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

NN = ZF.ideal([16, 4, 3*w^3 - 5*w^2 - 15*w + 6])

primes_array = [
[2, 2, w^3 - w^2 - 6*w - 2],\
[3, 3, -w^3 + 2*w^2 + 4*w - 2],\
[4, 2, w^3 - w^2 - 5*w - 2],\
[17, 17, -w^3 + w^2 + 6*w - 1],\
[17, 17, -w + 2],\
[29, 29, -w^3 + 2*w^2 + 4*w - 4],\
[29, 29, w^3 - 2*w^2 - 4*w],\
[31, 31, -w^2 + 2],\
[31, 31, -w^3 + 7*w + 5],\
[41, 41, -2*w^3 + 2*w^2 + 13*w - 2],\
[41, 41, 3*w^3 - 4*w^2 - 17*w + 5],\
[67, 67, 3*w^3 - 4*w^2 - 17*w + 1],\
[67, 67, -w^2 + 4*w + 2],\
[83, 83, 2*w^2 - 5*w - 2],\
[83, 83, -w^3 + 8*w + 6],\
[83, 83, w^3 - 2*w^2 - 2*w - 2],\
[83, 83, w^3 - 2*w^2 - 6*w],\
[97, 97, -3*w^3 + 2*w^2 + 17*w + 7],\
[97, 97, w^3 - 4*w^2 + 3*w + 1],\
[97, 97, -5*w^3 + 8*w^2 + 24*w - 8],\
[97, 97, 2*w^3 - 2*w^2 - 13*w - 4],\
[101, 101, -w^3 + 6*w + 6],\
[101, 101, -w^3 + 6*w + 2],\
[103, 103, -4*w^3 + 3*w^2 + 26*w + 12],\
[103, 103, -w^3 + 2*w^2 + 9*w + 3],\
[107, 107, 2*w^3 - 2*w^2 - 11*w],\
[107, 107, -w^3 + 2*w^2 + 3*w - 3],\
[107, 107, w^3 - w^2 - 4*w - 3],\
[107, 107, 2*w^3 - 3*w^2 - 10*w],\
[121, 11, -2*w^3 + 3*w^2 + 8*w + 2],\
[149, 149, -3*w^3 + 5*w^2 + 14*w - 3],\
[149, 149, -2*w^3 + 4*w^2 + 7*w - 4],\
[157, 157, 2*w^3 - 2*w^2 - 10*w - 1],\
[157, 157, 2*w^3 - 3*w^2 - 8*w + 2],\
[157, 157, 2*w^3 - 2*w^2 - 10*w - 3],\
[157, 157, -3*w^3 + 4*w^2 + 15*w + 1],\
[163, 163, -2*w^2 + 4*w + 5],\
[163, 163, 2*w^3 - 4*w^2 - 10*w + 5],\
[169, 13, 2*w^3 - 3*w^2 - 12*w + 2],\
[169, 13, -w^3 + 9*w + 3],\
[173, 173, 3*w^3 - 2*w^2 - 17*w - 9],\
[173, 173, -2*w^3 + 14*w + 7],\
[197, 197, 2*w^3 - 2*w^2 - 12*w + 1],\
[197, 197, 3*w^3 - 3*w^2 - 18*w - 1],\
[199, 199, 3*w^3 - 4*w^2 - 18*w + 8],\
[199, 199, 4*w^3 - 5*w^2 - 24*w + 6],\
[223, 223, w^3 - 2*w^2 - 3*w - 3],\
[223, 223, 2*w^3 - 3*w^2 - 10*w + 6],\
[227, 227, -w^3 + 2*w^2 + 6*w - 8],\
[227, 227, 3*w + 4],\
[227, 227, 3*w^3 - 3*w^2 - 18*w - 7],\
[227, 227, -2*w^3 + 3*w^2 + 12*w - 6],\
[229, 229, -w - 4],\
[229, 229, w^2 - 2*w + 2],\
[229, 229, -w^3 + w^2 + 6*w + 5],\
[229, 229, -w^3 + 2*w^2 + 5*w - 7],\
[233, 233, -2*w^3 + 3*w^2 + 8*w - 4],\
[233, 233, -3*w^3 + 4*w^2 + 15*w + 3],\
