/*
  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([-3, -7, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

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

primes_array = [
[3, 3, w],\
[3, 3, w + 1],\
[3, 3, w + 2],\
[8, 2, 2],\
[11, 11, -w^2 + 5],\
[13, 13, w^2 - w - 7],\
[17, 17, -w^2 + w + 4],\
[19, 19, -w^2 - w + 4],\
[29, 29, 2*w^2 - 2*w - 11],\
[31, 31, w^2 - 2*w - 4],\
[37, 37, -2*w^2 + 3*w + 7],\
[41, 41, w^2 - 2],\
[59, 59, 2*w^2 - 13],\
[61, 61, w^2 + w - 10],\
[67, 67, 2*w^2 - w - 11],\
[73, 73, -w^2 - 1],\
[83, 83, w^2 - w - 10],\
[89, 89, -w^2 + 4*w - 2],\
[97, 97, w^2 - 2*w - 7],\
[97, 97, -w^2 - 4*w - 5],\
[97, 97, 2*w^2 + w - 8],\
[101, 101, -4*w^2 + 2*w + 25],\
[103, 103, w^2 + w - 7],\
[107, 107, -3*w - 4],\
[109, 109, 5*w^2 - 3*w - 31],\
[121, 11, 2*w^2 - 3*w - 4],\
[125, 5, -5],\
[137, 137, -5*w^2 + 3*w + 34],\
[139, 139, w^2 + 2*w - 4],\
[151, 151, 3*w^2 - 23],\
[151, 151, w^2 - 3*w - 5],\
[151, 151, 3*w^2 - 3*w - 17],\
[157, 157, 4*w^2 - 9*w - 8],\
[157, 157, w^2 + 3*w - 2],\
[157, 157, 2*w^2 - w - 2],\
[167, 167, 2*w^2 + w - 11],\
[167, 167, -w^2 + 11],\
[167, 167, -3*w^2 + 19],\
[169, 13, w^2 - 4*w - 4],\
[173, 173, 2*w^2 - 3*w - 10],\
[179, 179, -3*w + 7],\
[191, 191, -w^2 + 5*w - 5],\
[193, 193, -w^2 + w - 2],\
[199, 199, 3*w - 2],\
[211, 211, w^2 - 3*w - 11],\
[223, 223, 3*w^2 - 20],\
[223, 223, w^2 - 5*w - 4],\
[223, 223, 2*w^2 - w - 8],\
[229, 229, 4*w^2 - 3*w - 26],\
[229, 229, 2*w^2 + w - 20],\
[229, 229, 2*w^2 - 2*w - 5],\
[239, 239, -6*w^2 + 3*w + 38],\
[241, 241, -w^2 + 4*w - 5],\
[257, 257, -2*w^2 + 5*w - 1],\
[263, 263, -2*w^2 - w + 17],\
[269, 269, -w^2 - w + 13],\
[269, 269, 3*w - 4],\
[269, 269, -4*w^2 + w + 31],\
[271, 271, 2*w^2 - 7],\
[271, 271, 3*w - 5],\
[271, 271, w^2 - 3*w - 8],\
[289, 17, 2*w^2 + w - 5],\
[307, 307, 2*w^2 - w - 5],\
[311, 311, 3*w^2 - 3*w - 19],\
[311, 311, -7*w^2 + 13*w + 19],\
[311, 311, w^2 - 2*w - 13],\
[313, 313, w^2 + 2*w - 7],\
[317, 317, -5*w^2 + w + 35],\
[331, 331, -w^2 + 3*w - 4],\
[337, 337, 3*w^2 - 3*w - 13],\
[343, 7, -7],\
[347, 347, 2*w^2 - 3*w - 13],\
[349, 349, 3*w^2 - 3*w - 20],\
[359, 359, -w^2 + 5*w - 2],\
[361, 19, -2*w^2 + 7*w - 1],\
[389, 389, -2*w^2 + w - 1],\
[397, 397, w^2 - 6*w + 10],\
[401, 401, 6*w^2 - 3*w - 44],\
[419, 419, -3*w^2 + 6*w + 10],\
[419, 419, 4*w^2 - 2*w - 31],\
[419, 419, -4*w^2 + 4*w + 19],\
[421, 421, 6*w^2 - 12*w - 13],\
[431, 431, 4*w^2 - 7*w - 10],\
[433, 433, w^2 - 4*w - 7],\
[443, 443, 2*w^2 + 4*w - 5],\
[443, 443, 5*w^2 - 11*w - 11],\
[443, 443, 3*w^2 - 14],\
[449, 449, -2*w^2 + 2*w - 1],\
[457, 457, w^2 - 6*w - 5],\
[457, 457, w^2 + 3*w - 5],\
[457, 457, 6*w^2 - 3*w - 37],\
[463, 463, w^2 + 5*w - 1],\
[467, 467, w^2 - w - 13],\
[479, 479, 5*w^2 - 9*w - 16],\
[487, 487, -6*w - 5],\
[499, 499, 3*w^2 - 3*w - 4],\
[499, 499, 3*w^2 + 3*w - 1],\
[499, 499, w^2 - 5*w + 8],\
[503, 503, 4*w^2 - 5*w - 16],\
[509, 509, 2*w^2 - 5*w - 8],\
[521, 521, 4*w^2 - w - 22],\
[523, 523, w^2 - 5*w - 16],\
[547, 547, 5*w^2 - 2*w - 38],\
[547, 547, 8*w^2 - 3*w - 58],\
[547, 547, 4*w^2 - 29],\
[557, 557, 7*w^2 - 4*w - 43],\
[557, 557, 3*w - 11],\
