/*
  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([9, -1, -7, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([1, 1, 1])

primes_array = [
[2, 2, -w^2 + w + 4],\
[3, 3, -w^2 + w + 3],\
[5, 5, w^2 - 4],\
[8, 2, w^3 - w^2 - 4*w + 5],\
[17, 17, -w^2 + w + 5],\
[27, 3, -w^3 + 4*w - 2],\
[29, 29, -w^2 + w + 1],\
[29, 29, -w^3 + 4*w + 2],\
[37, 37, -2*w^3 + 3*w^2 + 9*w - 13],\
[41, 41, w^3 - w^2 - 5*w + 2],\
[43, 43, -w^3 + w^2 + 3*w - 4],\
[61, 61, w^2 - 2*w - 2],\
[61, 61, 2*w^2 - w - 8],\
[67, 67, -4*w^3 + 5*w^2 + 18*w - 22],\
[73, 73, -w^2 - 2*w + 2],\
[79, 79, w^2 - 2*w - 4],\
[89, 89, w^2 + w - 5],\
[89, 89, 2*w^3 - 2*w^2 - 9*w + 8],\
[89, 89, w^3 - 5*w - 5],\
[89, 89, -w^3 + 2*w^2 + 4*w - 4],\
[97, 97, w^3 + w^2 - 5*w - 4],\
[101, 101, -2*w^3 + 2*w^2 + 8*w - 11],\
[103, 103, w^3 - 5*w + 1],\
[109, 109, 2*w - 1],\
[109, 109, -w^3 + 2*w^2 + 3*w - 7],\
[113, 113, 2*w^2 - 7],\
[125, 5, -w^3 + 3*w^2 + 3*w - 8],\
[127, 127, 2*w^3 - 4*w^2 - 11*w + 16],\
[131, 131, 3*w^2 + 2*w - 8],\
[131, 131, -2*w^3 + 4*w^2 + 8*w - 13],\
[137, 137, w^3 - 3*w^2 - 7*w + 10],\
[149, 149, -w^3 + 4*w - 4],\
[149, 149, w^2 - 2*w - 8],\
[157, 157, -w - 4],\
[163, 163, -w^3 + 2*w^2 + 6*w - 10],\
[167, 167, -w^3 + 2*w^2 + 5*w - 11],\
[167, 167, -3*w^2 - 2*w + 10],\
[167, 167, -w^2 - w + 7],\
[167, 167, 2*w^2 - 2*w - 5],\
[173, 173, -3*w^3 + 2*w^2 + 13*w - 11],\
[179, 179, w^3 - 3*w^2 - 5*w + 10],\
[191, 191, -2*w^3 - 3*w^2 + 10*w + 14],\
[191, 191, 3*w^2 + w - 7],\
[193, 193, w^2 - w + 1],\
[193, 193, -w^2 - w - 1],\
[227, 227, 4*w^3 - 4*w^2 - 17*w + 20],\
[227, 227, -3*w^3 + 4*w^2 + 13*w - 17],\
[229, 229, -w^3 + 6*w - 2],\
[233, 233, -3*w^3 + 3*w^2 + 13*w - 16],\
[241, 241, -w^2 + 2*w - 2],\
[263, 263, 2*w^3 - 5*w^2 - 12*w + 16],\
[263, 263, -3*w^3 + 4*w^2 + 13*w - 19],\
[269, 269, w^3 + 2*w^2 - 5*w - 7],\
[269, 269, -w^3 - 2*w^2 + w + 5],\
[281, 281, 2*w^3 - 2*w^2 - 8*w + 1],\
[281, 281, w^2 - 8],\
[289, 17, 2*w^3 - 10*w - 7],\
