/*
  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([4, 8, -7, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

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

primes_array = [
[2, 2, 1/2*w^2 + 1/2*w - 1],\
[2, 2, -1/2*w^2 + 3/2*w],\
[9, 3, -1/2*w^2 + 1/2*w + 2],\
[13, 13, 1/2*w^3 - w^2 - 5/2*w + 2],\
[13, 13, -1/2*w^3 + 1/2*w^2 + 3*w - 1],\
[23, 23, 1/2*w^3 - w^2 - 5/2*w],\
[23, 23, -1/2*w^3 + 1/2*w^2 + 3*w - 3],\
[37, 37, 1/2*w^3 - 1/2*w^2 - 3*w - 3],\
[37, 37, 1/2*w^3 - w^2 - 5/2*w + 6],\
[59, 59, -1/2*w^3 + 7/2*w],\
[59, 59, -1/2*w^3 + 3/2*w^2 + 2*w - 3],\
[61, 61, -w^3 + 2*w^2 + 7*w - 7],\
[61, 61, w^3 - w^2 - 6*w + 3],\
[61, 61, w^3 - 2*w^2 - 5*w + 3],\
[61, 61, w^3 - w^2 - 8*w + 1],\
[71, 71, -1/2*w^3 + 1/2*w^2 + 5*w - 5],\
[71, 71, -w^3 + 3/2*w^2 + 15/2*w - 6],\
[83, 83, -1/2*w^3 + 3/2*w^2 + 2*w - 7],\
[83, 83, 1/2*w^3 - 7/2*w - 4],\
[97, 97, 2*w - 1],\
[107, 107, 1/2*w^3 + 1/2*w^2 - 4*w - 1],\
[107, 107, 3/2*w^3 - 7/2*w^2 - 9*w + 13],\
[109, 109, -3/2*w^2 + 7/2*w + 4],\
[109, 109, 1/2*w^3 + 1/2*w^2 - 4*w - 5],\
[109, 109, -1/2*w^3 + 2*w^2 + 3/2*w - 8],\
[109, 109, 3/2*w^2 + 1/2*w - 6],\
[121, 11, w^2 - w - 5],\
[121, 11, w^2 - w - 3],\
[131, 131, -w^3 + 11*w - 1],\
[131, 131, -w^3 + 3*w^2 + 8*w - 9],\
[157, 157, -w^3 + w^2 + 6*w + 1],\
[157, 157, 1/2*w^3 - 2*w^2 - 11/2*w + 4],\
[167, 167, w^2 - 3*w - 3],\
[167, 167, 1/2*w^3 - 3/2*w^2 - 2*w + 1],\
[167, 167, 3/2*w^3 - 5/2*w^2 - 10*w + 9],\
[167, 167, w^2 + w - 5],\
[169, 13, w^2 - w - 9],\
[179, 179, w^3 - 7*w - 5],\
[179, 179, 1/2*w^3 + 1/2*w^2 - 2*w - 3],\
[181, 181, -1/2*w^3 + 1/2*w^2 + 3*w - 5],\
[181, 181, 1/2*w^3 - w^2 - 5/2*w - 2],\
[191, 191, -1/2*w^3 + 3/2*w + 2],\
[191, 191, 2*w - 5],\
[191, 191, -2*w - 3],\
[191, 191, -1/2*w^3 + 2*w^2 + 3/2*w - 10],\
[193, 193, -1/2*w^3 + w^2 + 9/2*w - 4],\
[193, 193, 1/2*w^3 - 5/2*w^2 + w + 5],\
[193, 193, -1/2*w^3 - w^2 + 5/2*w + 4],\
[193, 193, 1/2*w^3 - 1/2*w^2 - 5*w + 1],\
