/*
  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([-8, -14, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([2,2,w + 3])

primes_array = [
[2, 2, -1/2*w^2 + 1/2*w + 8],\
[2, 2, 1/2*w^2 - 3/2*w - 1],\
[2, 2, w + 3],\
[11, 11, -w^2 + 3*w + 7],\
[11, 11, -2*w - 1],\
[11, 11, -w^2 + w + 11],\
[27, 3, 3],\
[41, 41, 2*w + 5],\
[41, 41, -w^2 + 3*w + 3],\
[41, 41, -w^2 + w + 15],\
[43, 43, -3*w^2 + 11*w + 11],\
[47, 47, -w^2 - w + 5],\
[47, 47, w^2 - 5*w - 3],\
[47, 47, -2*w^2 + 4*w + 23],\
[59, 59, -w^2 + w + 13],\
[59, 59, w^2 - 3*w - 5],\
[59, 59, -2*w - 3],\
[97, 97, 7*w^2 - 11*w - 91],\
[97, 97, -2*w^2 - 8*w - 5],\
[97, 97, 5*w^2 - 19*w - 15],\
[107, 107, w^2 + 3*w - 1],\
[107, 107, -3*w^2 + 5*w + 37],\
[107, 107, w^2 - 5*w - 23],\
[113, 113, w^2 - 3*w - 1],\
[113, 113, w^2 - w - 17],\
[113, 113, -2*w - 7],\
[125, 5, -5],\
[127, 127, -w^2 - w - 1],\
[127, 127, 2*w^2 - 4*w - 29],\
[127, 127, -w^2 + 5*w - 3],\
[131, 131, -w^2 + 5*w - 1],\
[131, 131, 2*w^2 - 4*w - 27],\
[131, 131, w^2 + w - 1],\
[137, 137, 2*w^2 - 6*w - 5],\
[137, 137, -4*w - 11],\
[137, 137, 2*w^2 - 2*w - 31],\
[151, 151, -2*w^2 + 8*w + 3],\
[151, 151, 3*w^2 - 5*w - 41],\
[151, 151, -w^2 - 3*w - 3],\
[173, 173, 2*w^2 - 4*w - 25],\
[173, 173, w^2 + w - 3],\
[173, 173, -w^2 + 5*w + 1],\
[193, 193, -6*w - 17],\
[193, 193, 3*w^2 - 9*w - 7],\
[193, 193, 3*w^2 - 3*w - 47],\
[199, 199, w^2 + w - 7],\
[199, 199, 2*w^2 - 4*w - 21],\
[199, 199, -w^2 + 5*w + 5],\
[211, 211, 2*w - 7],\
[211, 211, w^2 - w - 3],\
[211, 211, -w^2 + 3*w + 15],\
[223, 223, w^2 - w - 7],\
[223, 223, w^2 - 3*w - 11],\
[223, 223, 2*w - 3],\
[257, 257, -2*w^2 - 4*w - 1],\
[257, 257, -3*w^2 + 13*w + 3],\
[257, 257, 5*w^2 - 9*w - 67],\
[269, 269, 2*w - 5],\
[269, 269, w^2 - 3*w - 13],\
[269, 269, w^2 - w - 5],\
[293, 293, -13*w^2 + 51*w + 35],\
[293, 293, -19*w^2 + 31*w + 247],\
[293, 293, 6*w^2 + 20*w + 9],\
[317, 317, 5*w^2 - 19*w - 17],\
[317, 317, 7*w^2 - 11*w - 89],\
[317, 317, 2*w^2 + 8*w + 3],\
[343, 7, -7],\
[379, 379, 3*w^2 - 11*w - 13],\
[379, 379, 4*w^2 - 6*w - 49],\
[379, 379, w^2 + 5*w + 1],\
[383, 383, -4*w - 5],\
[383, 383, -2*w^2 + 2*w + 25],\
[383, 383, 2*w^2 - 6*w - 11],\
[389, 389, 4*w^2 - 18*w - 13],\
[389, 389, -7*w^2 + 13*w + 83],\
[389, 389, 4*w^2 - 14*w - 11],\
