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

NN = ZF.ideal([8, 2, 2])

primes_array = [
[3, 3, -w - 2],\
[5, 5, w + 1],\
[7, 7, w - 3],\
[8, 2, 2],\
[9, 3, w^2 - 3*w - 2],\
[13, 13, w + 3],\
[13, 13, -w + 2],\
[19, 19, -w^2 + 2*w + 4],\
[23, 23, w^2 - 4*w + 1],\
[25, 5, w^2 - 2*w - 1],\
[31, 31, w^2 - 2*w - 9],\
[37, 37, -w^2 + 3*w + 3],\
[41, 41, w^2 - w - 1],\
[41, 41, w^2 - w - 5],\
[41, 41, w^2 - w - 10],\
[43, 43, -2*w - 3],\
[47, 47, -w^2 - w + 4],\
[49, 7, -w^2 + 5*w - 5],\
[59, 59, w^2 - w - 4],\
[59, 59, w^2 - 3],\
[59, 59, w - 5],\
[67, 67, 2*w^2 - 4*w - 9],\
[73, 73, 2*w^2 - 3*w - 12],\
[79, 79, -3*w^2 + 9*w + 5],\
[97, 97, 2*w^2 - w - 14],\
[97, 97, 2*w^2 - 2*w - 13],\
[97, 97, -2*w^2 + 5*w + 5],\
[101, 101, w^2 - w - 11],\
[131, 131, w - 6],\
[139, 139, -2*w^2 + 5*w + 4],\
[149, 149, -3*w + 11],\
[149, 149, -3*w - 4],\
[157, 157, w^2 - 3*w - 6],\
[173, 173, w^2 - 5*w + 3],\
[173, 173, -2*w^2 + 5*w + 8],\
[173, 173, 3*w^2 - 5*w - 17],\
[179, 179, 2*w^2 - w - 11],\
[179, 179, -3*w^2 + 7*w + 12],\
[179, 179, w^2 - 2*w - 12],\
[181, 181, w^2 + w - 16],\
[199, 199, 2*w^2 - 9],\
[211, 211, -w^2 + w - 2],\
[229, 229, -w^2 + 3*w - 3],\
[233, 233, w^2 + w - 7],\
[257, 257, 3*w^2 - 3*w - 20],\
[271, 271, w^2 - 4*w - 13],\
[271, 271, 3*w^2 - 4*w - 19],\
[271, 271, 2*w - 9],\
[277, 277, 3*w^2 - 5*w - 20],\
[281, 281, -w^2 - w + 13],\
[283, 283, -3*w + 8],\
[293, 293, -4*w - 7],\
[307, 307, w^2 + 2*w - 5],\
[311, 311, 2*w^2 - 4*w - 5],\
[313, 313, -w^2 - w - 3],\
[317, 317, -3*w^2 + 8*w + 5],\
[331, 331, -w^2 + 2*w - 3],\
[337, 337, -w - 7],\
[337, 337, w^2 + w - 9],\
[337, 337, 3*w - 5],\
[359, 359, 2*w^2 - 4*w - 15],\
[361, 19, 2*w^2 + w - 5],\
[367, 367, -w^2 - 3],\
[367, 367, 3*w^2 - 11*w + 4],\
[367, 367, -2*w^2 + 9*w - 6],\
[373, 373, 2*w^2 - w - 9],\
[379, 379, 3*w^2 - 5*w - 21],\
[383, 383, w - 8],\
[389, 389, -4*w^2 + 9*w + 15],\
[397, 397, 2*w^2 + w - 4],\
[397, 397, -4*w^2 + 13*w + 5],\
[397, 397, -w^2 - 2*w - 4],\
[401, 401, w^2 - 5*w - 4],\
[421, 421, 3*w^2 - 4*w - 18],\
[431, 431, w^2 - 4*w - 16],\
[433, 433, w^2 - 4*w - 7],\
[439, 439, w^2 + 2*w - 17],\
[457, 457, 2*w^2 - 3],\
