/*
  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([2, -1, -6, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([16, 2, 2])

primes_array = [
[2, 2, w],\
[5, 5, w - 1],\
[8, 2, -w^3 + w^2 + 6*w + 1],\
[9, 3, -w^3 + w^2 + 5*w + 1],\
[9, 3, -w^3 + 2*w^2 + 3*w - 1],\
[19, 19, -w^3 + 2*w^2 + 4*w - 1],\
[23, 23, w^2 - 2*w - 1],\
[29, 29, w^2 - 3*w - 1],\
[29, 29, -w^2 + w + 3],\
[37, 37, w^3 - w^2 - 5*w + 1],\
[43, 43, -w^3 + 2*w^2 + 3*w - 3],\
[47, 47, -w^2 + 2*w + 5],\
[47, 47, 2*w^3 - 2*w^2 - 11*w - 5],\
[53, 53, -w^3 + 2*w^2 + 2*w + 1],\
[59, 59, -w - 3],\
[73, 73, w^2 - w + 1],\
[73, 73, 2*w^3 - 2*w^2 - 12*w - 5],\
[79, 79, 4*w^3 - 4*w^2 - 22*w - 7],\
[97, 97, 2*w^3 - 3*w^2 - 9*w + 1],\
[101, 101, 2*w^2 - 2*w - 9],\
[103, 103, -w^3 + 2*w^2 + 3*w - 5],\
[107, 107, 2*w^3 - 4*w^2 - 6*w + 1],\
[113, 113, 2*w^2 - 4*w - 7],\
[113, 113, w^3 - w^2 - 7*w + 1],\
[125, 5, -2*w^3 + 5*w^2 + 7*w - 11],\
[137, 137, -w^2 + 4*w + 3],\
[137, 137, -w^3 + 3*w^2 + 3*w - 7],\
[139, 139, -3*w - 1],\
[151, 151, 2*w^3 - 2*w^2 - 11*w - 1],\
[157, 157, -2*w^3 + 3*w^2 + 11*w - 3],\
[163, 163, 2*w^3 - 3*w^2 - 8*w + 3],\
[163, 163, -w^3 + w^2 + 7*w + 1],\
[167, 167, 2*w^3 - 5*w^2 - 5*w + 1],\
[167, 167, -w^3 + 7*w + 3],\
[173, 173, -3*w^3 + 4*w^2 + 16*w - 3],\
[181, 181, 2*w - 3],\
[197, 197, -2*w^3 + 6*w^2 + 6*w - 17],\
[199, 199, 2*w^3 - 4*w^2 - 5*w + 1],\
[211, 211, w^3 - 2*w^2 - 6*w + 5],\
[227, 227, -w^3 + 4*w^2 + 2*w - 13],\
[227, 227, w^3 - 2*w^2 - 5*w - 3],\
[229, 229, 2*w^2 - w - 7],\
[233, 233, 2*w - 5],\
[233, 233, 3*w^3 - 4*w^2 - 16*w - 1],\
[239, 239, -2*w^3 + w^2 + 12*w + 11],\
[239, 239, -3*w^3 + 2*w^2 + 19*w + 9],\
[251, 251, 3*w^3 - 2*w^2 - 17*w - 7],\
[257, 257, -w^3 - w^2 + 7*w + 9],\
[257, 257, 3*w^3 - 4*w^2 - 15*w + 3],\
[269, 269, -3*w^3 + 4*w^2 + 14*w + 3],\
[271, 271, 2*w^3 - 14*w - 15],\
[277, 277, w^3 - 2*w^2 - w + 3],\
[277, 277, 2*w^3 - 6*w^2 + w + 5],\
[281, 281, w^3 - 2*w^2 - 3*w - 3],\
[281, 281, -2*w^3 + 3*w^2 + 10*w + 1],\
[289, 17, w^3 + 2*w^2 - 10*w - 17],\
