/*
  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([-78, -1, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([4, 2, 2])

primes_array = [
[2, 2, -3*w + 28],\
[2, 2, -3*w - 25],\
[3, 3, 26*w - 243],\
[3, 3, -26*w - 217],\
[11, 11, -2*w - 17],\
[11, 11, -2*w + 19],\
[13, 13, 2148*w + 17927],\
[13, 13, 2148*w - 20075],\
[19, 19, -292*w + 2729],\
[19, 19, 292*w + 2437],\
[25, 5, -5],\
[29, 29, 20*w + 167],\
[29, 29, 20*w - 187],\
[49, 7, -7],\
[71, 71, -240*w + 2243],\
[71, 71, 240*w + 2003],\
[79, 79, -3368*w + 31477],\
[79, 79, -3368*w - 28109],\
[83, 83, -84*w - 701],\
[83, 83, 84*w - 785],\
[97, 97, 38*w + 317],\
[97, 97, 38*w - 355],\
[103, 103, -136*w - 1135],\
[103, 103, 136*w - 1271],\
[107, 107, -1246*w - 10399],\
[107, 107, 1246*w - 11645],\
[113, 113, 6*w + 49],\
[113, 113, 6*w - 55],\
[137, 137, 558*w + 4657],\
[137, 137, -558*w + 5215],\
[139, 139, -2466*w + 23047],\
[139, 139, 2466*w + 20581],\
[151, 151, 16*w - 149],\
[151, 151, 16*w + 133],\
[157, 157, -396*w + 3701],\
[157, 157, 396*w + 3305],\
[163, 163, -4*w + 35],\
[163, 163, -4*w - 31],\
[173, 173, 1194*w - 11159],\
[173, 173, -1194*w - 9965],\
[181, 181, -12*w + 113],\
[181, 181, -12*w - 101],\
[197, 197, 6126*w - 57253],\
[197, 197, -6126*w - 51127],\
[199, 199, 66*w - 617],\
[199, 199, 66*w + 551],\
[241, 241, -506*w + 4729],\
[241, 241, 506*w + 4223],\
[263, 263, 2*w - 25],\
[263, 263, -2*w - 23],\
[269, 269, -4*w + 41],\
[269, 269, 4*w + 37],\
[277, 277, 2*w - 7],\
[277, 277, -2*w - 5],\
[281, 281, 448*w - 4187],\
[281, 281, 448*w + 3739],\
[289, 17, -17],\
[307, 307, 54*w + 451],\
[307, 307, 54*w - 505],\
[311, 311, 7982*w + 66617],\
[311, 311, 7982*w - 74599],\
[313, 313, 2*w - 1],\
[317, 317, -6*w + 53],\
[317, 317, -6*w - 47],\
[331, 331, -30*w + 281],\
[331, 331, -30*w - 251],\
[337, 337, 48*w - 449],\
[337, 337, 48*w + 401],\
[349, 349, -36*w - 301],\
[349, 349, -36*w + 337],\
[367, 367, 56*w + 467],\
[367, 367, 56*w - 523],\
[379, 379, -13180*w - 109999],\
[379, 379, -13180*w + 123179],\
[383, 383, -118*w - 985],\
[383, 383, -118*w + 1103],\
[389, 389, 18*w - 167],\
[389, 389, -18*w - 149],\
[401, 401, -246*w - 2053],\
[401, 401, 246*w - 2299],\
[409, 409, 350*w + 2921],\
[409, 409, -350*w + 3271],\
[421, 421, -2414*w - 20147],\
[421, 421, 2414*w - 22561],\
[457, 457, 34*w - 317],\
[457, 457, 34*w + 283],\
[463, 463, 6762*w + 56435],\
[463, 463, -6762*w + 63197],\
[479, 479, -5808*w + 54281],\
[479, 479, -5808*w - 48473],\
[487, 487, 3102*w + 25889],\
[487, 487, 3102*w - 28991],\
[509, 509, 102*w + 851],\
[509, 509, 102*w - 953],\
[521, 521, -2200*w - 18361],\
[521, 521, 2200*w - 20561],\
[523, 523, -4*w - 25],\
[523, 523, 4*w - 29],\
[529, 23, -23],\
[541, 541, -228*w - 1903],\
[541, 541, -228*w + 2131],\
[547, 547, -6*w + 61],\
[547, 547, 6*w + 55],\
[569, 569, -9520*w + 88973],\