[409, 409, 2*w - 11],\
[409, 409, w^2 - 3*w - 19],\
[409, 409, -w^2 + w - 1],\
[419, 419, 10*w^2 - 38*w - 31],\
[419, 419, -14*w^2 + 22*w + 181],\
[419, 419, 4*w^2 - 14*w - 17],\
[431, 431, -4*w^2 + 16*w + 13],\
[431, 431, 2*w^2 + 6*w - 1],\
[431, 431, -6*w^2 + 10*w + 75],\
[457, 457, -2*w^2 + 8*w + 9],\
[457, 457, 3*w^2 - 5*w - 35],\
[457, 457, w^2 + 3*w - 3],\
[557, 557, 5*w^2 - 9*w - 63],\
[557, 557, -3*w^2 + 13*w + 7],\
[557, 557, 2*w^2 + 4*w - 3],\
[563, 563, -9*w^2 + 35*w + 23],\
[563, 563, 13*w^2 - 21*w - 171],\
[563, 563, -4*w^2 - 14*w - 9],\
[601, 601, 5*w^2 - 17*w - 13],\
[601, 601, 6*w^2 - 8*w - 85],\
[601, 601, -w^2 - 9*w - 17],\
[613, 613, -2*w^2 + 15],\
[613, 613, 3*w^2 - 7*w - 31],\
[613, 613, 22*w^2 - 36*w - 285],\
[641, 641, 3*w^2 - 5*w - 43],\
[641, 641, -2*w^2 + 8*w + 1],\
[641, 641, -w^2 - 3*w - 5],\
[643, 643, 2*w^2 + 4*w - 1],\
[643, 643, 5*w^2 - 9*w - 65],\
[643, 643, -3*w^2 + 13*w + 5],\
[647, 647, 9*w^2 - 35*w - 27],\
[647, 647, 4*w^2 + 14*w + 5],\
[647, 647, 13*w^2 - 21*w - 167],\
[653, 653, -4*w^2 + 16*w + 7],\
[653, 653, 6*w^2 - 10*w - 81],\
[653, 653, -2*w^2 - 6*w - 5],\
[661, 661, -w^2 - 7*w - 13],\
[661, 661, -4*w^2 + 14*w + 9],\
[661, 661, 5*w^2 - 7*w - 71],\
[677, 677, -12*w^2 + 20*w + 155],\
[677, 677, -4*w^2 - 12*w - 3],\
[677, 677, -8*w^2 + 32*w + 21],\
[709, 709, -2*w^2 + 10*w - 3],\
[709, 709, 19*w^2 - 31*w - 249],\
[709, 709, 4*w^2 - 8*w - 55],\
[727, 727, -w^2 + 5*w + 21],\
[727, 727, -23*w^2 + 37*w + 301],\
[727, 727, -w^2 - w + 23],\
[733, 733, 2*w^2 + 2*w - 9],\
[733, 733, -2*w^2 + 10*w + 5],\
[733, 733, 4*w^2 - 8*w - 47],\
[739, 739, 2*w^2 - 6*w - 17],\
[739, 739, 2*w^2 - 2*w - 19],\
[739, 739, 4*w - 1],\
[773, 773, 8*w^2 - 12*w - 103],\
[773, 773, 6*w^2 - 22*w - 21],\
[773, 773, -2*w^2 - 10*w - 7],\
[809, 809, 4*w^2 + 16*w + 11],\
[809, 809, w^2 - 9*w - 7],\
[809, 809, 3*w^2 - w - 29],\
[821, 821, 3*w^2 + w - 23],\
[821, 821, 2*w^2 - 12*w - 9],\
[821, 821, 5*w^2 - 11*w - 51],\
[839, 839, 2*w^2 + 4*w - 7],\
[839, 839, 5*w^2 - 9*w - 59],\
[839, 839, 3*w^2 - 13*w - 11],\
[859, 859, w^2 + w - 11],\
[859, 859, 2*w^2 - 4*w - 17],\
[859, 859, w^2 - 5*w - 9],\
[881, 881, 4*w^2 - 14*w - 15],\
[881, 881, 5*w^2 - 7*w - 65],\
[881, 881, -w^2 - 7*w - 7],\
[887, 887, 5*w^2 - 7*w - 67],\