[227, 227, -1/2*w^3 + 3/2*w^2 + 4*w - 9],\
[227, 227, w^3 - 2*w^2 - 7*w + 5],\
[227, 227, w^3 - w^2 - 8*w + 3],\
[227, 227, 1/2*w^3 - 11/2*w - 4],\
[239, 239, 1/2*w^3 - 3/2*w^2 - 4*w + 3],\
[239, 239, 1/2*w^3 - 11/2*w + 2],\
[251, 251, 1/2*w^3 - w^2 - 1/2*w - 2],\
[251, 251, -1/2*w^3 + 1/2*w^2 + w - 3],\
[263, 263, -1/2*w^3 - w^2 + 17/2*w],\
[263, 263, 1/2*w^3 - 5/2*w^2 - 5*w + 7],\
[277, 277, w^3 - w^2 - 6*w + 1],\
[277, 277, -w^3 + 2*w^2 + 5*w - 5],\
[311, 311, -w^3 + 5/2*w^2 + 13/2*w - 8],\
[311, 311, -w^3 + 1/2*w^2 + 17/2*w],\
[337, 337, 2*w^3 - 3*w^2 - 13*w + 9],\
[337, 337, 5/2*w^2 + 3/2*w - 8],\
[347, 347, 2*w^3 - 7/2*w^2 - 25/2*w + 14],\
[347, 347, 2*w^3 - 5/2*w^2 - 27/2*w],\
[349, 349, w^3 - 3/2*w^2 - 11/2*w + 4],\
[349, 349, -w^3 + 3/2*w^2 + 11/2*w - 2],\
[359, 359, -1/2*w^3 + w^2 + 1/2*w + 4],\
[359, 359, 1/2*w^3 - 1/2*w^2 - w + 5],\
[373, 373, 1/2*w^3 - 3/2*w + 4],\
[373, 373, 1/2*w^3 - 3/2*w^2 - 3],\
[383, 383, w^3 - 3/2*w^2 - 15/2*w + 8],\
[383, 383, -2*w^2 + 1],\
[409, 409, 1/2*w^3 + w^2 - 13/2*w - 4],\
[409, 409, -1/2*w^3 + 5/2*w^2 + 3*w - 9],\
[431, 431, 3/2*w^3 - 5/2*w^2 - 10*w + 5],\
[431, 431, 1/2*w^3 - 5/2*w^2 - 3*w + 13],\
[431, 431, 1/2*w^3 + w^2 - 13/2*w - 8],\
[431, 431, 3/2*w^3 - 2*w^2 - 21/2*w + 6],\
[433, 433, 2*w^3 - 4*w^2 - 14*w + 13],\
[433, 433, w^3 - 1/2*w^2 - 9/2*w + 6],\
[443, 443, 2*w^3 - 11/2*w^2 - 21/2*w + 24],\
[443, 443, w^3 - 5*w - 3],\
[457, 457, -2*w^3 + 4*w^2 + 12*w - 13],\
[457, 457, -w^3 + 1/2*w^2 + 9/2*w],\
[467, 467, -3/2*w^3 + 7/2*w^2 + 7*w - 7],\
[467, 467, 3/2*w^2 + 1/2*w - 8],\
[467, 467, w^3 - 7/2*w^2 - 15/2*w + 12],\
[467, 467, 3/2*w^3 - w^2 - 19/2*w + 2],\
[503, 503, -2*w^2 + 11],\
[503, 503, 2*w^3 - 2*w^2 - 12*w + 7],\
[503, 503, 2*w^3 - 4*w^2 - 10*w + 5],\
[503, 503, -2*w^2 + 4*w + 9],\
[529, 23, 3/2*w^2 - 3/2*w - 8],\
[541, 541, -1/2*w^3 + w^2 + 1/2*w + 6],\
[541, 541, 1/2*w^3 - 1/2*w^2 - w + 7],\
[563, 563, 1/2*w^2 + 3/2*w - 6],\