[281, 281, -2*w^3 + 3*w^2 + 10*w + 2],\
[281, 281, -w^3 + 2*w^2 + 3*w - 5],\
[289, 17, -2*w^3 + 2*w^2 + 14*w + 5],\
[293, 293, w^2 - 6],\
[293, 293, w^3 - 7*w - 1],\
[331, 331, -6*w^3 + 10*w^2 + 27*w - 6],\
[331, 331, 2*w^3 - 2*w^2 - 12*w - 7],\
[347, 347, 3*w^3 - 2*w^2 - 18*w - 6],\
[347, 347, -4*w^3 + 2*w^2 + 23*w + 10],\
[347, 347, 2*w^3 - 6*w^2 - w + 4],\
[347, 347, -4*w^3 + 3*w^2 + 24*w + 8],\
[359, 359, -3*w^3 + 4*w^2 + 17*w + 1],\
[359, 359, w^2 - 4*w - 4],\
[359, 359, w^3 - 8*w],\
[359, 359, -w^3 + 2*w^2 + 6*w - 6],\
[367, 367, 2*w^2 - 3*w - 4],\
[367, 367, -w^3 + 3*w^2 + 4*w - 7],\
[379, 379, 3*w^3 - 3*w^2 - 16*w - 3],\
[379, 379, 2*w^3 - 2*w^2 - 9*w - 2],\
[461, 461, -2*w^3 + 5*w^2 + 2*w + 2],\
[461, 461, 4*w^3 - 4*w^2 - 26*w - 11],\
[463, 463, -2*w^3 + 13*w + 8],\
[463, 463, -w^3 - w^2 + 6*w + 7],\
[487, 487, 2*w^3 - 17*w - 8],\
[487, 487, -w^3 + 10*w + 4],\
[491, 491, -5*w^3 + 3*w^2 + 30*w + 13],\
[491, 491, -5*w^3 + 4*w^2 + 28*w + 14],\
[491, 491, 3*w^3 - 8*w^2 - 4*w + 4],\
[491, 491, 7*w^3 - 12*w^2 - 33*w + 15],\
[499, 499, -2*w^3 + 2*w^2 + 12*w - 5],\
[499, 499, 2*w^2 - 7*w],\
[529, 23, -w^3 + 3*w^2 + 4*w - 5],\
[529, 23, 2*w^2 - 3*w - 6],\
[557, 557, -2*w^3 + 3*w^2 + 12*w + 4],\
[557, 557, w^3 - 9*w - 9],\
[569, 569, 4*w^3 - 5*w^2 - 22*w + 4],\
[569, 569, -3*w^3 + 2*w^2 + 20*w + 4],\
[577, 577, 3*w^3 - 3*w^2 - 20*w + 3],\
[577, 577, -4*w^3 + 8*w^2 + 16*w - 9],\
[577, 577, 7*w^3 - 11*w^2 - 34*w + 9],\
[577, 577, -4*w^3 + 6*w^2 + 22*w - 11],\
[593, 593, -6*w^3 + 10*w^2 + 29*w - 14],\
[593, 593, -w^3 + 4*w^2 - w - 7],\
[619, 619, -3*w^3 + 5*w^2 + 12*w - 1],\
[619, 619, -4*w^3 + 8*w^2 + 18*w - 15],\
[625, 5, -5],\
[631, 631, -3*w^3 + 8*w^2 + 6*w - 4],\
[631, 631, -2*w^3 + 8*w^2 - 3*w - 8],\
[643, 643, 3*w^3 - 4*w^2 - 14*w + 2],\
[643, 643, -3*w^3 + 4*w^2 + 14*w + 2],\
[661, 661, w^3 - 8*w - 10],\
[661, 661, -3*w^3 + 2*w^2 + 21*w + 3],\
[661, 661, w^3 - 2*w^2 - 6*w - 4],\
[661, 661, 2*w^3 - 3*w^2 - 14*w + 4],\
[677, 677, -9*w^3 + 12*w^2 + 50*w - 10],\
[677, 677, 3*w^3 - 4*w^2 - 17*w - 7],\
[691, 691, 3*w^3 - 2*w^2 - 21*w - 9],\
