/*
  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, -6, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([11, 11, -w + 2])

primes_array = [
[2, 2, w + 1],\
[3, 3, -w^3 + w^2 + 5*w - 2],\
[11, 11, -w^3 + 5*w + 1],\
[11, 11, -w + 2],\
[13, 13, -w^3 + w^2 + 4*w + 1],\
[17, 17, -w^3 + w^2 + 6*w - 1],\
[17, 17, -w^2 - w + 3],\
[23, 23, w^3 - 6*w],\
[27, 3, w^3 + w^2 - 5*w - 4],\
[41, 41, -w^3 + w^2 + 5*w],\
[41, 41, 2*w^3 - 11*w - 4],\
[53, 53, 2*w^3 - 2*w^2 - 11*w + 6],\
[59, 59, 2*w^3 - 11*w - 2],\
[67, 67, w^3 - 7*w - 1],\
[71, 71, 3*w^3 - 2*w^2 - 15*w + 1],\
[79, 79, -3*w^3 + w^2 + 16*w + 3],\
[89, 89, -4*w^3 + 2*w^2 + 22*w - 3],\
[101, 101, -2*w^3 + 13*w + 6],\
[101, 101, w^3 + w^2 - 6*w - 3],\
[101, 101, -3*w^3 + 2*w^2 + 17*w - 3],\
[101, 101, w^3 - w^2 - 8*w - 3],\
[107, 107, 2*w^3 - 3*w^2 - 6*w],\
[109, 109, -w^3 + w^2 + 4*w - 3],\
[113, 113, w^3 - 8*w - 4],\
[113, 113, 7*w^3 - 4*w^2 - 39*w + 7],\
[121, 11, w^3 - w^2 - 6*w - 1],\
[127, 127, -2*w^3 + w^2 + 9*w + 3],\
[127, 127, 2*w^3 - 13*w],\
[131, 131, 6*w^3 - 3*w^2 - 34*w + 6],\
[139, 139, w^2 - 2*w - 4],\
[149, 149, 3*w^3 - w^2 - 18*w - 3],\
[151, 151, -2*w^3 + 2*w^2 + 9*w - 4],\
[163, 163, 3*w^3 - 2*w^2 - 16*w + 2],\
[163, 163, 2*w^3 - 10*w + 1],\
[167, 167, 3*w^3 - 18*w - 4],\
[173, 173, -3*w^3 + w^2 + 18*w - 3],\
[193, 193, 4*w^3 - 3*w^2 - 23*w + 9],\
[197, 197, 3*w^3 - w^2 - 17*w + 2],\
[197, 197, -3*w^3 - 2*w^2 + 15*w + 9],\
[199, 199, 2*w^3 - w^2 - 8*w - 6],\
[223, 223, -6*w^3 + 3*w^2 + 35*w - 7],\
[227, 227, 2*w^3 - 12*w - 1],\
[227, 227, -w^3 + w^2 + 7*w - 8],\
[227, 227, 3*w^3 - w^2 - 18*w - 1],\
[227, 227, -4*w^3 - 2*w^2 + 21*w + 14],\
[229, 229, -w^3 + 2*w^2 + 3*w - 5],\
[233, 233, w^3 - 3*w - 3],\
[233, 233, -3*w^3 + 3*w^2 + 15*w - 8],\
[241, 241, 4*w^2 - 7*w - 8],\
[241, 241, w^3 + w^2 - 4*w - 5],\
[251, 251, 2*w^3 - w^2 - 13*w - 1],\
[251, 251, 2*w^3 - 9*w - 6],\
[257, 257, w^3 - 8*w],\
[263, 263, -w^3 + 2*w^2 + 5*w - 7],\
[269, 269, 3*w^3 - 3*w^2 - 12*w - 1],\
[271, 271, 2*w^3 - w^2 - 12*w - 2],\
[271, 271, 9*w^3 - 5*w^2 - 50*w + 9],\
[277, 277, w^3 - 2*w^2 - 6*w + 6],\
