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

NN = ZF.ideal([13, 13, -2*w^3 + 3*w^2 + 10*w - 2])

primes_array = [
[5, 5, -w^3 + 2*w^2 + 3*w],\
[7, 7, w - 1],\
[11, 11, -w^3 + 2*w^2 + 4*w],\
[13, 13, -2*w^3 + 3*w^2 + 10*w - 2],\
[16, 2, 2],\
[17, 17, -w^3 + 2*w^2 + 5*w - 3],\
[17, 17, -w^3 + w^2 + 5*w],\
[17, 17, -w^2 + 2*w + 1],\
[19, 19, w^2 - w - 2],\
[29, 29, w^3 - 2*w^2 - 5*w],\
[29, 29, w^2 - w - 3],\
[31, 31, -w^3 + 2*w^2 + 3*w - 2],\
[43, 43, 2*w^3 - 3*w^2 - 11*w],\
[47, 47, w^3 - 7*w - 4],\
[53, 53, -2*w^3 + 3*w^2 + 9*w - 1],\
[61, 61, -w - 3],\
[73, 73, -w^3 + 2*w^2 + 3*w - 3],\
[73, 73, w^3 - w^2 - 7*w - 1],\
[81, 3, -3],\
[83, 83, -2*w^3 + 3*w^2 + 9*w + 1],\
[83, 83, -w^3 + 3*w^2 + 2*w - 3],\
[89, 89, w - 4],\
[103, 103, w^3 - 2*w^2 - 6*w + 3],\
[113, 113, 2*w^3 - w^2 - 14*w - 6],\
[121, 11, -w^2 + 2*w + 7],\
[125, 5, -3*w^3 + 4*w^2 + 15*w + 1],\
[139, 139, w^3 - 2*w^2 - 5*w - 3],\
[139, 139, w^3 - 5*w - 5],\
[151, 151, w^3 - 9*w - 7],\
[157, 157, -3*w^3 + 6*w^2 + 13*w - 6],\
[163, 163, w^3 - 3*w^2 - 2*w + 9],\
[167, 167, -w^3 + 7*w + 3],\
[167, 167, -3*w^3 + 4*w^2 + 15*w - 1],\
[173, 173, 2*w^3 - 3*w^2 - 7*w - 2],\
[181, 181, -w^3 + 3*w^2 + 4*w - 5],\
[191, 191, w^3 - w^2 - 4*w - 4],\
[193, 193, -w^3 + w^2 + 7*w + 6],\
[193, 193, 2*w^3 - 2*w^2 - 13*w - 5],\
[199, 199, -2*w^3 + 5*w^2 + 5*w - 5],\
[211, 211, w^3 - w^2 - 8*w - 3],\
[211, 211, -2*w^2 + 3*w + 8],\
[227, 227, -w^3 + 3*w^2 + 4*w - 4],\
[227, 227, w^2 - 3*w - 6],\
[229, 229, w^2 - 5],\
[229, 229, 3*w^3 - 4*w^2 - 17*w - 2],\
[233, 233, 3*w^3 - 6*w^2 - 13*w + 3],\
[233, 233, 3*w^3 - 4*w^2 - 16*w + 1],\
[241, 241, -2*w^3 + 4*w^2 + 8*w + 1],\
[251, 251, 2*w^3 - 2*w^2 - 9*w - 3],\
[277, 277, 3*w^3 - 4*w^2 - 17*w - 1],\
[281, 281, -2*w^2 + 5*w + 1],\
[283, 283, -4*w^3 + 6*w^2 + 19*w - 2],\
[293, 293, -3*w^3 + 6*w^2 + 12*w - 2],\
[307, 307, w^3 - w^2 - 4*w - 5],\
[307, 307, w^3 - w^2 - 8*w - 2],\
[311, 311, 2*w^2 - 3*w - 7],\
[313, 313, -2*w^3 + w^2 + 15*w + 3],\