[691, 691, 2*w^3 - 3*w^2 - 14*w - 2],\
[701, 701, -9*w^3 + 13*w^2 + 48*w - 15],\
[701, 701, 2*w^3 - 6*w^2 + w - 2],\
[709, 709, 2*w^2 - 2*w - 5],\
[709, 709, 2*w^2 - 2*w - 7],\
[709, 709, 4*w^3 - 5*w^2 - 20*w - 4],\
[709, 709, -3*w^3 + 6*w^2 + 14*w - 10],\
[727, 727, 7*w^3 - 10*w^2 - 36*w + 8],\
[727, 727, 3*w^2 - 6*w - 8],\
[743, 743, 9*w^3 - 12*w^2 - 51*w + 13],\
[743, 743, w^3 - 2*w^2 - w - 5],\
[743, 743, -4*w^3 + 4*w^2 + 27*w - 8],\
[743, 743, -3*w^3 + 4*w^2 + 21*w + 9],\
[751, 751, -w^3 + 10*w + 2],\
[751, 751, 3*w^3 - 4*w^2 - 18*w + 2],\
[761, 761, 4*w^3 - 4*w^2 - 23*w - 2],\
[761, 761, -3*w^3 + 2*w^2 + 19*w + 5],\
[809, 809, -3*w^3 + 5*w^2 + 10*w + 5],\
[809, 809, -w^3 - w^2 + 10*w + 3],\
[821, 821, -w^3 + 3*w^2 + 2*w - 9],\
[821, 821, 2*w^2 - 4*w - 9],\
[823, 823, w^3 - 6*w^2 + 9*w + 1],\
[823, 823, -3*w^3 + 2*w^2 + 15*w + 5],\
[827, 827, 3*w^3 - 3*w^2 - 18*w + 1],\
[827, 827, 2*w^3 - 16*w - 11],\
[827, 827, 3*w^3 - 24*w - 14],\
[827, 827, 3*w - 4],\
[829, 829, w^2 - 6*w - 2],\
[829, 829, 2*w^3 - 2*w^2 - 10*w - 9],\
[829, 829, 5*w^3 - 6*w^2 - 29*w - 1],\
[829, 829, w^3 + 2*w^2 - 14*w - 6],\
[841, 29, -2*w^3 + 2*w^2 + 14*w + 3],\
[857, 857, -3*w^3 + 6*w^2 + 11*w - 7],\
[857, 857, -4*w^3 + 7*w^2 + 18*w - 4],\
[859, 859, -w^3 + w^2 + 6*w - 5],\
[859, 859, w - 6],\
[883, 883, 5*w^3 - 5*w^2 - 26*w - 11],\
[883, 883, -3*w^3 + 3*w^2 + 14*w + 5],\
[887, 887, 5*w^3 - 2*w^2 - 34*w - 16],\
[887, 887, -w^3 + 4*w^2 - 10],\
[887, 887, 4*w^3 - 2*w^2 - 26*w - 11],\
[887, 887, 2*w^3 - 8*w^2 + 4*w + 7],\
[907, 907, 6*w^3 - 8*w^2 - 34*w + 7],\
[907, 907, -w^3 + w^2 + 8*w - 7],\
[941, 941, 2*w^3 - 2*w^2 - 8*w - 5],\
[941, 941, -3*w^3 + 23*w + 11],\
[953, 953, 2*w^3 - 10*w - 3],\
[953, 953, -4*w^3 + 2*w^2 + 24*w + 15],\
[961, 31, -2*w^3 + 2*w^2 + 14*w - 7],\
[991, 991, -3*w^3 + 4*w^2 + 13*w + 3],\
[991, 991, 4*w^3 - 5*w^2 - 20*w + 2]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - 13*x^4 + 46*x^2 - 32
