/*
  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, 5, -4, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([7,7,-w^2 + 2])

primes_array = [
[5, 5, w + 1],\
[5, 5, -w + 2],\
[7, 7, -w^2 + 2*w + 1],\
[7, 7, -w^2 + 2],\
[9, 3, w^3 - w^2 - 4*w],\
[16, 2, 2],\
[17, 17, -w^3 + 2*w^2 + 3*w - 2],\
[17, 17, -w^3 + w^2 + 4*w - 2],\
[25, 5, w^2 - w - 3],\
[37, 37, 2*w - 1],\
[43, 43, -w^3 + 2*w^2 + 2*w - 2],\
[43, 43, w^3 - w^2 - 3*w + 1],\
[59, 59, 2*w - 5],\
[59, 59, -2*w - 3],\
[79, 79, -w^3 + 2*w^2 + 5*w - 4],\
[79, 79, -w^3 + w^2 + 6*w - 2],\
[83, 83, -w^3 + 7*w],\
[83, 83, -w^3 + 2*w^2 + 4*w - 2],\
[83, 83, -w^3 + w^2 + 5*w - 3],\
[83, 83, w^2 - 2*w - 6],\
[89, 89, w^2 - 2*w - 4],\
[89, 89, w^2 - 5],\
[101, 101, -2*w^3 + 5*w^2 + 5*w - 11],\
[101, 101, w^2 + w - 4],\
[101, 101, w^2 - 3*w - 2],\
[101, 101, 2*w^3 - w^2 - 9*w - 3],\
[109, 109, w^3 - w^2 - 6*w - 3],\
[109, 109, -w^3 + 2*w^2 + 5*w - 9],\
[121, 11, w^3 - 8*w],\
[121, 11, w^3 - 3*w^2 - 5*w + 7],\
[131, 131, w^3 - 5*w - 6],\
[131, 131, -w^3 + 3*w^2 + 2*w - 10],\
[163, 163, 2*w^2 - 3*w - 6],\
[163, 163, 2*w^2 - w - 7],\
[167, 167, w^3 - w^2 - 3*w - 4],\
[167, 167, -w^3 + 2*w^2 + 2*w - 7],\
[193, 193, 2*w^3 - 3*w^2 - 7*w + 2],\
[193, 193, 2*w^3 - 3*w^2 - 7*w + 6],\
[227, 227, 2*w^3 - 3*w^2 - 8*w + 1],\
[227, 227, -2*w^3 + 3*w^2 + 8*w - 8],\
[251, 251, 2*w^2 - w - 9],\
[251, 251, 2*w^2 - 3*w - 8],\
[257, 257, w^3 + w^2 - 8*w - 7],\
[257, 257, -4*w^3 + 5*w^2 + 17*w - 1],\
[269, 269, 2*w^3 - 2*w^2 - 10*w + 3],\
[269, 269, 2*w^3 - 3*w^2 - 9*w + 2],\
[269, 269, 2*w^3 - 3*w^2 - 9*w + 8],\
[269, 269, -2*w^3 + 4*w^2 + 8*w - 7],\
[277, 277, w^3 - w^2 - 4*w - 4],\
[277, 277, -w^3 + 2*w^2 + 3*w - 8],\
[289, 17, 2*w^2 - 2*w - 3],\
[293, 293, -w^2 - 2*w + 5],\
[293, 293, -w^3 + 3*w^2 + 3*w - 4],\
[293, 293, w^3 - 6*w + 1],\
[293, 293, -2*w^3 + 4*w^2 + 7*w - 6],\
[311, 311, -w^3 + 3*w^2 + w - 7],\
[311, 311, w^3 - 4*w - 4],\
[331, 331, -2*w^3 + 4*w^2 + 9*w - 10],\
[331, 331, 2*w^3 - 2*w^2 - 11*w + 1],\
[337, 337, 2*w^2 - 3*w - 4],\
[337, 337, -w^3 + 2*w^2 + 6*w - 3],\