[557, 557, 2*w^2 - 4*w - 11],\
[569, 569, 5*w^2 - 4*w - 32],\
[569, 569, 3*w^2 - 3*w - 11],\
[569, 569, w^2 + 2*w - 16],\
[571, 571, w^2 - 4*w - 10],\
[593, 593, 2*w^2 - 6*w - 7],\
[601, 601, -w^2 - 3*w + 14],\
[607, 607, 3*w^2 + 3*w - 7],\
[619, 619, 3*w^2 - 6*w - 11],\
[631, 631, -w^2 - w - 5],\
[641, 641, 3*w^2 - 3*w - 5],\
[647, 647, -w^2 + 6*w - 1],\
[653, 653, -5*w^2 + 4*w + 35],\
[661, 661, 3*w^2 - 3*w - 10],\
[673, 673, w^2 + 4*w - 4],\
[677, 677, 5*w^2 - 6*w - 25],\
[677, 677, w^2 - 14],\
[677, 677, 4*w^2 - 3*w - 20],\
[691, 691, 5*w^2 - 2*w - 29],\
[701, 701, 5*w^2 - 10*w - 14],\
[719, 719, -3*w - 11],\
[727, 727, -2*w^2 + 9*w - 8],\
[733, 733, -w^2 - 4*w - 8],\
[739, 739, w^2 + 3*w - 8],\
[743, 743, 2*w^2 - 4*w - 17],\
[757, 757, -w^2 + w - 5],\
[769, 769, 3*w^2 - 3*w - 7],\
[769, 769, -w^2 - 3*w - 7],\
[769, 769, 5*w^2 - 3*w - 28],\
[773, 773, 7*w^2 - 6*w - 38],\
[787, 787, 2*w^2 - w - 20],\
[797, 797, -5*w^2 + 14*w - 1],\
[821, 821, -w^2 + 6*w - 4],\
[823, 823, 7*w^2 - 6*w - 41],\
[839, 839, 4*w^2 - 6*w - 11],\
[841, 29, 5*w^2 - 4*w - 26],\
[857, 857, 3*w^2 - 11],\
[863, 863, 2*w^2 + 2*w - 17],\
[881, 881, 5*w^2 - 5*w - 29],\
[883, 883, -4*w^2 + 8*w + 13],\
[907, 907, 4*w^2 - 9*w - 11],\
[911, 911, 2*w^2 + 3*w - 10],\
[947, 947, 2*w^2 + 5*w - 5],\
[953, 953, 2*w^2 - 5*w - 11],\
[961, 31, w^2 - 5*w - 13],\
[967, 967, 2*w^2 - 3*w - 22],\
[967, 967, 5*w^2 - 9*w - 10],\
[967, 967, 3*w^2 - 10],\
[977, 977, 5*w^2 - 37],\
[983, 983, 7*w^2 - 5*w - 40],\
[997, 997, -7*w^2 + 5*w + 49]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [-1, -1, e, -1, e, -e, e + 4, -2*e, -e - 4, -2*e - 4, -4, 0, 2*e - 8, -4*e - 6, -3*e + 4, 2*e - 10, 6*e + 4, 4*e + 2, -5*e - 8, 4*e - 2, -2*e - 8, 5*e + 8, 2*e + 8, -12, -e - 8, e - 4, 4*e + 2, -8, 4, 2*e + 12, -4*e + 4, -3*e + 4, -6*e - 12, -8*e - 14, -2*e + 8, 8, 0, 6*e + 8, -5*e - 8, 8*e + 8, -2*e + 4, -2*e, 14, -2*e, 4*e, 6*e + 8, 5*e - 8, -5*e - 4, -24, -6*e - 10, -2*e - 16, 3*e - 16, 26, 2, 6*e + 4, -4*e + 2, -8*e + 2, -4*e - 14, 13*e + 16, 6*e + 24, -3*e - 8, -24, 4*e, -2*e + 20, -12*e - 12, 2*e - 28, -4*e - 6, -4*e - 6, 14*e + 12, -4*e + 8, 4*e - 8, e + 4, -3*e + 4, 10*e + 8, -4*e - 10, 4*e + 18, 8*e + 10, -2*e + 10, -3*e + 4, -8*e - 20, 11*e + 12, 4*e + 4, 7*e - 4, -e + 20, -8*e - 20, 2*e + 8, e - 28, 4*e - 10, -e - 32, -14*e - 18, -11*e - 4, -8*e - 24, 2*e + 28, -e - 12, -8*e, 2*e - 12, 14*e + 16, -4*e + 4, -e + 12, -e + 4, 10*e + 20, 13*e + 24, 10*e + 4, -2*e + 4, 28, -9*e + 4, -6*e - 18, -14*e - 8, 4*e - 20, -7*e - 20, 4*e - 18, -e - 4, -4*e - 12, 2*e + 10, 6*e + 12, -2*e + 8, 8*e + 16, -9*e - 24, 10*e + 12, -8, -10*e + 12, 8*e + 14, 2*e + 32, -8*e + 14, e - 12, -7*e - 20, -3*e - 32, 12*e + 8, -6*e + 28, 12*e + 26, -12, -9*e, 8*e + 10, -4, -2*e + 18, 2*e - 44, -4*e + 20, -4*e - 36, 6*e + 38, 4*e - 2, 5*e - 16, e + 20, 22*e + 20, -8*e - 36, -10*e - 28, 6*e + 36, -4*e - 24, -32, 8*e + 16, -6*e - 16, -13*e - 12, 7*e + 52, -10*e + 20, -e - 8, 16*e - 4, 16*e + 18, 7*e - 32, 16*e + 4]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([3, 3, w])] = 1
AL_eigenvalues[ZF.ideal([3, 3, w + 1])] = 1

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