[887, 887, 4*w^2 - 14*w - 13],\
[887, 887, -w^2 - 7*w - 9],\
[907, 907, -3*w^2 - 11*w - 9],\
[907, 907, 10*w^2 - 16*w - 133],\
[907, 907, -7*w^2 + 27*w + 17],\
[911, 911, w^2 - 7*w - 7],\
[911, 911, 3*w^2 - 7*w - 27],\
[911, 911, 2*w^2 - 19],\
[919, 919, -3*w^2 - 5*w - 1],\
[919, 919, -4*w^2 + 18*w + 1],\
[919, 919, 7*w^2 - 13*w - 95],\
[947, 947, 4*w^2 - 6*w - 57],\
[947, 947, -3*w^2 + 11*w + 5],\
[947, 947, -w^2 - 5*w - 9],\
[967, 967, 8*w^2 - 30*w - 25],\
[967, 967, -3*w^2 - 13*w - 9],\
[967, 967, 11*w^2 - 17*w - 143],\
[991, 991, 2*w^2 - 8*w - 11],\
[991, 991, w^2 + 3*w - 5],\
[991, 991, 3*w^2 - 5*w - 33],\
[997, 997, 4*w^2 + 2*w - 27],\
[997, 997, 7*w^2 - 15*w - 75],\
[997, 997, -3*w^2 + 17*w + 11]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^5 - x^4 - 6*x^3 + 2*x^2 + 7*x - 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, e^3 - 2*e^2 - 3*e + 3, 1, e^4 - 2*e^3 - 4*e^2 + 6*e + 3, -2*e^4 + 3*e^3 + 8*e^2 - 5*e - 2, e^4 - 2*e^3 - 2*e^2 + 2*e - 3, -e^3 + 5*e + 2, 3*e^3 - 6*e^2 - 11*e + 10, -2*e^4 + 14*e^2 + 4*e - 10, -4*e^3 + 4*e^2 + 14*e - 4, e^3 - 7*e, 4*e^4 - 6*e^3 - 16*e^2 + 8*e + 6, -e^4 + 5*e^3 - 2*e^2 - 13*e + 13, -3*e^4 + 6*e^3 + 12*e^2 - 16*e - 7, 3*e^4 - 7*e^3 - 8*e^2 + 13*e + 1, -2*e^4 + 6*e^3 + 2*e^2 - 14*e + 8, e^4 - 4*e^3 - 2*e^2 + 10*e - 1, e^4 - 2*e^3 - 2*e^2 + 6*e - 5, -3*e^4 + 4*e^3 + 10*e^2 - 4*e + 3, -2*e^4 + 12*e^2, -e^4 + 5*e^3 - 2*e^2 - 17*e + 9, -4*e^4 + 6*e^3 + 16*e^2 - 6*e - 8, 2*e^4 - 8*e^3 + 6*e^2 + 18*e - 22, e^4 + 2*e^3 - 10*e^2 - 6*e + 11, 2*e^4 - 7*e^3 - 2*e^2 + 17*e - 6, 3*e^4 - 9*e^3 - 8*e^2 + 21*e + 9, -2*e^4 + 14*e^2 + 10*e - 20, -3*e^4 + 13*e^3 - 35*e + 11, -7*e^4 + 12*e^3 + 26*e^2 - 24*e - 19, -2*e^4 + 4*e^3 + 2*e^2 - 2*e + 18, -8*e^4 + 12*e^3 + 34*e^2 - 20*e - 18, -4*e^4 + 8*e^3 + 14*e^2 - 14*e + 4, 3*e^4 - 4*e^3 - 10*e^2 + 4*e - 1, 3*e^4 - 6*e^3 - 4*e^2 + 10*e - 13, -5*e^4 + 6*e^3 + 22*e^2 - 10*e - 15, 4*e^4 - 6*e^3 - 16*e^2 + 10*e + 14, -2*e^4 + 4*e^3 + 8*e^2 - 4*e - 6, -7*e^4 + 15*e^3 + 26*e^2 - 31*e - 17, -2*e^3 + 6*e^2 + 4*e - 12, 2*e^4 + e^3 - 14*e^2 - 7*e + 18, 3*e^4 - 3*e^3 - 16*e^2 + 11*e + 9, 4*e^4 - 6*e^3 - 20*e^2 + 10*e + 10, -2*e^4 - e^3 + 10*e^2 + 11*e - 6, -e^4 - 3*e^3 + 18*e^2 + 9*e - 23, 2*e^4 - 4*e^3 - 12, -8*e^4 + 14*e^3 + 30*e^2 - 30*e - 14, 3*e^4 - 11*e^3 - 4*e^2 + 33*e + 1, -6*e - 10, 6*e^4 - 7*e^3 - 24*e^2 + 3*e + 8, -e^4 + 4*e^3 - 4*e^2 - 4*e + 5, -3*e^4 + 4*e^3 + 16*e^2 - 25, -2*e^4 + 15*e^3 - 12*e^2 - 43*e + 28, -4*e^4 + 7*e^3 + 16*e^2 - 15*e - 6, -5*e^4 + 10*e^3 + 14*e^2 - 22*e - 5, 6*e^4 - 14*e^3 - 20*e^2 + 32*e + 6, -6*e^4 + 3*e^3 + 38*e^2 - e - 18, 8*e^4 - 14*e^3 - 26*e^2 + 22*e + 4, 4*e^4 - 6*e^3 - 18*e^2 + 8*e + 18, 2*e^4 - 4*e^3 - 6*e^2 + 10*e + 12, 2*e^4 - 4*e^3 + 4*e - 24, 8*e^4 - 17*e^3 - 26*e^2 + 45*e + 14, -2*e^4 - 4*e^3 + 14*e^2 + 20*e - 2, -2*e^4 + 8*e^3 - 20*e - 4, 6*e^4 - 6*e^3 - 28*e^2 + 2*e + 12, 5*e^4 - 6*e^3 - 18*e^2 + 9, -3*e^4 + 8*e^3 - 18*e + 19, -5*e^4 + 16*e^3 + 10*e^2 - 48*e - 9, -2*e^4 - 6*e^3 + 24*e^2 + 16*e - 16, 5*e^4 - 15*e^3 - 6*e^2 + 35*e - 13, 10*e^4 - 16*e^3 - 48*e^2 + 32*e + 34, -4*e^4 + 8*e^3 + 22*e^2 - 24*e - 14, -7*e^4 + 20*e^3 + 22*e^2 - 48*e - 23, e^4 - 8*e^3 + 2*e^2 + 42*e - 5, 5*e^4 - 4*e^3 - 24*e^2 - 8*e + 29, 8*e^3 - 14*e^2 - 36*e + 28, -6*e^4 + 36*e^2 + 20*e - 40, -3*e^4 + 7*e^3 + 10*e^2 - 17*e + 7, -2*e^3 + 12*e^2 + 2*e - 30, 9*e^4 - 20*e^3 - 32*e^2 + 60*e + 25, 6*e^4 - 10*e^3 - 26*e^2 + 20*e - 2, -4*e^4 + 8*e^3 + 16*e^2 - 18*e - 18, 13*e^4 - 19*e^3 - 56*e^2 + 29*e + 31, 2*e^4 - 14*e^2 - 4*e + 28, 3*e^4 + 4*e^3 - 34*e^2 - 12*e + 43, 12*e^4 - 26*e^3 - 38*e^2 + 62*e + 14, 15*e^4 - 28*e^3 - 54*e^2 + 52*e + 25, -e^4 + 3*e^3 - 6*e^2 - e + 21, 4*e^4 - 20*e^2 - 28*e + 18, -3*e^4 + 7*e^3 - 4*e^2 - 11*e + 35, e^4 - 4*e^3 - 6*e^2 + 12*e - 5, 4*e^4 - 2*e^3 - 30*e^2 - 2*e + 32, 18*e^4 - 34*e^3 - 64*e^2 + 66*e + 22, 2*e^4 + 6*e^3 - 24*e^2 - 34*e + 26, 8*e^4 - 12*e^3 - 36*e^2 + 28*e + 24, 9*e^4 - 16*e^3 - 36*e^2 + 30*e + 7, 14*e^4 - 14*e^3 - 70*e^2 + 16*e + 48, 4*e^4 - 8*e^3 - 12*e^2 - 4*e + 6, 7*e^4 - 10*e^3 - 34*e^2 + 26*e + 13, -4*e^2 + 12*e + 6, 4*e^4 - 13*e^3 + 2*e^2 + 31*e - 40, 2*e^4 - 10*e^3 - 6*e^2 + 42*e - 6, 8*e^4 - 10*e^3 - 36*e^2 + 14*e + 14, -5*e^4 + 32*e^2 + 8*e - 25, -4*e^3 - 2*e^2 + 16*e + 38, -6*e^4 + 14*e^3 + 14*e^2 - 26*e + 20, -2*e^3 - 6*e^2 + 20*e + 8, 8*e^4 - 21*e^3 - 32*e^2 + 63*e + 20, -9*e^4 + 12*e^3 + 28*e^2 - 12*e + 1, -18*e^4 + 31*e^3 + 72*e^2 - 49*e - 30, -4*e^4 + 20*e^3 - 10*e^2 - 62*e + 30, -e^4 + 6*e^3 - 14*e^2 - 12*e + 51, -9*e^4 + 26*e^3 + 14*e^2 - 52*e + 27, -12*e^4 + 31*e^3 + 22*e^2 - 