/*
  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([5, -6, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([17, 17, -w^2 + w + 3])

primes_array = [
[3, 3, w - 2],\
[5, 5, w],\
[5, 5, -w + 3],\
[8, 2, 2],\
[9, 3, w^2 + w - 4],\
[13, 13, w + 3],\
[17, 17, -w^2 + w + 3],\
[23, 23, w^2 - 2],\
[23, 23, w^2 - 3],\
[23, 23, -w^2 + 8],\
[29, 29, w - 4],\
[37, 37, w^2 + w - 8],\
[41, 41, w^2 + 2*w - 4],\
[47, 47, 2*w^2 + w - 8],\
[59, 59, -2*w^2 - 3*w + 6],\
[61, 61, -2*w - 1],\
[67, 67, -2*w - 3],\
[79, 79, 2*w^2 - 9],\
[109, 109, w^2 + 2*w - 6],\
[109, 109, 2*w^2 + w - 14],\
[109, 109, -w^2 - w - 1],\
[113, 113, 2*w^2 + w - 12],\
[127, 127, w^2 - 3*w - 2],\
[131, 131, -4*w^2 - 5*w + 9],\
[137, 137, w^2 + 2*w - 7],\
[137, 137, 3*w + 8],\
[137, 137, 2*w^2 + w - 17],\
[139, 139, 3*w^2 + 2*w - 11],\
[149, 149, w - 6],\
[151, 151, w^2 + 2*w - 9],\
[157, 157, -w^2 - 2*w + 13],\
[157, 157, 2*w^2 - 7],\
[163, 163, w^2 + 3*w + 3],\
[163, 163, 2*w^2 - 3*w - 3],\
[163, 163, 2*w^2 - w - 7],\
[167, 167, 3*w^2 + w - 13],\
[169, 13, 2*w^2 + 5*w - 1],\
[173, 173, w^2 + 2*w + 2],\
[179, 179, 4*w^2 + 2*w - 19],\
[181, 181, w^2 + 3*w - 6],\
[191, 191, 3*w^2 + 2*w - 14],\
[193, 193, w^2 - 3*w - 3],\
[197, 197, 3*w^2 - 2*w - 13],\
[211, 211, -w - 6],\
[223, 223, -w^2 + w - 3],\
[223, 223, -3*w - 2],\
[223, 223, 2*w^2 - 2*w - 3],\
[229, 229, 2*w^2 - w - 4],\
[233, 233, 2*w + 7],\
[239, 239, -3*w - 4],\
[239, 239, w^2 - 11],\
[239, 239, 4*w^2 - w - 21],\
[241, 241, 3*w^2 + w - 19],\
[251, 251, -w^2 - 4*w - 1],\
[257, 257, w - 7],\
[263, 263, 3*w^2 + w - 12],\
[269, 269, 3*w^2 - 4*w - 11],\
[271, 271, 4*w^2 + 6*w - 11],\
[277, 277, -4*w^2 - 3*w + 17],\
[283, 283, 5*w^2 + 4*w - 18],\
[289, 17, 3*w^2 + 2*w - 9],\
[307, 307, -2*w^2 - w + 18],\
[307, 307, w^2 + 2*w + 3],\
[307, 307, -w^2 - 3],\
[313, 313, 3*w^2 - 13],\
[317, 317, 3*w^2 - 2*w - 12],\
[331, 331, w^2 + 3*w - 9],\
[331, 331, w^2 + 3*w - 11],\
[331, 331, w^2 + 3*w + 4],\
[343, 7, -7],\
[347, 347, 2*w^2 + 2*w - 13],\
