/*
  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, 3, -1, -5, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([83, 83, -2*w^4 + 2*w^3 + 10*w^2 - 7*w - 6])

primes_array = [
[5, 5, -2*w^4 + w^3 + 9*w^2 - 2*w - 3],\
[17, 17, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],\
[17, 17, w^2 - 2],\
[23, 23, w^3 - 3*w],\
[29, 29, 2*w^4 - w^3 - 10*w^2 + 2*w + 4],\
[32, 2, 2],\
[37, 37, -w^4 + w^3 + 4*w^2 - 2*w],\
[41, 41, -3*w^4 + 2*w^3 + 14*w^2 - 5*w - 7],\
[43, 43, -3*w^4 + w^3 + 14*w^2 - 3*w - 6],\
[47, 47, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6],\
[53, 53, 2*w^4 - w^3 - 8*w^2 + 2*w + 1],\
[53, 53, -2*w^4 + w^3 + 10*w^2 - 4*w - 6],\
[59, 59, -w^4 + 4*w^2 + 1],\
[59, 59, -3*w^4 + 2*w^3 + 14*w^2 - 6*w - 8],\
[61, 61, 4*w^4 - 2*w^3 - 18*w^2 + 5*w + 7],\
[61, 61, 3*w^4 - w^3 - 15*w^2 + 2*w + 7],\
[73, 73, -4*w^4 + 2*w^3 + 18*w^2 - 7*w - 6],\
[83, 83, -2*w^4 + 9*w^2 + 2*w - 4],\
[83, 83, -2*w^4 + 2*w^3 + 10*w^2 - 7*w - 6],\
[97, 97, -2*w^4 + w^3 + 8*w^2 - 3*w + 1],\
[97, 97, -w^3 + w^2 + 4*w - 1],\
[101, 101, 2*w^4 - 9*w^2 - w + 5],\
[103, 103, 2*w^4 - 2*w^3 - 9*w^2 + 8*w + 2],\
[107, 107, -3*w^4 + w^3 + 15*w^2 - 4*w - 8],\
[109, 109, -3*w^4 + w^3 + 13*w^2 - 2*w - 4],\
[113, 113, 4*w^4 - 3*w^3 - 18*w^2 + 8*w + 8],\
[121, 11, -w^4 - w^3 + 5*w^2 + 4*w - 2],\
[131, 131, -4*w^4 + w^3 + 18*w^2 - 5],\
[139, 139, 2*w^3 - w^2 - 8*w + 1],\
[139, 139, -2*w^4 + w^3 + 9*w^2 - 3],\
[149, 149, 5*w^4 - 2*w^3 - 23*w^2 + 4*w + 11],\
[149, 149, w^4 - 4*w^2 - 3],\
[149, 149, 4*w^4 - 3*w^3 - 19*w^2 + 10*w + 8],\
[151, 151, 2*w^4 - w^3 - 9*w^2 + 5*w + 4],\
[157, 157, -2*w^4 - w^3 + 10*w^2 + 5*w - 4],\
[157, 157, 5*w^4 - 3*w^3 - 24*w^2 + 8*w + 13],\
[163, 163, -2*w^4 + 11*w^2 + w - 7],\
[167, 167, w^4 - 3*w^2 - 2*w - 3],\
[167, 167, -3*w^4 + 3*w^3 + 14*w^2 - 10*w - 9],\
[169, 13, 3*w^4 - 2*w^3 - 14*w^2 + 6*w + 3],\
[191, 191, -3*w^4 + w^3 + 15*w^2 - w - 6],\
[193, 193, 3*w^4 - 2*w^3 - 14*w^2 + 4*w + 5],\
[199, 199, -5*w^4 + 2*w^3 + 23*w^2 - 3*w - 10],\
[223, 223, -4*w^4 + w^3 + 20*w^2 - w - 9],\
[223, 223, -w^4 + 3*w^2 + 2],\
[233, 233, w^3 - 4*w - 4],\
[241, 241, -3*w^4 + 2*w^3 + 15*w^2 - 7*w - 7],\
[243, 3, -3],\
[251, 251, -3*w^4 + 3*w^3 + 13*w^2 - 8*w - 6],\
[257, 257, -4*w^4 + w^3 + 20*w^2 - 2*w - 9],\
