/*
  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([-28, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([8,8,w + 4])

primes_array = [
[2, 2, -w + 6],\
[2, 2, w + 5],\
[7, 7, 6*w - 35],\
[7, 7, -6*w - 29],\
[9, 3, 3],\
[11, 11, 4*w + 19],\
[11, 11, 4*w - 23],\
[13, 13, -2*w + 11],\
[13, 13, 2*w + 9],\
[25, 5, -5],\
[31, 31, 2*w - 13],\
[31, 31, -2*w - 11],\
[41, 41, -8*w - 39],\
[41, 41, 8*w - 47],\
[53, 53, -26*w - 125],\
[53, 53, 26*w - 151],\
[61, 61, -14*w + 81],\
[61, 61, -14*w - 67],\
[83, 83, 2*w - 15],\
[83, 83, -2*w - 13],\
[97, 97, 2*w - 5],\
[97, 97, -2*w - 3],\
[109, 109, 2*w - 3],\
[109, 109, -2*w - 1],\
[113, 113, 2*w - 1],\
[127, 127, 8*w - 45],\
[127, 127, 8*w + 37],\
[131, 131, -50*w + 291],\
[131, 131, 50*w + 241],\
[139, 139, 6*w - 37],\
[139, 139, -6*w - 31],\
[149, 149, 20*w + 97],\
[149, 149, 20*w - 117],\
[157, 157, -12*w - 59],\
[157, 157, 12*w - 71],\
[163, 163, 4*w - 19],\
[163, 163, 4*w + 15],\
[173, 173, 4*w + 23],\
[173, 173, 4*w - 27],\
[211, 211, 2*w - 19],\
[211, 211, -2*w - 17],\
[227, 227, -4*w - 13],\
[227, 227, 4*w - 17],\
[233, 233, 6*w + 25],\
[233, 233, -6*w + 31],\
[239, 239, -14*w - 69],\
[239, 239, 14*w - 83],\
[241, 241, 46*w + 221],\
[241, 241, -46*w + 267],\
[251, 251, 22*w - 129],\
[251, 251, 22*w + 107],\
[257, 257, -34*w + 197],\
[257, 257, -34*w - 163],\
[277, 277, 4*w - 29],\
[277, 277, -4*w - 25],\
[283, 283, 4*w - 15],\
[283, 283, -4*w - 11],\
[289, 17, -17],\
[307, 307, 68*w - 395],\
[307, 307, 68*w + 327],\
[311, 311, 10*w - 61],\
[311, 311, -10*w - 51],\
[313, 313, -32*w - 155],\
[313, 313, 32*w - 187],\
[317, 317, -18*w - 85],\
[317, 317, 18*w - 103],\
[331, 331, -4*w - 9],\
[331, 331, 4*w - 13],\
[337, 337, -16*w - 79],\
[337, 337, 16*w - 95],\
[347, 347, -12*w + 67],\
[347, 347, 12*w + 55],\
[353, 353, 14*w - 79],\
[353, 353, 14*w + 65],\
[361, 19, -19],\
[367, 367, -32*w - 153],\
[367, 367, -32*w + 185],\
[383, 383, 56*w + 269],\
[383, 383, 56*w - 325],\
[389, 389, -4*w - 27],\
[389, 389, 4*w - 31],\
[401, 401, 8*w - 51],\
[401, 401, -8*w - 43],\
[421, 421, 12*w - 73],\
[421, 421, -12*w - 61],\
[439, 439, -8*w - 33],\
[439, 439, 8*w - 41],\
[443, 443, -4*w - 1],\
[443, 443, 4*w - 5],\
[461, 461, -30*w + 173],\
[461, 461, 30*w + 143],\
[463, 463, 2*w - 25],\
[463, 463, -2*w - 23],\