[317, 317, -w^3 + 3*w^2 + w - 5],\
[331, 331, 2*w^3 - 3*w^2 - 12*w],\
[337, 337, w^3 - w^2 - 8*w - 1],\
[343, 7, -w^3 + 6*w + 8],\
[347, 347, 2*w^3 - 4*w^2 - 7*w + 5],\
[349, 349, 2*w^3 - 4*w^2 - 9*w],\
[353, 353, -3*w^3 + 4*w^2 + 16*w + 5],\
[353, 353, -3*w^3 + 4*w^2 + 14*w],\
[353, 353, -w^3 + 3*w^2 + 4*w - 8],\
[353, 353, -w^3 + 2*w^2 + 8*w - 3],\
[359, 359, -4*w^3 + 7*w^2 + 19*w - 3],\
[367, 367, -w^3 + 2*w^2 + 6*w - 5],\
[373, 373, 5*w^3 - 8*w^2 - 26*w + 3],\
[379, 379, -3*w^3 + 5*w^2 + 17*w - 4],\
[379, 379, -w^3 + 2*w^2 + 4*w - 6],\
[383, 383, w^2 - 6],\
[389, 389, w^3 - 2*w^2 - 3*w - 4],\
[389, 389, w^2 + w - 4],\
[397, 397, -4*w^3 + 5*w^2 + 23*w + 5],\
[397, 397, -3*w^3 + 5*w^2 + 12*w - 2],\
[419, 419, -3*w^3 + 4*w^2 + 17*w + 5],\
[419, 419, -3*w^3 + 5*w^2 + 15*w],\
[419, 419, 3*w^3 - 5*w^2 - 14*w - 1],\
[419, 419, 2*w^3 - 3*w^2 - 10*w - 6],\
[431, 431, 2*w^3 - 2*w^2 - 11*w],\
[433, 433, w^3 - 8*w - 1],\
[433, 433, 2*w^2 - 2*w - 5],\
[439, 439, 2*w^3 - 2*w^2 - 9*w - 6],\
[443, 443, 2*w^3 - 5*w^2 - 7*w],\
[443, 443, 2*w^3 - 5*w^2 - 9*w + 4],\
[457, 457, -2*w^3 + 5*w^2 + 7*w - 4],\
[457, 457, -w^3 + 2*w^2 - 3],\
[461, 461, -4*w^3 + 7*w^2 + 17*w - 7],\
[479, 479, 3*w^3 - 5*w^2 - 12*w + 6],\
[487, 487, -2*w^3 + 5*w^2 + 9*w - 9],\
[487, 487, w^3 - 10*w - 8],\
[499, 499, 2*w^3 - 2*w^2 - 12*w + 1],\
[503, 503, w^2 - 7],\
[503, 503, w^3 + w^2 - 7*w - 8],\
[529, 23, -3*w^3 + 5*w^2 + 12*w + 1],\
[529, 23, 3*w^3 - 4*w^2 - 14*w - 1],\
[541, 541, w^3 - 4*w^2 - 2*w + 11],\
[563, 563, -2*w^3 + w^2 + 15*w + 6],\
[569, 569, w^3 - 2*w^2 - 4*w - 4],\
[601, 601, 3*w^3 - 5*w^2 - 17*w + 1],\
[607, 607, -2*w^3 + 5*w^2 + 7*w - 5],\
[607, 607, -w^3 + 3*w^2 + 6*w - 5],\
[613, 613, -4*w^3 + 5*w^2 + 21*w + 1],\
[613, 613, -5*w^3 + 8*w^2 + 23*w - 4],\
[619, 619, w^3 + w^2 - 9*w - 5],\
[619, 619, -3*w^3 + 4*w^2 + 14*w + 2],\
[641, 641, -6*w^3 + 8*w^2 + 31*w + 3],\
[643, 643, 3*w^3 - 4*w^2 - 17*w - 6],\
[643, 643, -2*w^3 + 5*w^2 + 7*w - 6],\