[289, 17, 3*w^3 - 6*w^2 - 13*w + 13],\
[293, 293, 2*w^3 - w^2 - 13*w - 5],\
[311, 311, w^3 - w^2 - 3*w - 3],\
[311, 311, 2*w^3 - 3*w^2 - 11*w - 1],\
[313, 313, -2*w^3 + 4*w^2 + 9*w - 3],\
[317, 317, -3*w^3 + 3*w^2 + 17*w + 7],\
[331, 331, -2*w^3 + 3*w^2 + 8*w - 1],\
[337, 337, -4*w^3 + 6*w^2 + 19*w - 3],\
[347, 347, w^3 - 3*w^2 - 5*w + 1],\
[347, 347, -2*w^3 + 2*w^2 + 9*w - 3],\
[347, 347, 3*w^3 - 6*w^2 - 12*w + 11],\
[347, 347, w - 5],\
[349, 349, -w^3 + 5*w^2 - 3*w - 5],\
[349, 349, 2*w^3 - w^2 - 13*w - 11],\
[353, 353, w^3 - 2*w^2 - 4*w - 3],\
[353, 353, 2*w^2 - 6*w - 5],\
[359, 359, -2*w^3 + 3*w^2 + 6*w + 1],\
[367, 367, 2*w^3 - 2*w^2 - 8*w + 1],\
[367, 367, w^2 - 7],\
[373, 373, w^3 + w^2 - 7*w - 11],\
[373, 373, -2*w^3 + 2*w^2 + 12*w - 1],\
[373, 373, -2*w^3 + 15*w + 13],\
[373, 373, -3*w^3 + 3*w^2 + 15*w + 7],\
[379, 379, -2*w^3 + w^2 + 11*w + 9],\
[397, 397, -w^3 + 3*w^2 + 3*w - 11],\
[397, 397, -w^3 + 9*w - 1],\
[401, 401, -2*w^3 + 4*w^2 + 7*w - 7],\
[419, 419, -w^3 + 2*w^2 + 2*w - 5],\
[419, 419, -2*w^3 + 4*w^2 + 6*w + 3],\
[421, 421, 2*w^2 - 2*w - 7],\
[421, 421, 4*w^3 - 5*w^2 - 22*w - 1],\
[421, 421, w^2 - 3*w - 9],\
[421, 421, 3*w^2 - 5*w - 5],\
[431, 431, w^3 - 4*w^2 + 5],\
[431, 431, -w^3 + 6*w + 1],\
[433, 433, -2*w^3 + 4*w^2 + 10*w - 9],\
[443, 443, -w^3 + 2*w^2 + 4*w - 7],\
[443, 443, -3*w^3 + 6*w^2 + 10*w - 7],\
[479, 479, 2*w^2 - w - 9],\
[479, 479, 2*w^3 - 4*w^2 - 7*w - 3],\
[487, 487, 4*w^3 - 3*w^2 - 24*w - 9],\
[487, 487, -w^3 + 4*w^2 - 3*w - 3],\
[491, 491, -w^3 + 9*w + 3],\
[491, 491, -2*w^3 + 4*w^2 + 8*w - 1],\
[503, 503, 3*w^3 - 5*w^2 - 15*w + 3],\
[503, 503, w^3 - 2*w^2 - 8*w + 5],\
[509, 509, 2*w^3 - 2*w^2 - 8*w - 5],\
[509, 509, 2*w^2 - 4*w - 1],\
[521, 521, 3*w^3 - 4*w^2 - 15*w - 3],\
[541, 541, 3*w^3 - 2*w^2 - 18*w - 7],\
[541, 541, -2*w^3 + 5*w^2 + 6*w - 1],\
[547, 547, 2*w^3 - 2*w^2 - 9*w - 7],\
[557, 557, -2*w^2 + 4*w + 9],\
[563, 563, -2*w^3 + 5*w^2 + 7*w - 9],\
[571, 571, 3*w^3 - 7*w^2 - 7*w - 1],\
[571, 571, -2*w^3 + 5*w^2 + 7*w - 3],\
[577, 577, -3*w^3 + 4*w^2 + 16*w - 9],\