[353, 353, 4*w^2 + 5*w - 13],\
[359, 359, 3*w^2 + 6*w - 1],\
[379, 379, w^2 - 3*w - 6],\
[383, 383, 3*w^2 - 23],\
[383, 383, w^2 - 12],\
[383, 383, 3*w^2 + 3*w - 13],\
[401, 401, -6*w^2 - 7*w + 16],\
[409, 409, -4*w^2 + 3*w + 21],\
[421, 421, w^2 - 3*w - 9],\
[433, 433, w^2 - 3*w - 8],\
[433, 433, w^2 + 4*w - 8],\
[433, 433, 3*w^2 - w - 12],\
[439, 439, -4*w - 11],\
[449, 449, 2*w^2 + 3*w - 19],\
[457, 457, w^2 - 4*w - 3],\
[461, 461, 5*w^2 + 6*w - 16],\
[461, 461, -2*w^2 + 3*w + 16],\
[461, 461, 2*w^2 + 4*w - 9],\
[467, 467, 3*w^2 - 4*w - 12],\
[479, 479, 3*w^2 + w - 9],\
[487, 487, 5*w^2 + 2*w - 22],\
[491, 491, 3*w^2 - 11],\
[499, 499, 5*w^2 + w - 24],\
[503, 503, 3*w^2 + 2*w - 17],\
[503, 503, 3*w^2 + w - 7],\
[503, 503, 2*w^2 - 4*w - 7],\
[509, 509, 3*w - 11],\
[521, 521, 3*w^2 - 3*w - 16],\
[523, 523, -w - 8],\
[541, 541, 6*w^2 + 7*w - 19],\
[547, 547, 3*w^2 + 4*w - 12],\
[557, 557, 2*w^2 - 3*w - 12],\
[563, 563, 4*w^2 + 3*w - 12],\
[563, 563, 4*w^2 + w - 17],\
[563, 563, -w^2 - 5*w - 3],\
[569, 569, w^2 - 4*w - 4],\
[577, 577, w^2 - w - 13],\
[587, 587, 4*w^2 + 3*w - 8],\
[587, 587, 2*w^2 - 3*w - 13],\
[587, 587, 2*w^2 + 3*w - 18],\
[593, 593, 4*w - 13],\
[599, 599, w - 9],\
[613, 613, -w^2 + 3*w - 7],\
[613, 613, w^2 + 5*w + 2],\
[613, 613, 3*w^2 - 3*w - 17],\
[617, 617, w^2 + 5*w + 8],\
[641, 641, 4*w^2 + 3*w - 11],\
[641, 641, w^2 + w - 14],\
[641, 641, 3*w^2 - 2*w - 9],\
[647, 647, 2*w^2 + 3*w - 13],\
[653, 653, 3*w^2 - 8],\
[659, 659, 5*w^2 + 4*w - 21],\
[659, 659, 2*w^2 - 4*w - 19],\
[659, 659, -2*w^2 - 6*w + 9],\
[677, 677, -6*w - 13],\
[691, 691, 6*w - 11],\
[701, 701, 3*w^2 - 3*w - 19],\
[719, 719, 2*w^2 + 3*w - 16],\
[727, 727, 3*w^2 + 4*w - 13],\
[733, 733, 3*w^2 - w - 8],\
[739, 739, 2*w^2 + 4*w - 11],\
[739, 739, 4*w^2 - 3*w - 24],\
[739, 739, 3*w^2 - w - 6],\
[743, 743, 4*w^2 - 2*w - 17],\
[751, 751, -w - 9],\
[761, 761, 5*w^2 + 5*w - 19],\
[769, 769, w^2 - 4*w - 6],\
[769, 769, -5*w - 1],\
[769, 769, 2*w^2 - 19],\
[773, 773, 2*w^2 - 6*w - 3],\
[797, 797, 4*w^2 - 17],\