73*e + 20, 16*e^2 - 20*e - 30, -10*e^3 + 18*e^2 + 38*e - 28, -4*e^4 + 6*e^3 + 24*e^2 - 18*e - 6, 5*e^4 - 2*e^3 - 22*e^2 - 6*e + 11, 8*e^4 - 20*e^3 - 24*e^2 + 56*e + 10, 11*e^4 - 21*e^3 - 38*e^2 + 27*e + 1, 12*e^4 - 7*e^3 - 66*e^2 - 9*e + 54, -12*e^4 + 16*e^3 + 52*e^2 - 6*e - 36, 10*e^4 - 28*e^3 - 24*e^2 + 68*e + 2, -11*e^4 + 13*e^3 + 52*e^2 - 19*e - 17, -12*e^4 + 18*e^3 + 50*e^2 - 38*e - 34, -8*e^4 + 3*e^3 + 54*e^2 - 5*e - 44, -e^4 + 9*e^3 - 8*e^2 - 33*e + 21, -10*e^4 + 22*e^3 + 36*e^2 - 64*e - 18, 6*e^4 - 12*e^3 - 22*e^2 + 32*e - 4, -10*e^4 + 8*e^3 + 50*e^2 + 2*e - 34, -9*e^4 + 16*e^3 + 38*e^2 - 40*e - 21, -10*e^4 + 2*e^3 + 60*e^2 + 18*e - 52, -2*e^4 + 3*e^3 + 22*e^2 - 13*e - 26, 2*e^4 - 2*e^3 + 2*e^2 - 2*e - 30, 12*e^4 - 26*e^3 - 34*e^2 + 50*e + 12, 2*e^4 - 4*e^3 - 12*e^2 + 20*e + 44, 5*e^4 - 18*e^3 - 8*e^2 + 46*e - 11, 2*e^4 - 12*e^3 + 20*e^2 + 28*e - 40, -10*e^4 + 11*e^3 + 50*e^2 - 11*e - 32, -11*e^4 + 13*e^3 + 56*e^2 - 13*e - 53, 8*e^4 + e^3 - 56*e^2 - 9*e + 50, 4*e^4 - 5*e^3 - 8*e^2 + 9*e - 26, -7*e^4 + 6*e^3 + 36*e^2 + 10*e - 21, 6*e^4 - 18*e^3 + 2*e^2 + 20*e - 54, -20*e^4 + 36*e^3 + 76*e^2 - 68*e - 36, 2*e^4 - 6*e^3 + 2*e^2 + 10*e - 24, 10*e^4 - 22*e^3 - 38*e^2 + 54*e + 10, 6*e^4 - 3*e^3 - 34*e^2 + 11*e + 28, 13*e^4 - 31*e^3 - 44*e^2 + 79*e + 27, e^4 + 8*e^3 - 26*e^2 - 14*e + 31, -2*e^4 + 16*e^2 + 4*e - 18, -14*e^4 + 30*e^3 + 42*e^2 - 50*e - 4, -5*e^4 + 17*e^3 + 16*e^2 - 55*e + 1, 12*e^4 - 26*e^3 - 44*e^2 + 60*e + 18, 14*e^4 - 20*e^3 - 60*e^2 + 48*e + 26, 2*e^4 + 3*e^3 - 16*e^2 - 29*e + 26, -7*e^4 + 12*e^3 + 30*e^2 - 28*e - 19, -6*e^4 + 19*e^3 - 35*e + 40, 13*e^4 - 18*e^3 - 54*e^2 + 22*e + 13, 7*e^4 - 14*e^3 - 26*e^2 + 26*e - 25, 11*e^4 - 24*e^3 - 32*e^2 + 44*e - 3, -8*e^4 + 32*e^3 - 4*e^2 - 68*e + 60, 12*e^4 - 24*e^3 - 40*e^2 + 40*e + 28, 4*e^3 - 10*e^2 - 20*e + 2, 6*e^4 - 44*e^2 - 24*e + 38, -8*e^4 + 18*e^3 + 28*e^2 - 52*e - 30, 4*e^4 - 2*e^3 - 28*e^2 + 2*e + 32, -8*e^4 + 22*e^3 + 18*e^2 - 58*e - 6, -8*e^4 + 18*e^3 + 34*e^2 - 58*e - 42, 10*e^4 - 26*e^3 - 30*e^2 + 50*e + 16, -10*e^4 + 26*e^3 + 20*e^2 - 52*e + 18, 16*e^4 - 32*e^3 - 54*e^2 + 64*e + 24, -7*e^4 + 12*e^3 + 20*e^2 - 4*e + 9]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([2,2,w + 3])] = -1

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