/*
  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([19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w])

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

hecke_eigenvalues_array = [-1, e, e^3 + 2*e^2 - 4*e - 5, -e^3 + 5*e - 1, -e^3 - 3*e^2 + 4*e + 6, 2*e^3 + e^2 - 9*e + 2, 1, -e^3 + 3*e - 3, -e^3 - e^2 + 3*e - 2, -3*e - 3, -e^3 - e^2 + 6*e + 1, -3*e^3 - 5*e^2 + 11*e + 8, 2*e^3 + 3*e^2 - 5*e - 6, -e^3 - 3*e^2 + 3*e + 10, 3*e^3 + 3*e^2 - 18*e - 3, 4*e^3 + 3*e^2 - 16*e - 2, 4*e^3 + 2*e^2 - 19*e + 1, e^3 + e^2 - 2, 2*e^3 + 2*e^2 - 9*e + 4, -5*e^3 - 6*e^2 + 18*e + 9, e^3 - 10*e + 6, e^3 + 4*e^2 - 3*e - 11, 4*e^3 + 5*e^2 - 17*e - 13, 2*e^3 - 2*e^2 - 12*e + 7, -2*e^3 + e^2 + 16*e - 5, 3*e^3 + 3*e^2 - 10*e, 2*e^3 + 5*e^2 - 5*e - 13, -5*e^3 - 9*e^2 + 23*e + 12, -3*e^3 - e^2 + 14*e - 8, 7*e^3 + 10*e^2 - 24*e - 14, e^3 - 2*e^2 - 12*e + 14, 6*e^3 + 9*e^2 - 24*e - 5, e^2 + 2*e - 22, -4*e^3 + e^2 + 21*e - 9, 4*e^3 + 5*e^2 - 16*e - 17, -e^3 - 10*e^2 - 3*e + 23, -7*e^3 - 5*e^2 + 35*e, -4*e^3 - 5*e^2 + 13*e + 14, -9*e^3 - 11*e^2 + 30*e + 22, -e^3 + 6*e^2 + 8*e - 14, -6*e^3 - 8*e^2 + 27*e + 14, 2*e^2 + e - 6, -2*e^3 - 5*e^2 + 5*e + 3, 5*e^3 + 12*e^2 - 22*e - 24, -13*e^3 - 12*e^2 + 53*e + 2, e^3 - 6*e^2 - 14*e + 13, e^3 + 2*e^2 - 16, 7*e^3 + 2*e^2 - 32*e + 3, -8*e^3 - 10*e^2 + 30*e + 2, 2*e^2 - 3*e - 17, 5*e^3 + 8*e^2 - 16*e - 31, e^3 - 12*e - 9, -2*e^3 - 6*e^2 + 14*e + 18, -e^3 + e^2 - 6*e - 8, -8*e^3 - 9*e^2 + 25*e + 13, 8*e^3 + 3*e^2 - 39*e + 11, e^3 + 1, 5*e^3 + 8*e^2 - 16*e - 19, 8*e^3 + 13*e^2 - 34*e - 9, e^3 + 4*e^2 - 4*e - 27, -7*e^3 - 5*e^2 + 42*e, -2*e^3 - 8*e^2 + 10*e + 11, -8*e^3 - 6*e^2 + 39*e - 3, -5*e^3 - 6*e^2 + 19*e + 15, -7*e^3 - 7*e^2 + 33*e + 8, -6*e^3 - 9*e^2 + 26*e + 9, -10*e^3 - 5*e^2 + 52*e - 8, -5*e^3 + 43*e - 12, -2*e^3 - 5*e^2 + 7*e - 15, 4*e^3 + 5*e^2 - 15*e - 1, -2*e^3 - 5*e^2 + 19*e + 12, 4*e^3 + 7*e^2 - 21*e + 6, -8*e^3 - 7*e^2 + 36*e + 5, -4*e^3 - 6*e^2 + 21*e - 2, 7*e^3 + 13*e^2 - 31*e - 17, 4*e^3 - e^2 - 25*e + 15, 2*e^3 - 2*e^2 - 19*e + 10, -2*e^3 - 5*e^2 - 4*e + 19, -2*e^3 - 2*e^2 + 20*e, -4*e^3 - 5*e^2 + 19*e + 10, -4*e^3 - 4*e^2 + 18*e - 17, 7*e^3 - e^2 - 43*e, e^3 + 6*e^2 + 3*e - 17, 14*e^3 + 7*e^2 - 62*e + 5, -e^3 + 3*e^2 - e - 15, e^3 - 2*e^2 - e + 9, 7*e^3 + e^2 - 37*e - 3, 5*e^3 + 9*e^2 - 19*e - 26, 7*e^3 + 14*e^2 - 21*e - 33, -8*e^3 - 7*e^2 + 30*e + 9, 6*e^3 + 4*e^2 - 23*e + 19, -e^3 + 7*e^2 + 11*e - 7, -6*e^3 + 2*e^2 + 38*e - 5, 9*e^3 + 19*e^2 - 33*e - 23, 3*e^3 + e^2 - 9*e - 19, 5*e^3 - 6*e^2 - 26*e + 33, -e^3 - e^2 - 5*e - 5, 8*e^3 + 10*e^2 - 30*e - 37, 4*e^3 - 2*e^2 - 16*e + 10, 7*e^3 + 13*e^2 - 19*e - 25, -7*e^3 - 9*e^2 + 27*e - 11, -5*e^3 - 13*e^2 + 9*e + 32, 9*e^3 + 17*e^2 - 31*e - 24, -9*e^3 + e^2 + 49*e - 6, -14*e^3 - 22*e^2 + 52*e + 35, -13*e^3 - 17*e^2 + 48*e + 15, -3*e^3 + 5*e^2 + 30*e - 30, 6*e^3 + 10*e^2 - 14*e - 18, 3*e^3 + 12*e^2 - 9*e - 38, 7*e^3 + 8*e^2 - 30*e - 16, -3*e^3 - 3*e^2 + 22*e + 9, -7*e^3 - 6*e^2 + 37*e - 22, -e^3 - 9*e^2 - 10*e + 28, 2*e^3 - 6*e^2 - 25*e + 23, -5*e^3 - 15*e^2 + 12*e + 45, 9*e^3 + 23*e^2 - 35*e - 46, 6*e^3 + 11*e^2 - 15*e - 27, -3*e^3 - 14*e^2 - 4*e + 30, 7*e^3 + 11*e^2 - 13*e - 22, -9*e^3 - 15*e^2 + 37*e + 13, -7*e^3 + 3*e^2 + 38*e - 17, 8*e^3 + e^2 - 48*e + 4, -e^3 - 14*e^2 - 7*e + 31, 13*e^3 + 6*e^2 - 48*e + 24, -e^3 - 10*e^2 + 4*e + 28, 2*e^3 - 6*e^2 - 24*e + 9, 10*e^3 + 15*e^2 - 51*e - 11, -10*e^3 - 8*e^2 + 45*e - 7, 12*e^3 + 10*e^2 - 56*e + 6, 5*e^3 + 6*e^2 - 29*e, 5*e^3 + 5*e^2 - 19*e - 29, 8*e^3 + 10*e^2 - 30*e + 13, -6*e^3 + 4*e^2 + 26*e - 34, 4*e^3 + e^2 - 13*e - 5, 6*e^3 + 22*e^2 - 13*e - 54, -8*e^3 - 9*e^2 + 34*e - 5, -14*e^3 - 13*e^2 + 50*e - 6, 5*e^2 - 8*e - 18, -10*e^3 - e^2 + 49*e - 5, -6*e^3 - 5*e^2 + 33*e - 8, -7*e^3 - 14*e^2 + 28*e + 10, 5*e^3 - 4*e^2 - 16*e + 28, -12*e^3 - 4*e^2 + 53*e - 20, -2*e^3 + 10*e^2 + 17*e - 10, -2*e^3 + 3*e^2 + 21*e - 19, -7*e^3 - 4*e^2 + 43*e - 30, 10*e^3 + 17*e^2 - 36*e - 53, 11*e^3 + 14*e^2 - 42*e - 25, 3*e^3 + 5*e^2 - 25*e - 22, 14*e^3 + 13*e^2 - 58*e - 15, -e^3 - 7*e^2 + 10*e + 18, -19*e^3 - 20*e^2 + 87*e + 6, e^3 + 20*e^2 + 8*e - 48, e^3 + 5*e^2 + 13*e + 4, -12*e^3 - 7*e^2 + 55*e - 15, -e^3 - 6*e^2 - 11*e + 24, 9*e^2 + 7*e - 17, e^3 + 13*e^2 + 15*e - 42, -10*e^3 - 15*e^2 + 20*e + 35, 10*e^3 + 7*e^2 - 51*e - 11, 17*e^3 + 9*e^2 - 68*e + 12, -13*e^3 - 25*e^2 + 56*e + 36, 2*e^3 + 9*e + 8, -9*e^3 - 16*e^2 + 22*e + 47, -2*e^3 + e^2 + 4*e - 44, 2*e^3 + 16*e^2 + 9*e - 40]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w])] = -1

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