[457, 457, 4*w - 3],\
[457, 457, 2*w^2 - 3*w - 6],\
[463, 463, 3*w^2 - 12*w + 7],\
[479, 479, w^2 - 6*w + 6],\
[479, 479, -w^2 + 15],\
[479, 479, -6*w^2 + 19*w + 1],\
[487, 487, 2*w^2 - 7*w - 5],\
[487, 487, w^2 + 5*w + 8],\
[487, 487, 3*w^2 - 6*w - 17],\
[491, 491, -4*w + 15],\
[503, 503, 8*w^2 - 25*w - 5],\
[509, 509, 3*w^2 - 12*w + 4],\
[509, 509, -3*w^2 - 5*w + 6],\
[509, 509, w^2 - w - 14],\
[521, 521, -4*w^2 + 10*w + 11],\
[523, 523, 2*w^2 - 7*w + 4],\
[523, 523, 2*w^2 - 2*w - 7],\
[523, 523, -7*w^2 + 22*w + 8],\
[529, 23, 4*w^2 - 14*w + 3],\
[541, 541, w^2 + 4*w - 3],\
[541, 541, -5*w - 9],\
[541, 541, -5*w^2 + 11*w + 20],\
[557, 557, -w^2 + 5*w - 8],\
[563, 563, 2*w^2 - 15],\
[563, 563, -w^2 - 2*w + 20],\
[563, 563, w^2 - 3*w - 15],\
[569, 569, 2*w^2 - w - 5],\
[577, 577, 2*w^2 - 3*w - 21],\
[587, 587, -4*w^2 + 11*w + 6],\
[599, 599, 2*w^2 + 2*w - 9],\
[607, 607, -3*w - 10],\
[613, 613, 4*w^2 - 7*w - 25],\
[631, 631, -3*w^2 + 7*w + 7],\
[641, 641, 3*w^2 - 2*w - 17],\
[641, 641, -3*w^2 + 14*w - 12],\
[641, 641, -5*w^2 + 15*w + 9],\
[643, 643, -5*w - 8],\
[647, 647, 3*w^2 - w - 21],\
[653, 653, w^2 + 7*w + 9],\
[673, 673, -3*w^2 - 2*w + 9],\
[683, 683, 5*w^2 - 17*w + 2],\
[683, 683, 3*w^2 - 6*w - 11],\
[683, 683, 2*w^2 - 7*w - 6],\
[691, 691, w^2 + 3*w + 6],\
[701, 701, 2*w^2 - 5*w - 19],\
[709, 709, -3*w^2 + 9*w + 8],\
[727, 727, 3*w - 13],\
[739, 739, -w - 9],\
[743, 743, 5*w - 3],\
[743, 743, 2*w^2 - 5*w - 13],\
[743, 743, 3*w^2 - 3*w - 17],\
[757, 757, 3*w^2 - 4*w - 16],\
[761, 761, w^2 + 4*w - 4],\
[769, 769, 3*w^2 - w - 15],\
[773, 773, -w^2 + 2*w - 5],\
[787, 787, -2*w^2 + 10*w - 9],\
[797, 797, w^2 - 4*w - 17],\
[811, 811, 6*w^2 - 20*w + 1],\
[811, 811, -w^2 + 4*w - 7],\
[811, 811, 4*w - 9],\
[821, 821, 2*w^2 - 5*w - 14],\
[823, 823, -5*w^2 + 14*w + 8],\
[827, 827, 3*w^2 - w - 22],\
[829, 829, -3*w^2 + 6*w + 10],\
[859, 859, -9*w^2 + 29*w + 2],\
[859, 859, 4*w^2 - 13*w + 2],\
[859, 859, -w^2 + w - 5],\
[863, 863, 2*w^2 + w - 13],\
[877, 877, w^2 + 2*w - 11],\
[881, 881, 2*w^2 - 8*w + 7],\
[883, 883, 3*w^2 + 3*w - 4],\
[907, 907, w^2 + 2*w - 13],\
[929, 929, -3*w - 11],\
[937, 937, w^2 - 5*w - 12],\