[277, 277, w^3 + w^2 - 7*w - 2],\
[281, 281, 4*w^3 - 2*w^2 - 24*w + 5],\
[281, 281, -7*w^3 + 4*w^2 + 40*w - 12],\
[289, 17, -5*w^3 + 5*w^2 + 26*w - 15],\
[293, 293, -w^3 + 2*w^2 + 4*w - 4],\
[293, 293, -9*w^3 + 5*w^2 + 51*w - 12],\
[307, 307, -2*w^3 + 2*w^2 + 12*w - 7],\
[311, 311, -4*w^3 + 24*w + 11],\
[313, 313, 2*w^3 - 9*w - 4],\
[317, 317, 2*w^3 - 4*w^2 - 2*w - 3],\
[317, 317, 2*w^3 - 14*w - 7],\
[331, 331, -3*w^3 + 19*w + 3],\
[349, 349, -w^3 - 2*w^2 + 8*w + 8],\
[349, 349, w^3 + w^2 - 5*w - 8],\
[361, 19, -w^3 + w^2 + 7*w - 6],\
[361, 19, w^3 + w^2 - 3*w - 4],\
[373, 373, 2*w^3 - w^2 - 11*w - 3],\
[379, 379, -4*w^3 + 25*w + 14],\
[383, 383, -3*w^3 + w^2 + 15*w - 2],\
[383, 383, 3*w^3 - 4*w^2 - 10*w],\
[401, 401, 4*w^3 - w^2 - 24*w - 4],\
[409, 409, w^2 + 5*w + 5],\
[419, 419, w^2 - 2*w - 6],\
[419, 419, -6*w^3 + 3*w^2 + 34*w - 8],\
[433, 433, 3*w^3 - 16*w - 6],\
[433, 433, 3*w^3 - w^2 - 16*w - 5],\
[443, 443, -w^3 + 2*w^2 + w + 3],\
[449, 449, 2*w^2 - 2*w - 5],\
[457, 457, -4*w^3 + 4*w^2 + 20*w - 9],\
[457, 457, w^2 + w - 7],\
[463, 463, w^3 - 7*w - 7],\
[463, 463, w^3 + 2*w^2 - 5*w - 9],\
[487, 487, w^3 + 2*w^2 - 5*w - 5],\
[487, 487, -w^3 + w^2 + 6*w - 7],\
[509, 509, -2*w^3 - 2*w^2 + 13*w + 10],\
[509, 509, 3*w^3 - 17*w - 3],\
[509, 509, -5*w^3 + 3*w^2 + 30*w - 11],\
[509, 509, -w^3 - 2*w^2 + 4*w + 6],\
[521, 521, -2*w^3 - w^2 + 9*w + 7],\
[541, 541, -2*w^3 - 2*w^2 + 12*w + 9],\
[541, 541, -2*w^3 + 12*w + 9],\
[547, 547, 2*w^3 + w^2 - 7*w - 3],\
[547, 547, 5*w^3 - 3*w^2 - 30*w + 3],\
[557, 557, w^3 + 2*w^2 - 5*w - 11],\
[557, 557, -w^3 + w^2 + 3*w - 4],\
[563, 563, 3*w^3 - w^2 - 15*w - 4],\
[569, 569, 2*w^2 - w - 8],\
[577, 577, 4*w^3 - 22*w - 7],\
[593, 593, -3*w^3 + w^2 + 17*w + 4],\
[607, 607, -w^3 - 3*w^2 + w + 4],\
[613, 613, 3*w^3 - 2*w^2 - 14*w],\
[613, 613, w^2 - 3*w - 5],\
[619, 619, -w^3 + 2*w^2 + 4*w - 10],\
[625, 5, -5],\
[641, 641, -3*w^3 - 3*w^2 + 12*w + 7],\
[643, 643, 2*w^3 + 2*w^2 - 14*w - 13],\
[643, 643, 3*w^3 - 19*w - 7],\
[647, 647, w^3 - 5*w - 7],\
[653, 653, -3*w^3 + 2*w^2 + 14*w - 2],\
[659, 659, -6*w^3 + 2*w^2 + 34*w - 1],\