[467, 467, 34*w + 165],\
[467, 467, 34*w - 199],\
[503, 503, -26*w - 127],\
[503, 503, 26*w - 153],\
[509, 509, 4*w - 33],\
[509, 509, -4*w - 29],\
[521, 521, -10*w + 53],\
[521, 521, 10*w + 43],\
[529, 23, -23],\
[547, 547, 14*w - 85],\
[547, 547, -14*w - 71],\
[557, 557, 66*w + 317],\
[557, 557, 66*w - 383],\
[563, 563, 2*w - 27],\
[563, 563, -2*w - 25],\
[569, 569, -64*w + 373],\
[569, 569, 64*w + 309],\
[587, 587, 12*w + 53],\
[587, 587, -12*w + 65],\
[593, 593, 8*w + 45],\
[593, 593, 8*w - 53],\
[601, 601, -26*w - 123],\
[601, 601, 26*w - 149],\
[617, 617, 6*w - 23],\
[617, 617, -6*w - 17],\
[647, 647, 24*w - 137],\
[647, 647, -24*w - 113],\
[653, 653, 28*w - 165],\
[653, 653, -28*w - 137],\
[677, 677, -22*w - 103],\
[677, 677, 22*w - 125],\
[691, 691, 20*w - 113],\
[691, 691, 20*w + 93],\
[709, 709, -10*w - 41],\
[709, 709, 10*w - 51],\
[719, 719, -8*w + 37],\
[719, 719, -8*w - 29],\
[727, 727, -22*w - 109],\
[727, 727, 22*w - 131],\
[739, 739, 46*w - 269],\
[739, 739, -46*w - 223],\
[761, 761, -6*w - 13],\
[761, 761, 6*w - 19],\
[769, 769, 110*w - 639],\
[769, 769, 110*w + 529],\
[773, 773, 4*w - 37],\
[773, 773, -4*w - 33],\
[787, 787, 2*w - 31],\
[787, 787, -2*w - 29],\
[809, 809, 56*w + 271],\
[809, 809, -56*w + 327],\
[821, 821, 6*w - 17],\
[821, 821, -6*w - 11],\
[823, 823, 38*w - 223],\
[823, 823, -38*w - 185],\
[827, 827, 132*w - 767],\
[827, 827, 132*w + 635],\
[841, 29, -29],\
[853, 853, -76*w + 443],\
[853, 853, 76*w + 367],\
[863, 863, 14*w - 87],\
[863, 863, -14*w - 73],\
[911, 911, 2*w - 33],\
[911, 911, -2*w - 31],\
[919, 919, -6*w - 41],\
[919, 919, 6*w - 47],\
[929, 929, 50*w + 239],\
[929, 929, 50*w - 289],\
[953, 953, 6*w - 11],\
[953, 953, -6*w - 5],\
[967, 967, -8*w - 25],\
[967, 967, 8*w - 33],\
[991, 991, 16*w + 71],\
[991, 991, -16*w + 87]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - x^5 - 10*x^4 + 7*x^3 + 22*x^2 - 2*x - 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, e, -e^5 + e^4 + 9*e^3 - 7*e^2 - 16*e + 2, -1/2*e^5 + 5*e^3 + 1/2*e^2 - 21/2*e - 5/2, -1/2*e^5 + e^4 + 5*e^3 - 13/2*e^2 - 23/2*e + 1/2, 1/2*e^5 - e^4 - 5*e^3 + 13/2*e^2 + 21/2*e + 1/2, e^3 + e^2 - 7*e - 3, -1/2*e^5 + 4*e^3 + 1/2*e^2 - 9/2*e - 7/2, -e^5 + e^4 + 10*e^3 - 7*e^2 - 22*e + 3, e^3 - 5*e - 2, 3/2*e^5 - 14*e^3 + 1/2*e^2 + 53/2*e + 3/2, -e^5 + e^4 + 10*e^3 - 7*e^2 - 23*e, -2*e^2 + e + 9, -e^5 + 2*e^4 + 12*e^3 - 13*e^2 - 34*e + 2, 3/2*e^5 - e^4 - 14*e^3 + 17/2*e^2 + 47/2*e - 13/2, -e^5 + 8*e^3 - e^2 - 12*e + 6, e^5 - 10*e^3 + 25*e + 2, -e^5 + 10*e^3 + 2*e^2 - 25*e - 8, -e^5 + e^4 + 8*e^3 - 6*e^2 - 9*e + 7, -3/2*e^5 + 14*e^3 - 3/2*e^2 - 53/2*e + 7/2, -2*e^5 + 3*e^4 + 21*e^3 - 20*e^2 - 47*e + 3, 2*e^5 - e^4 - 19*e^3 + 8*e^2 + 33*e - 1, 3/2*e^5 - 14*e^3 + 3/2*e^2 + 55/2*e - 25/2, -e^4 + 7*e^2 - e - 5, -3*e^5 + 28*e^3 - e^2 - 55*e + 1, -7/2*e^5 + 4*e^4 + 36*e^3 - 59/2*e^2 - 165/2*e + 7/2, -e^5 - e^4 + 12*e^3 + 6*e^2 - 31*e - 5, 3*e^5 - 3*e^4 - 31*e^3 + 20*e^2 + 66*e - 3, e^4 + e^3 - 7*e^2 - 5*e - 2, -1/2*e^5 + 5*e^3 - 3/2*e^2 - 21/2*e + 23/2, -3*e^5 + 3*e^4 + 27*e^3 - 21*e^2 - 46*e + 8, -e^5 - e^4 + 7*e^3 + 10*e^2 - 8*e - 17, -e^4 - 3*e^3 + 7*e^2 + 19*e, 3/2*e^5 - 2*e^4 - 15*e^3 + 29/2*e^2 + 65/2*e - 23/2, 3*e^5 - 5*e^4 - 33*e^3 + 37*e^2 + 81*e - 11, 2*e^5 - e^4 - 18*e^3 + 8*e^2 + 36*e - 7, -e^4 + 4*e^2 + 9, -e^4 + 7*e^2 + 3*e + 7, -1/2*e^5 + 2*e^3 - 9/2*e^2 + 11/2*e + 35/2, -4*e^5 + 3*e^4 + 41*e^3 - 20*e^2 - 91*e - 5, e^5 + e^4 - 9*e^3 - 5*e^2 + 12*e + 4, -3*e^5 + 4*e^4 + 28*e^3 - 29*e^2 - 47*e + 19, 2*e^3 - 2*e^2 - 6*e + 10, 2*e^3 - 2*e^2 - 5*e + 13, 4*e^5 - 4*e^4 - 38*e^3 + 30*e^2 + 71*e - 7, -3/2*e^5 + 3*e^4 + 18*e^3 - 43/2*e^2 - 93/2*e + 17/2, 2*e^4 + 2*e^3 - 17*e^2 - 8*e + 25, -5/2*e^5 + 2*e^4 + 27*e^3 - 33/2*e^2 - 143/2*e + 3/2, -7/2*e^5 + 4*e^4 + 39*e^3 - 55/2*e^2 - 197/2*e + 5/2, -3*e^5 + 3*e^4 + 30*e^3 - 19*e^2 - 63*e + 8, -e^5 + e^4 + 14*e^3 - 5*e^2 - 43*e - 18, -4*e^4 - 4*e^3 + 27*e^2 + 24*e - 9, -2*e^5 + 4*e^4 + 24*e^3 - 27*e^2 - 70*e + 9, 4*e^5 - 5*e^4 - 36*e^3 + 36*e^2 + 64*e - 13, e^5 - e^4 - 6*e^3 + 5*e^2 - 5*e, -2*e^4 - 2*e^3 + 12*e^2 + 8*e, 3/2*e^5 - e^4 - 16*e^3 + 13/2*e^2 + 73/2*e + 17/2, -2*e^5 + e^4 + 18*e^3 - 11*e^2 - 30*e + 10, 5*e^5 - 7*e^4 - 54*e^3 + 47*e^2 + 139*e - 6, 2*e^5 + e^4 - 18*e^3 - 6*e^2 + 24*e + 9, -5/2*e^5 + e^4 + 20*e^3 - 17/2*e^2 - 55/2*e + 19/2, 3*e^5 - 2*e^4 - 26*e^3 + 18*e^2 + 35*e - 24, -1/2*e^5 - e^3 + 7/2*e^2 + 53/2*e - 17/2, -3/2*e^5 - 2*e^4 + 15*e^3 + 25/2*e^2 - 57/2*e + 1/2, 1/2*e^5 + 3*e^4 + e^3 - 33/2*e^2 - 45/2*e - 35/2, -3*e^5 + 2*e^4 + 27*e^3 - 13*e^2 - 53*e, -2*e^5 + 3*e^4 + 26*e^3 - 24*e^2 - 78*e + 19, -1/2*e^5 + 2*e^3 + 1/2*e^2 + 17/2*e + 19/2, 2*e^5 - 3*e^4 - 20*e^3 + 20*e^2 + 43*e + 14, -2*e^5 + e^4 + 24*e^3 - 4*e^2 - 69*e - 6, 4*e^5 - 5*e^4 - 39*e^3 + 35*e^2 + 79*e - 6, e^5 + e^4 - 13*e^3 - 2*e^2 + 40*e - 7, -5*e^5 + 2*e^4 + 49*e^3 - 16*e^2 - 103*e + 9, 5*e^5 - 6*e^4 - 47*e^3 + 42*e^2 + 95*e - 9, 3/2*e^5 - e^4 - 15*e^3 + 21/2*e^2 + 69/2*e - 37/2, 3*e^5 - 3*e^4 - 26*e^3 + 18*e^2 + 39*e - 7, -4*e^5 + e^4 + 38*e^3 - 7*e^2 - 74*e + 6, -e^5 + 2*e^4 + 9*e^3 - 14*e^2 - 26*e + 2, 7/2*e^5 + e^4 - 33*e^3 - 17/2*e^2 + 135/2*e + 19/2, -3/2*e^5 + 3*e^4 + 14*e^3 - 43/2*e^2 - 59/2*e - 1/2, 3*e^5 - 2*e^4 - 32*e^3 + 14*e^2 + 76*e - 3, 1/2*e^5 + e^4 - e^3 - 25/2*e^2 - 21/2*e + 53/2, 7/2*e^5 - e^4 - 33*e^3 + 17/2*e^2 + 133/2*e - 1/2, 2*e^4 - e^3 - 10*e^2 + 7*e - 4, e^5 - 2*e^4 - 7*e^3 + 11*e^2 + 6*e - 7, 3*e^5 - 2*e^4 - 29*e^3 + 13*e^2 + 58*e - 11, -e^5 + 9*e^3 + e^2 - 24*e + 7, e^5 + 2*e^4 - 6*e^3 - 14*e^2 - 5*e + 6, -3/2*e^5 + e^4 + 14*e^3 - 19/2*e^2 - 61/2*e + 13/2, -e^4 - 2*e^3 + 12*e^2 + 10*e - 5, 2*e^5 - 5*e^4 - 22*e^3 + 34*e^2 + 52*e + 1, 2*e^5 - 2*e^4 - 16*e^3 + 17*e^2 + 10*e - 11, 3*e^5 - 2*e^4 - 26*e^3 + 18*e^2 + 47*e - 16, 9/2*e^5 - 2*e^4 - 47*e^3 + 27/2*e^2 + 217/2*e + 29/2, e^5 + 3*e^4 - 13*e^3 - 25*e^2 + 42*e + 32, 6*e^5 - 6*e^4 - 57*e^3 + 37*e^2 + 107*e + 9, 5*e^5 - 4*e^4 - 49*e^3 + 28*e^2 + 104*e - 4, -2*e^5 + 2*e^4 + 18*e^3 - 16*e^2 - 24*e + 8, -6*e^4 - 2*e^3 + 42*e^2 + 14*e - 38, -2*e^5 + 6*e^4 + 23*e^3 - 36*e^2 - 65*e - 8, 2*e^5 - 2*e^4 - 17*e^3 + 8*e^2 + 19*e + 20, 4*e^5 - 5*e^4 - 38*e^3 + 30*e^2 + 79*e + 2, 3/2*e^5 - 17*e^3 + 7/2*e^2 + 99/2*e + 13/2, 4*e^5 - 5*e^4 - 37*e^3 + 33*e^2 + 71*e - 10, -5/2*e^5 + 23*e^3 - 3/2*e^2 - 67/2*e - 27/2, 5*e^5 - 3*e^4 - 51*e^3 + 23*e^2 + 109*e - 11, -3*e^5 + 26*e^3 + 2*e^2 - 47*e + 2, 2*e^5 - 12*e^3 - e^2 - 2*e + 9, -7*e^5 + 9*e^4 + 72*e^3 - 59*e^2 - 161*e, -5*e^5 + 5*e^4 + 52*e^3 - 33*e^2 - 115*e - 2, e^4 + 5*e^3 - 5*e^2 - 25*e + 4, -e^5 - 5*e^4 + 3*e^3 + 40*e^2 + 14*e - 39, -e^5 + 3*e^4 + 11*e^3 - 20*e^2 - 22*e - 13, e^5 - 5*e^4 - 13*e^3 + 38*e^2 + 36*e - 35, 2*e^5 + 4*e^4 - 20*e^3 - 27*e^2 + 46*e + 9, 8*e^5 - 8*e^4 - 76*e^3 + 55*e^2 + 152*e - 5, -5*e^5 + 3*e^4 + 51*e^3 - 17*e^2 - 113*e + 9, 5*e^5 - 9*e^4 - 49*e^3 + 61*e^2 + 113*e - 17, -3*e^5 + 6*e^4 + 29*e^3 - 37*e^2 - 54*e + 11, -e^5 - 4*e^4 + 7*e^3 + 29*e^2 - 2*e - 