/*
  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, -4, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([9, 3, w^3 - 2*w^2 - 3*w + 1])

primes_array = [
[2, 2, -w^3 + 2*w^2 + 4*w - 1],\
[9, 3, w^3 - 2*w^2 - 3*w + 1],\
[11, 11, -w^3 + 2*w^2 + 4*w],\
[11, 11, -w + 2],\
[13, 13, w + 2],\
[13, 13, w^3 - 2*w^2 - 4*w + 4],\
[13, 13, -w^2 + 2*w + 2],\
[37, 37, -w^3 + 2*w^2 + 5*w - 3],\
[37, 37, w^3 - 2*w^2 - 5*w + 1],\
[47, 47, -2*w^3 + 2*w^2 + 10*w + 5],\
[47, 47, w^2 - w - 5],\
[59, 59, -w^3 + 3*w^2 + 3*w - 6],\
[59, 59, w^3 - 3*w^2 - 3*w + 2],\
[71, 71, -2*w^3 + 4*w^2 + 7*w],\
[71, 71, w^3 - 2*w^2 - 2*w - 2],\
[73, 73, w^3 - 2*w^2 - 4*w - 2],\
[73, 73, w - 4],\
[83, 83, w^3 - w^2 - 6*w + 1],\
[83, 83, w^2 - 3*w - 3],\
[97, 97, -3*w^3 + 5*w^2 + 12*w + 1],\
[97, 97, -2*w^3 + 5*w^2 + 3*w - 1],\
[107, 107, -2*w^3 + 3*w^2 + 9*w - 1],\
[107, 107, 2*w - 3],\
[107, 107, -w^3 + 3*w^2 - 3],\
[107, 107, 2*w^3 - 4*w^2 - 8*w + 1],\
[109, 109, -3*w^3 + 5*w^2 + 12*w - 1],\
[109, 109, 2*w^3 - 5*w^2 - 3*w + 3],\
[121, 11, -2*w^3 + 4*w^2 + 6*w - 3],\
[131, 131, 2*w^2 - 6*w - 3],\
[131, 131, -3*w^3 + 5*w^2 + 13*w + 2],\
[131, 131, 2*w^3 - 4*w^2 - 5*w - 2],\
[131, 131, -2*w^3 + 2*w^2 + 12*w + 5],\
[167, 167, -2*w^2 + 4*w + 9],\
[167, 167, -w^2 + 3*w + 5],\
[179, 179, -2*w^3 + 5*w^2 + 4*w - 6],\
[179, 179, 2*w^3 - 4*w^2 - 8*w - 1],\
[179, 179, -3*w^3 + 3*w^2 + 17*w + 8],\
[179, 179, -2*w^3 + 3*w^2 + 8*w - 2],\
[193, 193, 6*w^3 - 8*w^2 - 30*w - 7],\
[193, 193, -2*w^3 + w^2 + 13*w + 7],\
[227, 227, 3*w^3 - 4*w^2 - 15*w - 1],\
[227, 227, -3*w^3 + 7*w^2 + 8*w - 3],\
[229, 229, -3*w^3 + w^2 + 20*w + 15],\
[229, 229, -w^3 + 5*w^2 - 4*w - 5],\
[239, 239, -4*w^3 + 5*w^2 + 20*w + 4],\
[239, 239, -5*w^3 + 7*w^2 + 24*w + 3],\
[241, 241, -2*w^3 + 4*w^2 + 9*w - 4],\
[241, 241, -w^3 + 2*w^2 + 6*w - 2],\
[263, 263, -w^3 + 4*w^2 - w - 7],\
[263, 263, -2*w^3 + 5*w^2 + 3*w - 5],\
[263, 263, -w^3 + 7*w + 1],\
[263, 263, -3*w^3 + 5*w^2 + 12*w - 3],\
[277, 277, -2*w^3 + w^2 + 12*w + 8],\
[277, 277, 3*w^3 - 5*w^2 - 14*w + 1],\
[277, 277, 2*w^2 - 2*w - 3],\
