/*
  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([2, 0, -6, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([20, 10, -w^2 + 2*w + 4])

primes_array = [
[2, 2, w],\
[5, 5, -w^3 + 3*w^2 + 2*w - 1],\
[5, 5, w - 1],\
[11, 11, w^3 - 2*w^2 - 5*w - 1],\
[13, 13, -w^3 + 3*w^2 + 3*w - 1],\
[17, 17, 2*w^3 - 5*w^2 - 9*w + 5],\
[23, 23, -w^3 + 3*w^2 + 4*w - 5],\
[25, 5, -w^2 + 3*w + 1],\
[29, 29, -2*w^3 + 6*w^2 + 5*w - 1],\
[31, 31, -w^2 + 2*w + 1],\
[43, 43, -w^3 + 3*w^2 + 2*w - 3],\
[43, 43, -w^3 + 2*w^2 + 6*w - 3],\
[59, 59, 2*w^3 - 6*w^2 - 7*w + 7],\
[73, 73, 3*w^3 - 6*w^2 - 16*w - 5],\
[79, 79, -5*w^3 + 14*w^2 + 20*w - 17],\
[81, 3, -3],\
[83, 83, -w - 3],\
[83, 83, w^2 - 3*w - 7],\
[89, 89, w^2 - 2*w - 7],\
[101, 101, -2*w^3 + 5*w^2 + 8*w - 3],\
[101, 101, 5*w^3 - 14*w^2 - 19*w + 17],\
[103, 103, -w^3 + w^2 + 8*w + 3],\
[103, 103, 2*w^3 - 5*w^2 - 7*w - 1],\
[109, 109, 3*w^3 - 7*w^2 - 14*w + 1],\
[109, 109, -w^2 + 4*w + 1],\
[127, 127, -3*w^3 + 8*w^2 + 11*w - 7],\
[127, 127, -w^3 + 2*w^2 + 7*w + 1],\
[137, 137, -2*w^3 + 5*w^2 + 9*w - 1],\
[139, 139, -w^3 + 3*w^2 + 2*w - 5],\
[149, 149, w^3 - 3*w^2 - w - 1],\
[149, 149, -2*w^3 + 4*w^2 + 10*w + 1],\
[163, 163, w^3 - 4*w^2 - w + 11],\
[163, 163, 2*w^3 - 4*w^2 - 10*w + 1],\
[173, 173, -4*w^3 + 11*w^2 + 15*w - 13],\
[179, 179, 2*w^2 - 6*w - 3],\
[179, 179, -w^3 + 2*w^2 + 7*w - 1],\
[181, 181, w^3 - 10*w - 9],\
[181, 181, -3*w^3 + 8*w^2 + 14*w - 7],\
[199, 199, w^3 - 8*w - 9],\
[211, 211, 2*w - 3],\
[223, 223, -w^3 + 4*w^2 + w - 5],\
[227, 227, w^3 - 2*w^2 - 5*w - 5],\
[227, 227, -3*w^3 + 7*w^2 + 14*w - 5],\
[227, 227, -4*w^3 + 10*w^2 + 19*w - 7],\
[227, 227, w^2 - 4*w - 5],\
[239, 239, -2*w^3 + 5*w^2 + 7*w - 3],\
[251, 251, -w^3 + 4*w^2 + 2*w - 11],\
[251, 251, 2*w^3 - 6*w^2 - 6*w + 1],\
[257, 257, -3*w^3 + 6*w^2 + 16*w + 3],\
[263, 263, -w^3 + 3*w^2 + 3*w - 7],\
[263, 263, w^3 - 4*w^2 + 7],\
[269, 269, -2*w^3 + 6*w^2 + 6*w - 11],\
[269, 269, -w^3 + 4*w^2 - w - 5],\
[271, 271, -3*w^3 + 9*w^2 + 11*w - 11],\
[271, 271, w^2 - 5*w - 3],\
[277, 277, w^3 - 4*w^2 - 2*w + 7],\
