2 + 4/5*w - 3],\
[439, 439, -w^3 + 3*w^2 + 5*w - 16],\
[439, 439, -4/5*w^3 - 6/5*w^2 + 29/5*w + 7],\
[439, 439, -3/5*w^3 + 13/5*w^2 + 3/5*w - 11],\
[439, 439, 4/5*w^3 + 6/5*w^2 - 39/5*w - 10],\
[449, 449, 2/5*w^3 - 12/5*w^2 - 2/5*w + 5],\
[449, 449, 4/5*w^3 - 24/5*w^2 - 4/5*w + 15],\
[461, 461, -1/5*w^3 - 4/5*w^2 - 4/5*w + 4],\
[461, 461, 2*w^2 - 2*w - 11],\
[461, 461, 3/5*w^3 + 2/5*w^2 - 33/5*w - 4],\
[479, 479, -w^3 + 3*w^2 + 6*w - 14],\
[479, 479, -3/5*w^3 + 3/5*w^2 + 28/5*w + 3],\
[499, 499, 2/5*w^3 + 8/5*w^2 - 17/5*w - 6],\
[499, 499, -3/5*w^3 + 13/5*w^2 + 13/5*w - 17],\
[509, 509, -w^3 + 2*w^2 + 4*w - 6],\
[509, 509, 6/5*w^3 - 1/5*w^2 - 46/5*w - 4],\
[521, 521, -2/5*w^3 + 2/5*w^2 + 7/5*w - 7],\
[521, 521, -3/5*w^3 + 3/5*w^2 + 23/5*w - 8],\
[529, 23, -7/5*w^3 + 17/5*w^2 + 42/5*w - 15],\
[529, 23, 2/5*w^3 + 13/5*w^2 - 27/5*w - 21],\
[541, 541, 1/5*w^3 - 11/5*w^2 - 1/5*w + 10],\
[541, 541, 3/5*w^3 - 8/5*w^2 - 13/5*w + 13],\
[541, 541, -2*w^2 + w + 13],\
[541, 541, 3/5*w^3 + 2/5*w^2 - 23/5*w + 2],\
[569, 569, w^3 - 2*w^2 - 5*w + 7],\
[569, 569, -1/5*w^3 + 1/5*w^2 + 1/5*w - 6],\
[571, 571, w^2 + w - 9],\
[571, 571, -2/5*w^3 + 12/5*w^2 + 7/5*w - 12],\
[571, 571, 1/5*w^3 + 9/5*w^2 - 11/5*w - 11],\
[571, 571, -2/5*w^3 + 7/5*w^2 + 17/5*w - 4],\
[601, 601, -3/5*w^3 + 13/5*w^2 + 18/5*w - 11],\
[601, 601, -4/5*w^3 + 4/5*w^2 + 34/5*w + 1],\
[631, 631, 2/5*w^3 + 18/5*w^2 - 27/5*w - 30],\
[631, 631, 3/5*w^3 - 23/5*w^2 - 3/5*w + 15],\
[661, 661, -2*w^3 + 17*w + 12],\
[661, 661, -9/5*w^3 + 4/5*w^2 + 74/5*w + 1],\
[691, 691, -7/5*w^3 - 18/5*w^2 + 72/5*w + 35],\
[691, 691, 6/5*w^3 - 31/5*w^2 - 6/5*w + 19],\
[701, 701, -1/5*w^3 + 6/5*w^2 + 16/5*w - 6],\
[701, 701, 2/5*w^3 - 7/5*w^2 - 22/5*w + 8],\
[709, 709, -3/5*w^3 + 13/5*w^2 + 13/5*w - 18],\
[709, 709, 1/5*w^3 + 14/5*w^2 - 11/5*w - 20],\
[709, 709, -2/5*w^3 - 8/5*w^2 + 17/5*w + 5],\
[709, 709, 1/5*w^3 + 4/5*w^2 - 21/5*w - 10],\
[719, 719, -3/5*w^3 + 8/5*w^2 + 13/5*w - 12],\
[719, 719, 3/5*w^3 + 2/5*w^2 - 23/5*w + 1],\
[719, 719, -7/5*w^3 + 17/5*w^2 + 32/5*w - 14],\
[719, 719, 7/5*w^3 + 3/5*w^2 - 52/5*w - 8],\
[739, 739, 1/5*w^3 - 1/5*w^2 - 11/5*w - 5],\