[647, 647, 3*w^3 - 4*w^2 - 13*w - 10],\
[653, 653, -3*w^3 + 5*w^2 + 14*w + 2],\
[653, 653, w^3 - 2*w^2 - w - 3],\
[661, 661, 5*w^3 - 9*w^2 - 24*w + 5],\
[683, 683, 2*w^3 - 3*w^2 - 13*w - 2],\
[683, 683, 4*w^3 - 7*w^2 - 18*w + 1],\
[719, 719, -w^3 + 2*w^2 + 3*w - 7],\
[719, 719, 3*w^3 - 5*w^2 - 16*w - 1],\
[727, 727, -2*w^3 + 5*w^2 + 8*w - 7],\
[727, 727, 2*w^3 - 5*w^2 - 7*w + 13],\
[733, 733, -2*w^2 + 7],\
[733, 733, -3*w^3 + 5*w^2 + 11*w + 2],\
[739, 739, -6*w^3 + 9*w^2 + 29*w - 3],\
[739, 739, -2*w^3 + 3*w^2 + 7*w + 5],\
[751, 751, 4*w^3 - 5*w^2 - 23*w - 3],\
[757, 757, -w^3 + 10*w + 4],\
[757, 757, 2*w^3 - w^2 - 14*w - 4],\
[769, 769, -w^3 + 4*w^2 + 4*w - 8],\
[769, 769, 4*w^3 - 6*w^2 - 21*w - 2],\
[787, 787, 3*w^3 - 3*w^2 - 19*w - 6],\
[787, 787, -2*w^2 + 6*w + 7],\
[839, 839, -3*w^3 + 3*w^2 + 19*w + 5],\
[841, 29, 3*w^3 - 4*w^2 - 13*w - 4],\
[853, 853, -3*w^3 + 5*w^2 + 14*w + 3],\
[853, 853, 2*w^2 - w - 9],\
[853, 853, 3*w^3 - 5*w^2 - 11*w + 1],\
[853, 853, w - 6],\
[857, 857, 2*w^3 - 2*w^2 - 9*w - 7],\
[857, 857, -4*w^3 + 8*w^2 + 13*w - 1],\
[859, 859, 2*w^3 - 4*w^2 - 13*w + 2],\
[859, 859, w^3 - w^2 - 9*w - 2],\
[863, 863, -2*w^3 - w^2 + 17*w + 11],\
[877, 877, 5*w^3 - 6*w^2 - 28*w - 9],\
[877, 877, -4*w^3 + 5*w^2 + 20*w + 6],\
[881, 881, 4*w^3 - 5*w^2 - 18*w - 8],\
[911, 911, -w^3 + 4*w^2 - w - 8],\
[911, 911, 2*w^3 - 3*w^2 - 13*w - 1],\
[919, 919, w^3 - w^2 - 7*w + 5],\
[929, 929, 2*w^3 - 3*w^2 - 13*w + 1],\
[929, 929, -2*w^3 + 5*w^2 + 6*w - 11],\
[941, 941, -3*w^3 + 7*w^2 + 12*w - 5],\
[953, 953, -2*w^3 + 5*w^2 + 5*w - 7],\
[971, 971, w^3 - w^2 - 3*w - 5],\
[971, 971, 3*w^3 - 7*w^2 - 12*w + 11],\
[977, 977, 3*w^3 - 5*w^2 - 11*w],\
[997, 997, -2*w^3 + 3*w^2 + 6*w - 5],\
[997, 997, -4*w^3 + 7*w^2 + 16*w - 2]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^5 + 4*x^4 - 4*x^3 - 21*x^2 - x + 20