[277, 277, -2*w^3 + 7*w^2 + 8*w - 9],\
[281, 281, 2*w^2 - 4*w - 5],\
[281, 281, 2*w^3 - 4*w^2 - 11*w + 1],\
[283, 283, -3*w^3 + 8*w^2 + 14*w - 11],\
[293, 293, w^3 - 2*w^2 - 6*w + 5],\
[307, 307, -2*w^3 + 5*w^2 + 7*w - 1],\
[313, 313, 2*w^3 - 4*w^2 - 9*w + 3],\
[313, 313, -2*w^3 + 5*w^2 + 6*w - 3],\
[317, 317, -2*w^3 + 5*w^2 + 9*w + 1],\
[347, 347, 4*w^3 - 9*w^2 - 21*w + 3],\
[347, 347, 4*w^3 - 10*w^2 - 18*w + 5],\
[349, 349, -2*w^3 + 6*w^2 + 7*w - 3],\
[353, 353, -w^3 + 12*w + 5],\
[353, 353, w^2 - w - 7],\
[359, 359, -2*w^3 + 5*w^2 + 6*w - 7],\
[359, 359, -w^3 + 4*w^2 - 11],\
[373, 373, -w^3 + 2*w^2 + 6*w + 5],\
[379, 379, -3*w^3 + 8*w^2 + 14*w - 5],\
[397, 397, 6*w^3 - 14*w^2 - 30*w + 7],\
[401, 401, 3*w^3 - 5*w^2 - 20*w - 7],\
[409, 409, 3*w^3 - 7*w^2 - 12*w - 3],\
[409, 409, 2*w^2 - 4*w - 7],\
[421, 421, w^3 - 11*w - 11],\
[421, 421, -w^3 + 3*w^2 + w - 7],\
[431, 431, 2*w^3 - 3*w^2 - 14*w - 3],\
[431, 431, -4*w^3 + 11*w^2 + 15*w - 15],\
[433, 433, 2*w^3 - 5*w^2 - 6*w + 1],\
[439, 439, -4*w^3 + 11*w^2 + 13*w - 9],\
[439, 439, -2*w^3 + 6*w^2 + 8*w - 11],\
[443, 443, 2*w - 5],\
[443, 443, -6*w^3 + 17*w^2 + 24*w - 21],\
[449, 449, w^3 - 2*w^2 - 8*w + 3],\
[449, 449, 4*w^3 - 12*w^2 - 15*w + 17],\
[457, 457, -2*w^3 + 4*w^2 + 9*w + 5],\
[461, 461, -3*w^3 + 6*w^2 + 15*w - 1],\
[461, 461, -3*w^3 + 7*w^2 + 17*w + 1],\
[463, 463, -4*w^3 + 10*w^2 + 17*w - 7],\
[463, 463, -3*w^3 + 10*w^2 + 9*w - 17],\
[467, 467, 4*w^3 - 8*w^2 - 21*w - 7],\
[467, 467, w^3 - 6*w^2 + 3*w + 19],\
[479, 479, -w^3 + w^2 + 8*w + 1],\
[487, 487, -4*w^3 + 11*w^2 + 14*w - 15],\
[487, 487, 3*w^3 - 6*w^2 - 15*w - 5],\
[491, 491, w^2 - 5],\
[503, 503, 2*w^3 - 5*w^2 - 11*w + 3],\
[523, 523, -3*w^3 + 7*w^2 + 13*w + 1],\
[547, 547, 2*w^3 - 3*w^2 - 12*w - 3],\
[547, 547, 2*w^3 - 3*w^2 - 12*w - 9],\
[557, 557, -3*w^3 + 9*w^2 + 9*w - 11],\
[557, 557, -6*w^3 + 15*w^2 + 27*w - 13],\
[563, 563, -3*w^3 + 7*w^2 + 14*w - 7],\
[571, 571, -5*w^3 + 11*w^2 + 26*w - 3],\
[593, 593, -6*w^3 + 15*w^2 + 26*w - 7],\
[599, 599, 7*w^3 - 18*w^2 - 29*w + 17],\
[599, 599, -2*w^3 + 6*w^2 + 8*w - 3],\
[601, 601, -7*w^3 + 20*w^2 + 28*w - 23],\