[563, 563, 1/2*w^2 - 5/2*w - 4],\
[577, 577, -1/2*w^3 + 5/2*w^2 + w - 11],\
[577, 577, -1/2*w^3 - w^2 + 9/2*w + 8],\
[587, 587, -w^3 + 1/2*w^2 + 13/2*w - 2],\
[587, 587, -w^3 + 5/2*w^2 + 9/2*w - 4],\
[599, 599, 1/2*w^3 + w^2 - 9/2*w - 12],\
[599, 599, -1/2*w^3 + 5/2*w^2 + w - 15],\
[601, 601, 1/2*w^3 - 1/2*w^2 - 3*w - 5],\
[601, 601, -1/2*w^3 + w^2 + 5/2*w - 8],\
[625, 5, -5],\
[659, 659, 3/2*w^3 - 1/2*w^2 - 8*w + 3],\
[659, 659, 3/2*w^3 - 4*w^2 - 9/2*w + 4],\
[673, 673, 2*w^3 - 2*w^2 - 16*w - 1],\
[673, 673, w^3 - 3/2*w^2 - 7/2*w - 8],\
[673, 673, w^3 - 3/2*w^2 - 7/2*w + 12],\
[673, 673, 2*w^3 - 4*w^2 - 14*w + 17],\
[683, 683, -1/2*w^3 - w^2 + 9/2*w - 2],\
[683, 683, -w^3 + 5/2*w^2 + 9/2*w - 12],\
[683, 683, -w^3 + 1/2*w^2 + 13/2*w + 6],\
[683, 683, 3/2*w^3 - w^2 - 11/2*w + 10],\
[709, 709, -1/2*w^3 + w^2 + 9/2*w - 10],\
[709, 709, 1/2*w^3 - 1/2*w^2 - 5*w - 5],\
[719, 719, -3/2*w^3 + 3/2*w^2 + 11*w - 3],\
[719, 719, -3/2*w^3 + 3*w^2 + 19/2*w - 8],\
[733, 733, w^3 - 3*w^2 - 8*w + 13],\
[733, 733, -w^3 + 7/2*w^2 + 7/2*w - 12],\
[733, 733, w^3 + 1/2*w^2 - 15/2*w - 6],\
[733, 733, -3/2*w^3 + 9/2*w^2 + 8*w - 19],\
[757, 757, w^3 - w^2 - 4*w + 1],\
[757, 757, -2*w^3 + 7/2*w^2 + 25/2*w - 10],\
[769, 769, -2*w^2 + 4*w + 7],\
[769, 769, -2*w^2 + 9],\
[827, 827, -1/2*w^3 - 1/2*w^2 + 6*w - 1],\
[827, 827, -1/2*w^3 + 2*w^2 + 7/2*w - 4],\
[829, 829, 1/2*w^3 + 2*w^2 - 7/2*w - 12],\
[829, 829, 7/2*w^3 - 8*w^2 - 41/2*w + 32],\
[829, 829, 3/2*w^3 - w^2 - 15/2*w - 2],\
[829, 829, -w^3 + w^2 + 8*w + 5],\
[839, 839, 3/2*w^3 - w^2 - 23/2*w],\
[839, 839, -3/2*w^3 + 7/2*w^2 + 9*w - 11],\
[853, 853, 1/2*w^3 - 3/2*w^2 - 2*w + 11],\
[853, 853, -1/2*w^3 + 7/2*w + 8],\
[863, 863, 2*w^3 - 11/2*w^2 - 25/2*w + 26],\
[863, 863, 3/2*w^3 - 5/2*w^2 - 10*w + 13],\
[887, 887, 2*w^3 - 7/2*w^2 - 29/2*w + 8],\
[887, 887, 2*w^3 - 5/2*w^2 - 31/2*w + 8],\
[911, 911, 2*w^3 - w^2 - 15*w - 3],\
[911, 911, 2*w^3 - 5*w^2 - 11*w + 17],\