45, 8*e^5 - 5*e^4 - 73*e^3 + 35*e^2 + 137*e - 12, -6*e^5 + 3*e^4 + 55*e^3 - 23*e^2 - 93*e + 2, 2*e^5 - 12*e^3 + 7*e^2 - 39, -e^5 + 14*e^3 - 45*e - 26, e^5 + 3*e^4 - 11*e^3 - 18*e^2 + 22*e - 9, 2*e^5 - 3*e^4 - 25*e^3 + 25*e^2 + 71*e - 18, -7*e^5 + 12*e^4 + 70*e^3 - 82*e^2 - 156*e + 27, 3/2*e^5 - e^4 - 20*e^3 + 3/2*e^2 + 115/2*e + 57/2, 17/2*e^5 - 9*e^4 - 77*e^3 + 123/2*e^2 + 261/2*e - 53/2, 2*e^5 - 6*e^4 - 25*e^3 + 44*e^2 + 73*e - 20, -3/2*e^5 + 12*e^3 - 7/2*e^2 - 57/2*e + 59/2, -4*e^5 + 7*e^4 + 46*e^3 - 45*e^2 - 132*e + 4, 7/2*e^5 - 6*e^4 - 32*e^3 + 89/2*e^2 + 125/2*e - 73/2, 2*e^5 - 7*e^4 - 20*e^3 + 44*e^2 + 48*e - 19, 3/2*e^5 - 7*e^4 - 14*e^3 + 99/2*e^2 + 59/2*e - 63/2, -15/2*e^5 + 7*e^4 + 66*e^3 - 99/2*e^2 - 227/2*e + 11/2, -4*e^5 + 34*e^3 - 3*e^2 - 49*e - 10, 3*e^5 - 6*e^4 - 42*e^3 + 38*e^2 + 140*e + 3, 5*e^5 - 3*e^4 - 51*e^3 + 23*e^2 + 118*e - 26, -2*e^5 + e^4 + 19*e^3 - 12*e^2 - 23*e + 27, -2*e^5 + 6*e^4 + 17*e^3 - 45*e^2 - 25*e + 33, 7/2*e^5 - 5*e^4 - 41*e^3 + 79/2*e^2 + 215/2*e - 41/2, -2*e^5 + 9*e^4 + 21*e^3 - 62*e^2 - 58*e + 28, -3*e^5 + 7*e^4 + 29*e^3 - 53*e^2 - 49*e + 29, -5*e^5 + 6*e^4 + 44*e^3 - 43*e^2 - 73*e + 17, 2*e^5 - 2*e^4 - 26*e^3 + 20*e^2 + 76*e - 36, -8*e^5 + 7*e^4 + 73*e^3 - 53*e^2 - 133*e + 18, -5*e^5 + 5*e^4 + 47*e^3 - 36*e^2 - 86*e + 27, 4*e^5 - 43*e^3 + 3*e^2 + 113*e + 3, 3/2*e^5 - 3*e^4 - 17*e^3 + 39/2*e^2 + 83/2*e + 35/2, 3*e^5 - 3*e^4 - 30*e^3 + 21*e^2 + 75*e - 8, 13/2*e^5 - 10*e^4 - 66*e^3 + 129/2*e^2 + 301/2*e - 43/2, e^5 + 5*e^4 - 12*e^3 - 34*e^2 + 30*e + 34, -4*e^5 + 4*e^4 + 37*e^3 - 19*e^2 - 65*e - 25, -e^5 - 2*e^4 + 5*e^3 + 12*e^2 + 12*e + 6, -8*e^5 + 11*e^4 + 84*e^3 - 73*e^2 - 192*e + 2, -3*e^5 + 3*e^4 + 32*e^3 - 14*e^2 - 85*e - 29, 11*e^5 - 7*e^4 - 98*e^3 + 49*e^2 + 167*e - 18, -2*e^5 + 3*e^4 + 16*e^3 - 24*e^2 - 18*e + 25, 2*e^5 + 4*e^4 - 17*e^3 - 30*e^2 + 29*e + 42, -4*e^5 + 8*e^4 + 43*e^3 - 52*e^2 - 109*e, -9/2*e^5 + 3*e^4 + 39*e^3 - 43/2*e^2 - 139/2*e + 35/2, 9/2*e^5 - 3*e^4 - 45*e^3 + 27/2*e^2 + 187/2*e + 41/2, 4*e^5 - 4*e^4 - 37*e^3 + 39*e^2 + 69*e - 47, -e^5 + 2*e^4 + 19*e^3 - 10*e^2 - 78*e - 4, 11*e^5 - 12*e^4 - 101*e^3 + 87*e^2 + 190*e - 31, 7*e^5 - 2*e^4 - 67*e^3 + 19*e^2 + 128*e + 3]
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 + 6])] = 1

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