[809, 809, -7*w^2 - 4*w + 29],\
[811, 811, 6*w^2 + 3*w - 29],\
[823, 823, 4*w^2 + 2*w - 13],\
[823, 823, -w^2 - 4*w - 7],\
[823, 823, 4*w^2 + 4*w - 17],\
[827, 827, -2*w^2 - 3*w - 2],\
[829, 829, -w^2 + w - 6],\
[829, 829, -w^2 + 4*w - 9],\
[829, 829, -2*w^2 + 3*w + 21],\
[839, 839, 6*w^2 - 2*w - 31],\
[841, 29, w^2 + 3*w + 6],\
[853, 853, -7*w^2 - 6*w + 27],\
[859, 859, 6*w - 1],\
[881, 881, w^2 + 5*w - 16],\
[907, 907, 2*w^2 + 7*w + 7],\
[919, 919, w^2 - 4*w - 9],\
[953, 953, -4*w^2 - w + 32],\
[967, 967, 5*w^2 - 5*w - 18],\
[971, 971, 3*w^2 - 4*w - 24],\
[977, 977, 2*w^2 + 4*w + 3],\
[997, 997, w^2 + 5*w - 12],\
[997, 997, 5*w^2 - 3*w - 22],\
[997, 997, 3*w^2 + 8*w - 2]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - 5*x^5 - 2*x^4 + 31*x^3 - 2*x^2 - 54*x - 13
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -e^2 + 2*e + 3, 1/2*e^5 - 2*e^4 - e^3 + 15/2*e^2 + 1/2*e - 5/2, -e + 2, -1/2*e^5 + e^4 + 5*e^3 - 9/2*e^2 - 31/2*e - 5/2, -e^4 + 4*e^3 + e^2 - 12*e + 2, 1, -e^3 + 2*e^2 + 3*e, -e^4 + 3*e^3 + 4*e^2 - 9*e - 6, e^5 - 2*e^4 - 10*e^3 + 8*e^2 + 31*e + 10, -e^5 + 4*e^4 + 2*e^3 - 13*e^2 - 3*e + 1, 2*e^4 - 7*e^3 - 7*e^2 + 24*e + 12, e^3 - e^2 - 5*e + 3, -1/2*e^5 + 3*e^4 - 2*e^3 - 21/2*e^2 + 17/2*e + 9/2, -e^5 + 4*e^4 + 2*e^3 - 15*e^2 - e + 11, -e^5 + 2*e^4 + 9*e^3 - 8*e^2 - 24*e - 9, e^5 - 4*e^4 - 2*e^3 + 16*e^2 - e - 12, -1/2*e^5 - e^4 + 13*e^3 - 5/2*e^2 - 73/2*e - 3/2, e^4 - 3*e^3 - 5*e^2 + 10*e + 9, 2*e^5 - 9*e^4 + 2*e^3 + 27*e^2 - 19*e - 8, 3*e^3 - 9*e^2 - 7*e + 13, -1/2*e^5 - e^4 + 12*e^3 - 5/2*e^2 - 65/2*e + 7/2, 1/2*e^5 + e^4 - 13*e^3 + 3/2*e^2 + 83/2*e + 15/2, 3*e^4 - 10*e^3 - 13*e^2 + 32*e + 29, -1/2*e^5 - e^4 + 11*e^3 + 11/2*e^2 - 75/2*e - 33/2, -2*e^5 + 10*e^4 - 4*e^3 - 33*e^2 + 24*e + 17, e^5 - e^4 - 12*e^3 + 3*e^2 + 35*e + 16, -e^3 - 2*e^2 + 11*e + 16, 2*e^5 - 4*e^4 - 21*e^3 + 24*e^2 + 55*e - 2, -e^5 + 4*e^4 + 4*e^3 - 19*e^2 - 6*e + 4, -1/2*e^5 + 3*e^4 - 5*e^3 - 19/2*e^2 + 57/2*e + 25/2, -3*e^5 + 11*e^4 + 11*e^3 - 40*e^2 - 25*e - 1, -e^4 + 3*e^3 + 8*e^2 - 16*e - 12, 1/2*e^5 - e^4 - 2*e^3 - 5/2*e^2 + 7/2*e + 25/2, -e^5 + 2*e^4 + 5*e^3 - 9*e - 15, 2*e^5 - 5*e^4 - 17*e^3 + 20*e^2 + 46*e + 28, -e^5 + 3*e^4 + 8*e^3 - 19*e^2 - 19*e + 8, -e^5 - 2*e^4 + 24*e^3 + 6*e^2 - 81*e - 38, -e^4 + 11*e^2 - 2*e - 18, 3*e^5 - 6*e^4 - 30*e^3 + 32*e^2 + 83*e + 8, 2*e^4 - 7*e^3 - 13*e^2 + 37*e + 23, -2*e^5 + 6*e^4 + 12*e^3 - 21*e^2 - 30*e - 13, -2*e^5 + 4*e^4 + 16*e^3 - 8*e^2 - 50*e - 35, 3*e^2 - 8*e - 13, -3/2*e^5 + e^4 + 20*e^3 - 9/2*e^2 - 111/2*e - 53/2, -e^5 - e^4 + 25*e^3 - 8*e^2 - 79*e - 15, 2*e^5 - 5*e^4 - 16*e^3 + 22*e^2 + 38*e + 9, 2*e^5 - 2*e^4 - 26*e^3 + 12*e^2 + 76*e + 16, -e^5 - e^4 + 21*e^3 - 6*e^2 - 56*e - 5, -2*e^5 + 6*e^4 + 12*e^3 - 19*e^2 - 37*e - 26, -7*e^4 + 24*e^3 + 21*e^2 - 74*e - 28, -e^5 + 5*e^4 - 25*e^2 + 12*e + 28, 3/2*e^5 - e^4 - 23*e^3 + 17/2*e^2 + 145/2*e + 49/2, -3*e^5 + 12*e^4 + 3*e^3 - 32*e^2 + e - 15, 6*e^4 - 22*e^3 - 12*e^2 + 65*e + 7, -3*e^5 + 14*e^4 + e^3 - 52*e^2 + 8*e + 26, 7/2*e^5 - 11*e^4 - 18*e^3 + 79/2*e^2 + 77/2*e + 33/2, -e^5 + 3*e^4 + 4*e^3 - 11*e^2 - 7*e + 14, -e^5 + 4*e^4 + 3*e^3 - 19*e^2 - 2*e + 13, -e^5 + 3*e^4 + 6*e^3 - 8*e^2 - 17*e - 11, e^3 + 3*e^2 - 13*e - 1, e^5 + e^4 - 20*e^3 - e^2 + 65*e + 20, 2*e^5 - 11*e^4 + 8*e^3 + 32*e^2 - 36*e - 13, 3*e^5 - 7*e^4 - 24*e^3 + 28*e^2 + 61*e + 15, -2*e^5 + 8*e^4 + 3*e^3 - 24*e^2 - 7*e - 16, 4*e^4 - 17*e^3 - 11*e^2 + 60*e + 32, 3*e^5 - 11*e^4 - 9*e^3 + 33*e^2 + 22*e + 14, 4*e^5 - 15*e^4 - 16*e^3 + 65*e^2 + 28*e - 30, e^5 - 19*e^3 + 11*e^2 + 52*e - 3, -1/2*e^5 + 2*e^4 - 2*e^3 + 11/2*e^2 + 11/2*e - 35/2, e^5 + e^4 - 23*e^3 + 7*e^2 + 68*e + 2, -2*e^5 + 8*e^4 - 2*e^3 - 13*e^2 + 14*e - 15, -3/2*e^5 + e^4 + 25*e^3 - 29/2*e^2 - 157/2*e - 5/2, e^5 - e^4 - 12*e^3 + 10*e^2 + 29*e - 17, -2*e^5 + 9*e^4 + 4*e^3 - 40*e^2 - 2*e + 25, e^5 - 3*e^4 - 4*e^3 + 8*e^2 + 11*e + 3, -e^5 + 9*e^4 - 17*e^3 - 24*e^2 + 66*e + 15, -3*e^5 + 2*e^4 + 40*e^3 - 2*e^2 - 133*e - 57, -4*e^5 + 12*e^4 + 23*e^3 - 44*e^2 - 57*e - 8, 4*e^4 - 15*e^3 - 16*e^2 + 49*e + 44, -3*e^4 + 