[257, 257, -3*w^4 + 2*w^3 + 15*w^2 - 5*w - 11],\
[269, 269, 2*w^4 - 11*w^2 - w + 6],\
[271, 271, w^4 + w^3 - 5*w^2 - 6*w + 2],\
[271, 271, 3*w^4 - w^3 - 15*w^2 + 8],\
[277, 277, -w^4 - w^3 + 6*w^2 + 3*w - 5],\
[283, 283, -5*w^4 + 3*w^3 + 23*w^2 - 10*w - 10],\
[293, 293, -w^4 + w^3 + 6*w^2 - 3*w - 8],\
[307, 307, -w^4 - w^3 + 6*w^2 + 4*w - 2],\
[311, 311, -w^3 + 2*w^2 + 3*w - 2],\
[311, 311, 2*w^4 - w^3 - 10*w^2 + 5*w + 7],\
[313, 313, -5*w^4 + 4*w^3 + 23*w^2 - 12*w - 11],\
[313, 313, -w^4 - w^3 + 5*w^2 + 3*w - 3],\
[347, 347, -w^4 + 2*w^3 + 3*w^2 - 7*w + 1],\
[353, 353, -3*w^4 + 14*w^2 + 3*w - 7],\
[353, 353, -3*w^4 + 3*w^3 + 13*w^2 - 9*w - 7],\
[367, 367, -w^4 + w^3 + 5*w^2 - 6*w - 4],\
[367, 367, -3*w^4 + w^3 + 15*w^2 - 3*w - 5],\
[379, 379, 5*w^4 - 2*w^3 - 23*w^2 + 6*w + 9],\
[383, 383, 4*w^4 - 2*w^3 - 19*w^2 + 7*w + 11],\
[397, 397, w^4 - 3*w^2 - w - 3],\
[397, 397, -2*w^3 + w^2 + 8*w],\
[401, 401, 2*w^4 - 2*w^3 - 8*w^2 + 5*w],\
[421, 421, -4*w^4 + 2*w^3 + 17*w^2 - 5*w - 6],\
[421, 421, -w^4 + 2*w^3 + 6*w^2 - 8*w - 5],\
[431, 431, 5*w^4 - 2*w^3 - 22*w^2 + 5*w + 7],\
[431, 431, -w^4 - w^3 + 6*w^2 + 5*w - 4],\
[439, 439, 4*w^4 - 2*w^3 - 19*w^2 + 7*w + 10],\
[443, 443, 4*w^4 - w^3 - 19*w^2 + 3*w + 8],\
[457, 457, -3*w^4 + 3*w^3 + 15*w^2 - 10*w - 9],\
[461, 461, -w^3 + 6*w],\
[461, 461, 3*w^4 - 3*w^3 - 14*w^2 + 9*w + 8],\
[463, 463, 5*w^4 - 3*w^3 - 23*w^2 + 7*w + 8],\
[499, 499, 6*w^4 - 3*w^3 - 27*w^2 + 7*w + 11],\
[509, 509, 2*w^4 - w^3 - 8*w^2 + w - 1],\
[509, 509, 3*w^4 - 2*w^3 - 13*w^2 + 6*w + 7],\
[521, 521, w^3 - 6*w + 1],\
[523, 523, 4*w^4 - 2*w^3 - 18*w^2 + 3*w + 7],\
[541, 541, w^2 + w - 5],\
[547, 547, 5*w^4 - 2*w^3 - 25*w^2 + 4*w + 12],\
[547, 547, -6*w^4 + 3*w^3 + 27*w^2 - 10*w - 9],\
[557, 557, -4*w^4 + 3*w^3 + 18*w^2 - 10*w - 9],\
[557, 557, -4*w^4 + w^3 + 19*w^2 - 3*w - 6],\
[557, 557, 3*w^4 - 2*w^3 - 14*w^2 + 8*w + 7],\
[563, 563, -w^4 + 3*w^3 + 4*w^2 - 11*w + 1],\
[569, 569, w^2 + 3*w - 2],\
[577, 577, -5*w^4 + 2*w^3 + 23*w^2 - 6*w - 8],\
[587, 587, -2*w^4 + 10*w^2 + 3*w - 6],\
[593, 593, 4*w^4 - 3*w^3 - 19*w^2 + 8*w + 11],\
[601, 601, -4*w^4 + w^3 + 18*w^2 - 3*w - 6],\
[601, 601, -5*w^4 + 3*w^3 + 24*w^2 - 10*w - 13],\
[601, 601, -w^4 + 5*w^2 + 3*w - 3],\
[607, 607, -3*w^4 + 2*w^3 + 15*w^2 - 8*w - 9],\
