/*
  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([31, 31, -w^3 + 2*w^2 + 3*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^4 - x^3 - 15*x^2 + 20*x + 4
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 1/3*e^3 - 4*e + 2/3, 1/3*e^3 - 4*e + 8/3, -2, 1/3*e^3 - 5*e + 11/3, 2, 1/6*e^3 + 1/2*e^2 - 3/2*e - 11/3, 1/3*e^3 - 6*e + 14/3, 1/3*e^3 - 5*e + 2/3, 1/3*e^3 - 5*e + 20/3, -e + 2, -1, -1/3*e^3 + 4*e - 20/3, e^2 - 4, -e^2 - e + 12, -1/3*e^3 + 2*e - 2/3, -1/6*e^3 - 1/2*e^2 + 9/2*e + 17/3, 1/6*e^3 - 1/2*e^2 - 9/2*e + 43/3, 1/6*e^3 + 1/2*e^2 - 5/2*e - 5/3, 1/6*e^3 - 3/2*e^2 - 9/2*e + 49/3, -1/6*e^3 + 3/2*e^2 + 1/2*e - 49/3, -1/6*e^3 - 1/2*e^2 + 11/2*e + 11/3, 2/3*e^3 - 8*e - 2/3, -1/3*e^3 + 7*e - 8/3, -1/2*e^3 + 1/2*e^2 + 17/2*e - 5, -2/3*e^3 + e^2 + 12*e - 22/3, -1/6*e^3 - 1/2*e^2 + 7/2*e + 29/3, -e^3 + 13*e - 14, e^2 - 8, 5/6*e^3 + 1/2*e^2 - 19/2*e + 29/3, 3/2*e^3 + 1/2*e^2 - 43/2*e + 9, 1/3*e^3 - 4*e - 16/3, 2/3*e^3 - 11*e - 2/3, -5/6*e^3 - 1/2*e^2 + 19/2*e - 5/3, -4/3*e^3 + e^2 + 22*e - 26/3, 1/3*e^3 + 2*e^2 - 4*e - 40/3, -1/2*e^3 - 7/2*e^2 + 11/2*e + 25, -2/3*e^3 - e^2 + 6*e + 14/3, -2*e^2 - 4*e + 26, -2/3*e^3 + 2*e^2 + 10*e - 40/3, -e^3 + 12*e - 8, -4/3*e^3 - e^2 + 19*e - 2/3, -e^2 + e + 6, -e^3 - e^2 + 10*e - 6, -1/6*e^3 - 1/2*e^2 - 5/2*e + 35/3, 2/3*e^3 + e^2 - 6*e - 14/3, -2/3*e^3 - e^2 + 9*e - 28/3, -2/3*e^3 + 13*e - 16/3, -11/6*e^3 + 3/2*e^2 + 55/2*e - 59/3, 8/3*e^3 - 38*e + 58/3, 4/3*e^3 + e^2 - 19*e - 4/3, 1/3*e^3 + e^2 - 2*e - 52/3, 4/3*e^3 - 24*e + 62/3, -e^3 - e^2 + 14*e - 4, -e^3 + 14*e + 14, -5/6*e^3 - 7/2*e^2 + 19/2*e + 73/3, -e^3 - e^2 + 12*e - 10, e^3 + 2*e^2 - 16*e - 14, -2*e^3 - 2*e^2 + 28*e - 8, -4/3*e^3 + 18*e + 22/3, 2*e^3 - e^2 - 31*e + 14, 2/3*e^3 + 2*e^2 - 4*e - 38/3, -2*e^3 + 2*e^2 + 28*e - 14, -2/3*e^3 + e^2 + 11*e - 40/3, -e^3 - 2*e^2 + 7*e + 20, 4/3*e^3 - 3*e^2 - 19*e + 92/3, 3/2*e^3 + 1/2*e^2 - 45/2*e + 9, 5/3*e^3 - e^2 - 28*e + 70/3, -8/3*e^3 + 2*e^2 + 37*e - 70/3, -3/2*e^3 + 1/2*e^2 + 47/2*e - 15, 2*e^3 + e^2 - 29*e + 6, 5/3*e^3 + 2*e^2 - 23*e + 22/3, 2*e^2 + 3*e - 2, 1/2*e^3 - 3/2*e^2 - 11/2*e + 15, 5/2*e^3 + 1/2*e^2 - 65/2*e + 17, -13/6*e^3 - 1/2*e^2 + 55/2*e - 37/3, 5/3*e^3 - 2*e^2 - 21*e + 58/3, 5/6*e^3 - 7/2*e^2 - 17/2*e + 101/3, 3*e^2 + 4*e - 36, -7/6*e^3 - 3/2*e^2 + 27/2*e - 19/3, -2/3*e^3 + 8*e - 10/3, 3/2*e^3 - 3/2*e^2 - 47/2*e + 1, -4/3*e^3 + 18*e - 62/3, e^2 + e - 8, -7/3*e^3 - 2*e^2 + 24*e + 28/3, -2/3*e^3 - 3*e^2 + 9*e + 2/3, 5/6*e^3 + 5/2*e^2 - 23/2*e - 73/3, -1/3*e^3 - 3*e - 2/3, 17/6*e^3 - 1/2*e^2 - 81/2*e + 95/3, -e^3 + 15*e - 26, -7/6*e^3 - 1/2*e^2 + 21/2*e + 11/3, e^2 - 2*e - 6, e^3 + 2*e^2 - 6*e - 8, -1/3*e^3 + 2*e^2 + 5*e - 2/3, -2*e^3 - e^2 + 24*e + 4, 11/6*e^3 + 5/2*e^2 - 33/2*e - 43/3, e^2 + e + 12, 2*e^3 - 2*e^2 - 23*e + 24, -7/3*e^3 + 2*e^2 + 35*e - 92/3, -7/6*e^3 + 1/2*e^2 + 27/2*e - 7/3, 2/3*e^3 - e^2 - 6*e + 118/3, -1/3*e^3 - 38/3, 2/3*e^3 - 2*e^2 - 2*e + 88/3, -5/2*e^3 + 1/2*e^2 + 67/2*e - 29, e^3 - e^2 - 16*e - 6, 1/3*e^3 - e^2 - 3*e - 28/3, 4/3*e^3 + e^2 - 10*e + 2/3, -e^3 - 4*e^2 + 9*e + 22, -11/6*e^3 + 3/2*e^2 + 67/2*e - 77/3, -17/6*e^3 + 1/2*e^2 + 91/2*e - 95/3, -8/3*e^3 - 3*e^2 + 33*e + 2/3, -2/3*e^3 - 2*e^2 + 14*e + 56/3, -8/3*e^3 + 37*e - 40/3, -e^3 + 19*e + 8, e^2 - e + 20, -5/3*e^3 + 22*e - 34/3, 4/3*e^3 - 13*e + 62/3, -7/3*e^3 - 2*e^2 + 28*e + 28/3, -5*e^3 + 61*e - 30, -1/2*e^3 - 5/2*e^2 + 17/2*e - 3, -1/3*e^3 - 2*e^2 + 4*e + 52/3, 5/6*e^3 + 3/2*e^2 - 31/2*e + 5/3, -4*e^3 + 3*e^2 + 53*e - 36, -5/3*e^3 + 2*e^2 + 26*e - 130/3, -1/3*e^3 + e^2 + 5*e + 34/3, -5/3*e^3 + 20*e + 20/3, 5/6*e^3 + 3/2*e^2 - 29/2*e + 23/3, -1/6*e^3 - 1/2*e^2 - 7/2*e - 19/3, -2/3*e^3 + 16*e - 10/3, 1/2*e^3 - 3/2*e^2 + 1/2*e + 23, -4/3*e^3 + e^2 + 15*e - 74/3, -1/6*e^3 - 1/2*e^2 + 25/2*e + 71/3, 13/3*e^3 + 2*e^2 - 65*e + 86/3, -1/6*e^3 + 7/2*e^2 + 9/2*e - 115/3, -1/2*e^3 - 1/2*e^2 + 37/2*e + 3, -1/3*e^3 + 5*e^2 + 7*e - 122/3, -1/2*e^3 - 1/2*e^2 + 13/2*e - 17, -2*e^3 + 3*e^2 + 33*e - 26, -11/6*e^3 - 9/2*e^2 + 39/2*e + 127/3, 5/6*e^3 - 9/2*e^2 - 19/2*e + 89/3, 17/6*e^3 + 1/2*e^2 - 73/2*e + 125/3, -5/3*e^3 - 3*e^2 + 18*e + 20/3, -2/3*e^3 - e^2 + 7*e + 2/3, 2*e^3 + 5*e^2 - 23*e - 22, -1/3*e^3 - 4*e^2 + 4*e + 94/3, 7/3*e^3 - 2*e^2 - 37*e + 92/3, 3/2*e^3 - 1/2*e^2 - 57/2*e + 19, 17/6*e^3 + 1/2*e^2 - 69/2*e + 89/3, 4/3*e^3 - 21*e - 10/3, -2*e^3 + e^2 + 17*e - 16, 4/3*e^3 - 13*e + 26/3, 2/3*e^3 + 3*e^2 - 86/3, 7/6*e^3 + 3/2*e^2 - 7/2*e - 59/3, -4/3*e^3 - e^2 + 21*e - 26/3, 10/3*e^3 + 4*e^2 - 38*e - 52/3, 1/6*e^3 + 9/2*e^2 - 7/2*e - 71/3, 5/3*e^3 - e^2 - 15*e + 88/3, e^3 - 4*e^2 - 16*e + 26]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([31, 31, -w^3 + 2*w^2 + 3*w - 2])] = 1

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