[601, 601, 3*w^3 - 7*w^2 - 12*w + 5],\
[607, 607, 5*w^3 - 12*w^2 - 24*w + 9],\
[607, 607, w^3 - 5*w^2 + 15],\
[613, 613, -w^3 + 5*w^2 - 2*w - 9],\
[617, 617, w^3 + w^2 - 13*w - 15],\
[617, 617, 2*w^3 - 4*w^2 - 8*w - 1],\
[619, 619, 5*w^3 - 10*w^2 - 27*w - 7],\
[619, 619, 2*w^3 - 4*w^2 - 10*w + 3],\
[643, 643, w^3 - 5*w^2 + 5*w + 5],\
[643, 643, 4*w^3 - 9*w^2 - 20*w - 1],\
[643, 643, 3*w^3 - 7*w^2 - 14*w - 3],\
[643, 643, -3*w^3 + 7*w^2 + 13*w - 3],\
[647, 647, -3*w^3 + 8*w^2 + 12*w - 3],\
[653, 653, -w^3 + 9*w + 13],\
[653, 653, -w^3 + 4*w^2 + 3*w - 9],\
[653, 653, -2*w^3 + 7*w^2 + 4*w - 7],\
[653, 653, w^2 - 4*w - 9],\
[673, 673, -w^3 + 5*w^2 + w - 13],\
[683, 683, w^3 - 3*w^2 - 3*w - 3],\
[701, 701, 5*w^3 - 13*w^2 - 23*w + 9],\
[709, 709, 2*w^3 - w^2 - 19*w - 15],\
[719, 719, -5*w^3 + 15*w^2 + 20*w - 19],\
[727, 727, -w - 5],\
[751, 751, -4*w^3 + 10*w^2 + 15*w - 9],\
[751, 751, 6*w^3 - 17*w^2 - 22*w + 21],\
[757, 757, 3*w^3 - 6*w^2 - 18*w - 1],\
[757, 757, -w^3 + 4*w^2 + 2*w + 1],\
[769, 769, w^2 - 2*w + 3],\
[773, 773, 5*w^3 - 12*w^2 - 23*w + 9],\
[787, 787, 2*w^3 - 4*w^2 - 5*w + 1],\
[797, 797, -3*w^3 + 8*w^2 + 10*w - 3],\
[809, 809, 2*w^2 - 5*w - 1],\
[809, 809, 4*w^3 - 6*w^2 - 28*w - 11],\
[827, 827, 4*w^3 - 8*w^2 - 22*w - 1],\
[827, 827, 2*w^2 - 6*w - 9],\
[829, 829, -w^3 + w^2 + 10*w + 1],\
[839, 839, -2*w^3 + 2*w^2 + 16*w + 7],\
[877, 877, 6*w^3 - 16*w^2 - 25*w + 13],\
[877, 877, w^3 - w^2 - 7*w + 1],\
[881, 881, 2*w^3 - 2*w^2 - 15*w - 7],\
[881, 881, -4*w^3 + 7*w^2 + 26*w + 9],\
[887, 887, 2*w^3 - 4*w^2 - 13*w - 1],\
[907, 907, 5*w^3 - 11*w^2 - 26*w - 1],\
[907, 907, 5*w^3 - 13*w^2 - 22*w + 9],\
[907, 907, w^3 - 5*w^2 + 2*w + 13],\
[907, 907, -2*w^3 + 7*w^2 + 5*w - 7],\
[911, 911, -3*w^3 + 8*w^2 + 9*w - 5],\
[919, 919, -2*w^3 + 7*w^2 + 6*w - 3],\
[919, 919, -3*w^3 + 8*w^2 + 15*w - 7],\
[947, 947, -2*w^3 + 5*w^2 + 12*w - 9],\
[947, 947, -w^2 + w - 3],\
[947, 947, 3*w^3 - 6*w^2 - 14*w - 7],\
[947, 947, -w^3 + 5*w^2 - w - 17],\
[953, 953, 3*w^3 - 9*w^2 - 9*w + 13],\
[953, 953, -4*w^3 + 13*w^2 + 11*w - 23],\