[659, 659, -2*w^3 + w^2 + 17*w + 9],\
[683, 683, w^3 + 2*w^2 - 7*w - 7],\
[691, 691, 3*w^3 - 15*w - 7],\
[701, 701, w^3 + 2*w^2 - 6*w - 10],\
[701, 701, -w^3 + 3*w^2 + 6*w - 13],\
[701, 701, 5*w^3 - 2*w^2 - 30*w + 4],\
[701, 701, 4*w^3 - 2*w^2 - 21*w + 2],\
[727, 727, -2*w^3 - 3*w^2 + 9*w + 7],\
[743, 743, 3*w - 4],\
[751, 751, w^3 - 9*w - 3],\
[757, 757, w^3 - 5*w^2 + 4*w + 7],\
[757, 757, 4*w^3 - 5*w^2 - 11*w - 5],\
[761, 761, 4*w^3 - 5*w^2 - 20*w + 16],\
[769, 769, -3*w^3 + 16*w + 12],\
[773, 773, 2*w^3 + 2*w^2 - 11*w - 10],\
[773, 773, w^3 - 3*w - 5],\
[773, 773, -w^3 + 2*w^2 + 9*w + 3],\
[773, 773, 5*w^3 - w^2 - 27*w],\
[787, 787, -4*w^3 + 3*w^2 + 22*w - 4],\
[787, 787, -2*w^3 + w^2 + 14*w - 6],\
[797, 797, 2*w^3 - w^2 - 14*w + 2],\
[797, 797, -3*w^3 + 2*w^2 + 19*w - 7],\
[811, 811, 2*w^3 + w^2 - 14*w - 12],\
[811, 811, 2*w^3 + 2*w^2 - 8*w - 3],\
[829, 829, -2*w^3 + 14*w + 11],\
[857, 857, -3*w^3 + 3*w^2 + 14*w - 7],\
[859, 859, 3*w^3 - w^2 - 15*w],\
[859, 859, -11*w^3 + 7*w^2 + 60*w - 15],\
[863, 863, 2*w^2 - 2*w - 13],\
[863, 863, 2*w^3 + w^2 - 12*w - 4],\
[877, 877, -6*w^3 - 2*w^2 + 32*w + 19],\
[887, 887, 3*w^3 + w^2 - 14*w - 7],\
[887, 887, 2*w^3 - w^2 - 15*w - 5],\
[911, 911, 3*w^3 - 18*w - 2],\
[919, 919, w^3 - w^2 - 6*w - 5],\
[919, 919, -4*w^3 + w^2 + 24*w - 2],\
[941, 941, 3*w^3 - 3*w^2 - 13*w],\
[947, 947, 2*w^3 - 2*w^2 - 10*w - 1],\
[947, 947, -3*w^3 + 2*w^2 + 14*w + 6],\
[953, 953, -4*w^3 + 3*w^2 + 21*w - 3],\
[953, 953, 4*w^3 - w^2 - 23*w + 1],\
[961, 31, -2*w^3 + 3*w^2 + 12*w - 8],\
[961, 31, 3*w^2 - 4*w - 6],\
[967, 967, -5*w^3 - 3*w^2 + 23*w + 14],\
[971, 971, 5*w^3 - w^2 - 27*w - 8],\
[971, 971, -9*w^3 + 6*w^2 + 50*w - 12],\
[983, 983, -w^3 + w^2 + 10*w + 3],\
[997, 997, 2*w^3 + w^2 - 11*w - 3],\
[997, 997, w^3 + 2*w^2 - 6*w - 8],\
[997, 997, -5*w^3 + 4*w^2 + 29*w - 15]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 5*x^2 + 2