[947, 947, -w^3 + 3*w^2 + 6*w - 9],\
[947, 947, w^3 - 9*w - 1],\
[961, 31, -3/2*w^3 + 3*w^2 + 23/2*w - 8],\
[961, 31, 3/2*w^3 - 3/2*w^2 - 13*w + 5],\
[983, 983, -1/2*w^3 + 2*w^2 - 1/2*w - 6],\
[983, 983, 1/2*w^3 + 1/2*w^2 - 2*w - 5],\
[997, 997, w^3 - 13*w + 7],\
[997, 997, 1/2*w^3 - 5/2*w^2 - w + 9],\
[997, 997, 1/2*w^3 + w^2 - 9/2*w - 6],\
[997, 997, -w^3 + 3*w^2 + 10*w - 5]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - 9*x^4 + 20*x^2 - 11
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -e, 1, e^4 - 8*e^2 + 8, e^4 - 8*e^2 + 8, e^3 - 3*e, -e^3 + 3*e, -4*e^4 + 27*e^2 - 30, -4*e^4 + 27*e^2 - 30, -3*e^5 + 23*e^3 - 33*e, 3*e^5 - 23*e^3 + 33*e, -2*e^2 + 5, e^4 - 4*e^2 - 5, e^4 - 4*e^2 - 5, -2*e^2 + 5, 3*e^5 - 22*e^3 + 25*e, -3*e^5 + 22*e^3 - 25*e, 2*e^5 - 14*e^3 + 17*e, -2*e^5 + 14*e^3 - 17*e, -2*e^4 + 14*e^2 - 28, -4*e^5 + 31*e^3 - 40*e, 4*e^5 - 31*e^3 + 40*e, e^4 - 9*e^2 + 10, -e^4 + 10*e^2 - 15, -e^4 + 10*e^2 - 15, e^4 - 9*e^2 + 10, e^4 - 4*e^2 - 18, -2*e^4 + 13*e^2 - 28, e^3 - 8*e, -e^3 + 8*e, -3*e^4 + 20*e^2 - 21, -3*e^4 + 20*e^2 - 21, -3*e^3 + 18*e, 3*e^3 - 8*e, -3*e^3 + 8*e, 3*e^3 - 18*e, -e^4 + 4*e^2 + 15, 3*e^5 - 25*e^3 + 34*e, -3*e^5 + 25*e^3 - 34*e, 5*e^4 - 37*e^2 + 43, 5*e^4 - 37*e^2 + 43, -e^5 + 7*e^3 - e, -3*e^5 + 25*e^3 - 36*e, 3*e^5 - 25*e^3 + 36*e, e^5 - 7*e^3 + e, e^4 - 5*e^2 - 7, 3*e^2 - 7, 3*e^2 - 7, e^4 - 5*e^2 - 7, -3*e^5 + 20*e^3 - 12*e, 2*e^5 - 22*e^3 + 49*e, -2*e^5 + 22*e^3 - 49*e, 3*e^5 - 20*e^3 + 12*e, 2*e^5 - 18*e^3 + 29*e, -2*e^5 + 18*e^3 - 29*e, -3*e^5 + 20*e^3 - 14*e, 3*e^5 - 20*e^3 + 14*e, -3*e^5 + 22*e^3 - 31*e, 3*e^5 - 22*e^3 + 31*e, -3*e^4 + 31*e^2 - 54, -3*e^4 + 31*e^2 - 54, e^5 - 9*e^3 + 11*e, -e^5 + 9*e^3 - 11*e, -4*e^4 + 27*e^2 - 22, -4*e^4 + 27*e^2 - 22, 8*e^5 - 57*e^3 + 67*e, -8*e^5 + 57*e^3 - 67*e, 4*e^4 - 29*e^2 + 28, 4*e^4 - 29*e^2 + 28, -4*e^5 + 25*e^3 - 12*e, 4*e^5 - 25*e^3 + 12*e, e^2 + 1, e^2 + 1, -5*e^5 + 35*e^3 - 36*e, 