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 2*e^4 + 5*e^3 - 15*e^2 - 17*e + 23, -e^2 - 2*e + 4, -1, -2*e^4 - 5*e^3 + 14*e^2 + 16*e - 19, -3*e^4 - 7*e^3 + 24*e^2 + 25*e - 34, -e^2 - e + 4, e^4 + 3*e^3 - 6*e^2 - 10*e + 9, -2*e^4 - 5*e^3 + 15*e^2 + 20*e - 20, -e^4 - 2*e^3 + 9*e^2 + 6*e - 16, 2*e^4 + 4*e^3 - 18*e^2 - 12*e + 31, -2*e^4 - 5*e^3 + 14*e^2 + 17*e - 17, e^4 + 2*e^3 - 9*e^2 - 7*e + 12, 3*e^4 + 7*e^3 - 24*e^2 - 22*e + 40, 3*e^4 + 7*e^3 - 25*e^2 - 25*e + 44, -3*e^4 - 6*e^3 + 29*e^2 + 21*e - 54, 3*e^4 + 8*e^3 - 20*e^2 - 25*e + 32, e^4 + 3*e^3 - 8*e^2 - 14*e + 14, 5*e^4 + 11*e^3 - 40*e^2 - 35*e + 65, -4*e^4 - 8*e^3 + 35*e^2 + 27*e - 54, 2*e^4 + 7*e^3 - 9*e^2 - 24*e + 11, -4*e^4 - 9*e^3 + 33*e^2 + 29*e - 52, -4*e^4 - 9*e^3 + 30*e^2 + 27*e - 44, -5*e^4 - 14*e^3 + 32*e^2 + 48*e - 47, 3*e^4 + 7*e^3 - 26*e^2 - 28*e + 40, e^4 + e^3 - 14*e^2 - 8*e + 34, 5*e^4 + 9*e^3 - 46*e^2 - 28*e + 78, -6*e^4 - 14*e^3 + 48*e^2 + 45*e - 74, -2*e^4 - 7*e^3 + 11*e^2 + 27*e - 12, -e^4 - 4*e^3 + 3*e^2 + 16*e + 7, 5*e^4 + 13*e^3 - 35*e^2 - 45*e + 50, -2*e^4 - 5*e^3 + 15*e^2 + 21*e - 32, 4*e^4 + 7*e^3 - 41*e^2 - 26*e + 74, 3*e^4 + 9*e^3 - 16*e^2 - 35*e + 14, 5*e^4 + 12*e^3 - 38*e^2 - 40*e + 61, -12*e^4 - 26*e^3 + 99*e^2 + 85*e - 158, -4*e^4 - 11*e^3 + 27*e^2 + 43*e - 35, -2*e^4 - 5*e^3 + 14*e^2 + 9*e - 24, -6*e^4 - 13*e^3 + 51*e^2 + 50*e - 74, -11*e^4 - 26*e^3 + 86*e^2 + 87*e - 124, 8*e^4 + 18*e^3 - 61*e^2 - 56*e + 87, -2*e^4 - e^3 + 27*e^2 - 4*e - 47, 12*e^4 + 28*e^3 - 93*e^2 - 92*e + 140, 5*e^4 + 11*e^3 - 36*e^2 - 28*e + 48, -7*e^4 - 17*e^3 + 51*e^2 + 54*e - 85, -10*e^4 - 22*e^3 + 83*e^2 + 77*e - 134, e^4 + 2*e^3 - 5*e^2 + 3*e + 5, 13*e^4 + 30*e^3 - 102*e^2 - 105*e + 151, -e^4 - 4*e^3 - e^2 + 10*e + 15, -3*e^4 - 6*e^3 + 27*e^2 + 22*e - 52, 2*e^4 + 9*e^3 - 4*e^2 - 32*e + 2, -4*e^4 - 12*e^3 + 24*e^2 + 41*e - 29, 10*e^4 + 26*e^3 - 72*e^2 - 95*e + 98, 4*e^4 + 7*e^3 - 42*e^2 - 34*e + 81, -8*e^4 - 17*e^3 + 65*e^2 + 58*e - 86, 8*e^4 + 18*e^3 - 69*e^2 - 68*e + 106, -11*e^4 - 25*e^3 + 86*e^2 + 72*e - 141, -11*e^4 - 25*e^3 + 90*e^2 + 87*e - 126, -10*e^4 - 25*e^3 + 77*e^2 + 88*e - 