[277, 277, -2*w^3 + 6*w^2 + 4*w - 9],\
[289, 17, -w^3 + 4*w^2 - 4],\
[289, 17, 2*w^2 - 3*w - 6],\
[313, 313, -w - 4],\
[313, 313, -2*w^3 + 3*w^2 + 11*w + 1],\
[313, 313, -w^3 + 2*w^2 + 4*w - 6],\
[313, 313, -2*w^3 + 14*w + 13],\
[337, 337, 2*w^3 - 3*w^2 - 11*w + 3],\
[337, 337, -5*w^3 + 6*w^2 + 27*w + 7],\
[337, 337, w^3 - w^2 - 8*w - 1],\
[337, 337, -6*w^3 + 6*w^2 + 33*w + 14],\
[349, 349, -w^3 + 4*w^2 - 6],\
[349, 349, 2*w^2 - 3*w - 4],\
[359, 359, -4*w^3 + 7*w^2 + 16*w + 2],\
[359, 359, -9*w^3 + 13*w^2 + 42*w + 9],\
[373, 373, w^3 + w^2 - 10*w - 5],\
[373, 373, 2*w^2 - 7*w],\
[373, 373, 3*w^2 - 7*w - 5],\
[373, 373, 3*w^3 - 4*w^2 - 16*w - 2],\
[383, 383, 2*w^3 - 6*w^2 - 3*w + 2],\
[383, 383, w^3 - 6*w - 8],\
[397, 397, -3*w^3 + 7*w^2 + 9*w - 12],\
[397, 397, -2*w^3 + 2*w^2 + 13*w],\
[409, 409, -2*w^3 + 4*w^2 + 5*w - 8],\
[409, 409, -3*w^3 + 6*w^2 + 10*w - 10],\
[419, 419, -w^3 + 3*w^2 + 4*w - 7],\
[419, 419, w^3 - 2*w^2 - w - 3],\
[419, 419, 3*w^3 - 6*w^2 - 11*w + 1],\
[419, 419, 2*w^3 - 5*w^2 - 7*w + 3],\
[421, 421, -w^3 + 4*w^2 + w - 7],\
[421, 421, -w^3 + 4*w^2 + w - 5],\
[431, 431, w^2 - 6],\
[431, 431, -2*w^3 + 5*w^2 + 6*w - 2],\
[443, 443, 2*w^3 - 5*w^2 - 7*w - 1],\
[443, 443, 6*w^3 - 6*w^2 - 33*w - 16],\
[443, 443, w^3 + w^2 - 11*w - 6],\
[443, 443, w^3 + w^2 - 11*w - 2],\
[457, 457, 2*w^3 - 5*w^2 - 3*w - 3],\
[457, 457, -3*w^3 + 5*w^2 + 12*w + 5],\
[479, 479, -2*w^2 + 5*w + 6],\
[479, 479, -w^3 + 8*w],\
[503, 503, -2*w^3 + 5*w^2 + 9*w - 9],\
[503, 503, -3*w^3 + 7*w^2 + 12*w - 5],\
[503, 503, -w^3 + 5*w^2 - 4*w - 7],\
[503, 503, -2*w^3 + 6*w^2 + 6*w - 11],\
[529, 23, 3*w^3 - 6*w^2 - 9*w + 1],\
[529, 23, -3*w^3 + 6*w^2 + 9*w - 5],\
[541, 541, -2*w^3 + w^2 + 12*w + 6],\
[541, 541, 2*w^3 - 7*w^2 + 6],\
[563, 563, -3*w^3 + 7*w^2 + 9*w - 4],\
[563, 563, 2*w^2 - 2*w - 9],\
[563, 563, w^3 - w^2 - 3*w - 4],\
[563, 563, -3*w^3 + 8*w^2 + 6*w - 10],\
[577, 577, -6*w^3 + 9*w^2 + 26*w + 2],\
[577, 577, w^3 - 3*w^2 - 2*w + 9],\
[587, 587, -w^3 + 3*w^2 - w - 4],\
[587, 587, -3*w^3 + 5*w^2 + 13*w - 4],\
[599, 599, -w^3 + 2*w^2 + 7*w - 7],\