[587, 587, w^2 + w - 5],\
[593, 593, 4*w^3 - 5*w^2 - 20*w - 3],\
[593, 593, 3*w^2 - 5*w - 11],\
[601, 601, -2*w^2 + 6*w + 3],\
[601, 601, 3*w^3 - 6*w^2 - 11*w + 3],\
[607, 607, -w - 5],\
[613, 613, w^3 - 6*w + 1],\
[619, 619, -2*w^3 + 3*w^2 + 6*w - 1],\
[647, 647, -3*w^2 + 6*w + 5],\
[653, 653, 2*w^3 - w^2 - 11*w - 11],\
[661, 661, -2*w^3 + 3*w^2 + 6*w - 5],\
[661, 661, -2*w^3 + 3*w^2 + 12*w + 3],\
[673, 673, w^2 - 2*w + 3],\
[683, 683, 2*w^3 - 2*w^2 - 8*w - 1],\
[691, 691, 2*w^3 - 3*w^2 - 12*w + 1],\
[691, 691, 3*w^3 - 4*w^2 - 14*w - 1],\
[691, 691, -3*w^3 + 6*w^2 + 10*w + 1],\
[691, 691, -2*w^3 + 2*w^2 + 10*w - 3],\
[727, 727, -w^3 - 2*w^2 + 9*w + 15],\
[733, 733, 2*w^3 - 5*w^2 - 3*w + 5],\
[733, 733, 2*w^3 - 2*w^2 - 5*w + 1],\
[761, 761, -w^3 + 3*w^2 + w - 7],\
[769, 769, -2*w^3 + 4*w^2 + 5*w - 5],\
[773, 773, -2*w^3 + 2*w^2 + 13*w + 3],\
[787, 787, -w^3 + 4*w^2 + w - 5],\
[809, 809, 6*w^3 - 9*w^2 - 30*w + 5],\
[809, 809, 2*w^2 - 4*w - 11],\
[809, 809, -3*w^2 + 4*w + 13],\
[809, 809, w^3 - 8*w + 1],\
[811, 811, -5*w^3 + 11*w^2 + 11*w - 1],\
[823, 823, 3*w^3 - 6*w^2 - 7*w + 1],\
[827, 827, 3*w^3 - 20*w - 17],\
[829, 829, 4*w^3 - 4*w^2 - 23*w - 3],\
[839, 839, -w^3 + 6*w + 11],\
[841, 29, 4*w^3 - 4*w^2 - 22*w - 11],\
[853, 853, -w^2 - 3],\
[853, 853, 2*w^3 - w^2 - 11*w - 3],\
[857, 857, 2*w^3 - 2*w^2 - 12*w + 3],\
[863, 863, w^2 - w - 9],\
[877, 877, 4*w^3 - 5*w^2 - 20*w - 1],\
[877, 877, w^3 - w^2 - 3*w - 5],\
[881, 881, -w^3 + 2*w^2 + w - 5],\
[887, 887, -3*w^2 + 6*w + 11],\
[919, 919, -3*w^3 + 22*w + 19],\
[919, 919, 5*w^3 - 5*w^2 - 27*w - 9],\
[937, 937, 4*w^3 - 4*w^2 - 23*w - 9],\
[947, 947, w^3 + w^2 - 7*w - 13],\
[967, 967, 3*w^3 - 3*w^2 - 13*w + 1],\
[971, 971, -w^3 + 6*w^2 - 5*w - 7],\
[971, 971, 4*w^3 - 4*w^2 - 21*w - 5],\
[977, 977, -2*w^3 - 2*w^2 + 17*w + 21],\
[983, 983, 3*w^3 - 3*w^2 - 17*w - 1],\
[991, 991, -w^3 + 4*w^2 + 4*w - 5],\
[991, 991, -3*w^3 + 6*w^2 + 13*w - 5]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 + 3*x - 9