[337, 337, w^3 - w^2 - 7*w + 4],\
[337, 337, 2*w^2 - w - 5],\
[353, 353, -3*w^3 + 4*w^2 + 13*w - 2],\
[353, 353, 3*w^3 - 5*w^2 - 12*w + 12],\
[373, 373, -2*w^3 + 6*w^2 + 3*w - 11],\
[373, 373, -w^3 - w^2 + 6*w + 7],\
[373, 373, -w^3 + 4*w^2 + w - 11],\
[373, 373, 2*w^3 - 9*w - 4],\
[379, 379, w^3 - w^2 - 7*w + 1],\
[379, 379, 3*w^3 - 7*w^2 - 7*w + 11],\
[379, 379, 3*w^3 - 2*w^2 - 12*w],\
[379, 379, -w^3 + 2*w^2 + 6*w - 6],\
[383, 383, -3*w^3 + 4*w^2 + 12*w - 8],\
[383, 383, 3*w^3 - 5*w^2 - 11*w + 5],\
[421, 421, 2*w^3 - 2*w^2 - 9*w - 4],\
[421, 421, -2*w^3 + 4*w^2 + 7*w - 13],\
[457, 457, -w^3 + 2*w^2 + 6*w - 5],\
[457, 457, w^3 - w^2 - 7*w + 2],\
[461, 461, -w^3 + 3*w^2 + 4*w - 4],\
[461, 461, -w^3 + 7*w - 2],\
[463, 463, -w^3 + 4*w^2 + 3*w - 10],\
[463, 463, w^3 + w^2 - 8*w - 4],\
[467, 467, -w^3 + 6*w - 2],\
[467, 467, -w^3 + 3*w^2 + 3*w - 3],\
[479, 479, -w^3 + w^2 + w - 4],\
[479, 479, w^3 - 2*w^2 - 3],\
[487, 487, -3*w^3 + 4*w^2 + 12*w - 3],\
[487, 487, -3*w^3 + 5*w^2 + 11*w - 10],\
[499, 499, 3*w^3 - 4*w^2 - 11*w + 8],\
[499, 499, 3*w^3 - 5*w^2 - 10*w + 4],\
[503, 503, -w^3 + 2*w^2 + w - 5],\
[503, 503, w^3 - w^2 - 2*w - 3],\
[521, 521, 2*w^3 - 4*w^2 - 6*w + 3],\
[521, 521, -w^3 + 10*w - 8],\
[521, 521, w^3 - 3*w^2 - 7*w + 1],\
[521, 521, -2*w^3 + 2*w^2 + 8*w - 5],\
[541, 541, 2*w - 7],\
[541, 541, 2*w + 5],\
[547, 547, w^3 - 4*w^2 + 7],\
[547, 547, w^3 + w^2 - 5*w - 4],\
[563, 563, -2*w^3 + 4*w^2 + 7*w - 5],\
[563, 563, -2*w^3 + 2*w^2 + 9*w - 4],\
[587, 587, -w^3 - w^2 + 4*w + 6],\
[587, 587, w^3 - 4*w^2 + w + 8],\
[593, 593, w^2 + w - 7],\
[593, 593, 2*w^3 - 5*w^2 - 4*w + 5],\
[593, 593, -2*w^3 + w^2 + 8*w - 2],\
[593, 593, w^2 - 3*w - 5],\
[613, 613, 2*w^3 - 3*w^2 - 11*w + 2],\
[613, 613, w^3 + w^2 - 6*w - 3],\
[613, 613, 3*w^3 - 4*w^2 - 12*w + 4],\
[613, 613, 2*w^3 - 3*w^2 - 11*w + 10],\
[631, 631, -w^3 - w^2 + 7*w + 6],\
[631, 631, -w^3 + 4*w^2 + 2*w - 11],\
[647, 647, w^2 - 4*w - 3],\
[647, 647, w^2 + 2*w - 6],\
[677, 677, w^2 + w - 8],\
[677, 677, -w^3 + 7*w - 3],\
[677, 677, -w^3 + 3*w^2 + 4*w - 3],\
