/*
  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([-34, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([6,6,-w + 2])

primes_array = [
[2, 2, w - 6],\
[3, 3, w + 1],\
[3, 3, w + 2],\
[5, 5, w + 2],\
[5, 5, w + 3],\
[11, 11, w + 1],\
[11, 11, w + 10],\
[17, 17, -3*w + 17],\
[29, 29, w + 11],\
[29, 29, w + 18],\
[37, 37, w + 16],\
[37, 37, w + 21],\
[47, 47, -w - 9],\
[47, 47, w - 9],\
[49, 7, -7],\
[61, 61, w + 20],\
[61, 61, w + 41],\
[89, 89, 2*w - 15],\
[89, 89, -2*w - 15],\
[103, 103, -14*w + 81],\
[103, 103, 4*w - 21],\
[107, 107, w + 26],\
[107, 107, w + 81],\
[109, 109, w + 19],\
[109, 109, w + 90],\
[127, 127, 2*w - 3],\
[127, 127, -2*w - 3],\
[131, 131, w + 54],\
[131, 131, w + 77],\
[137, 137, -3*w - 13],\
[137, 137, 3*w - 13],\
[139, 139, w + 27],\
[139, 139, w + 112],\
[151, 151, 8*w - 45],\
[151, 151, -10*w + 57],\
[163, 163, w + 69],\
[163, 163, w + 94],\
[169, 13, -13],\
[173, 173, w + 42],\
[173, 173, w + 131],\
[181, 181, w + 45],\
[181, 181, w + 136],\
[191, 191, -w - 15],\
[191, 191, w - 15],\
[197, 197, w + 25],\
[197, 197, w + 172],\
[211, 211, w + 33],\
[211, 211, w + 178],\
[223, 223, -3*w - 23],\
[223, 223, 3*w - 23],\
[227, 227, w + 48],\
[227, 227, w + 179],\
[239, 239, -5*w + 33],\
[239, 239, -23*w + 135],\
[257, 257, -3*w - 7],\
[257, 257, 3*w - 7],\
[263, 263, -24*w + 139],\
[263, 263, 6*w - 31],\
[269, 269, w + 29],\
[269, 269, w + 240],\
[271, 271, -9*w + 55],\
[271, 271, -15*w + 89],\
[277, 277, w + 119],\
[277, 277, w + 158],\
[281, 281, 3*w - 5],\
[281, 281, -3*w - 5],\
[283, 283, w + 113],\
[283, 283, w + 170],\
[317, 317, w + 44],\
[317, 317, w + 273],\
[347, 347, w + 46],\
[347, 347, w + 301],\
[353, 353, 9*w - 49],\
[353, 353, -21*w + 121],\
[359, 359, -25*w + 147],\
[359, 359, -7*w + 45],\
[361, 19, -19],\
[379, 379, w + 105],\
[379, 379, w + 274],\
[383, 383, 6*w - 29],\
[383, 383, -36*w + 209],\
[397, 397, w + 35],\
[397, 397, w + 362],\
[409, 409, -5*w - 21],\
[409, 409, 5*w - 21],\
[419, 419, w + 156],\
[419, 419, w + 263],\
[433, 433, -12*w + 73],\
[433, 433, -18*w + 107],\
[457, 457, 6*w + 41],\
[457, 457, -6*w + 41],\
[463, 463, -4*w - 9],\
[463, 463, 4*w - 9],\
[499, 499, w + 212],\
[499, 499, w + 287],\
[529, 23, -23],\
[541, 541, w + 121],\
[541, 541, w + 420],\
[547, 547, w + 157],\
[547, 547, w + 390],\
[569, 569, -28*w + 165],\
[569, 569, -10*w + 63],\
[571, 571, w + 264],\
[571, 571, w + 307],\
[577, 577, -47*w + 273],\
[577, 577, 7*w - 33],\
[593, 593, 2*w - 27],\
[593, 593, -2*w - 27],\
[599, 599, -6*w - 25],\
[599, 599, 6*w - 25],\
[619, 619, w + 167],\
[619, 619, w + 452],\
[631, 631, -15*w + 91],\
[631, 631, -21*w + 125],\
[643, 643, w + 57],\
[643, 643, w + 586],\
[647, 647, -11*w + 69],\
[647, 647, -29*w + 171],\