[967, 967, 8*w^3 - 20*w^2 - 33*w + 19]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - 24*x^4 + 128*x^2 - 64
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, e, 1, -1/4*e^3 + 3*e, -1/16*e^5 + 5/4*e^3 - 5*e, -1/16*e^5 + 5/4*e^3 - 5*e, -1/8*e^4 + 5/2*e^2 - 4, 1/16*e^5 - 5/4*e^3 + 3*e, 1/8*e^5 - 3*e^3 + 15*e, -1/4*e^3 + 3*e, -1/8*e^4 + 7/2*e^2 - 12, -1/8*e^5 + 13/4*e^3 - 17*e, -1/8*e^4 + 3/2*e^2 + 4, -1/16*e^5 + 5/4*e^3 - 5*e, 1/4*e^4 - 5*e^2 + 16, -1/4*e^4 + 5*e^2 - 14, 3/8*e^4 - 13/2*e^2 + 20, -e^2 + 8, -1/4*e^4 + 3*e^2 + 2, -1/16*e^5 + 5/4*e^3 - 5*e, -1/4*e^4 + 4*e^2 - 6, 1/8*e^5 - 9/4*e^3 + 5*e, 1/4*e^5 - 11/2*e^3 + 22*e, 1/4*e^5 - 5*e^3 + 17*e, -1/8*e^5 + 3*e^3 - 13*e, e^3 - 12*e, 4*e, 1/4*e^5 - 5*e^3 + 17*e, 1/8*e^4 - 3/2*e^2 + 4, 1/2*e^4 - 8*e^2 + 14, -1/16*e^5 + 5/4*e^3 - e, 1/8*e^4 - 3/2*e^2 + 12, -1/4*e^5 + 11/2*e^3 - 22*e, 1/4*e^4 - 6*e^2 + 18, -1/8*e^5 + 11/4*e^3 - 11*e, 1/4*e^3 - 3*e, 1/2*e^4 - 8*e^2 + 14, -1/8*e^5 + 3*e^3 - 9*e, -1/8*e^4 + 5/2*e^2 + 12, e^2 + 8, 1/2*e^4 - 8*e^2 + 16, 1/4*e^4 - 4*e^2 + 8, -3/2*e^3 + 18*e, 4*e, 5/8*e^4 - 23/2*e^2 + 36, 1/4*e^5 - 23/4*e^3 + 29*e, -3/8*e^4 + 9/2*e^2 + 4, -1/4*e^5 + 6*e^3 - 32*e, 1/8*e^5 - 4*e^3 + 31*e, 1/4*e^4 - 5*e^2 + 8, 1/8*e^4 - 5/2*e^2 + 12, 1/2*e^4 - 7*e^2 + 10, -1/4*e^4 + 3*e^2 - 2, -1/8*e^4 - 3/2*e^2 + 28, 1/8*e^5 - 9/4*e^3 + e, 1/4*e^4 - 5*e^2 + 14, 1/4*e^4 - 8*e^2 + 34, -1/4*e^4 + e^2 + 26, -1/8*e^5 + 7/2*e^3 - 27*e, -e^2 + 8, 3*e^2 - 30, -1/2*e^5 + 11*e^3 - 48*e, 1/4*e^5 - 11/2*e^3 + 15*e, 1/4*e^5 - 13/2*e^3 + 35*e, -7/16*e^5 + 39/4*e^3 - 49*e, 1/4*e^5 - 25/4*e^3 + 35*e, 1/2*e^5 - 21/2*e^3 + 38*e, -1/4*e^5 + 9/2*e^3 - 13*e, 3/16*e^5 - 19/4*e^3 + 35*e, -2*e^2 + 10, -1/8*e^4 + 5/2*e^2 - 4, 1/4*e^4 - 5*e^2 + 24, 1/4*e^4 - 3*e^2 + 6, -3/4*e^3 + 5*e, 1/4*e^5 - 6*e^3 + 33*e, 7/16*e^5 - 35/4*e^3 + 35*e, -3/16*e^5 + 19/4*e^3 - 25*e, 1/4*e^4 - 2*e^2 - 10, 1/4*e^4 - 4*e^2 + 2, 1/4*e^4 - 3*e^2 - 26, 1/2*e^5 - 39/4*e^3 + 33*e, -1/8*e^4 + 1/2*e^2 + 28, -3/16*e^5 + 23/4*e^3 - 41*e, -1/4*e^5 + 5*e^3 - 16*e, 3/4*e^4 - 13*e^2 + 16, -1/8*e^4 + 7/2*e^2 + 4, -3*e^2 + 8, 5/16*e^5 - 