5*e^5 - 35*e^3 + 36*e, 8*e^4 - 50*e^2 + 33, 8*e^4 - 50*e^2 + 33, -2*e^5 + 12*e^3 - 3*e, 4*e^5 - 26*e^3 + 17*e, -4*e^5 + 26*e^3 - 17*e, 2*e^5 - 12*e^3 + 3*e, -9*e^2 + 36, -9*e^2 + 36, -7*e^5 + 52*e^3 - 66*e, 7*e^5 - 52*e^3 + 66*e, -5*e^4 + 33*e^2 - 33, -5*e^4 + 33*e^2 - 33, e^5 - 8*e^3 + 7*e, 4*e^5 - 32*e^3 + 58*e, -4*e^5 + 32*e^3 - 58*e, -e^5 + 8*e^3 - 7*e, 4*e^5 - 34*e^3 + 66*e, 7*e^5 - 51*e^3 + 59*e, -7*e^5 + 51*e^3 - 59*e, -4*e^5 + 34*e^3 - 66*e, -3*e^4 + 23*e^2 - 63, 10*e^4 - 77*e^2 + 78, 10*e^4 - 77*e^2 + 78, -7*e^3 + 31*e, 7*e^3 - 31*e, 7*e^4 - 42*e^2 + 36, 7*e^4 - 42*e^2 + 36, 7*e^5 - 52*e^3 + 69*e, -7*e^5 + 52*e^3 - 69*e, e^5 - 11*e^3 + 29*e, -e^5 + 11*e^3 - 29*e, 12*e^2 - 35, 12*e^2 - 35, e^4 + 3*e^2 - 5, -9*e^5 + 63*e^3 - 71*e, 9*e^5 - 63*e^3 + 71*e, 8*e^4 - 62*e^2 + 69, -7*e^4 + 46*e^2 - 58, -7*e^4 + 46*e^2 - 58, 8*e^4 - 62*e^2 + 69, -3*e^5 + 25*e^3 - 63*e, 4*e^5 - 36*e^3 + 67*e, -4*e^5 + 36*e^3 - 67*e, 3*e^5 - 25*e^3 + 63*e, 4*e^4 - 33*e^2 + 25, 4*e^4 - 33*e^2 + 25, -e^5 + 8*e^3 + 2*e, e^5 - 8*e^3 - 2*e, 14*e^4 - 102*e^2 + 121, 5*e^4 - 42*e^2 + 53, 5*e^4 - 42*e^2 + 53, 14*e^4 - 102*e^2 + 121, 3*e^4 - 9*e^2 - 1, 3*e^4 - 9*e^2 - 1, 2*e^4 - 26*e^2 + 48, 2*e^4 - 26*e^2 + 48, -2*e^5 + 17*e^3 - 48*e, 2*e^5 - 17*e^3 + 48*e, 7*e^4 - 54*e^2 + 30, -12*e^4 + 99*e^2 - 115, -12*e^4 + 99*e^2 - 115, 7*e^4 - 54*e^2 + 30, -2*e^5 + 5*e^3 + 35*e, 2*e^5 - 5*e^3 - 35*e, 5*e^4 - 40*e^2 + 37, 5*e^4 - 40*e^2 + 37, -e^5 + 9*e^3 - 5*e, e^5 - 9*e^3 + 5*e, -2*e^5 + 14*e^3 - 4*e, 2*e^5 - 14*e^3 + 4*e, 7*e^5 - 56*e^3 + 88*e, -7*e^5 + 56*e^3 - 88*e, -11*e^5 + 82*e^3 - 104*e, 11*e^5 - 82*e^3 + 104*e, 10*e^4 - 58*e^2 + 47, 10*e^4 - 58*e^2 + 47, -5*e^5 + 40*e^3 - 60*e, 5*e^5 - 40*e^3 + 60*e, -5*e^4 + 50*e^2 - 96, 2*e^4 - 7*e^2 - 19, 2*e^4 - 7*e^2 - 19, -5*e^4 + 50*e^2 - 96]
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, -1/2*w^2 + 1/2*w + 2])] = -1

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