[739, 739, w - 6],\
[761, 761, -4/5*w^3 + 9/5*w^2 + 19/5*w - 11],\
[761, 761, 4/5*w^3 + 1/5*w^2 - 29/5*w],\
[769, 769, 3/5*w^3 - 13/5*w^2 - 13/5*w + 8],\
[769, 769, -2/5*w^3 - 8/5*w^2 + 17/5*w + 15],\
[769, 769, 7/5*w^3 - 17/5*w^2 - 37/5*w + 18],\
[769, 769, -6/5*w^3 - 4/5*w^2 + 41/5*w + 5],\
[809, 809, -1/5*w^3 + 6/5*w^2 - 4/5*w - 14],\
[809, 809, -2/5*w^3 - 3/5*w^2 + 22/5*w - 4],\
[809, 809, -6/5*w^3 + 6/5*w^2 + 41/5*w - 5],\
[809, 809, w^3 - w^2 - 5*w + 4],\
[821, 821, -3/5*w^3 - 17/5*w^2 + 33/5*w + 30],\
[821, 821, -4/5*w^3 + 24/5*w^2 + 9/5*w - 15],\
[829, 829, 7/5*w^3 - 12/5*w^2 - 52/5*w + 10],\
[829, 829, -1/5*w^3 + 11/5*w^2 - 4/5*w - 19],\
[829, 829, -13/5*w^3 + 3/5*w^2 + 108/5*w + 10],\
[829, 829, -w^3 + 4*w^2 + 5*w - 23],\
[839, 839, 4/5*w^3 - 9/5*w^2 - 24/5*w + 4],\
[839, 839, 3/5*w^3 + 2/5*w^2 - 18/5*w - 8],\
[841, 29, -w^3 + w^2 + 6*w - 4],\
[859, 859, 6/5*w^3 - 11/5*w^2 - 36/5*w + 9],\
[859, 859, w^3 - 6*w - 3],\
[881, 881, 4/5*w^3 - 4/5*w^2 - 9/5*w - 1],\
[881, 881, 9/5*w^3 - 24/5*w^2 - 19/5*w + 14],\
[881, 881, -w^3 + 6*w + 2],\
[881, 881, 1/5*w^3 - 1/5*w^2 - 1/5*w - 6],\
[911, 911, w^3 - w^2 - 5*w + 1],\
[911, 911, 6/5*w^3 - 6/5*w^2 - 41/5*w + 2],\
[919, 919, -7/5*w^3 + 12/5*w^2 + 42/5*w - 7],\
[919, 919, -6/5*w^3 + 1/5*w^2 + 36/5*w + 5],\
[929, 929, -2/5*w^3 - 8/5*w^2 + 17/5*w + 16],\
[929, 929, -3/5*w^3 + 13/5*w^2 + 13/5*w - 7],\
[961, 31, w^3 - w^2 - 6*w + 2],\
[971, 971, 8/5*w^3 - 13/5*w^2 - 48/5*w + 6],\
[971, 971, -2/5*w^3 - 3/5*w^2 + 22/5*w],\
[971, 971, 7/5*w^3 - 7/5*w^2 - 57/5*w + 7],\
[971, 971, -1/5*w^3 + 6/5*w^2 - 4/5*w - 10],\
[991, 991, 2/5*w^3 + 8/5*w^2 - 22/5*w - 7],\
[991, 991, -2/5*w^3 + 12/5*w^2 + 2/5*w - 15]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 + x^3 - 14*x^2 - x + 33
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 0, -1/2*e^3 - e^2 + 4*e + 9/2, -1/2*e^3 - e^2 + 4*e + 9/2, -1/2*e^3 - e^2 + 4*e - 1/2, -1/2*e^3 - 2*e^2 + 3*e + 19/2, 1/2*e^3 + 2*e^2 - 3*e - 19/2, 1/2*e^3 + 2*e^2 - 3*e - 19/2, -1/2*e^3 - 2*e^2 + 3*e + 19/2, -e^2 + 6, e^2 - 6, e^2 + e - 5, e^2 + e - 5, e + 1, e + 1, -1/2*e^3 - e^2 + 5*e + 7/2, -1/2*e^3 - e^2 + 5*e + 7/2, 2*e^2 + 2*e - 12, 2*e^2 + 2*e - 12, 1/2*e^3 - 3*e + 5/2, -1/2*e^3 + 3*e - 5/2, e^3 + 5*e^2 - 5*e - 34, -1/2*e^3 - 3*e^2 - e + 33/2, -1/2*e^3 - 3*e^2 - e + 33/2, -1/2*e^3 + 8*e + 19/2, e^3 + 3*e^2 - 5*e - 12, e^3 + 3*e^2 - 5*e - 12, -5/2*e^3 - 7*e^2 + 15*e + 69/2, 5/2*e^3 + 7*e^2 - 15*e - 69/2, 2*e^3 + 5*e^2 - 15*e - 25, 2*e^3 + 5*e^2 - 15*e - 25, 2*e^3 + 5*e^2 - 17*e - 13, 2*e^3 + 5*e^2 - 17*e - 13, e^3 + e^2 - 11*e - 6, -e^3 - e^2 + 11*e + 6, 1/2*e^3 - e^2 - 8*e + 31/2, 1/2*e^3 - e^2 - 8*e + 31/2, -2*e^3 - 7*e^2 + 12*e + 37, 2*e^3 + 4*e^2 - 17*e - 14, 2*e^3 + 7*e^2 - 12*e - 37, -2*e^3 - 4*e^2 + 17*e + 14, -1/2*e^3 - 5*e^2 - 2*e + 45/2, 1/2*e^3 + 5*e^2 + 2*e - 45/2, 3/2*e^3 + 4*e^2 - 18*e - 31/2, 3/2*e^3 + 4*e^2 - 18*e - 31/2, -3/2*e^3 - 4*e^2 + 15*e + 45/2, -3/2*e^3 - 4*e^2 + 15*e + 45/2, -3/2*e^3 - e^2 + 19*e - 3/2, 3/2*e^3 + e^2 - 19*e + 3/2, e^3 + 3*e^2 - 9*e + 2, e^3 + 3*e^2 - 9*e + 2, e^3 + 3*e^2 - 6*e - 6, 3*e^2 + 6*e - 12, 3*e^2 + 6*e - 12, e^3 + 3*e^2 - 6*e - 6, e^3 + 3*e^2 - 6*e + 2, -e^3 - 3*e^2 + 6*e - 2, e^3 - e^2 - 15*e + 12, -3/2*e^3 - 6*e^2 + 15*e + 69/2, e^3 - e^2 - 15*e + 12, -3/2*e^3 - 6*e^2 + 15*e + 69/2, -7/2*e^3 - 8*e^2 + 26*e + 61/2, 7/2*e^3 + 8*e^2 - 26*e - 61/2, e^3 + 5*e^2 - e - 30, -1/2*e^3 - e^2 - 2*e - 3/2, -e^3 - 5*e^2 + e + 30, 1/2*e^3 + e^2 + 2*e + 3/2, 2*e^3 + 9*e^2 - 7*e - 59, -2*e^3 - 9*e^2 + 7*e + 59, 1/2*e^3 + e^2 - 3*e + 3/2, -1/2*e^3 - e^2 + 3*e - 3/2, -e^2 + 6*e + 12, -e^2 + 6*e + 12, 1/2*e^3 + 2*e^2 - 8*e - 49/2, -1/2*e^3 - 2*e^2 + 8*e + 49/2, e^3 + 7*e^2 - e - 42, -e^3 - 7*e^2 + e + 42, 1/2*e^3 + 2*e^2 - 2*e - 7/2, 1/2*e^3 + 2*e^2 - 2*e - 7/2, -7/2*e^3 - 6*e^2 + 31*e + 37/2, 7/2*e^3 + 6*e^2 - 31*e - 37/2, 2*e^3 + 7*e^2 - 7*e - 47, -2*e^3 - 7*e^2 + 7*e + 47, -2*e^3 - 5*e^2 + 22*e + 30, 2*e^3 + 5*e^2 - 22*e - 30, -2*e^3 - 6*e^2 + 7*e + 33, 2*e^3 + 2*e^2 - 10*e + 18, -2*e^3 - 6*e^2 + 7*e + 33, 2*e^3 + 7*e^2 - 17*e - 27, -2*e^3 - 7*e^2 + 17*e + 27, -e^3 - e^2 + 11*e - 4, e^3 + e^2 - 11*e + 4, -1/2*e^3 + 2*e^2 + 8*e - 39/2, 1/2*e^3 - 2*e^2 - 8*e + 39/2, 2*e^3 + 6*e^2 - 15*e - 36, 2*e^3 + 6*e^2 - 15*e - 36, -3/2*e^3 - 2*e^2 + 