[677, 677, w^2 - 3*w - 6],\
[709, 709, w^3 - 7*w - 8],\
[709, 709, w^3 - 3*w^2 - 4*w + 14],\
[739, 739, w^3 - w^2 - 4*w - 5],\
[739, 739, -3*w^3 + 3*w^2 + 16*w - 3],\
[739, 739, -3*w^3 + 6*w^2 + 13*w - 13],\
[739, 739, -w^3 + 2*w^2 + 3*w - 9],\
[751, 751, -w - 5],\
[751, 751, w^3 + w^2 - 7*w - 4],\
[751, 751, -w^3 + 4*w^2 + 2*w - 9],\
[751, 751, w - 6],\
[757, 757, -w^3 + 4*w^2 + 2*w - 10],\
[757, 757, w^3 + w^2 - 7*w - 5],\
[761, 761, -3*w^3 + 2*w^2 + 17*w - 1],\
[761, 761, -3*w^3 + 6*w^2 + 9*w - 17],\
[761, 761, 2*w^3 - 5*w^2 - 9*w + 9],\
[761, 761, 2*w^3 - 2*w^2 - 7*w - 6],\
[773, 773, -3*w^3 + 8*w^2 + 6*w - 12],\
[773, 773, w^3 - 4*w - 6],\
[773, 773, -w^3 + 3*w^2 + w - 9],\
[773, 773, -3*w^3 + w^2 + 13*w + 1],\
[797, 797, 3*w^3 - 2*w^2 - 13*w + 1],\
[797, 797, 3*w^3 - 7*w^2 - 8*w + 11],\
[823, 823, 2*w^3 - 3*w^2 - 6*w - 4],\
[823, 823, -2*w^3 + 3*w^2 + 7*w + 3],\
[823, 823, w^3 - 2*w^2 - 7*w - 3],\
[823, 823, -3*w^3 + 5*w^2 + 9*w - 2],\
[857, 857, 2*w^3 - w^2 - 10*w + 1],\
[857, 857, -2*w^3 + 5*w^2 + 6*w - 8],\
[883, 883, 3*w^3 - 3*w^2 - 12*w + 1],\
[883, 883, 3*w^3 - 6*w^2 - 9*w + 11],\
[887, 887, -2*w^3 + 5*w^2 + 7*w - 9],\
[887, 887, -4*w^3 + 6*w^2 + 17*w - 7],\
[887, 887, -2*w^3 + 5*w^2 + 9*w - 16],\
[887, 887, 2*w^3 - w^2 - 11*w + 1],\
[907, 907, -3*w^3 + 3*w^2 + 16*w - 2],\
[907, 907, -3*w^3 + 6*w^2 + 13*w - 14],\
[919, 919, 2*w^3 - 2*w^2 - 7*w + 9],\
[919, 919, -2*w^3 + 2*w^2 + 12*w - 5],\
[929, 929, -2*w^3 + 4*w^2 + 8*w - 5],\
[929, 929, 3*w^3 - 4*w^2 - 11*w + 12],\
[929, 929, 3*w^3 - 5*w^2 - 10*w],\
[929, 929, -2*w^3 + 2*w^2 + 10*w - 5],\
[967, 967, 3*w^3 - 5*w^2 - 9*w + 11],\
[967, 967, -3*w^3 + 7*w^2 + 6*w - 10],\
[971, 971, 2*w^3 - 2*w^2 - 7*w - 5],\
[971, 971, -2*w^3 + w^2 + 8*w - 3],\
[971, 971, 2*w^3 - 5*w^2 - 4*w + 4],\
[971, 971, -2*w^3 + 4*w^2 + 5*w - 12],\
[983, 983, 4*w - 7],\
[983, 983, 2*w^2 - 4*w - 7],\
[983, 983, 2*w^2 - 9],\
[983, 983, 4*w + 3],\
[991, 991, 2*w^3 - 6*w^2 - 5*w + 14],\
[991, 991, -2*w^3 + 11*w + 5]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - 12*x^4 + 31*x^2 - 4