[307, 307, w^3 + 2*w^2 - 6*w - 8],\
[307, 307, -2*w^3 + 2*w^2 + 8*w - 7],\
[313, 313, 5*w^3 - 6*w^2 - 24*w + 28],\
[313, 313, 2*w^3 - 2*w^2 - 10*w + 5],\
[337, 337, w^3 - 2*w - 2],\
[347, 347, -w^3 + 4*w^2 + w - 13],\
[349, 349, -2*w^3 + w^2 + 9*w - 7],\
[353, 353, 2*w^3 + w^2 - 7*w + 1],\
[359, 359, -3*w^2 - 3*w + 5],\
[359, 359, 3*w^3 - 5*w^2 - 13*w + 16],\
[367, 367, 3*w^3 - 6*w^2 - 15*w + 23],\
[367, 367, -w^3 + 2*w^2 + 6*w - 4],\
[367, 367, -2*w^3 + 2*w^2 + 8*w - 5],\
[367, 367, 2*w^3 - 3*w^2 - 12*w + 16],\
[373, 373, -2*w^3 + 4*w^2 + 12*w - 17],\
[373, 373, 3*w^2 + 4*w - 8],\
[383, 383, -w^3 + 2*w^2 + 3*w - 13],\
[383, 383, 2*w^2 - 2*w - 11],\
[389, 389, 5*w^3 - 6*w^2 - 24*w + 26],\
[389, 389, -w^3 + 3*w - 5],\
[401, 401, w^3 + 2*w^2 - 4*w - 10],\
[401, 401, 2*w^3 - w^2 - 7*w + 5],\
[409, 409, -2*w^2 + 3*w + 10],\
[419, 419, w^2 - 3*w - 1],\
[419, 419, -w^3 + 5*w^2 + 9*w - 14],\
[421, 421, 2*w^3 - 8*w - 7],\
[431, 431, -3*w^2 + 16],\
[431, 431, -3*w^3 + 11*w - 7],\
[433, 433, 4*w^3 - 7*w^2 - 18*w + 26],\
[439, 439, 3*w - 2],\
[439, 439, -w^3 + w^2 + 5*w - 8],\
[443, 443, 2*w^3 - w^2 - 6*w + 4],\
[443, 443, w^2 + 3*w - 1],\
[449, 449, -2*w^3 + 5*w^2 + 7*w - 17],\
[457, 457, 4*w^3 - 6*w^2 - 19*w + 22],\
[479, 479, 3*w^2 - 3*w - 13],\
[479, 479, -4*w^3 + 8*w^2 + 22*w - 31],\
[479, 479, 3*w^3 - 7*w^2 - 17*w + 26],\
[479, 479, -6*w^3 + 7*w^2 + 28*w - 32],\
[487, 487, -2*w^3 + 4*w^2 + 7*w - 8],\
[491, 491, 3*w^2 - w - 11],\
[491, 491, 4*w^3 - 5*w^2 - 18*w + 20],\
[521, 521, -3*w^3 + 11*w - 5],\
[521, 521, 3*w^3 - 3*w^2 - 15*w + 14],\
[523, 523, -2*w^3 + 5*w^2 + 8*w - 14],\
[529, 23, -2*w^3 + 3*w^2 + 8*w - 14],\
[529, 23, 3*w^3 - 2*w^2 - 14*w + 4],\
[541, 541, -w^3 + 2*w^2 + 7*w - 11],\
[541, 541, -w^3 - w^2 + 5*w - 2],\
[547, 547, -3*w - 2],\
[547, 547, w^3 - 2*w^2 - 7*w + 7],\
[547, 547, w^3 - 4*w^2 - 8*w + 10],\
[547, 547, 2*w^3 - 3*w^2 - 7*w + 11],\
[563, 563, 2*w^3 - 4*w^2 - 12*w + 13],\
[563, 563, -w^3 + 4*w^2 + w - 7],\