[569, 569, -9520*w - 79453],\
[577, 577, -11642*w + 108805],\
[577, 577, 11642*w + 97163],\
[587, 587, 2*w - 31],\
[587, 587, -2*w - 29],\
[593, 593, 1616*w - 15103],\
[593, 593, -1616*w - 13487],\
[599, 599, -17794*w + 166301],\
[599, 599, -17794*w - 148507],\
[601, 601, -552*w + 5159],\
[601, 601, 552*w + 4607],\
[607, 607, -3952*w - 32983],\
[607, 607, -3952*w + 36935],\
[613, 613, -12*w - 103],\
[613, 613, -12*w + 115],\
[617, 617, 8*w + 71],\
[617, 617, 8*w - 79],\
[653, 653, -28*w - 235],\
[653, 653, -28*w + 263],\
[659, 659, -12*w - 97],\
[659, 659, -12*w + 109],\
[661, 661, 1090*w + 9097],\
[661, 661, 1090*w - 10187],\
[683, 683, 108*w - 1009],\
[683, 683, 108*w + 901],\
[701, 701, -6*w - 43],\
[701, 701, 6*w - 49],\
[709, 709, -3420*w + 31963],\
[709, 709, -3420*w - 28543],\
[733, 733, 154*w - 1439],\
[733, 733, 154*w + 1285],\
[739, 739, -1350*w + 12617],\
[739, 739, -1350*w - 11267],\
[743, 743, -818*w - 6827],\
[743, 743, -818*w + 7645],\
[769, 769, -10*w + 89],\
[769, 769, -10*w - 79],\
[773, 773, -4*w - 43],\
[773, 773, 4*w - 47],\
[787, 787, -7028*w + 65683],\
[787, 787, -7028*w - 58655],\
[797, 797, -604*w + 5645],\
[797, 797, 604*w + 5041],\
[811, 811, 4*w - 23],\
[811, 811, -4*w - 19],\
[823, 823, 222*w + 1853],\
[823, 823, -222*w + 2075],\
[839, 839, -14400*w + 134581],\
[839, 839, -14400*w - 120181],\
[863, 863, -18696*w - 156035],\
[863, 863, -18696*w + 174731],\
[881, 881, 6*w - 47],\
[881, 881, -6*w - 41],\
[887, 887, -46*w + 431],\
[887, 887, -46*w - 385],\
[907, 907, 52*w - 485],\
[907, 907, 52*w + 433],\
[937, 937, -14*w - 113],\
[937, 937, -14*w + 127],\
[941, 941, 332*w + 2771],\
[941, 941, -332*w + 3103],\
[947, 947, -5172*w - 43165],\
[947, 947, -5172*w + 48337],\
[961, 31, -31],\
[971, 971, 158*w - 1477],\
[971, 971, 158*w + 1319],\
[977, 977, -64*w - 535],\
[977, 977, 64*w - 599],\
[983, 983, 2*w - 37],\
[983, 983, -2*w - 35],\
[991, 991, -870*w + 8131],\
[991, 991, 870*w + 7261],\
[997, 997, -1668*w - 13921],\
[997, 997, 1668*w - 15589]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 - 9*x + 4
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1, 1, e, e, -1/2*e^2 - 1/2*e + 6, -1/2*e^2 - 1/2*e + 6, -1/2*e^2 - 1/2*e, -1/2*e^2 - 1/2*e, -e^2 + 4, -e^2 + 4, -e^2 + 10, 1/2*e^2 + 3/2*e - 8, 1/2*e^2 + 3/2*e - 8, 3/2*e^2 + 3/2*e - 4, 1/2*e^2 - 5/2*e - 6, 1/2*e^2 - 5/2*e - 6, 1/2*e^2 - 1/2*e - 6, 1/2*e^2 - 1/2*e - 6, e^2 + 4*e - 12, e^2 + 4*e - 12, e^2 + 2*e + 2, e^2 + 2*e + 2, e + 4, e + 4, 5/2*e^2 + 1/2*e - 14, 5/2*e^2 + 1/2*e - 14, -5*e - 2, -5*e - 2, e - 2, e - 2, -5/2*e^2 - 7/2*e + 18, -5/2*e^2 - 7/2*e + 18, -3/2*e^2 - 3/2*e + 6, -3/2*e^2 - 3/2*e + 6, e^2 + 14, e^2 + 14, 3*e^2 + 3*e - 24, 3*e^2 + 3*e - 24, -4*e^2 - 5*e + 26, -4*e^2 - 5*e + 26, 5/2*e^2 - 9/2*e - 16, 5/2*e^2 - 9/2*e - 16, e^2 + 7*e - 6, e^2 + 7*e - 6, -1/2*e^2 - 5/2*e + 10, -1/2*e^2 - 5/2*e + 10, e^2 - e + 6, e^2 - e + 6, -3*e^2 - 4*e + 8, -3*e^2 - 4*e + 8, e^2 - 2*e + 6, e^2 - 2*e + 6, -3/2*e^2 - 15/2*e + 20, -3/2*e^2 - 15/2*e + 20, 3*e^2 + 7*e - 26, 3*e^2 + 7*e - 26, -e^2 + 8*e + 18, -e^2 - 8*e + 4, -e^2 - 8*e + 4, -2*e^2 + 2*e + 16, -2*e^2 + 2*e + 16, -2*e^2 + 8*e + 26, -e^2 + 2*e + 6, -e^2 + 2*e + 6, 5/2*e^2 + 15/2*e - 10, 5/2*e^2 + 15/2*e - 10, -1/2*e^2 + 13/2*e + 16, -1/2*e^2 + 13/2*e + 16, 6*e^2 + 6*e - 26, 6*e^2 + 6*e - 26, e^2 + 8*e, e^2 + 8*e, -7/2*e^2 - 7/2*e + 18, -7/2*e^2 - 7/2*e + 18, e^2 - 16, e^2 - 16, 11/2*e^2 + 9/2*e - 28, 11/2*e^2 + 9/2*e - 28, -1/2*e^2 + 11/2*e + 12, -1/2*e^2 + 11/2*e + 12, -9/2*e^2 - 9/2*e + 28, -9/2*e^2 - 9/2*e + 28, -5/2*e^2 - 7/2*e + 12, -5/2*e^2 - 7/2*e + 12, -e^2 - 5*e + 14, -e^2 - 5*e + 14, 3/2*e^2 + 11/2*e - 14, 3/2*e^2 + 11/2*e - 14, 7/2*e^2 - 5/2*e - 30, 7/2*e^2 - 5/2*e - 30, 4*e - 8, 4*e - 8, 9/2*e^2 - 13/2*e - 32, 9/2*e^2 - 13/2*e - 32, 15/2*e^2 + 9/2*e - 48, 15/2*e^2 + 9/2*e - 48, -4*e - 4, -4*e - 4, -5*e^2 + 4*e + 42, 4*e^2 - e - 6, 4*e^2 - e - 6, 2*e^2 + 7*e - 24, 2*e^2 + 7*e - 24, -7/2*e^2 - 3/2*e + 8, -7/2*e^2 - 3/2*e + 8, -2*e^2 - 4*e + 18, -2*e^2 - 4*e + 18, -11/2*e^2 - 5/2*e + 30, -11/2*e^2 - 5/2*e + 30, -3/2*e^2 - 25/2*e + 4, -3/2*e^2 - 25/2*e + 4, 5*e^2 - 6*e - 32, 5*e^2 - 6*e - 32, 7/2*e^2 - 7/2*e - 40, 7/2*e^2 - 7/2*e - 40, -3/2*e^2 + 7/2*e + 26, -3/2*e^2 + 7/2*e + 26, 1/2*e^2 - 7/2*e - 12, 1/2*e^2 - 7/2*e - 12, 3/2*e^2 - 5/2*e - 28, 3/2*e^2 - 5/2*e - 28, 3*e^2 - 6*e - 18, 3*e^2 - 6*e - 18, 2*e^2 - 6*e - 20, 2*e^2 - 6*e - 20, 6*e^2 + 3*e - 30, 6*e^2 + 3*e - 30, 4*e^2 + 7*e - 48, 4*e^2 + 7*e - 48, -9/2*e^2 + 7/2*e + 8, -9/2*e^2 + 7/2*e + 8, -e^2 - 14*e + 6, -e^2 - 14*e + 6, 9*e^2 + 8*e - 58, 9*e^2 + 8*e - 58, 9/2*e^2 + 7/2*e - 34, 9/2*e^2 + 7/2*e - 34, -13/2*e^2 - 11/2*e + 22, -13/2*e^2 - 11/2*e + 22, -7*e^2 + 2*e + 42, -7*e^2 + 2*e + 42, 7*e^2 + 15*e - 46, 7*e^2 + 15*e - 46, -9/2*e^2 - 3/2*e + 50, -9/2*e^2 - 3/2*e + 50, -7*e^2 - 4*e + 30, -7*e^2 - 4*e + 30, -11/2*e^2 - 5/2*e + 30, -11/2*e^2 - 5/2*e + 30, -1/2*e^2 - 17/2*e - 14, -1/2*e^2 - 17/2*e - 14, -4*e^2 - 11*e + 20, -4*e^2 - 11*e + 20, 13/2*e^2 - 3/2*e - 50, 13/2*e^2 - 3/2*e - 50, 1/2*e^2 + 13/2*e + 24, 1/2*e^2 + 13/2*e + 24, -e^2 - 2*e - 16, -e^2 - 2*e - 16, 9/2*e^2 + 9/2*e - 30, 9/2*e^2 + 9/2*e - 30, -19/2*e^2 + 7/2*e + 60, -19/2*e^2 + 7/2*e + 60, -4*e^2 - 11*e + 50, -4*e^2 - 11*e + 50, -5*e + 40, -5*e + 40, -5/2*e^2 - 7/2*e + 64, 5/2*e^2 + 23/2*e + 6, 5/2*e^2 + 23/2*e + 6, -e^2 + e - 10, -e^2 + e - 10, 1/2*e^2 - 17/2*e - 30, 1/2*e^2 - 17/2*e - 30, -4*e^2 - 8*e + 64, -4*e^2 - 8*e + 64, -3*e^2 + 22, -3*e^2 + 22]
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, -3*w + 28])] = -1
AL_eigenvalues[ZF.ideal([2, 2, -3*w - 25])] = -1

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