[599, 599, 3*w^3 - 6*w^2 - 13*w + 1],\
[599, 599, -6*w^3 + 10*w^2 + 26*w + 1],\
[599, 599, -4*w^3 + 7*w^2 + 17*w + 1],\
[601, 601, -2*w^3 + 4*w^2 + 11*w - 8],\
[601, 601, 5*w^3 - 7*w^2 - 25*w - 4],\
[601, 601, -4*w^3 + 9*w^2 + 11*w - 9],\
[601, 601, -3*w^3 + 2*w^2 + 18*w + 10],\
[613, 613, -4*w^3 + 7*w^2 + 18*w - 2],\
[613, 613, -w^3 - w^2 + 8*w + 9],\
[625, 5, -5],\
[647, 647, 3*w^3 - 2*w^2 - 17*w - 13],\
[647, 647, w^2 + w - 5],\
[647, 647, -2*w^3 + 2*w^2 + 9*w + 6],\
[647, 647, -3*w^3 + 7*w^2 + 10*w - 5],\
[661, 661, -2*w^3 + 4*w^2 + 8*w - 9],\
[661, 661, -2*w - 5],\
[683, 683, -2*w^3 + 5*w^2 + 8*w - 2],\
[683, 683, -2*w^3 + 5*w^2 + 8*w - 10],\
[709, 709, -3*w^3 + 6*w^2 + 8*w],\
[709, 709, 3*w^2 - 6*w - 4],\
[733, 733, w^3 - 2*w^2 - 4*w - 4],\
[733, 733, w - 6],\
[743, 743, -3*w^3 + 4*w^2 + 14*w],\
[743, 743, -2*w^3 + 6*w^2 + w - 6],\
[769, 769, -2*w^3 + 4*w^2 + 10*w - 3],\
[769, 769, -2*w^3 + 4*w^2 + 10*w - 5],\
[827, 827, 2*w^2 - 6*w - 5],\
[827, 827, 2*w^3 - 2*w^2 - 12*w + 1],\
[839, 839, -3*w^3 + 7*w^2 + 12*w - 13],\
[839, 839, -w^3 + 4*w^2 - w - 11],\
[841, 29, -4*w^3 + 6*w^2 + 16*w + 1],\
[841, 29, -3*w^2 + 5*w + 15],\
[853, 853, -3*w^3 + 7*w^2 + 6*w - 3],\
[853, 853, -4*w^3 + 7*w^2 + 15*w - 1],\
[863, 863, -3*w^3 + 8*w^2 + 6*w],\
[863, 863, 2*w^3 - 2*w^2 - 9*w - 10],\
[877, 877, -3*w^3 + 8*w^2 + 4*w + 2],\
[877, 877, -2*w^3 + w^2 + 16*w + 4],\
[947, 947, -3*w^3 + 3*w^2 + 13*w + 6],\
[947, 947, -5*w^3 + 13*w^2 + 11*w - 10],\
[983, 983, w^3 - 6*w^2 + 6*w + 10],\
[983, 983, -4*w^3 + 6*w^2 + 16*w + 7],\
[997, 997, w^3 - 5*w^2 - 2*w + 9],\
[997, 997, -4*w^3 + 9*w^2 + 13*w - 15],\
[997, 997, 4*w^3 - 11*w^2 - 11*w + 13],\
[997, 997, 4*w^3 - 5*w^2 - 22*w - 2]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 10*x^2 + 17
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -1, -1/2*e^3 + 9/2*e, -1/2*e^3 + 9/2*e, -1/2*e^2 + 5/2, -1/2*e^2 + 5/2, -e^2 + 1, -e^2 + 9, -e^2 + 9, 1/2*e^3 - 5/2*e, 1/2*e^3 - 5/2*e, e^3 - 11*e, e^3 - 11*e, -2*e^3 + 12*e, -2*e^3 + 12*e, 3/2*e^2 - 19/2, 3/2*e^2 - 19/2, 2*e^3 - 14*e, 2*e^3 - 14*e, 3*e^2 - 21, 3*e^2 - 