/*
  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, 6, 0, -8, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([11, 11, 2*w^4 + w^3 - 16*w^2 - 9*w + 9])

primes_array = [
[2, 2, w],\
[3, 3, 2*w^4 + w^3 - 15*w^2 - 7*w + 7],\
[11, 11, 2*w^4 + w^3 - 16*w^2 - 9*w + 9],\
[17, 17, w^4 + w^3 - 8*w^2 - 6*w + 5],\
[29, 29, w^4 - 8*w^2 + 3],\
[43, 43, 6*w^4 + 3*w^3 - 46*w^2 - 24*w + 21],\
[43, 43, 2*w^4 + w^3 - 15*w^2 - 6*w + 7],\
[47, 47, 3*w^4 + w^3 - 23*w^2 - 9*w + 11],\
[47, 47, w^4 + w^3 - 8*w^2 - 7*w + 3],\
[49, 7, -w^4 + 8*w^2 + 2*w - 5],\
[53, 53, -6*w^4 - 3*w^3 + 46*w^2 + 23*w - 23],\
[53, 53, 2*w^4 + w^3 - 15*w^2 - 7*w + 5],\
[59, 59, 13*w^4 + 7*w^3 - 101*w^2 - 54*w + 55],\
[67, 67, 5*w^4 + 3*w^3 - 39*w^2 - 23*w + 19],\
[67, 67, 4*w^4 + 2*w^3 - 32*w^2 - 16*w + 19],\
[71, 71, 4*w^4 + 3*w^3 - 31*w^2 - 22*w + 15],\
[71, 71, 9*w^4 + 5*w^3 - 70*w^2 - 38*w + 39],\
[71, 71, 4*w^4 + w^3 - 31*w^2 - 8*w + 17],\
[81, 3, -5*w^4 - 2*w^3 + 38*w^2 + 17*w - 19],\
[83, 83, -3*w^4 - w^3 + 24*w^2 + 9*w - 13],\
[89, 89, 7*w^4 + 3*w^3 - 54*w^2 - 24*w + 27],\
[89, 89, 4*w^4 + 2*w^3 - 30*w^2 - 15*w + 11],\
[107, 107, 4*w^4 + 2*w^3 - 32*w^2 - 15*w + 19],\
[109, 109, 4*w^4 + 2*w^3 - 30*w^2 - 16*w + 11],\
[113, 113, 2*w^4 + w^3 - 16*w^2 - 8*w + 13],\
[113, 113, -3*w^4 - 2*w^3 + 23*w^2 + 15*w - 13],\
[137, 137, -2*w^4 - w^3 + 16*w^2 + 6*w - 11],\
[139, 139, 5*w^4 + 3*w^3 - 39*w^2 - 22*w + 21],\
[139, 139, w^2 + w - 3],\
[149, 149, 2*w^4 + w^3 - 16*w^2 - 10*w + 9],\
[151, 151, w^4 + w^3 - 8*w^2 - 9*w + 5],\
[157, 157, -7*w^4 - 3*w^3 + 54*w^2 + 24*w - 25],\
[167, 167, 2*w^4 + w^3 - 16*w^2 - 9*w + 11],\
[167, 167, -14*w^4 - 7*w^3 + 109*w^2 + 54*w - 59],\
[173, 173, 6*w^4 + 3*w^3 - 46*w^2 - 23*w + 25],\
[179, 179, w^3 - 6*w + 1],\
[179, 179, 10*w^4 + 4*w^3 - 77*w^2 - 32*w + 39],\
[179, 179, -3*w^4 - 2*w^3 + 24*w^2 + 16*w - 13],\
[191, 191, 7*w^4 + 4*w^3 - 54*w^2 - 30*w + 29],\
[193, 193, -3*w^4 - w^3 + 24*w^2 + 8*w - 15],\
[197, 197, 7*w^4 + 3*w^3 - 54*w^2 - 26*w + 29],\
[197, 197, -w^4 - w^3 + 8*w^2 + 6*w - 7],\