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 0, 2*e^3 - 10*e, 1, -e^3 + 4*e, -2*e^3 + 8*e, -2, e^2 - 6, -e^3 + 8*e, -2*e^3 + 10*e, e^3 - 4*e, 4*e^3 - 20*e, -3*e^2 + 6, -e^2 - 6, 4*e^3 - 18*e, -4*e^3 + 14*e, -7*e^3 + 26*e, -8*e^2 + 22, 2*e^2 - 18, -4*e^3 + 18*e, 2*e^3 - 10*e, 4*e^3 - 14*e, 3*e^3 - 14*e, e^2 + 4, -2*e^3, 4*e^3 - 20*e, 2*e, e^2 - 14, -5*e^2 + 18, 8*e^3 - 34*e, -8*e^3 + 34*e, 5*e^3 - 14*e, 8*e^3 - 28*e, 2*e^2 - 8, -4*e^2 + 8, -6*e^2 + 6, 6*e^2 - 6, -e^2 - 8, -3*e^2 - 4, 10*e^2 - 28, -9*e^2 + 30, e^2 + 6, 4*e^2 - 12, -12, 6*e^2 - 16, -7*e^3 + 28*e, 13*e^2 - 32, -3*e^3 + 8*e, -4*e^3 + 24*e, 2*e^2 + 10, -7*e^2 + 14, -4, -14, -6*e^3 + 14*e, -8*e^3 + 28*e, 5*e^3 - 16*e, 8*e^3 - 30*e, -2*e^3 + 24*e, 2, 8*e^2 - 34, -3*e^2 + 16, -8*e^3 + 30*e, -6*e^3 + 18*e, 8*e^2 - 22, -4*e^2 + 4, -2*e, -10*e^3 + 48*e, -6*e^2 + 6, -2*e^2 + 14, 10*e^2 - 32, -3*e^3 + 14*e, -12*e^2 + 38, 14*e^2 - 38, 2*e^2 + 14, -2*e^3 + 6*e, 7*e^3 - 30*e, -8*e^3 + 42*e, 8*e^3 - 30*e, -4*e^2 + 2, 2*e^3 - 14*e, -12*e^3 + 52*e, -5*e^2 + 10, -14*e, 4*e^3 - 4*e, -6*e^2 + 24, 0, 18*e^3 - 70*e, 12*e^2 - 14, -24, 4*e^3 - 8*e, 11*e^3 - 38*e, 14*e^2 - 44, -2*e^3 + 8*e, -22, -e^2 + 16, 4*e^3 - 36*e, 5*e^2 - 12, -15*e^3 + 58*e, -e^2 + 4, 13*e^3 - 50*e, -10*e^2 + 8, 7*e^2 - 20, -12*e^3 + 52*e, e^3 - 18*e, -5*e^3 + 32*e, -13*e^3 + 50*e, 4*e^3 - 14*e, 9*e^3 - 34*e, 4*e^3 - 40*e, 4*e^3 - 8*e, 2*e^3, 9*e^2 - 52, -2, 16*e^3 - 58*e, -4*e^2 - 20, 6*e^2 - 28, -8*e^3 + 28*e, 11*e^2 - 30, -12, 14*e, 8*e^3 - 28*e, -6*e^3 + 30*e, -10*e^2 + 34, -12*e^2 + 46, 3*e^3 - 36*e, 13*e^3 - 36*e, 11*e^2 - 10, -7*e^2 + 26, -6*e^3 + 44*e, -10*e^2 + 18, 4*e^3 - 40*e, 4*e^2 - 6, -12*e^3 + 54*e, -5*e^2 - 32, -2*e^3 - 12*e, -13*e^2 + 20, 2*e^3 + 2*e, -e^2 - 6, 15*e^2 - 52, -16*e^2 + 50, 10*e^3 - 26*e, -16*e^2 + 60, 4*e^2 + 22, 16*e^3 - 88*e, 18*e^3 - 80*e, -11*e^3 + 48*e, 17*e^3 - 84*e, 26*e^2 - 68, 4*e^3 - 6*e, -12*e^3 + 76*e, 8*e^2 - 16, -6*e^2 + 4, -13*e^2 + 46, -8*e^2, -19*e^3 + 78*e, 11*e^3 - 32*e, 4*e^3 - 18*e, -2*e^3 - 10*e, 26*e^2 - 70, 8*e^3 - 30*e, -e^3 + 26*e, -10*e^3 + 30*e, -23*e^3 + 94*e, -e^3 + 14*e, 17*e^2 - 54, 9*e^2 - 36, -7*e^3 + 60*e, 6*e^2 - 46]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([11, 11, -w + 2])] = -1

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