[941, 941, 5*w - 4],\
[947, 947, 4*w^2 - 7*w - 20],\
[953, 953, 2*w^2 + 6*w - 3],\
[961, 31, 5*w^2 - 8*w - 29],\
[967, 967, -5*w^2 + 14*w + 7],\
[971, 971, w^2 - 5*w - 10],\
[977, 977, 2*w^2 - 4*w - 23],\
[977, 977, w^2 - 8*w + 17],\
[977, 977, 3*w^2 - 19],\
[991, 991, 3*w^2 - 4*w - 3]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 + x - 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -e + 2, -2*e - 1, -1, e + 2, -2*e - 5, 2*e + 3, 4*e + 4, 6*e + 4, -e, -7, -2*e - 7, -3*e - 6, 5*e + 5, 5*e - 2, -5*e - 7, -8*e - 2, -5, e - 3, 6*e + 1, 6*e - 1, -13, -5*e - 3, -5*e + 5, 6*e, -7, -10*e - 7, -7*e - 15, -9*e - 2, -6*e + 7, 8*e - 9, 18*e + 11, -3*e + 19, -8*e + 11, 5*e - 11, -6*e + 15, -15*e - 7, -7*e + 5, 13*e + 13, 7*e + 14, -7*e + 4, -4*e - 17, -4*e + 13, -2*e - 22, -6*e - 17, -11*e - 11, -8*e - 8, -16*e - 13, 12*e + 3, 2*e - 16, 10*e - 6, 15*e + 3, 11, 18*e + 2, -13*e - 20, 15*e + 14, e + 16, -12*e - 9, 6*e - 15, 5*e + 16, 15*e - 11, 3*e + 3, 10*e + 5, -12*e + 7, -10*e - 11, 18*e + 24, 14*e + 23, -6*e + 3, 15*e + 21, -16*e - 11, -10*e + 8, -e - 32, -10*e + 2, -3*e, 10*e + 25, 2*e - 25, 21*e - 1, -12*e + 13, -14*e - 13, 15*e + 17, -e - 24, 31*e + 11, 13*e + 2, -17*e, -7*e - 25, 5*e - 19, -18*e - 8, -16*e - 16, -27*e - 4, 18*e + 10, 7*e + 10, -e, 9*e - 12, 8*e + 10, -2*e - 37, 14*e + 9, -14*e + 20, 36*e + 16, -10*e - 5, 31, -19*e + 15, -9*e - 17, 22*e + 28, 27*e + 20, -23*e - 19, -20*e - 11, -26*e - 9, 9*e - 24, 14*e + 21, 12*e + 12, -10*e - 15, 9*e + 29, 5*e - 22, 16*e + 19, 19*e + 26, 12*e - 1, -4*e + 26, -10*e - 11, -21*e - 13, -3*e + 1, 11*e + 21, -11*e - 18, -6*e + 31, -27*e - 6, -2*e + 6, -7*e - 2, 19, -12*e - 39, 23*e + 33, 35*e + 14, 2*e + 47, -12*e - 1, -28*e - 38, -16*e - 25, -34*e - 15, 6*e + 15, -8*e - 32, -16*e + 16, -24*e - 11, 20*e, -9*e - 45, 18*e + 29, -6*e - 2, 22*e + 3, -9*e - 28, -13*e + 15, -3*e - 51, 12*e - 7, 14*e - 27, -15*e + 3, -13*e - 32, -30*e + 7, 30*e + 17, -14*e + 19, -18*e - 23, -5*e + 44, 22*e + 44, 9*e - 9, -26*e + 3, -5*e - 54, 38*e + 27, 17*e + 34]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([8, 2, 2])] = 1

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