10*e^3 + 8*e^2 - 28*e - 15, 1/2*e^5 - 8*e^4 + 23*e^3 + 39/2*e^2 - 147/2*e - 41/2, -2*e^5 + 9*e^4 + 2*e^3 - 37*e^2 + 8*e + 38, 5/2*e^5 - 9*e^4 - 9*e^3 + 87/2*e^2 + 5/2*e - 79/2, 5*e^5 - 16*e^4 - 24*e^3 + 56*e^2 + 52*e + 23, -2*e^5 + 9*e^4 - e^3 - 24*e^2 + 9*e - 13, 5*e^5 - 13*e^4 - 39*e^3 + 54*e^2 + 112*e + 19, 5*e^5 - 14*e^4 - 31*e^3 + 50*e^2 + 84*e + 16, e^5 - 3*e^4 - 6*e^3 + 13*e^2 + 19*e - 2, 3*e^5 - 8*e^4 - 18*e^3 + 22*e^2 + 51*e + 40, 3*e^5 - 8*e^4 - 25*e^3 + 43*e^2 + 60*e - 15, -2*e^5 + e^4 + 29*e^3 - 5*e^2 - 97*e - 32, 1/2*e^5 + 5*e^4 - 31*e^3 - 5/2*e^2 + 207/2*e + 15/2, 4*e^5 - 9*e^4 - 36*e^3 + 38*e^2 + 104*e + 21, 3*e^5 - 14*e^4 - 2*e^3 + 48*e^2 - 3*e + 3, -1/2*e^5 + 8*e^4 - 20*e^3 - 41/2*e^2 + 119/2*e + 49/2, -2*e^5 + 14*e^4 - 17*e^3 - 48*e^2 + 66*e + 54, 7/2*e^5 - 12*e^4 - 10*e^3 + 55/2*e^2 + 37/2*e + 83/2, -2*e^5 + 2*e^4 + 27*e^3 - 8*e^2 - 83*e - 46, 3*e^5 - 3*e^4 - 39*e^3 + 19*e^2 + 110*e + 27, -13/2*e^5 + 23*e^4 + 25*e^3 - 171/2*e^2 - 95/2*e + 17/2, -3*e^5 + 5*e^4 + 37*e^3 - 30*e^2 - 116*e - 8, -1/2*e^5 + 8*e^4 - 20*e^3 - 47/2*e^2 + 133/2*e + 21/2, e^5 - 5*e^4 - 2*e^3 + 28*e^2 + 5*e - 37, e^5 - 8*e^4 + 18*e^3 + 9*e^2 - 60*e + 15, -3/2*e^5 + 4*e^4 + 8*e^3 - 7/2*e^2 - 61/2*e - 67/2, e^5 + 2*e^4 - 25*e^3 + 9*e^2 + 72*e - 29, -6*e^5 + 21*e^4 + 24*e^3 - 67*e^2 - 62*e - 36, -3*e^5 + 8*e^4 + 24*e^3 - 27*e^2 - 85*e - 21, -2*e^5 + 9*e^4 + 5*e^3 - 41*e^2 + 8, -9/2*e^5 + 13*e^4 + 31*e^3 - 115/2*e^2 - 159/2*e - 11/2, 2*e^5 - 9*e^4 - 3*e^3 + 30*e^2 + 13*e + 15, 5*e^5 - 19*e^4 - 13*e^3 + 60*e^2 + 28*e + 27, 4*e^5 - 12*e^4 - 24*e^3 + 53*e^2 + 54*e - 15, 7*e^4 - 23*e^3 - 26*e^2 + 71*e + 63, e^5 - 7*e^4 + 14*e^3 + 6*e^2 - 40*e + 22, 3*e^5 - 12*e^4 - 2*e^3 + 31*e^2 - 3*e + 9, -7/2*e^5 + 12*e^4 + 14*e^3 - 79/2*e^2 - 85/2*e - 3/2, 2*e^5 - 12*e^4 + 9*e^3 + 43*e^2 - 47*e - 37, -5*e^5 + 16*e^4 + 28*e^3 - 69*e^2 - 60*e + 19, -2*e^5 + 17*e^4 - 29*e^3 - 47*e^2 + 95*e + 22, 2*e^5 - 9*e^4 - 4*e^3 + 38*e^2 - 6*e - 1, e^5 - 6*e^4 + 6*e^3 + 26*e^2 - 35*e - 60, e^5 - 9*e^4 + 11*e^3 + 