120, 17*e^4 + 39*e^3 - 135*e^2 - 128*e + 206, -5*e^4 - 12*e^3 + 43*e^2 + 39*e - 94, e^4 + 4*e^3 - 5*e^2 - 12*e + 22, 9*e^4 + 18*e^3 - 80*e^2 - 52*e + 135, -3*e^4 - 3*e^3 + 36*e^2 + 12*e - 69, 16*e^4 + 36*e^3 - 128*e^2 - 119*e + 194, 6*e^4 + 16*e^3 - 44*e^2 - 56*e + 65, -3*e^4 - 7*e^3 + 25*e^2 + 31*e - 50, -7*e^4 - 21*e^3 + 45*e^2 + 80*e - 60, -10*e^4 - 22*e^3 + 85*e^2 + 76*e - 150, -11*e^4 - 24*e^3 + 92*e^2 + 87*e - 148, -7*e^4 - 18*e^3 + 49*e^2 + 50*e - 64, -6*e^4 - 12*e^3 + 54*e^2 + 32*e - 89, 3*e^4 + 5*e^3 - 29*e^2 - 21*e + 44, 15*e^4 + 34*e^3 - 122*e^2 - 117*e + 197, -16*e^4 - 35*e^3 + 131*e^2 + 111*e - 216, 16*e^4 + 35*e^3 - 133*e^2 - 114*e + 212, -12*e^4 - 28*e^3 + 93*e^2 + 90*e - 132, 7*e^4 + 17*e^3 - 55*e^2 - 58*e + 84, -19*e^4 - 44*e^3 + 153*e^2 + 148*e - 240, 17*e^4 + 40*e^3 - 131*e^2 - 136*e + 200, -9*e^4 - 19*e^3 + 80*e^2 + 70*e - 125, 8*e^4 + 19*e^3 - 60*e^2 - 60*e + 72, -14*e^4 - 34*e^3 + 102*e^2 + 111*e - 158, -11*e^4 - 27*e^3 + 79*e^2 + 81*e - 112, -4*e^4 - 9*e^3 + 32*e^2 + 32*e - 48, -10*e^4 - 24*e^3 + 74*e^2 + 71*e - 112, -4*e^2 - 8*e + 34, 12*e^4 + 30*e^3 - 86*e^2 - 102*e + 124, 7*e^4 + 13*e^3 - 66*e^2 - 47*e + 98, 14*e^4 + 31*e^3 - 118*e^2 - 108*e + 174, -16*e^4 - 36*e^3 + 131*e^2 + 122*e - 214, -10*e^4 - 24*e^3 + 77*e^2 + 74*e - 131, 4*e^4 + 8*e^3 - 37*e^2 - 30*e + 51, -e^4 - 4*e^3 + 7*e^2 + 26*e - 23, e^4 + 4*e^3 - 3*e - 14, e^4 + 3*e^3 - 7*e^2 - 7*e + 2, -4*e^4 - 10*e^3 + 27*e^2 + 35*e - 28, -15*e^4 - 34*e^3 + 120*e^2 + 123*e - 172, -19*e^4 - 47*e^3 + 141*e^2 + 159*e - 210, 9*e^4 + 24*e^3 - 67*e^2 - 90*e + 101, -6*e^4 - 13*e^3 + 52*e^2 + 47*e - 100, -2*e^4 - 3*e^3 + 19*e^2 + 10*e - 17, 17*e^4 + 36*e^3 - 144*e^2 - 114*e + 237, 15*e^4 + 36*e^3 - 117*e^2 - 120*e + 178, 16*e^4 + 35*e^3 - 132*e^2 - 114*e + 231, 19*e^4 + 44*e^3 - 154*e^2 - 155*e + 239, 20*e^4 + 50*e^3 - 141*e^2 - 159*e + 191, 2*e^4 + 6*e^3 - 16*e^2 - 28*e + 31, 14*e^4 + 33*e^3 - 109*e^2 - 103*e + 191, 7*e^4 + 19*e^3 - 48*e^2 - 78*e + 52, 11*e^4 + 