21, -3*e^3 + 17*e, e^3 - 5*e, -3*e^3 + 17*e, e^3 - 5*e, -7/2*e^2 + 35/2, -7/2*e^2 + 35/2, 6*e^2 - 30, -e^3 + e, 2*e, -e^3 + e, 2*e, e^3 - 5*e, e^3 - 5*e, 3*e^3 - 17*e, e^3 - 7*e, e^3 - 7*e, 3*e^3 - 17*e, -3*e^2 + 21, -3*e^2 + 21, 7/2*e^3 - 39/2*e, 7/2*e^3 - 39/2*e, e^2 + 11, e^2 + 11, -3*e^3 + 19*e, -3*e^3 + 19*e, 11/2*e^2 - 59/2, 11/2*e^2 - 59/2, e^3 - 9*e, -e^3 + 5*e, e^3 - 9*e, -e^3 + 5*e, 11/2*e^2 - 63/2, 11/2*e^2 - 63/2, -10*e^2 + 50, -10*e^2 + 50, 3*e^2 - 1, 3*e^2 - 1, 3*e^2 + 3, -e^2 + 27, 3*e^2 + 3, -e^2 + 27, -7/2*e^2 + 47/2, -13/2*e^2 + 69/2, -7/2*e^2 + 47/2, -13/2*e^2 + 69/2, -18, -18, -1/2*e^3 - 11/2*e, -1/2*e^3 - 11/2*e, -5*e^2 + 19, 3*e^2 - 19, -5*e^2 + 19, 3*e^2 - 19, -4*e^3 + 20*e, -4*e^3 + 20*e, -5*e^2 + 39, -5*e^2 + 39, 7*e^2 - 25, 7*e^2 - 25, -e^3 + e, 7/2*e^3 - 31/2*e, 7/2*e^3 - 31/2*e, -e^3 + e, 7/2*e^2 + 5/2, 7/2*e^2 + 5/2, 1/2*e^3 - 37/2*e, 1/2*e^3 - 37/2*e, e^3 - 13*e, e^3 - 13*e, 7/2*e^3 - 63/2*e, 7/2*e^3 - 63/2*e, -4*e^2 + 34, -4*e^2 + 34, 3*e^3 - 19*e, 3*e^3 - 19*e, -3*e^3 + 27*e, -3*e^3 + 27*e, 3/2*e^3 - 39/2*e, 3/2*e^3 - 39/2*e, 4*e^2 + 14, -4*e^2 + 54, 11*e^2 - 59, 11*e^2 - 59, -e^3 + 17*e, 5/2*e^3 - 61/2*e, -e^3 + 17*e, 5/2*e^3 - 61/2*e, -7*e^2 + 41, -7*e^2 + 41, 3*e^3 - 13*e, 3*e^3 - 13*e, -3*e^3 + 19*e, -3*e^3 + 19*e, -13/2*e^3 + 81/2*e, -13/2*e^3 + 81/2*e, -5*e^2 + 27, 6*e^2 - 20, -5*e^2 + 27, 6*e^2 - 20, -6*e^2 + 32, -6*e^2 + 32, -8*e^2 + 50, -2*e^3 + 24*e, -1/2*e^3 + 37/2*e, -2*e^3 + 24*e, -1/2*e^3 + 37/2*e, 3*e^2 - 43, 3*e^2 - 43, 1/2*e^3 - 25/2*e, 1/2*e^3 - 25/2*e, 3/2*e^2 - 87/2, 3/2*e^2 - 87/2, -1/2*e^2 + 61/2, -1/2*e^2 + 61/2, -3*e^3 + 35*e, -3*e^3 + 35*e, 13*e^2 - 75, 13*e^2 - 75, -10*e, -10*e, 3*e^3 - 19*e, 3*e^3 - 19*e, -5/2*e^2 + 77/2, -5/2*e^2 + 77/2, 2*e^2 - 56, 2*e^2 - 56, e^3 - 13*e, e^3 - 13*e, -4*e^2 + 6, -4*e^2 + 6, 3*e^3 - 13*e, 3*e^3 - 13*e, -5*e^3 + 41*e, -5*e^3 + 41*e, 16*e^2 - 86, -7*e^2 - 5, 16*e^2 - 86, -7*e^2 - 5]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

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

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