/*
  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([-28, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([14, 14, 3*w + 14])

primes_array = [
[2, 2, -w + 6],\
[2, 2, w + 5],\
[7, 7, 6*w - 35],\
[7, 7, -6*w - 29],\
[9, 3, 3],\
[11, 11, 4*w + 19],\
[11, 11, 4*w - 23],\
[13, 13, -2*w + 11],\
[13, 13, 2*w + 9],\
[25, 5, -5],\
[31, 31, 2*w - 13],\
[31, 31, -2*w - 11],\
[41, 41, -8*w - 39],\
[41, 41, 8*w - 47],\
[53, 53, -26*w - 125],\
[53, 53, 26*w - 151],\
[61, 61, -14*w + 81],\
[61, 61, -14*w - 67],\
[83, 83, 2*w - 15],\
[83, 83, -2*w - 13],\
[97, 97, 2*w - 5],\
[97, 97, -2*w - 3],\
[109, 109, 2*w - 3],\
[109, 109, -2*w - 1],\
[113, 113, 2*w - 1],\
[127, 127, 8*w - 45],\
[127, 127, 8*w + 37],\
[131, 131, -50*w + 291],\
[131, 131, 50*w + 241],\
[139, 139, 6*w - 37],\
[139, 139, -6*w - 31],\
[149, 149, 20*w + 97],\
[149, 149, 20*w - 117],\
[157, 157, -12*w - 59],\
[157, 157, 12*w - 71],\
[163, 163, 4*w - 19],\
[163, 163, 4*w + 15],\
[173, 173, 4*w + 23],\
[173, 173, 4*w - 27],\
[211, 211, 2*w - 19],\
[211, 211, -2*w - 17],\
[227, 227, -4*w - 13],\
[227, 227, 4*w - 17],\
[233, 233, 6*w + 25],\
[233, 233, -6*w + 31],\
[239, 239, -14*w - 69],\
[239, 239, 14*w - 83],\
[241, 241, 46*w + 221],\
[241, 241, -46*w + 267],\
[251, 251, 22*w - 129],\
[251, 251, 22*w + 107],\
[257, 257, -34*w + 197],\
[257, 257, -34*w - 163],\
[277, 277, 4*w - 29],\
[277, 277, -4*w - 25],\
[283, 283, 4*w - 15],\
[283, 283, -4*w - 11],\
[289, 17, -17],\
[307, 307, 68*w - 395],\
[307, 307, 68*w + 327],\
[311, 311, 10*w - 61],\
[311, 311, -10*w - 51],\
[313, 313, -32*w - 155],\
[313, 313, 32*w - 187],\
[317, 317, -18*w - 85],\
[317, 317, 18*w - 103],\
[331, 331, -4*w - 9],\
[331, 331, 4*w - 13],\
[337, 337, -16*w - 79],\
[337, 337, 16*w - 95],\
[347, 347, -12*w + 67],\
[347, 347, 12*w + 55],\
[353, 353, 14*w - 79],\
[353, 353, 14*w + 65],\
[361, 19, -19],\
[367, 367, -32*w - 153],\
[367, 367, -32*w + 185],\
[383, 383, 56*w + 269],\
[383, 383, 56*w - 325],\
[389, 389, -4*w - 27],\
[389, 389, 4*w - 31],\
[401, 401, 8*w - 51],\
[401, 401, -8*w - 43],\
[421, 421, 12*w - 73],\
[421, 421, -12*w - 61],\
[439, 439, -8*w - 33],\
[439, 439, 8*w - 41],\
[443, 443, -4*w - 1],\
[443, 443, 4*w - 5],\
[461, 461, -30*w + 173],\
[461, 461, 30*w + 143],\
[463, 463, 2*w - 25],\
[463, 463, -2*w - 23],\
[467, 467, 34*w + 165],\
[467, 467, 34*w - 199],\
[503, 503, -26*w - 127],\
[503, 503, 26*w - 153],\
[509, 509, 4*w - 33],\
[509, 509, -4*w - 29],\
[521, 521, -10*w + 53],\
[521, 521, 10*w + 43],\
[529, 23, -23],\
[547, 547, 14*w - 85],\
[547, 547, -14*w - 71],\
[557, 557, 66*w + 317],\
[557, 557, 66*w - 383],\
[563, 563, 2*w - 27],\
[563, 563, -2*w - 25],\
[569, 569, -64*w + 373],\
[569, 569, 64*w + 309],\
[587, 587, 12*w + 53],\
[587, 587, -12*w + 65],\
[593, 593, 8*w + 45],\
[593, 593, 8*w - 53],\
[601, 601, -26*w - 123],\
[601, 601, 26*w - 149],\
[617, 617, 6*w - 23],\
[617, 617, -6*w - 17],\
[647, 647, 24*w - 137],\
[647, 647, -24*w - 113],\