23*e^3 - 96*e^2 - 82*e + 152, -6*e^4 - 12*e^3 + 50*e^2 + 34*e - 88, 8*e^4 + 21*e^3 - 54*e^2 - 57*e + 76, 8*e^4 + 22*e^3 - 50*e^2 - 70*e + 74, 3*e^4 + 8*e^3 - 24*e^2 - 36*e + 39, 9*e^4 + 22*e^3 - 63*e^2 - 67*e + 91, -6*e^4 - 20*e^3 + 32*e^2 + 71*e - 54, -14*e^4 - 32*e^3 + 117*e^2 + 116*e - 181, -9*e^4 - 23*e^3 + 60*e^2 + 75*e - 84, -10*e^4 - 23*e^3 + 84*e^2 + 82*e - 127, -6*e^4 - 17*e^3 + 43*e^2 + 64*e - 60, -10*e^4 - 24*e^3 + 73*e^2 + 80*e - 71, 2*e^4 + 4*e^3 - 13*e^2 - 5*e + 9, -3*e^4 - 6*e^3 + 22*e^2 + 12*e - 15, -12*e^4 - 27*e^3 + 98*e^2 + 85*e - 149, 25*e^4 + 56*e^3 - 209*e^2 - 189*e + 334, 2*e^4 + 6*e^3 - 2*e^2 - 5*e - 18, -19*e^4 - 43*e^3 + 160*e^2 + 153*e - 246, -5*e^4 - 10*e^3 + 45*e^2 + 41*e - 65, -23*e^4 - 58*e^3 + 171*e^2 + 196*e - 255, 12*e^4 + 30*e^3 - 85*e^2 - 92*e + 127, 9*e^4 + 25*e^3 - 63*e^2 - 90*e + 84, -e^4 - 2*e^3 + 8*e^2 + 7*e - 30, -17*e^4 - 41*e^3 + 134*e^2 + 138*e - 200, 3*e^4 + 7*e^3 - 19*e^2 - 9*e + 7, 19*e^4 + 47*e^3 - 145*e^2 - 169*e + 207, 4*e^4 + 6*e^3 - 45*e^2 - 25*e + 78, -11*e^4 - 25*e^3 + 83*e^2 + 77*e - 125, -6*e^4 - 16*e^3 + 43*e^2 + 64*e - 34, 15*e^4 + 35*e^3 - 121*e^2 - 116*e + 211, 5*e^4 + 16*e^3 - 25*e^2 - 59*e + 32, -11*e^4 - 22*e^3 + 98*e^2 + 72*e - 164, 28*e^4 + 64*e^3 - 225*e^2 - 221*e + 351, 8*e^4 + 23*e^3 - 48*e^2 - 73*e + 49, 10*e^4 + 21*e^3 - 80*e^2 - 62*e + 130, -6*e^4 - 18*e^3 + 35*e^2 + 57*e - 25, -10*e^4 - 22*e^3 + 82*e^2 + 58*e - 146, 2*e^4 + 3*e^3 - 25*e^2 - 2*e + 68, 12*e^4 + 29*e^3 - 93*e^2 - 109*e + 130, 6*e^4 + 18*e^3 - 34*e^2 - 60*e + 34, -13*e^4 - 30*e^3 + 102*e^2 + 106*e - 147, 3*e^4 + 7*e^3 - 22*e^2 - 32*e + 5, -3*e^4 - 9*e^3 + 20*e^2 + 40*e - 54, -6*e^4 - 15*e^3 + 43*e^2 + 58*e - 51, 6*e^4 + 17*e^3 - 40*e^2 - 57*e + 77, -e^3 - 4*e^2 + 8*e + 23, -4*e^4 - 4*e^3 + 50*e^2 + 7*e - 102, -15*e^4 - 40*e^3 + 101*e^2 + 131*e - 117]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([13, 13, -2*w^3 + 3*w^2 + 10*w - 2])] = 1

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