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 1/2*e^4 - 7/2*e^2 + 2, 0, -1/4*e^5 + 5/4*e^3 + 5/2*e, -1/4*e^5 + 5/4*e^3 + 5/2*e, -3/4*e^5 + 31/4*e^3 - 33/2*e, -3/4*e^5 + 31/4*e^3 - 33/2*e, 1/2*e^4 - 11/2*e^2 + 12, 1/2*e^4 - 11/2*e^2 + 12, 1/4*e^5 - 13/4*e^3 + 15/2*e, 1/4*e^5 - 13/4*e^3 + 15/2*e, -1/2*e^4 + 7/2*e^2 - 6, -1/2*e^4 + 7/2*e^2 - 6, -e^5 + 7*e^3 - 4*e, 1/2*e^5 - 9/2*e^3 + 9*e, -e^5 + 7*e^3 - 4*e, 1/2*e^5 - 9/2*e^3 + 9*e, 5/2*e^4 - 43/2*e^2 + 28, 5/2*e^4 - 43/2*e^2 + 28, 1/2*e^4 - 3/2*e^2 - 2, 1/2*e^4 - 3/2*e^2 - 2, 5/4*e^5 - 49/4*e^3 + 59/2*e, 5/4*e^5 - 49/4*e^3 + 59/2*e, e^4 - 9*e^2 + 18, e^4 - 9*e^2 + 18, -e^5 + 11*e^3 - 24*e, -1/2*e^5 + 13/2*e^3 - 23*e, -e^5 + 11*e^3 - 24*e, -1/2*e^5 + 13/2*e^3 - 23*e, -e^4 + 5*e^2 + 16, -1/4*e^5 + 21/4*e^3 - 35/2*e, -1/4*e^5 + 21/4*e^3 - 35/2*e, -1/2*e^4 + 7/2*e^2 - 12, -1/2*e^4 + 7/2*e^2 - 12, -1/2*e^4 + 7/2*e^2 - 12, -1/2*e^4 + 7/2*e^2 - 12, -4*e^2 + 16, -4*e^2 + 16, 3*e^4 - 23*e^2 + 24, 3*e^4 - 23*e^2 + 24, -1/4*e^5 - 3/4*e^3 + 41/2*e, -1/4*e^5 - 3/4*e^3 + 41/2*e, -1/4*e^5 - 3/4*e^3 + 17/2*e, -1/4*e^5 - 3/4*e^3 + 17/2*e, -e^4 + 13*e^2 - 26, -e^4 + 13*e^2 - 26, -5/2*e^4 + 31/2*e^2 - 4, -5/2*e^4 + 31/2*e^2 - 4, -1/2*e^5 + 9/2*e^3 - e, 4*e^3 - 28*e, 4*e^3 - 28*e, -1/2*e^5 + 9/2*e^3 - e, -7/2*e^4 + 57/2*e^2 - 40, 1/2*e^4 + 5/2*e^2 - 26, -7/2*e^4 + 57/2*e^2 - 40, 1/2*e^4 + 5/2*e^2 - 26, -1/4*e^5 + 13/4*e^3 - 31/2*e, -1/4*e^5 + 13/4*e^3 - 31/2*e, -3/4*e^5 + 23/4*e^3 - 13/2*e, -3/4*e^5 + 23/4*e^3 - 13/2*e, 3*e^4 - 23*e^2 + 28, 3/4*e^5 - 31/4*e^3 + 33/2*e, 3/4*e^5 - 31/4*e^3 + 33/2*e, -1/2*e^4 - 5/2*e^2 + 32, -1/2*e^4 - 5/2*e^2 + 32, -2*e^5 + 20*e^3 - 34*e, 1/2*e^5 - 21/2*e^3 + 43*e, 1/2*e^5 - 21/2*e^3 + 43*e, -2*e^5 + 20*e^3 - 34*e, 5/2*e^5 - 49/2*e^3 + 59*e, 5/2*e^5 - 49/2*e^3 + 59*e, -3/2*e^5 + 27/2*e^3 - 19*e, -3/2*e^5 + 27/2*e^3 - 19*e, 3/2*e^4 - 29/2*e^2 + 18, 3/2*e^4 - 29/2*e^2 + 18, -5/2*e^4 + 39/2*e^2 - 16, -5/2*e^4 + 39/2*e^2 - 16, 5/4*e^5 - 41/4*e^3 + 23/2*e, 5/4*e^5 - 41/4*e^3 + 23/2*e, 1/2*e^4 - 23/2*e^2 + 38, 1/2*e^4 - 23/2*e^2 + 38, 9/2*e^4 - 91/2*e^2 + 76, 9/2*e^4 - 91/2*e^2 + 76, 2*e^5 - 20*e^3 + 42*e, -1/2*e^5 + 9/2*e^3 - e, 2*e^5 - 20*e^3 + 42*e, -1/2*e^5 + 9/2*e^3 - e, 4*e^2 - 24, 4*e^2 - 24, 3/2*e^4 - 21/2*e^2 + 16, 3/2*e^4 - 21/2*e^2 + 16, -7/4*e^5 + 59/4*e^3 - 57/2*e, -7/4*e^5 + 59/4*e^3 - 57/2*e, -7/4*e^5 + 83/4*e^3 - 117/2*e, -7/4*e^5 + 83/4*e^3 - 117/2*e, -1/2*e^4 + 3/2*e^2 + 2, -5/2*e^4 + 39/2*e^2 - 2, -5/2*e^4 + 39/2*e^2 - 2, -1/2*e^4 + 3/2*e^2 + 2, 5/4*e^5 - 49/4*e^3 + 27/2*e, 5/4*e^5 - 49/4*e^3 + 27/2*e, -5/2*e^4 + 47/2*e^2 - 40, -5/2*e^4 + 47/2*e^2 - 40, -9/2*e^4 + 59/2*e^2 + 10, -3/2*e^4 + 13/2*e^2 + 18, -3/2*e^4 + 13/2*e^2 + 18, -3/2*e^4 + 25/2*e^2 - 8, -3/2*e^4 + 25/2*e^2 - 8, -5/2*e^4 + 51/2*e^2 - 36, -5/2*e^4 + 35/2*e^2 - 12, -5/2*e^4 + 51/2*e^2 - 36, -5/2*e^4 + 35/2*e^2 - 12, -5/4*e^5 + 57/4*e^3 - 55/2*e, -5/4*e^5 + 57/4*e^3 - 55/2*e, -7/2*e^4 + 57/2*e^2 - 62, -7/2*e^4 + 57/2*e^2 - 62, 5/4*e^5 - 33/4*e^3 - 5/2*e, 5/4*e^5 - 33/4*e^3 - 5/2*e, 9/2*e^4 - 83/2*e^2 + 86, 9/2*e^4 - 83/2*e^2 + 86, 1/2*e^4 - 15/2*e^2 + 16, 1/2*e^4 - 15/2*e^2 + 16, 9/2*e^4 - 59/2*e^2 + 20, 9/2*e^4 - 59/2*e^2 + 20, -2*e^5 + 24*e^3 - 70*e, 4*e^5 - 38*e^3 + 78*e, -2*e^5 + 24*e^3 - 70*e, 4*e^5 - 38*e^3 + 78*e, 11/2*e^4 - 105/2*e^2 + 76, 11/2*e^4 - 105/2*e^2 + 76, 11/4*e^5 - 103/4*e^3 + 97/2*e, 11/4*e^5 - 103/4*e^3 + 97/2*e, 3/4*e^5 - 7/4*e^3 - 51/2*e, 3/4*e^5 - 7/4*e^3 - 51/2*e, 7/4*e^5 - 83/4*e^3 + 117/2*e, 7/4*e^5 - 83/4*e^3 + 117/2*e, 7/2*e^4 - 41/2*e^2 - 22, 7/2*e^4 - 41/2*e^2 - 22, 2*e^5 - 16*e^3 + 22*e, 3/2*e^5 - 23/2*e^3 + 5*e, 3/2*e^5 - 23/2*e^3 + 5*e, 2*e^5 - 16*e^3 + 22*e, 9/2*e^4 - 87/2*e^2 + 80, -7/2*e^4 + 61/2*e^2 - 18, 9/2*e^4 - 87/2*e^2 + 80, -7/2*e^4 + 61/2*e^2 - 18, -e^4 + 21*e^2 - 44, 15/4*e^5 - 139/4*e^3 + 125/2*e, 15/4*e^5 - 139/4*e^3 + 125/2*e, 13/2*e^4 - 83/2*e^2 + 14, 13/2*e^4 - 83/2*e^2 + 14, 4*e^4 - 20*e^2 - 24, 4*e^4 - 20*e^2 - 24, 1/2*e^5 - 25/2*e^3 + 57*e, -1/2*e^5 + 17/2*e^3 - 45*e, 1/2*e^5 - 25/2*e^3 + 57*e, -1/2*e^5 + 17/2*e^3 - 45*e, 3*e^4 - 31*e^2 + 50, 3*e^4 - 31*e^2 + 50, -3/4*e^5 + 39/4*e^3 - 53/2*e, -3/4*e^5 + 39/4*e^3 - 53/2*e, 11/4*e^5 - 71/4*e^3 - 7/2*e, 11/4*e^5 - 71/4*e^3 - 7/2*e, 7/2*e^4 - 61/2*e^2 + 86, -9*e^4 + 81*e^2 - 122, -9*e^4 + 81*e^2 - 122]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

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

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