25/4*e^3 + 15*e, 1/4*e^4 - 3*e^2 - 22, -1/2*e^4 + 10*e^2 - 22, -1/16*e^5 + 9/4*e^3 - 21*e, -1/8*e^5 + 3*e^3 - 9*e, -3/8*e^5 + 33/4*e^3 - 33*e, 8, 1/2*e^5 - 23/2*e^3 + 50*e, 1/8*e^4 + 1/2*e^2 - 4, 1/4*e^3 - 7*e, -e^4 + 16*e^2 - 24, 1/8*e^5 - 15/4*e^3 + 35*e, -e^2, -1/4*e^5 + 13/2*e^3 - 38*e, 1/4*e^5 - 6*e^3 + 40*e, 5/2*e^3 - 34*e, -1/8*e^4 - 1/2*e^2 + 12, -1/2*e^4 + 6*e^2 - 2, 1/16*e^5 + 3/4*e^3 - 21*e, -1/8*e^5 + 11/4*e^3 - 19*e, 1/8*e^5 - 13/4*e^3 + 13*e, -1/16*e^5 + 9/4*e^3 - 21*e, 5/8*e^5 - 59/4*e^3 + 67*e, 1/2*e^5 - 21/2*e^3 + 34*e, -1/2*e^4 + 5*e^2 + 14, 3/16*e^5 - 15/4*e^3 + 3*e, -1/4*e^5 + 11/2*e^3 - 26*e, 5/8*e^4 - 17/2*e^2 + 4, -1/4*e^4 + 5*e^2 - 34, -1/2*e^4 + 8*e^2 + 2, -1/8*e^5 + 4*e^3 - 33*e, -1/2*e^3 + 6*e, -1/4*e^5 + 11/2*e^3 - 18*e, 1/8*e^5 - 17/4*e^3 + 33*e, -1/2*e^5 + 23/2*e^3 - 54*e, 1/4*e^3 - 7*e, -3/8*e^5 + 43/4*e^3 - 71*e, -1/4*e^5 + 21/4*e^3 - 23*e, 3/4*e^4 - 11*e^2 + 6, 1/4*e^4 - e^2 - 26, -2*e^2 + 22, 1/4*e^4 - 3*e^2 + 14, -1/2*e^4 + 10*e^2 - 22, 3/2*e^4 - 25*e^2 + 56, 3/4*e^5 - 33/2*e^3 + 71*e, -e^4 + 20*e^2 - 42, 3/8*e^4 - 15/2*e^2 - 12, -3/8*e^4 + 19/2*e^2 - 44, -1/4*e^5 + 11/2*e^3 - 26*e, -3/8*e^4 + 15/2*e^2 - 28, 3/4*e^5 - 17*e^3 + 77*e, -2*e^2 - 18, -1/4*e^4 + 3*e^2 - 22, -1/8*e^5 + 4*e^3 - 25*e, -1/8*e^5 + 9/4*e^3 + 3*e, 5/8*e^5 - 27/2*e^3 + 57*e, 1/16*e^5 - 5/4*e^3 + 3*e, 1/2*e^3 - 13*e, 1/2*e^5 - 43/4*e^3 + 45*e, -7/8*e^4 + 33/2*e^2 - 44, -3/16*e^5 + 27/4*e^3 - 49*e, -7/8*e^5 + 75/4*e^3 - 79*e, 1/4*e^5 - 6*e^3 + 45*e, -5/16*e^5 + 41/4*e^3 - 77*e, 1/8*e^5 - 3*e^3 + 19*e, -5/16*e^5 + 21/4*e^3 - 13*e, -3/4*e^5 + 63/4*e^3 - 65*e, 3/2*e^3 - 26*e, -3/8*e^5 + 31/4*e^3 - 31*e, -3/8*e^4 + 9/2*e^2 + 28, -7/8*e^4 + 33/2*e^2 - 28, 3/4*e^5 - 31/2*e^3 + 58*e, 1/8*e^5 - 19/4*e^3 + 31*e, 5/8*e^5 - 59/4*e^3 + 83*e, -e^2 + 32, 3/8*e^4 - 9/2*e^2 - 20, 3/8*e^4 - 9/2*e^2 + 20, -7/8*e^4 + 33/2*e^2 - 36, 1/4*e^4 - 4*e^2 + 22, -3/4*e^4 + 7*e^2 + 42, 7/8*e^5 - 69/4*e^3 + 57*e]
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])] = -1
AL_eigenvalues[ZF.ideal([5, 5, w - 1])] = -1

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