[613, 613, 3*w^4 - 14*w^2 - 2*w + 8],\
[617, 617, -4*w^4 + w^3 + 19*w^2 - 3*w - 7],\
[619, 619, w^3 + w^2 - 3*w - 5],\
[625, 5, 4*w^4 - 2*w^3 - 19*w^2 + 3*w + 8],\
[631, 631, 3*w^4 - w^3 - 12*w^2 - 1],\
[641, 641, -w^4 + 2*w^3 + 3*w^2 - 7*w],\
[647, 647, 2*w^2 - w - 5],\
[647, 647, -2*w^3 + w^2 + 8*w - 5],\
[647, 647, 5*w^4 - 3*w^3 - 23*w^2 + 9*w + 12],\
[661, 661, -8*w^4 + 3*w^3 + 39*w^2 - 7*w - 19],\
[677, 677, -7*w^4 + 4*w^3 + 34*w^2 - 11*w - 17],\
[691, 691, w^2 + 2*w - 4],\
[691, 691, -5*w^4 + 2*w^3 + 23*w^2 - 3*w - 8],\
[691, 691, -7*w^4 + 5*w^3 + 32*w^2 - 14*w - 15],\
[701, 701, w - 4],\
[709, 709, -3*w^4 + 3*w^3 + 14*w^2 - 8*w - 7],\
[709, 709, 5*w^4 - 2*w^3 - 24*w^2 + 5*w + 9],\
[709, 709, -w^4 + w^3 + 5*w^2 - 3*w - 7],\
[719, 719, -3*w^4 + 13*w^2 + w - 7],\
[719, 719, -3*w^4 + 14*w^2 - 4],\
[727, 727, -7*w^4 + 4*w^3 + 32*w^2 - 10*w - 16],\
[727, 727, -3*w^4 + w^3 + 14*w^2 - 5*w - 8],\
[733, 733, -3*w^4 + 2*w^3 + 15*w^2 - 8*w - 10],\
[733, 733, -3*w^4 + 3*w^3 + 13*w^2 - 10*w - 7],\
[733, 733, 8*w^4 - 4*w^3 - 38*w^2 + 9*w + 15],\
[739, 739, 4*w^4 - 3*w^3 - 20*w^2 + 9*w + 11],\
[739, 739, -w^4 + w^3 + 5*w^2 - 3*w + 1],\
[743, 743, -5*w^4 + 4*w^3 + 24*w^2 - 13*w - 16],\
[743, 743, w^4 - 7*w^2 - 2*w + 5],\
[751, 751, -w^4 + 2*w^3 + 3*w^2 - 7*w - 1],\
[757, 757, 5*w^4 - 2*w^3 - 22*w^2 + 5*w + 8],\
[757, 757, w^4 + w^3 - 6*w^2 - 5*w + 2],\
[761, 761, 8*w^4 - 3*w^3 - 38*w^2 + 8*w + 18],\
[761, 761, 3*w^4 - w^3 - 14*w^2 + 2*w + 10],\
[761, 761, -4*w^4 + 3*w^3 + 19*w^2 - 7*w - 10],\
[769, 769, -w^4 + 2*w^3 + 5*w^2 - 8*w - 1],\
[773, 773, 9*w^4 - 4*w^3 - 41*w^2 + 9*w + 15],\
[773, 773, 4*w^4 - 2*w^3 - 18*w^2 + 3*w + 6],\
[773, 773, 2*w^2 - 5],\
[787, 787, -w^4 + 6*w^2 - w - 7],\
[787, 787, w^4 - w^3 - 6*w^2 + w + 6],\
[821, 821, 2*w^4 - w^3 - 9*w^2 + 4*w + 8],\
[823, 823, 6*w^4 - w^3 - 27*w^2 - w + 8],\
[827, 827, -3*w^4 + 2*w^3 + 16*w^2 - 6*w - 9],\
[827, 827, -8*w^4 + 5*w^3 + 37*w^2 - 16*w - 17],\
[839, 839, 2*w^4 - 10*w^2 + w + 8],\
[853, 853, 2*w^4 - 2*w^3 - 11*w^2 + 7*w + 7],\
[857, 857, 5*w^4 - 2*w^3 - 24*w^2 + 7*w + 14],\
[877, 877, -w^4 + 2*w^3 + 5*w^2 - 10*w - 6],\
[883, 883, 3*w^4 - 3*w^3 - 15*w^2 + 12*w + 10],\
[887, 887, -4*w^4 + w^3 + 17*w^2 - w - 7],\
[907, 907, 2*w^4 - 11*w^2 - w + 2],\