[199, 199, -2*w^4 - w^3 + 15*w^2 + 6*w - 5],\
[199, 199, 2*w^4 + 2*w^3 - 15*w^2 - 15*w + 3],\
[227, 227, 3*w^4 + w^3 - 23*w^2 - 10*w + 15],\
[229, 229, -7*w^4 - 4*w^3 + 53*w^2 + 31*w - 21],\
[229, 229, -4*w^4 - 2*w^3 + 31*w^2 + 17*w - 17],\
[229, 229, -10*w^4 - 6*w^3 + 78*w^2 + 46*w - 43],\
[233, 233, -w^4 + 7*w^2 + w + 1],\
[239, 239, w^2 - 3],\
[241, 241, 8*w^4 + 4*w^3 - 62*w^2 - 30*w + 31],\
[257, 257, -2*w^4 - 2*w^3 + 15*w^2 + 14*w - 3],\
[269, 269, 7*w^4 + 4*w^3 - 55*w^2 - 30*w + 31],\
[269, 269, -5*w^4 - 3*w^3 + 38*w^2 + 23*w - 19],\
[269, 269, 6*w^4 + 2*w^3 - 47*w^2 - 16*w + 27],\
[277, 277, -6*w^4 - 2*w^3 + 47*w^2 + 18*w - 25],\
[281, 281, -10*w^4 - 4*w^3 + 77*w^2 + 30*w - 41],\
[307, 307, 5*w^4 + 2*w^3 - 38*w^2 - 16*w + 19],\
[313, 313, 18*w^4 + 9*w^3 - 139*w^2 - 69*w + 71],\
[313, 313, -16*w^4 - 8*w^3 + 123*w^2 + 62*w - 57],\
[317, 317, -8*w^4 - 3*w^3 + 61*w^2 + 24*w - 27],\
[317, 317, 10*w^4 + 5*w^3 - 77*w^2 - 40*w + 41],\
[331, 331, 5*w^4 + 2*w^3 - 39*w^2 - 16*w + 19],\
[331, 331, -19*w^4 - 10*w^3 + 147*w^2 + 77*w - 73],\
[337, 337, 4*w^4 + 2*w^3 - 30*w^2 - 16*w + 9],\
[343, 7, -13*w^4 - 6*w^3 + 101*w^2 + 47*w - 53],\
[347, 347, -5*w^4 - 3*w^3 + 39*w^2 + 24*w - 21],\
[347, 347, -9*w^4 - 5*w^3 + 70*w^2 + 39*w - 39],\
[349, 349, w^4 - 9*w^2 + w + 9],\
[353, 353, 2*w^4 - 16*w^2 - 2*w + 9],\
[367, 367, 9*w^4 + 4*w^3 - 70*w^2 - 30*w + 37],\
[383, 383, 5*w^4 + 3*w^3 - 39*w^2 - 21*w + 21],\
[389, 389, -w^4 + 8*w^2 + 2*w - 7],\
[397, 397, -8*w^4 - 4*w^3 + 63*w^2 + 31*w - 35],\
[397, 397, -6*w^4 - 3*w^3 + 47*w^2 + 25*w - 25],\
[401, 401, w^2 - 5],\
[419, 419, -4*w^4 - 2*w^3 + 32*w^2 + 18*w - 19],\
[419, 419, 6*w^4 + 3*w^3 - 46*w^2 - 22*w + 21],\
[433, 433, 6*w^4 + 2*w^3 - 48*w^2 - 16*w + 33],\
[433, 433, 2*w^4 + 2*w^3 - 14*w^2 - 15*w - 1],\
[433, 433, -11*w^4 - 7*w^3 + 84*w^2 + 53*w - 37],\
[457, 457, -8*w^4 - 5*w^3 + 62*w^2 + 37*w - 31],\
[457, 457, -13*w^4 - 7*w^3 + 101*w^2 + 54*w - 53],\
[457, 457, w^4 + w^3 - 7*w^2 - 8*w + 3],\
[463, 463, w^4 - 8*w^2 - w + 9],\
[487, 487, -8*w^4 - 3*w^3 + 62*w^2 + 26*w - 33],\