9*e - 31/2, 3/2*e^3 + 2*e^2 - 9*e + 31/2, -7/2*e^3 - 12*e^2 + 22*e + 131/2, -2*e^3 - 5*e^2 + 18*e + 31, -7/2*e^3 - 12*e^2 + 22*e + 131/2, -2*e^3 - 5*e^2 + 18*e + 31, 7/2*e^3 + 12*e^2 - 26*e - 99/2, -7/2*e^3 - 12*e^2 + 26*e + 99/2, -5/2*e^3 - 7*e^2 + 16*e + 41/2, -2*e^3 - 5*e^2 + 19*e + 35, -2*e^3 - 5*e^2 + 19*e + 35, -5/2*e^3 - 7*e^2 + 16*e + 41/2, 3*e^3 + 10*e^2 - 17*e - 59, 3*e^3 + 10*e^2 - 17*e - 59, 11/2*e^3 + 17*e^2 - 38*e - 155/2, 11/2*e^3 + 17*e^2 - 38*e - 155/2, -e^3 - e^2 + 2*e - 8, -e^3 - e^2 + 2*e - 8, -4*e^3 - 11*e^2 + 31*e + 41, -4*e^3 - 11*e^2 + 31*e + 41, -e^3 - 6*e^2 + 8*e + 51, -e^3 - 6*e^2 + 8*e + 51, 3/2*e^3 + 9*e^2 - 9*e - 113/2, 9/2*e^3 + 14*e^2 - 32*e - 113/2, -3/2*e^3 - 9*e^2 + 9*e + 113/2, -9/2*e^3 - 14*e^2 + 32*e + 113/2, -1/2*e^3 - 4*e^2 + 3*e + 63/2, 1/2*e^3 + 4*e^2 - 3*e - 63/2, -2*e^3 - 10*e^2 + 2*e + 60, 2*e^3 + 10*e^2 - 2*e - 60, -5/2*e^3 - e^2 + 20*e - 23/2, 5/2*e^3 + e^2 - 20*e + 23/2, 9/2*e^3 + 14*e^2 - 36*e - 111/2, 9/2*e^3 + 14*e^2 - 36*e - 111/2, -e^3 + 21*e + 10, e^3 - 21*e - 10, e^3 + 2*e^2 - e - 17, -e^3 - 2*e^2 + e + 17, 5*e^3 + 13*e^2 - 40*e - 48, -5*e^3 - 13*e^2 + 40*e + 48, 7/2*e^3 + 5*e^2 - 31*e - 15/2, -7/2*e^3 - 5*e^2 + 31*e + 15/2, -2*e^3 - 4*e^2 + 28*e, -2*e^3 - 4*e^2 + 28*e, e^3 + 3*e^2 - 16*e - 23, e^3 + e^2 - 21*e + 14, -e^3 - e^2 + 21*e - 14, -e^3 - 3*e^2 + 16*e + 23, -5*e^3 - 11*e^2 + 35*e + 36, 5*e^3 + 11*e^2 - 35*e - 36, 5/2*e^3 + 11*e^2 - 14*e - 145/2, e^3 + 3*e^2 - 21*e - 28, -e^3 - 3*e^2 + 21*e + 28, -5/2*e^3 - 7*e^2 + 15*e + 63/2, -5/2*e^3 - 7*e^2 + 15*e + 63/2, -5/2*e^3 - 6*e^2 + 12*e + 27/2, -5/2*e^3 - 6*e^2 + 12*e + 27/2, 2*e^3 + 7*e^2 - 21*e - 21, 2*e^3 + 7*e^2 - 21*e - 21, -e^3 + 16*e + 5, e^3 - 16*e - 5, -11/2*e^3 - 16*e^2 + 43*e + 117/2, 11/2*e^3 + 16*e^2 - 43*e - 117/2, 4*e^3 + 7*e^2 - 24*e - 10, -4*e^3 - 14*e^2 + 34*e + 66, 3*e^3 + 13*e^2 - 11*e - 78, -4*e^3 - 14*e^2 + 34*e + 66, 3*e^3 + 13*e^2 - 11*e - 78, 1/2*e^3 - 3*e^2 - 12*e + 23/2, 1/2*e^3 - 3*e^2 - 12*e + 23/2]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([5,5,1/5*w^3 - 6/5*w^2 - 1/5*w + 4])] = 1

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