[571, 571, -w^3 + 2*w^2 + 2*w - 8],\
[571, 571, 3*w^2 - 10],\
[577, 577, 2*w^3 + w^2 - 7*w - 5],\
[599, 599, 2*w^3 - 7*w + 2],\
[599, 599, -4*w^3 + 7*w^2 + 21*w - 29],\
[617, 617, -2*w^3 + w^2 + 11*w + 1],\
[631, 631, 3*w^2 + w - 13],\
[631, 631, 3*w^2 - 4*w - 10],\
[647, 647, 3*w^3 - 4*w^2 - 13*w + 11],\
[677, 677, -2*w^3 + 4*w^2 + 8*w - 17],\
[691, 691, -w^3 + 3*w^2 + 3*w - 14],\
[733, 733, -w^3 + 4*w^2 + 3*w - 11],\
[733, 733, 3*w^3 - w^2 - 15*w - 2],\
[733, 733, -w^3 + w^2 + 7*w - 2],\
[733, 733, 3*w^3 - 4*w^2 - 16*w + 16],\
[739, 739, -3*w^3 + w^2 + 11*w - 8],\
[743, 743, -3*w^3 + 2*w^2 + 12*w - 14],\
[751, 751, -2*w^3 + w^2 + 8*w - 2],\
[761, 761, -2*w^3 + 2*w^2 + 7*w - 10],\
[761, 761, 2*w^3 - 4*w^2 - 3*w + 8],\
[769, 769, -w^3 + w^2 + 7*w - 4],\
[769, 769, 2*w^3 + 2*w^2 - 8*w - 11],\
[787, 787, -2*w^3 + 3*w^2 + 6*w - 10],\
[787, 787, -2*w^2 - 3*w + 8],\
[797, 797, -2*w^3 + w^2 + 7*w - 1],\
[797, 797, -2*w^3 - 2*w^2 + 11*w + 10],\
[809, 809, -2*w^3 + 5*w^2 + 7*w - 19],\
[811, 811, 2*w^3 - 2*w^2 - 7*w - 2],\
[811, 811, 3*w^3 - 2*w^2 - 11*w - 1],\
[827, 827, 3*w^3 - 2*w^2 - 14*w + 10],\
[829, 829, 2*w^3 - 2*w^2 - 9*w + 2],\
[829, 829, 4*w^2 + 2*w - 11],\
[841, 29, 2*w^3 + w^2 - 6*w - 2],\
[863, 863, 2*w^3 - 8*w - 1],\
[883, 883, 3*w^3 - 4*w^2 - 15*w + 13],\
[887, 887, 3*w^3 - 2*w^2 - 12*w + 10],\
[907, 907, 2*w^3 - 3*w^2 - 9*w + 7],\
[911, 911, 2*w^3 + w^2 - 11*w - 5],\
[919, 919, -2*w^3 + 5*w - 8],\
[929, 929, -w^3 + 4*w^2 + 3*w - 13],\
[937, 937, w^3 - 5*w - 7],\
[937, 937, -2*w^3 + 3*w^2 + 10*w - 8],\
[971, 971, -2*w^3 + 4*w^2 + 11*w - 20],\
[971, 971, -4*w^3 + 4*w^2 + 19*w - 20],\
[977, 977, -4*w^3 + 6*w^2 + 19*w - 28],\
[983, 983, -4*w^2 - 4*w + 7],\
[991, 991, -5*w^3 + 8*w^2 + 26*w - 32],\
[991, 991, -6*w^3 + 6*w^2 + 26*w - 31],\