[907, 907, -2*w^4 - w^3 + 10*w^2 + 4*w - 5],\
[911, 911, -9*w^4 + 4*w^3 + 43*w^2 - 9*w - 19],\
[919, 919, 4*w^4 - w^3 - 20*w^2 + w + 8],\
[937, 937, w^4 - 7*w^2 - 2*w + 4],\
[947, 947, 2*w^4 + 2*w^3 - 10*w^2 - 9*w + 3],\
[953, 953, -2*w^3 + 5*w + 4],\
[961, 31, -4*w^4 + w^3 + 19*w^2 - w - 4],\
[967, 967, 8*w^4 - 5*w^3 - 37*w^2 + 13*w + 18],\
[967, 967, 3*w^4 - w^3 - 13*w^2 - 2*w + 3],\
[971, 971, -4*w^4 + 3*w^3 + 21*w^2 - 10*w - 15],\
[971, 971, -3*w^4 + 12*w^2 + 3*w - 4],\
[971, 971, w^4 - 6*w^2 + 3*w + 6],\
[983, 983, -7*w^4 + 2*w^3 + 34*w^2 - 5*w - 16],\
[983, 983, w^4 - 7*w^2 - w + 8],\
[991, 991, -4*w^4 + 4*w^3 + 19*w^2 - 15*w - 12],\
[991, 991, -8*w^4 + 3*w^3 + 37*w^2 - 5*w - 15],\
[991, 991, -w^4 + w^3 + 7*w^2 - 3*w - 7],\
[997, 997, -8*w^4 + 5*w^3 + 37*w^2 - 16*w - 16]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 - 5*x^2 - 4*x + 29
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -e^2 + e + 7, -2/3*e^2 + e + 26/3, e^2 - 12, -4/3*e^2 + 2*e + 34/3, 2*e^2 - 4*e - 15, -4/3*e^2 + 4*e + 22/3, -1/3*e^2 - 2/3, -2*e^2 + 3*e + 16, -1/3*e^2 + 16/3, 4/3*e^2 - 2*e - 28/3, 4/3*e^2 - 2*e - 28/3, -e^2 + 14, 4/3*e^2 - 2*e - 10/3, 1/3*e^2 - 3*e - 1/3, 2/3*e^2 - 50/3, -8, -5/3*e^2 + 2*e + 68/3, 1, -2/3*e^2 + 38/3, -2/3*e^2 + 38/3, 8/3*e^2 - 4*e - 74/3, -1/3*e^2 + 3*e + 7/3, 5/3*e^2 - 7*e - 17/3, -e^2 + e + 17, e^2 - 4, 2/3*e^2 - 4*e - 44/3, -5/3*e^2 + 3*e + 71/3, -4*e, 4/3*e^2 - e - 34/3, -4*e + 10, 11/3*e^2 - 9*e - 71/3, e^2 - 8*e - 2, 2/3*e^2 - 26/3, -6, -3*e^2 + 7*e + 19, 11/3*e^2 - 3*e - 137/3, -2/3*e^2 - 2*e + 38/3, 8/3*e^2 - 7*e - 68/3, 10/3*e^2 - 8*e - 34/3, 8/3*e^2 - 4*e - 50/3, 7/3*e^2 - e - 103/3, -e^2 + 5*e + 1, -8/3*e^2 + 6*e + 32/3, -7/3*e^2 + 3*e + 91/3, -19/3*e^2 + 13*e + 127/3, 4*e^2 - 8*e - 18, 11/3*e^2 - 7*e - 71/3, -8/3*e^2 + 10*e + 44/3, 8/3*e^2 - 5*e - 80/3, -17/3*e^2 + 12*e + 104/3, -4/3*e^2 + e + 34/3, -4/3*e^2 + 58/3, 6*e - 10, 5/3*e^2 - 2/3, 16/3*e^2 - 10*e - 118/3, -4/3*e^2 + 7*e - 2/3, -11/3*e^2 - e + 149/3, 2*e^2 - 13*e - 10, -2*e^2 + 6*e + 20, 1/3*e^2 - 10*e + 20/3, 4*e^2 - 10*e - 16, -11/3*e^2 + 128/3, 3*e^2 - e - 35, -4/3*e^2 + 6*e - 50/3, 17/3*e^2 - 9*e - 125/3, -7*e, -2*e - 18, -8/3*e^2 + 4*e + 56/3, 10/3*e^2 - 10*e - 52/3, -8/3*e^2 + 4*e + 98/3, -2/3*e^2 - 2*e + 2/3, -2/3*e^2 - 2*e + 62/3, 4/3*e^2 - 8*e - 46/3, -7/3*e^2 + 6*e + 46/3, -14/3*e^2 + 4*e + 170/3, 8/3*e^2 - 7*e + 10/3, -4/3*e^2 + 2*e + 46/3, 2/3*e^2 - 4*e - 20/3, -2/3*e^2 + 6*e + 38/3, 14/3*e^2 - 12*e - 50/3, 17/3*e^2 - 5*e - 155/3, 8/3*e^2 - 2*e - 146/3, -4/3*e^2 + 5*e - 56/3, 5/3*e^2 + 2*e - 80/3, -2/3*e^2 + 2*e + 44/3, -3*e^2 + 6*e + 22, -2/3*e^2 - 8*e + 38/3, 19/3*e^2 - 12*e - 136/3, -20/3*e^2 + 8*e + 170/3, 4*e^2 - 4*e - 54, 2*e^2 - 4*e + 18, -16/3*e^2 + 12*e + 148/3, 4*e^2 - 5*e - 42, 6*e^2 - 15*e - 48, 7/3*e^2 + 2*e - 112/3, 2*e^2 - 8*e - 10, -20/3*e^2 + 16*e + 164/3, -13/3*e^2 + 7*e + 139/3, -8/3*e^2 + 44/3, e^2 - 9*e + 5, -2*e^2 - 3*e + 42, 10/3*e^2 + 4*e - 148/3, -26/3*e^2 + 20*e + 134/3, 4*e^2 - 7*e - 56, -10/3*e^2 + 18*e + 28/3, -26/3*e^2 + 19*e + 194/3, 3*e^2 - 2*e - 28, -3*e^2 + 7*e + 17, 2/3*e^2 - 3*e + 40/3, 2*e^2 - 7*e - 38, -8*e^2 + 7*e + 88, -1/3*e^2 - 2*e - 26/3, -5*e^2 + 16*e + 38, 5*e^2 - 11*e - 37, -16/3*e^2 + 4*e + 202/3, 10/3*e^2 - 14*e - 52/3, -2*e^2 + 4*e + 20, -9*e + 26, 8*e^2 - 12*e - 80, -6*e + 18, -28/3*e^2 + 20*e + 184/3, -2*e^2 + 7*e + 20, 2/3*e^2 - e - 2/3, -8/3*e^2 + 4*e + 122/3, 14/3*e^2 - 19*e - 92/3, -4*e^2 + 7*e + 16, 2*e^2 + 4*e - 12, 4*e^2 - 8*e - 12, 4*e^2 - 4*e - 32, -16/3*e^2 + 16*e + 142/3, 8*e^2 - 4*e - 86, -16/3*e^2 + 12*e + 10/3, -22/3*e^2 + 13*e + 208/3, -5*e^2 + 7*e + 39, 2/3*e^2 - 3*e - 110/3, 8/3*e^2 - 4*e - 62/3, 2*e^2 - 4*e - 16, 2/3*e^2 - 8*e + 16/3, -8/3*e^2 + 6*e - 40/3, -2*e - 34, 2*e^2 - 3*e - 4, -38/3*e^2 + 26*e + 230/3, -2*e^2 + 8*e + 26, -22/3*e^2 + 16*e + 106/3, -8/3*e^2 + 8*e + 86/3, 2*e^2 + 8*e - 30, 8/3*e^2 - 20*e - 32/3, -2*e^2 + 5*e - 6, 7/3*e^2 - 19*e - 1/3, 16/3*e^2 - 3*e - 184/3, -19/3*e^2 + 2*e + 184/3, 7/3*e^2 - 4*e - 58/3, 14/3*e^2 - 14*e - 152/3, -14/3*e^2 + 194/3, -8/3*e^2 + 10*e + 56/3, -26/3*e^2 + 15*e + 146/3, 8*e - 12, -2*e^2 - 2*e + 48, -23/3*e^2 + 6*e + 302/3, -10/3*e^2 - 2*e + 166/3, 4/3*e^2 - 4*e + 44/3, 2*e^2 + 6*e - 42, -22/3*e^2 + 15*e + 76/3, 10/3*e^2 - 2*e - 196/3, 3*e^2 - 18*e - 12, 5*e^2 - 14*e - 14, -10*e^2 + 18*e + 88, -4/3*e^2 + 16*e - 38/3, 14/3*e^2 - 17*e - 92/3, 6*e^2 - 16*e - 30, -7/3*e^2 - 7*e + 133/3]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([83, 83, -2*w^4 + 2*w^3 + 10*w^2 - 7*w - 6])] = -1

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