[491, 491, 6*w^4 + 2*w^3 - 47*w^2 - 17*w + 27],\
[491, 491, 13*w^4 + 6*w^3 - 100*w^2 - 47*w + 49],\
[499, 499, -12*w^4 - 5*w^3 + 94*w^2 + 40*w - 53],\
[499, 499, -w^4 + 8*w^2 - w - 5],\
[503, 503, -4*w^4 - 2*w^3 + 30*w^2 + 18*w - 17],\
[509, 509, -5*w^4 - 2*w^3 + 40*w^2 + 15*w - 25],\
[509, 509, -9*w^4 - 3*w^3 + 71*w^2 + 24*w - 45],\
[521, 521, 4*w^4 + 2*w^3 - 32*w^2 - 16*w + 23],\
[529, 23, 4*w^4 + 3*w^3 - 31*w^2 - 21*w + 17],\
[541, 541, w^4 + 2*w^3 - 9*w^2 - 12*w + 5],\
[547, 547, -7*w^4 - 3*w^3 + 55*w^2 + 24*w - 29],\
[547, 547, -12*w^4 - 6*w^3 + 94*w^2 + 47*w - 53],\
[563, 563, -5*w^4 - 3*w^3 + 38*w^2 + 23*w - 13],\
[563, 563, 2*w^4 + w^3 - 17*w^2 - 6*w + 13],\
[563, 563, -11*w^4 - 6*w^3 + 85*w^2 + 45*w - 45],\
[569, 569, 6*w^4 + 3*w^3 - 47*w^2 - 22*w + 23],\
[577, 577, -2*w^4 + 13*w^2 + 5*w - 3],\
[587, 587, -7*w^4 - 4*w^3 + 55*w^2 + 31*w - 31],\
[587, 587, 7*w^4 + 4*w^3 - 54*w^2 - 29*w + 29],\
[593, 593, 14*w^4 + 7*w^3 - 109*w^2 - 53*w + 61],\
[593, 593, -4*w^4 - w^3 + 31*w^2 + 11*w - 15],\
[599, 599, -9*w^4 - 5*w^3 + 70*w^2 + 40*w - 35],\
[607, 607, -13*w^4 - 6*w^3 + 102*w^2 + 46*w - 61],\
[607, 607, 8*w^4 + 4*w^3 - 63*w^2 - 33*w + 35],\
[607, 607, -6*w^4 - 2*w^3 + 46*w^2 + 15*w - 21],\
[613, 613, -6*w^4 - 3*w^3 + 48*w^2 + 20*w - 25],\
[613, 613, -w^4 + 8*w^2 - w - 7],\
[617, 617, 2*w^4 + 2*w^3 - 15*w^2 - 13*w + 7],\
[619, 619, -w^4 + w^3 + 8*w^2 - 7*w - 5],\
[643, 643, -3*w^4 - 2*w^3 + 24*w^2 + 14*w - 17],\
[647, 647, -9*w^4 - 5*w^3 + 70*w^2 + 38*w - 41],\
[653, 653, -7*w^4 - 2*w^3 + 53*w^2 + 20*w - 29],\
[659, 659, 6*w^4 + 3*w^3 - 47*w^2 - 23*w + 23],\
[677, 677, w^4 + w^3 - 8*w^2 - 5*w + 1],\
[677, 677, 4*w^4 + 3*w^3 - 30*w^2 - 23*w + 11],\
[677, 677, -9*w^4 - 4*w^3 + 71*w^2 + 31*w - 43],\
[683, 683, -14*w^4 - 6*w^3 + 108*w^2 + 47*w - 55],\
[701, 701, 13*w^4 + 7*w^3 - 100*w^2 - 53*w + 51],\
[709, 709, 10*w^4 + 6*w^3 - 77*w^2 - 44*w + 37],\
[719, 719, -9*w^4 - 4*w^3 + 69*w^2 + 32*w - 35],\
[719, 719, 7*w^4 + 4*w^3 - 54*w^2 - 28*w + 31],\
[719, 719, 12*w^4 + 6*w^3 - 93*w^2 - 45*w + 49],\
[727, 727, -7*w^4 - 3*w^3 + 53*w^2 + 25*w - 25],\