[997, 997, -3*w^3 + 8*w^2 + 18*w - 26]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [e, e^3 - e^2 - 5*e + 1, -e^3 + 5*e + 1, e^3 - e^2 - 4*e + 3, e - 2, -e^3 + 5*e + 6, e^3 - e^2 - 2*e + 1, e^3 - 6*e + 4, 2*e^3 - e^2 - 8*e, -4*e^3 + 20*e + 5, -3*e^3 + 17*e + 6, 2*e^2 - 7, e^2 - e + 5, e^3 - 9*e, e^3 + 3*e^2 - 6*e - 13, -2*e^3 + 10*e + 2, -3*e^3 + 2*e^2 + 14*e - 8, e^3 - e^2 - 4*e - 1, -3*e^2 - 4*e + 10, 3*e^3 - e^2 - 11*e, -e^3 + 2*e^2 + 6*e + 4, 2*e^2 - 4*e - 6, 2*e^3 - 5*e^2 - 8*e + 17, -2*e^3 + 3*e^2 + 7*e - 3, -5*e^3 - e^2 + 23*e + 15, -2*e^3 + 4*e^2 + 7*e - 8, 2*e^3 - 4*e^2 - 8*e + 13, 3*e^3 - 5*e^2 - 19*e + 13, 5*e^3 - 29*e - 18, -7*e^3 + 5*e^2 + 35*e + 1, 2*e^3 - 2*e^2 - 6*e + 1, 5*e^3 - 2*e^2 - 31*e - 10, e^3 + 3*e - 5, -5*e^3 + 32*e + 10, -2*e^3 - 3*e^2 + 10*e + 11, e^3 + 3*e^2 - 3*e - 13, -3*e^3 - e^2 + 23*e + 19, -3*e^2 + 6*e + 7, -2*e^3 + 5*e^2 + 8*e - 17, e^2 - 5, 7*e^3 - 2*e^2 - 31*e, e^3 - 3*e^2 - 5*e + 3, -2*e^3 - 3*e^2 + 8*e + 11, -7*e^3 - e^2 + 32*e + 7, -4*e^3 - 3*e^2 + 24*e + 19, 2*e^3 - 10*e, 3*e^3 + e^2 - 7*e - 11, 2*e^3 + 5*e^2 - 14*e - 14, e^3 - 3*e^2 - 7*e + 17, -4*e^3 + 7*e^2 + 20*e - 11, 3*e^3 - 5*e^2 - 17*e + 7, e^3 - e^2 - 5*e - 11, -8*e^3 + 4*e^2 + 36*e + 2, -e^2 - 8*e + 14, -10*e^3 + 8*e^2 + 52*e + 1, 7*e^3 + 2*e^2 - 37*e - 18, -2*e^3 + 6*e^2 + 6*e - 17, e^3 - 2*e^2 - 3*e + 2, -4*e^3 + 7*e^2 + 12*e - 23, -e^3 + 2*e^2 - 5*e - 2, 3*e^3 + 3*e^2 - 14*e - 7, -6*e^3 + 3*e^2 + 31*e - 11, 4*e^3 + 2*e^2 - 20*e - 24, -2*e^3 - 3*e^2 + 6*e + 5, -4*e^3 + 2*e^2 + 20*e - 7, -5*e^3 - 3*e^2 + 19*e + 23, -7*e^3 - 3*e^2 + 37*e + 23, e^3 + 3*e^2 - 13*e - 21, 10*e^3 - e^2 - 58*e - 25, e^2 + 6*e + 1, -6*e^3 + 38*e + 8, -3*e^3 + 7*e^2 + 13*e - 38, 4*e^3 - 4*e^2 - 10*e + 9, 3*e^3 + 3*e^2 - 21*e - 35, -3*e^3 - 2*e^2 + 31*e + 16, 6*e^3 + 4*e^2 - 28*e - 13, -5*e^3 + 11*e^2 + 22*e - 31, 2*e^3 - 6*e^2 - 5*e + 26, e^3 - 6*e^2 - 6*e + 22, 7*e^3 - 49*e - 23, -6*e^3 + 2*e^2 + 28*e + 4, -7*e^3 + 7*e^2 + 37*e - 7, 3*e^2 - 3*e - 21, 4*e^3 - 8*e^2 - 24*e + 26, 8*e^3 + 4*e^2 - 44*e - 28, e^3 - 3*e^2 - 11*e + 25, 6*e^3 + 4*e^2 - 40*e - 12, 9*e^3 + 3*e^2 - 57*e - 33, -e^3 - 6*e^2 + 9*e + 22, 6*e^3 - 4*e^2 - 24*e + 16, 4*e^3 + e^2 - 28*e + 9, -e^3 + 2*e^2 + 3*e - 18, -4*e^3 + 2*e^2 + 18*e + 6, -11*e^3 - 4*e^2 + 57*e + 30, -e^3 + 11*e^2 + e - 37, e^3 - e^2 - 11*e - 15, -3*e^3 + 6*e^2 + 19*e - 18, -6*e^3 - 6*e^2 + 36*e + 22, 7*e^3 + 3*e^2 - 45*e - 31, -2*e^3 + 5*e^2 + 9*e - 23, e^3 + 9*e^2 - 5*e - 30, -11*e^3 + 4*e^2 + 61*e + 24, 7*e^3 + 4*e^2 - 37*e - 9, -e^3 + 12*e^2 + 6*e - 28, 15*e^3 - 11*e^2 - 75*e + 2, 5*e^3 - 4*e^2 - 23*e + 2, -11*e^3 + 55*e + 34, 6*e^3 - 5*e^2 - 24*e + 27, -7*e^3 + e^2 + 29*e - 5, -8*e^3 + 6*e^2 + 28*e - 20, 8*e^3 + 3*e^2 - 52*e - 33, -7*e^3 - 8*e^2 + 37*e + 34, -10*e^3 + e^2 + 50*e + 3, -7*e^3 + e^2 + 35*e + 1, 12*e^3 - 60*e - 8, -5*e^3 - e^2 + 17*e + 13, -4*e^3 - 10*e^2 + 18*e + 30, -6*e^3 + 7*e^2 + 22*e - 29, -8*e^3 - 5*e^2 + 36*e + 37, 2*e^3 + 6*e^2 - 20*e - 28, e^3 + 9*e^2 - 3*e - 29, 2*e^3 - 4*e^2 - 11*e + 30, 6*e^3 + 2*e^2 - 38*e - 20, 9*e^3 - 6*e^2 - 38*e + 30, -17*e^3 + e^2 + 77*e + 18, -8*e^3 + 10*e^2 + 41*e - 28, 4*e^3 - 12*e^2 - 16*e + 17, 7*e^3 + e^2 - 35*e + 1, 6*e + 12, 7*e^3 - 8*e^2 - 37*e - 2, -8*e^3 + 6*e^2 + 36*e, -13*e^3 - 6*e^2 + 64*e + 40, -5*e^3 + 2*e^2 + 19*e - 20, e^3 + e^2 - 5*e - 16, -8*e^3 - 5*e^2 + 48*e + 21, 7*e^3 + 7*e^2 - 49*e - 31, 3*e^3 - 8*e^2 + 6*e + 40, -4*e^3 + 20*e - 4, 3*e^3 - 4*e^2 - 6*e + 18, -6*e^3 + 10*e^2 + 22*e - 32, -11*e^3 + 5*e^2 + 63*e + 25, -8*e^3 - 6*e^2 + 38*e + 32, 6*e^3 - 6*e^2 - 43*e + 18, 11*e^3 + 3*e^2 - 65*e - 13, 9*e^3 + 6*e^2 - 47*e - 28, 9*e^3 - 11*e^2 - 51*e + 23, 5*e^3 + 5*e^2 - 29*e - 25, -3*e^3 - e^2 + 7*e + 5, -7*e^3 + 13*e^2 + 29*e - 25, 9*e^3 + 4*e^2 - 45*e - 14, -9*e^3 + 2*e^2 + 27*e + 10, -9*e^3 - 5*e^2 + 35*e + 42, -6*e^3 + 5*e^2 + 23*e + 17, -2*e^3 + 5*e^2 - 2*e - 33, -21*e^3 + e^2 + 95*e + 21, 6*e^3 - 4*e^2 - 38*e - 26, -6*e^3 + 11*e^2 + 36*e - 36, -9*e^3 - e^2 + 65*e + 21, -11*e^3 + 18*e^2 + 41*e - 50, -2*e^3 + 8*e^2 + 16*e - 26, 12*e^2 - 22*e - 47]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}

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