34*e^2 - 27*e - 25, -3*e^5 + 16*e^4 - 7*e^3 - 54*e^2 + 33*e + 13, -2*e^5 + 17*e^4 - 32*e^3 - 51*e^2 + 126*e + 63, -9/2*e^5 + 19*e^4 + 10*e^3 - 147/2*e^2 - 19/2*e + 51/2, 3*e^5 - 15*e^4 + 3*e^3 + 47*e^2 - 16*e + 18, -6*e^5 + 24*e^4 + 16*e^3 - 88*e^2 - 38*e + 24, -2*e^5 + e^4 + 31*e^3 - 6*e^2 - 100*e - 46, 3*e^4 - 8*e^3 - 18*e^2 + 36*e + 17, -e^5 + 5*e^4 - 7*e^3 - 9*e^2 + 37*e + 4, 11*e^4 - 42*e^3 - 31*e^2 + 147*e + 69, e^5 - 3*e^4 - 2*e^3 + 7*e^2 - 5*e - 8, 2*e^5 - 8*e^4 - 7*e^3 + 35*e^2 + 3*e + 4, -e^5 - 2*e^4 + 29*e^3 - 15*e^2 - 84*e + 7, -2*e^5 - e^4 + 32*e^3 + 11*e^2 - 98*e - 75, e^5 - 5*e^4 - 2*e^3 + 21*e^2 + 5*e - 16, 2*e^5 - 7*e^4 - 9*e^3 + 36*e^2 + 9*e - 17, 3*e^5 - 15*e^4 - e^3 + 64*e^2 - 21*e - 31, -3*e^5 + 13*e^4 - 46*e^2 + 23*e + 29, -5*e^5 + 19*e^4 + 21*e^3 - 82*e^2 - 45*e + 25, -e^5 + 3*e^4 + 7*e^3 - 9*e^2 - 36*e - 18, 4*e^5 - 16*e^4 - 7*e^3 + 52*e^2 + 7*e - 18, -5*e^5 + 19*e^4 + 16*e^3 - 76*e^2 - 17*e + 20, 1/2*e^5 - 10*e^4 + 34*e^3 + 25/2*e^2 - 219/2*e - 13/2, -2*e^5 + 11*e^4 - 9*e^3 - 32*e^2 + 45*e + 18, -e^5 + 24*e^3 - 18*e^2 - 83*e + 12, -2*e^5 + 10*e^4 - 37*e^2 - 2*e + 27, 7/2*e^5 - 13*e^4 - 17*e^3 + 119/2*e^2 + 95/2*e - 71/2, 5*e^5 - 9*e^4 - 59*e^3 + 62*e^2 + 164*e - 1, 3*e^5 - 3*e^4 - 42*e^3 + 18*e^2 + 133*e + 19, -7/2*e^5 + 20*e^4 - 13*e^3 - 113/2*e^2 + 77/2*e - 17/2, e^4 - e^3 - 5*e^2 - 7*e - 11, 15/2*e^5 - 29*e^4 - 20*e^3 + 209/2*e^2 + 59/2*e - 43/2, -4*e^5 + 21*e^4 - 13*e^3 - 68*e^2 + 65*e + 41, -1/2*e^5 - 3*e^4 + 16*e^3 + 39/2*e^2 - 111/2*e - 63/2, e^5 + 2*e^4 - 16*e^3 - 18*e^2 + 52*e + 4, -1/2*e^5 + e^4 + 10*e^3 - 15/2*e^2 - 105/2*e + 29/2, 4*e^5 - 6*e^4 - 46*e^3 + 32*e^2 + 136*e + 10, 3*e^5 - 14*e^4 - 4*e^3 + 61*e^2 - 3*e - 27, -e^5 + 5*e^4 + e^3 - 16*e^2 - 16*e + 10, -1/2*e^5 + 5*e^4 - 5*e^3 - 41/2*e^2 + 15/2*e + 13/2, 4*e^5 - 14*e^4 - 18*e^3 + 49*e^2 + 56*e + 3, 4*e^5 - 21*e^4 + 4*e^3 + 92*e^2 - 36*e - 65]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([17, 17, -w^2 + w + 3])] = -1

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