[733, 733, 6*w^4 + 3*w^3 - 46*w^2 - 25*w + 19],\
[751, 751, -9*w^4 - 4*w^3 + 69*w^2 + 31*w - 37],\
[761, 761, -13*w^4 - 5*w^3 + 101*w^2 + 39*w - 55],\
[761, 761, 13*w^4 + 8*w^3 - 101*w^2 - 61*w + 53],\
[773, 773, -8*w^4 - 4*w^3 + 62*w^2 + 30*w - 37],\
[773, 773, -w^4 - w^3 + 7*w^2 + 9*w - 1],\
[773, 773, -10*w^4 - 6*w^3 + 77*w^2 + 46*w - 39],\
[809, 809, -8*w^4 - 5*w^3 + 63*w^2 + 36*w - 33],\
[809, 809, -7*w^4 - 3*w^3 + 55*w^2 + 25*w - 33],\
[809, 809, -5*w^4 - 4*w^3 + 38*w^2 + 30*w - 13],\
[811, 811, 9*w^4 + 4*w^3 - 70*w^2 - 33*w + 35],\
[821, 821, -6*w^4 - 3*w^3 + 45*w^2 + 23*w - 21],\
[823, 823, -6*w^4 - 3*w^3 + 47*w^2 + 25*w - 29],\
[827, 827, 12*w^4 + 6*w^3 - 92*w^2 - 45*w + 45],\
[841, 29, 3*w^4 + w^3 - 24*w^2 - 11*w + 15],\
[841, 29, -2*w^4 - 2*w^3 + 15*w^2 + 12*w - 7],\
[853, 853, -3*w^4 - w^3 + 22*w^2 + 7*w - 5],\
[857, 857, w^4 - w^3 - 8*w^2 + 6*w + 3],\
[859, 859, -7*w^4 - 4*w^3 + 55*w^2 + 28*w - 31],\
[859, 859, -5*w^4 - 2*w^3 + 40*w^2 + 16*w - 27],\
[877, 877, -14*w^4 - 7*w^3 + 109*w^2 + 55*w - 61],\
[877, 877, 2*w^4 + 3*w^3 - 15*w^2 - 22*w + 3],\
[881, 881, -8*w^4 - 3*w^3 + 61*w^2 + 23*w - 29],\
[883, 883, 7*w^4 + 3*w^3 - 53*w^2 - 25*w + 21],\
[929, 929, -4*w^4 - 2*w^3 + 30*w^2 + 19*w - 17],\
[947, 947, -9*w^4 - 6*w^3 + 71*w^2 + 45*w - 43],\
[947, 947, -6*w^4 - 4*w^3 + 48*w^2 + 27*w - 27],\
[953, 953, 5*w^4 + 2*w^3 - 40*w^2 - 16*w + 23],\
[961, 31, -3*w^4 - 3*w^3 + 23*w^2 + 23*w - 9]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [e, 0, -1, -2*e - 2, 2*e - 2, e^2 + 2*e, -e^2 - 2*e + 4, -2*e^2 + 4, -3*e^2 - 2*e + 12, -2*e^2 + 6, -2*e - 6, -e^2 + 4*e + 10, e^2 + 2*e, -4*e - 4, e^2 + 2*e - 8, -e^2 + 2*e + 4, 4*e - 4, 2*e^2 - 8, -2*e^2 - 2*e - 2, 2*e^2 + 4*e - 12, e^2 - 2*e - 14, e^2 - 6*e - 6, -4*e - 4, -e^2 - 8*e + 6, 5*e^2 + 2*e - 14, -3*e^2 + 14, -2*e + 14, 2*e^2 - 4*e - 8, -e^2 - 6*e, 10, 2*e^2 - 24, 2*e^2 - 2*e - 6, 3*e^2 + 2*e - 16, -4*e + 4, -4*e^2 - 2*e + 6, 12, 2*e^2 + 4*e + 4, 4*e + 12, e^2 + 2*e + 4, 4*e^2 - 2*e - 14, -2*e^2 + 10, 2*e^2 - 6*e - 14, 6*e^2 - 16, 6*e^2 - 20, -2*e^2 - 4, 2*e^2 - 4*e - 26, -2*e^2 + 2*e - 2, e^2 + 4*e - 22, -2*e^2 + 8*e + 14, e^2 + 6*e - 8, -6*e^2 - 2*e + 26, 2*e^2 + 4*e + 6, -2*e^2 - 2*e + 14, 2*e^2 - 2*e - 18, e^2 - 8*e - 18, 4*e - 14, 7*e^2 + 4*e - 30, 3*e^2 + 2*e, 2*e^2 + 6, 3*e^2 - 4*e - 22, -10, e^2 - 6*e - 22, 4*e^2 - 8*e - 32, -8*e^2 - 4*e + 28, 8*e^2 - 2*e - 30, 4*e^2 + 8*e - 16, -4*e^2 + 4*e + 12, -6*e^2 - 4*e + 16, 14, -2, -e^2 + 2*e - 8, 6*e^2 - 24, -3*e^2 + 2*e + 14, -4*e^2 + 4*e + 18, -6*e^2 - 2*e + 22, -2*e^2 - 2*e - 14, 2*e^2 - 4*e + 12, -4*e^2 - 4*e + 20, -6*e^2 - 2*e + 10, -7*e^2 - 6*e + 34, 7*e^2 - 2*e - 34, -4*e^2 + 8*e + 26, -10*e + 2, 4*e^2 + 2*e - 38, 7*e^2 - 2*e - 36, -4*e^2 + 4*e, -3*e^2 - 6*e + 4, -2*e^2 + 4*e - 12, -e^2 + 10*e, -10*e^2 - 4*e + 36, -3*e^2 - 2*e + 20, 8*e^2 - 4*e - 38, -6*e^2 - 2*e + 26, -6*e^2 + 4*e + 30, -4*e^2 + 4*e + 14, -9*e^2 + 6*e + 42, 3*e^2 + 14*e - 20, -2*e^2 + 12*e + 12, -2*e^2 - 4*e + 20, -2*e^2 - 4*e - 8, e^2 - 2*e - 24, 6*e^2 + 2*e - 34, -e^2 - 2*e - 2, 12*e^2 + 8*e - 44, -8*e^2 + 28, -2*e^2 - 4*e + 22, -2*e^2 - 10*e + 2, -4*e^2 - 16*e + 24, 12*e + 4, 4*e - 28, -e^2 - 2*e - 4, -2*e^2 + 10*e + 30, -3*e^2 + 10*e + 26, -5*e^2 - 8*e + 6, 4*e^2 - 8*e - 4, 12*e - 12, -32, 4*e^2 + 16*e - 30, -3*e^2 - 18*e + 24, e^2 - 22, -2, -6*e^2 - 4*e + 14, 12*e^2 + 8*e - 40, -8*e^2 - 10*e + 42, -e^2 + 16*e + 6, -3*e^2 - 6*e + 28, 4*e - 20, 2*e^2 + 4*e - 40, 14*e^2 + 8*e - 44, -16*e + 2, e^2 - 2*e, 4*e^2 + 16*e - 6, -8*e^2 + 38, -8*e - 10, -2*e^2 + 4*e - 10, 3*e^2 + 2*e + 14, -6*e^2 - 22*e + 34, -6*e^2 + 18*e + 42, -4*e^2 - 8*e + 10, -10*e^2 + 4*e + 40, 5*e^2 + 4*e - 10, -4*e^2 + 12*e + 32, 2*e^2 + 4*e - 28, 4*e^2 + 18*e - 38, 4*e + 30, 2*e^2 + 6*e + 18, -2*e - 10, 8*e^2 - 44, -2*e^2 + 12*e + 16, -3*e^2 + 4*e + 30, -12*e^2 - 8*e + 34, 2*e^2 - 6*e - 42, 6*e^2 + 16*e - 20, 6*e^2 - 2*e - 14, 7*e^2 - 18*e - 40, 8*e^2 + 4*e - 28, 10*e^2 + 2*e - 46, -2*e^2 - 4*e + 18]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([11, 11, 2*w^4 + w^3 - 16*w^2 - 9*w + 9])] = 1

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