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [-1, -2, 1, -1/3*e + 2, e, 1/3*e + 3, -2*e - 4, 2/3*e + 2, 2/3*e, -2, 1/3*e - 3, 2/3*e, -2/3*e - 2, -2*e - 2, -1/3*e - 8, 5/3*e - 3, -5/3*e - 3, 8/3*e + 8, -1/3*e - 15, 4/3*e + 4, -2*e - 12, -3*e - 5, -e - 10, 5/3*e + 11, 10/3*e - 4, -13/3*e - 4, -5/3*e - 4, -13/3*e - 8, -4/3*e - 16, -2/3*e, 5/3*e + 9, 4*e + 12, -8/3*e, 2*e - 8, 2/3*e - 16, -6*e - 6, 4*e - 6, 16/3*e + 6, 17/3*e + 9, -7/3*e + 15, 1/3*e - 15, -14/3*e - 4, 7/3*e - 7, 7/3*e + 5, -14/3*e - 10, -2/3*e - 10, -1/3*e - 3, 11/3*e + 8, -7/3*e - 26, -2*e + 6, -4/3*e + 14, -2/3*e - 14, 26/3*e + 12, 3*e + 11, -4*e - 2, -5/3*e - 30, 11/3*e + 2, 10/3*e + 18, 2*e + 22, 20/3*e + 18, -3*e + 6, -26/3*e - 18, -11/3*e + 6, 5/3*e - 7, 7*e + 5, -7/3*e + 11, 1/3*e + 12, e, -6*e - 8, 22/3*e + 8, -3*e - 8, 7*e + 16, -2/3*e + 24, -4/3*e + 12, -8/3*e - 24, -22/3*e - 10, -20/3*e - 16, 28/3*e + 8, -20/3*e - 6, -7/3*e - 16, -10/3*e, 28/3*e + 20, 16/3*e + 2, -16/3*e - 20, -7*e - 19, 2*e + 4, 16/3*e + 8, -22/3*e - 14, -2/3*e - 8, -4/3*e - 32, -8/3*e + 12, -e + 4, 9*e + 15, 4/3*e - 12, 8*e - 4, -10, -2/3*e + 4, 4/3*e + 36, 25/3*e + 27, 3*e + 30, 4*e + 28, -16/3*e - 14, -2/3*e - 36, 8/3*e - 22, -7*e + 7, 2*e + 10, 2*e, -3*e + 6, -6*e - 30, 13*e + 21, -1/3*e + 21, -8*e + 4, -19/3*e - 16, e - 5, -1/3*e - 21, -11/3*e + 14, 4*e + 26, -1/3*e - 28, -4/3*e - 6, -2/3*e + 26, -1/3*e + 23, -2*e - 36, -4/3*e - 18, 4, -4*e - 22, 4*e - 26, 4/3*e + 12, -7/3*e - 44, 3*e + 3, 11/3*e, 1/3*e - 6, -10*e - 26, 26/3*e + 12, 4/3*e + 10, 37/3*e + 11, -4/3*e - 34, 4/3*e - 46, 13/3*e + 27, 8/3*e + 26, -29/3*e - 33, -e + 24, e, 5/3*e - 40, -2*e + 32, -e - 42, 6*e - 2, 2*e + 4, -13/3*e - 10, 10*e + 26, -22/3*e - 24, 4*e + 22, 2*e + 24, -6*e - 20, 2*e - 28, e + 22, 2*e - 40, -38/3*e - 34, -6*e - 16, -23/3*e - 32, 19/3*e - 30, 4/3*e + 32, 7*e - 8, 4*e + 36, 10, 20/3*e - 26, -4/3*e + 16, -4*e - 28]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([2, 2, w])] = 1
AL_eigenvalues[ZF.ideal([8, 2, -w^3 + w^2 + 6*w + 1])] = -1

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