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 1/2*e^5 - 11/2*e^3 + 11*e, e^2 - 4, -1, 2, 1, -e^3 + 7*e, 1/2*e^5 - 11/2*e^3 + 13*e, -1/2*e^4 + 11/2*e^2 - 8, -e^2 + 2, -e^4 + 10*e^2 - 12, 1/2*e^4 - 7/2*e^2 + 6, -1/2*e^5 + 11/2*e^3 - 16*e, e^5 - 11*e^3 + 22*e, -e^4 + 10*e^2 - 8, 1/2*e^4 - 9/2*e^2 + 6, -1/2*e^5 + 13/2*e^3 - 24*e, -e^5 + 11*e^3 - 22*e, e^5 - 14*e^3 + 46*e, 1/2*e^5 - 11/2*e^3 + 16*e, 3/2*e^5 - 35/2*e^3 + 41*e, 1/2*e^5 - 15/2*e^3 + 25*e, 1/2*e^5 - 17/2*e^3 + 31*e, 3/2*e^5 - 35/2*e^3 + 43*e, -e^5 + 12*e^3 - 31*e, -e^5 + 13*e^3 - 35*e, 1/2*e^4 - 15/2*e^2 + 16, -1/2*e^4 + 13/2*e^2 - 8, 1/2*e^4 - 11/2*e^2 + 20, -e^4 + 7*e^2 - 2, 1/2*e^5 - 11/2*e^3 + 8*e, -2*e, e^4 - 11*e^2 + 24, -2*e^4 + 16*e^2 - 12, 5/2*e^5 - 51/2*e^3 + 46*e, 1/2*e^5 - 13/2*e^3 + 18*e, -1/2*e^4 + 11/2*e^2 - 8, 1/2*e^4 - 13/2*e^2 + 24, 3/2*e^5 - 33/2*e^3 + 28*e, 1/2*e^5 - 19/2*e^3 + 40*e, -1/2*e^5 + 17/2*e^3 - 36*e, 2*e^5 - 26*e^3 + 74*e, 2*e^5 - 25*e^3 + 71*e, e^5 - 16*e^3 + 59*e, e^3 - 7*e, -3*e^5 + 32*e^3 - 67*e, 1/2*e^5 - 3/2*e^3 - 17*e, -3/2*e^5 + 31/2*e^3 - 29*e, -2*e^4 + 14*e^2 - 10, 1/2*e^4 - 3/2*e^2 - 16, 1/2*e^4 - 1/2*e^2 - 8, -e^5 + 12*e^3 - 31*e, 2*e^5 - 21*e^3 + 45*e, 3*e^5 - 34*e^3 + 81*e, 3*e^3 - 19*e, -2*e^5 + 24*e^3 - 60*e, -1/2*e^5 + 13/2*e^3 - 18*e, -3/2*e^4 + 19/2*e^2 + 10, e^4 - 7*e^2, 1/2*e^4 - 21/2*e^2 + 32, 3/2*e^4 - 19/2*e^2 - 12, -5/2*e^4 + 43/2*e^2 - 16, -2*e^4 + 18*e^2 - 6, -e^5 + 14*e^3 - 53*e, 3/2*e^5 - 33/2*e^3 + 41*e, 1/2*e^4 - 9/2*e^2 - 4, 1/2*e^4 - 5/2*e^2 - 20, e^2 + 2, 22, -2*e^4 + 18*e^2 - 12, 1/2*e^4 - 13/2*e^2 + 26, -3/2*e^4 + 17/2*e^2 + 22, -2*e^4 + 18*e^2 - 12, -5/2*e^5 + 61/2*e^3 - 86*e, e^5 - 9*e^3 + 4*e, e^4 - 5*e^2 - 22, -e^4 + 10*e^2 - 26, -e^4 + 3*e^2 + 30, 3*e^4 - 29*e^2 + 30, -5/2*e^5 + 57/2*e^3 - 57*e, 3/2*e^5 - 31/2*e^3 + 23*e, 5/2*e^4 - 41/2*e^2 + 6, 3/2*e^4 - 13/2*e^2 - 18, -5/2*e^5 + 61/2*e^3 - 72*e, -1/2*e^5 + 19/2*e^3 - 44*e, 3*e^5 - 39*e^3 + 116*e, -3/2*e^5 + 47/2*e^3 - 82*e, 7/2*e^4 - 61/2*e^2 + 22, 2*e^2 + 16, 1/2*e^4 - 5/2*e^2 + 2, 1/2*e^4 - 1/2*e^2 + 10, -3*e^5 + 34*e^3 - 60*e, 7/2*e^5 - 81/2*e^3 + 98*e, -6*e^3 + 39*e, -e^5 + 10*e^3 - e, 5/2*e^5 - 65/2*e^3 + 85*e, -4*e^5 + 47*e^3 - 117*e, 7/2*e^4 - 61/2*e^2 + 44, -3*e^4 + 25*e^2 - 14, 3/2*e^4 - 39/2*e^2 + 46, 4*e^4 - 34*e^2 + 44, -e^5 + 10*e^3 - 18*e, 3/2*e^5 - 33/2*e^3 + 44*e, 4*e^5 - 44*e^3 + 98*e, 2*e^5 - 24*e^3 + 50*e, -9/2*e^5 + 105/2*e^3 - 123*e, 1/2*e^5 - 3/2*e^3 - 11*e, 3/2*e^5 - 43/2*e^3 + 61*e, -4*e^5 + 42*e^3 - 77*e, -1/2*e^4 + 5/2*e^2, 1/2*e^4 - 3/2*e^2 - 32, -e^4 + 17*e^2 - 30, 5/2*e^4 - 27/2*e^2 - 16, e^4 - 9*e^2 + 20, 3/2*e^4 - 47/2*e^2 + 50, 3/2*e^5 - 43/2*e^3 + 70*e, -11/2*e^5 + 125/2*e^3 - 130*e, -1/2*e^5 + 17/2*e^3 - 29*e, 2*e^5 - 27*e^3 + 69*e, -9/2*e^5 + 97/2*e^3 - 93*e, -2*e^5 + 30*e^3 - 95*e, 2*e^4 - 18*e^2 + 6, -1/2*e^4 - 3/2*e^2 + 16, e^4 - 10*e^2 - 20, -e^4 + 9*e^2 - 8, -7*e^2 + 8, -3*e^4 + 25*e^2 - 8, -e^4 + 13*e^2 - 44, -7/2*e^4 + 51/2*e^2 - 18, -e^4 + 13*e^2 - 12, -3*e^4 + 31*e^2 - 36, -2*e^4 + 18*e^2 - 34, -3*e^2 + 42, 5/2*e^5 - 63/2*e^3 + 73*e, -4*e^5 + 47*e^3 - 109*e, -e^5 + 8*e^3 + 7*e, -1/2*e^5 + 9/2*e^3 - 15*e, 6*e^5 - 68*e^3 + 145*e, -1/2*e^5 + 23/2*e^3 - 57*e, -3/2*e^5 + 31/2*e^3 - 25*e, -7/2*e^5 + 73/2*e^3 - 77*e, e^3 - 7*e, e^3 - 15*e, e^4 - 9*e^2 - 12, 6*e^2 - 32, -4*e^2 + 24, -3*e^4 + 25*e^2 - 4, -e^5 + 12*e^3 - 33*e, -4*e^5 + 47*e^3 - 125*e, -e^4 + 7*e^2 - 16, -2*e^4 + 18*e^2 - 52, -e^5 + 13*e^3 - 24*e, -3/2*e^5 + 39/2*e^3 - 46*e, -4*e^5 + 40*e^3 - 60*e, -3*e^5 + 40*e^3 - 132*e, 7/2*e^4 - 67/2*e^2 + 30, 3*e^4 - 19*e^2, 5/2*e^4 - 45/2*e^2 + 6, -5*e^4 + 50*e^2 - 64, -e^3 + 7*e, -e^5 + 12*e^3 - 37*e, -6*e^5 + 69*e^3 - 169*e, -6*e^5 + 64*e^3 - 133*e, -7/2*e^4 + 69/2*e^2 - 54, -7/2*e^4 + 51/2*e^2 + 6, -3/2*e^5 + 41/2*e^3 - 76*e, 9/2*e^5 - 111/2*e^3 + 156*e, -5/2*e^5 + 69/2*e^3 - 116*e, -e^5 + 13*e^3 - 26*e, -3/2*e^5 + 35/2*e^3 - 22*e, -7/2*e^5 + 71/2*e^3 - 62*e, -11/2*e^5 + 123/2*e^3 - 150*e, -e^5 + 9*e^3, 5/2*e^4 - 39/2*e^2 + 26, 3/2*e^4 - 41/2*e^2 + 38]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([7,7,-w^2 + 2])] = 1

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