[653, 653, 28*w - 165],\
[653, 653, -28*w - 137],\
[677, 677, -22*w - 103],\
[677, 677, 22*w - 125],\
[691, 691, 20*w - 113],\
[691, 691, 20*w + 93],\
[709, 709, -10*w - 41],\
[709, 709, 10*w - 51],\
[719, 719, -8*w + 37],\
[719, 719, -8*w - 29],\
[727, 727, -22*w - 109],\
[727, 727, 22*w - 131],\
[739, 739, 46*w - 269],\
[739, 739, -46*w - 223],\
[761, 761, -6*w - 13],\
[761, 761, 6*w - 19],\
[769, 769, 110*w - 639],\
[769, 769, 110*w + 529],\
[773, 773, 4*w - 37],\
[773, 773, -4*w - 33],\
[787, 787, 2*w - 31],\
[787, 787, -2*w - 29],\
[809, 809, 56*w + 271],\
[809, 809, -56*w + 327],\
[821, 821, 6*w - 17],\
[821, 821, -6*w - 11],\
[823, 823, 38*w - 223],\
[823, 823, -38*w - 185],\
[827, 827, 132*w - 767],\
[827, 827, 132*w + 635],\
[841, 29, -29],\
[853, 853, -76*w + 443],\
[853, 853, 76*w + 367],\
[863, 863, 14*w - 87],\
[863, 863, -14*w - 73],\
[911, 911, 2*w - 33],\
[911, 911, -2*w - 31],\
[919, 919, -6*w - 41],\
[919, 919, 6*w - 47],\
[929, 929, 50*w + 239],\
[929, 929, 50*w - 289],\
[953, 953, 6*w - 11],\
[953, 953, -6*w - 5],\
[967, 967, -8*w - 25],\
[967, 967, 8*w - 33],\
[991, 991, 16*w + 71],\
[991, 991, -16*w + 87]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [-1, e, 1, -e + 1, 0, e - 1, e + 1, -e - 5, -2, 2*e, -2*e - 4, e + 9, -3*e - 5, -6, 3*e - 1, 2, -e + 5, 2*e - 4, -3*e + 5, -4*e - 10, -4*e - 2, -5*e - 7, -6*e - 6, -2, e + 3, 6*e + 4, -4*e + 4, e - 3, -2*e - 10, -6*e + 4, -4*e + 8, 2*e - 12, 6*e + 12, 4*e - 4, 5*e - 1, -4*e, -4*e - 12, 5*e + 3, -4*e - 12, -10*e - 10, -2*e - 2, -3*e - 15, 10*e + 10, 4*e + 2, -2, 4*e - 14, 10*e + 6, 2*e + 2, -e - 5, -8*e - 4, 8*e, -4*e - 14, 6*e - 4, 3*e - 23, 3*e + 5, 8*e + 12, 12*e + 6, 4*e - 6, 4*e + 12, 12*e + 8, -6*e - 12, 5*e + 25, -2*e + 26, -4*e - 16, 4*e - 24, -4*e + 22, -4*e + 24, 12*e - 2, 4*e - 10, 8*e - 6, -e - 9, -e + 27, e - 29, -6*e - 12, -4, e - 19, 10*e + 18, 5*e + 13, 8*e + 6, 6*e + 26, -6*e + 20, -2*e - 6, 9*e - 3, 6*e - 12, -8*e - 22, 7*e + 3, -4*e, -12*e - 8, -14*e - 4, 2*e + 8, 8*e + 14, 3*e + 3, 11*e - 9, -5*e - 27, 6*e + 2, -9*e + 3, 2*e - 22, -4*e - 10, e - 9, 12*e - 10, -10*e - 16, 3*e - 7, -11*e + 7, 3*e + 35, 6*e + 6, 14*e - 8, -2*e + 18, 3*e + 31, -16*e - 14, e + 23, 2*e + 6, -13*e + 7, -16*e - 6, -4*e - 18, -22, -38, -7*e + 7, 16*e + 10, 15*e + 3, 10*e + 26, -5*e - 31, -12*e - 2, -3*e + 27, -10*e - 28, 5*e + 9, -6*e + 6, 8*e + 14, -14*e + 14, 15*e + 21, 14*e + 10, 14*e + 8, -8*e - 20, 2, 2*e + 30, 6*e + 14, 4*e - 20, 8*e - 18, -10*e - 12, -3*e + 3, -12*e + 10, 14*e + 18, -6*e + 8, 6*e - 24, 2*e + 16, 22*e + 4, -4*e - 34, -16, -10*e + 22, -11*e + 5, 19*e + 13, 12*e + 18, -23*e - 11, -6, -12*e - 20, -14*e - 30, 20*e + 4, -2*e + 46, -14*e + 10, 14*e + 6, -8*e + 26, 9*e - 17, 9*e + 29, 9*e + 9, 2*e + 18, 6*e + 30, -4*e, -10*e + 14]
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 + 6])] = 1
AL_eigenvalues[ZF.ideal([7, 7, 6*w - 35])] = -1

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