[653, 653, w + 120],\
[653, 653, w + 533],\
[677, 677, w + 64],\
[677, 677, w + 613],\
[683, 683, w + 248],\
[683, 683, w + 435],\
[691, 691, w + 322],\
[691, 691, w + 369],\
[709, 709, w + 265],\
[709, 709, w + 444],\
[727, 727, -9*w + 59],\
[727, 727, -39*w + 229],\
[761, 761, -27*w + 155],\
[761, 761, 15*w - 83],\
[769, 769, 5*w - 9],\
[769, 769, -5*w - 9],\
[787, 787, w + 63],\
[787, 787, w + 724],\
[811, 811, w + 70],\
[811, 811, w + 741],\
[821, 821, w + 149],\
[821, 821, w + 672],\
[827, 827, w + 193],\
[827, 827, w + 634],\
[853, 853, w + 253],\
[853, 853, w + 600],\
[863, 863, -6*w - 19],\
[863, 863, 6*w - 19],\
[877, 877, w + 339],\
[877, 877, w + 538],\
[907, 907, w + 74],\
[907, 907, w + 833],\
[919, 919, -3*w - 35],\
[919, 919, 3*w - 35],\
[937, 937, 7*w - 27],\
[937, 937, -7*w - 27],\
[941, 941, w + 144],\
[941, 941, w + 797],\
[947, 947, w + 292],\
[947, 947, w + 655],\
[953, 953, 2*w - 33],\
[953, 953, -2*w - 33],\
[961, 31, -31],\
[967, 967, 9*w - 61],\
[967, 967, -9*w - 61],\
[977, 977, -4*w - 39],\
[977, 977, 4*w - 39],\
[997, 997, w + 55],\
[997, 997, w + 942]]
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, 0, -e, -e - 4, 2*e, -2, 4, e - 4, -2*e - 4, -e - 4, 2*e + 4, -2*e + 4, -2*e - 6, -12, e - 4, 4*e + 6, -2, 4*e + 8, 2*e - 4, -e - 4, 2*e - 8, -3*e, 6*e + 8, 4*e, 8*e + 8, -8*e - 8, -5*e + 4, 10, -2*e - 10, 3*e - 4, 2*e - 8, 4*e + 8, 2*e - 12, -e + 8, 2*e - 4, -4*e - 6, 2*e + 4, 3*e + 20, -4*e - 16, -5*e, -4*e + 8, -8*e, 6*e + 8, 5*e, -e + 4, -16, -2*e + 4, -2*e - 20, e - 24, -4, 2*e + 12, 2*e + 20, 4*e + 2, 2, -2*e + 12, -4*e - 24, 2*e - 20, -e + 4, -8*e - 16, 12*e + 8, -2*e - 24, e + 16, 22, 2*e + 10, -3*e + 12, 4*e, -2*e - 12, -9*e - 12, -12*e - 12, -9*e - 8, -12*e - 18, 8*e + 14, 8*e + 16, 12*e + 8, 4*e + 10, -7*e - 8, 2*e - 4, 12*e + 16, -8*e - 24, -6*e - 4, -5*e - 4, 2*e + 2, -8*e - 26, e, 12*e + 12, -2, -8*e - 18, -18, 8*e + 22, -12*e, -24, -11*e + 4, 10*e + 16, 10*e + 2, 8*e + 24, 7*e - 8, -9*e - 12, -4*e + 24, -12*e - 22, 8*e + 26, 20, -9*e - 24, -6, 8*e + 26, -2*e - 26, 14*e + 22, -2*e + 20, 8*e, -6*e - 4, 5*e - 24, 12*e, 0, 16*e + 28, -11*e - 16, -4*e + 24, 4*e, -4*e - 24, -e, 12, -3*e - 44, -7*e + 8, -6*e - 4, -4*e - 24, -5*e - 36, 17*e + 24, -6*e + 24, 4*e - 16, -20*e - 24, 6*e + 6, 8*e - 6, -4*e - 6, 4*e + 2, e + 36, 6*e + 8, 4*e + 28, -5*e - 8, -6*e + 4, 3*e + 44, -15*e - 28, -8*e + 24, e, -10*e - 16, 8*e + 16, -8*e + 8, 8, 7*e - 32, -15*e - 12, 12*e + 24, 4*e + 8, 12*e + 24, 8*e + 6, -2*e - 30, -3*e - 4, 8*e + 12, 7*e + 4, 12*e + 16, 8*e + 38, 4*e - 18, -34, -12*e - 16, -8*e - 40, 10*e - 10, 6*e - 18, 2*e + 32, 17*e + 24]
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([3,3,-w + 2])] = -1

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