/*
  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, -2, -5, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([35, 35, -w^3 + w^2 + 4*w - 4])

primes_array = [
[4, 2, -w^3 + 4*w + 1],\
[5, 5, w - 1],\
[7, 7, -w^2 + w + 2],\
[13, 13, -w^3 + w^2 + 4*w],\
[13, 13, -w^2 + 3],\
[17, 17, -w^2 + 2],\
[19, 19, -w^3 + 5*w],\
[31, 31, -w^2 + 2*w + 3],\
[37, 37, -2*w^3 + w^2 + 8*w - 1],\
[43, 43, -w - 3],\
[53, 53, -w^3 + 2*w^2 + 3*w - 2],\
[53, 53, w^3 - 6*w - 2],\
[59, 59, 2*w^3 - w^2 - 10*w - 2],\
[61, 61, 2*w^3 - w^2 - 10*w],\
[61, 61, 2*w^3 - w^2 - 8*w],\
[71, 71, 2*w^3 - 9*w - 2],\
[73, 73, -w^3 - w^2 + 6*w + 3],\
[81, 3, -3],\
[83, 83, -w^3 + 2*w^2 + 3*w - 3],\
[101, 101, 2*w^3 - 8*w - 3],\
[125, 5, w^3 + w^2 - 4*w - 6],\
[127, 127, 2*w^3 - w^2 - 7*w - 2],\
[127, 127, w^3 - 7*w - 7],\
[131, 131, w^3 - 3*w - 4],\
[131, 131, 3*w^3 - 13*w - 7],\
[137, 137, w^2 + w - 4],\
[137, 137, -w^3 + 2*w^2 + 4*w - 4],\
[149, 149, 3*w + 5],\
[157, 157, -2*w^3 - w^2 + 7*w + 2],\
[157, 157, -2*w^3 + 2*w^2 + 9*w - 1],\
[163, 163, -2*w^3 + w^2 + 7*w],\
[163, 163, -3*w^3 + w^2 + 15*w + 3],\
[163, 163, 2*w^3 - 7*w - 1],\
[163, 163, 2*w^3 - 11*w - 5],\
[167, 167, -2*w^3 + w^2 + 9*w - 4],\
[167, 167, w^3 - w^2 - 7*w + 1],\
[167, 167, w^3 - w^2 - 7*w - 1],\
[167, 167, w^2 - 2*w - 5],\
[169, 13, w^3 + w^2 - 6*w - 2],\
[173, 173, -3*w^3 + 13*w + 3],\
[181, 181, 2*w^2 + w - 4],\
[191, 191, 2*w^3 - 10*w - 1],\
[191, 191, -2*w^3 + 2*w^2 + 6*w - 3],\
[193, 193, 3*w^3 - 13*w - 6],\
[197, 197, -w^3 + w^2 + 2*w - 3],\
[197, 197, w^3 + w^2 - 7*w - 7],\
[199, 199, -w^2 + 2*w + 6],\
[223, 223, -2*w^3 + w^2 + 11*w + 2],\
[227, 227, -3*w^3 + 2*w^2 + 15*w - 1],\
[229, 229, -w^3 + w^2 + 6*w - 5],\
[233, 233, 2*w^3 - 11*w - 4],\
[239, 239, -2*w^3 + 11*w + 3],\
[251, 251, 2*w^3 - w^2 - 10*w - 4],\
[257, 257, -2*w^3 + w^2 + 6*w + 1],\
[257, 257, w^3 - 7*w - 2],\
[263, 263, 2*w^3 - 9*w],\
[269, 269, w^3 - 4*w - 6],\
[269, 269, -2*w^3 + 8*w - 3],\
[271, 271, 2*w^2 - w - 5],\
[277, 277, -3*w^3 + 2*w^2 + 11*w - 2],\
[277, 277, 3*w^3 - w^2 - 15*w - 5],\
[281, 281, 2*w^2 - 5],\
[283, 283, -3*w^3 + 2*w^2 + 12*w + 2],\
[283, 283, -2*w^3 + w^2 + 6*w + 6],\
[293, 293, -2*w^3 - w^2 + 12*w + 7],\
[293, 293, -2*w^3 + 9*w - 1],\
[311, 311, -3*w^3 + w^2 + 12*w + 4],\
[313, 313, -w^3 + w^2 + 2*w - 4],\
[313, 313, -4*w^3 + 2*w^2 + 16*w - 3],\
[331, 331, 3*w^3 - w^2 - 16*w - 5],\
[331, 331, 2*w^3 - 2*w^2 - 8*w - 1],\
[343, 7, 2*w^3 - 3*w^2 - 9*w + 6],\
[359, 359, -2*w^3 + 10*w - 1],\
[367, 367, -w^3 + 2*w^2 + w - 4],\
[367, 367, 3*w^3 - 11*w - 1],\
[373, 373, -w^3 + w^2 + 2*w - 5],\
[373, 373, 2*w^3 - 2*w^2 - 11*w - 1],\
[389, 389, -2*w^3 + 3*w^2 + 8*w - 6],\
[409, 409, 3*w^3 - 2*w^2 - 13*w - 1],\
[419, 419, 3*w^3 - w^2 - 12*w - 1],\
[419, 419, w^3 - w^2 - 4*w - 4],\
[439, 439, w^2 + 2*w - 4],\
[439, 439, 3*w^3 + w^2 - 11*w - 5],\
[443, 443, 2*w^3 + 2*w^2 - 7*w - 8],\
[449, 449, 4*w^3 - w^2 - 16*w - 8],\
[457, 457, 2*w^3 + w^2 - 11*w - 4],\
[461, 461, -5*w^3 + 2*w^2 + 24*w + 2],\
[463, 463, -w^3 + 3*w^2 + 6*w - 5],\
[467, 467, w^3 - 3*w^2 - 6*w + 4],\
[467, 467, 3*w^3 - w^2 - 12*w - 2],\
[479, 479, -w^3 + w^2 + 4*w - 6],\
[479, 479, -w^3 + 2*w + 8],\
[491, 491, -w^3 + 2*w^2 + 2*w - 6],\
[491, 491, w - 5],\
[499, 499, 2*w^3 + w^2 - 10*w - 4],\
[509, 509, -4*w^3 + w^2 + 20*w + 5],\
[521, 521, w^3 - w + 6],\
[523, 523, -4*w^3 + 18*w + 5],\
[523, 523, -2*w^3 - 2*w^2 + 9*w + 12],\
[557, 557, w^2 - 3*w - 6],\
[571, 571, 3*w^3 - 11*w - 8],\
[577, 577, 2*w^3 + 2*w^2 - 11*w - 7],\
[577, 577, -3*w^3 + 4*w^2 + 11*w - 6],\
[587, 587, -w^3 - w^2 + 7*w + 11],\
[593, 593, w^3 + w^2 - 8*w - 8],\
[593, 593, -2*w^3 + 4*w^2 + 6*w - 9],\
[599, 599, 4*w^3 - 17*w - 9],\
[599, 599, w^3 + w^2 - 6*w - 10],\
[601, 601, 3*w^2 - 10],\
[607, 607, 4*w^3 - w^2 - 18*w - 8],\
[619, 619, w^2 - w - 8],\
[641, 641, -w^3 + 3*w^2 + 2*w - 10],\
[643, 643, 3*w^3 - w^2 - 16*w + 1],\
[653, 653, -2*w^3 + 2*w^2 + 9*w + 3],\
[677, 677, w^3 + w^2 - 8*w - 11],\
[677, 677, 2*w^3 - w^2 - 12*w - 3],\
[683, 683, 3*w^3 - 2*w^2 - 13*w - 2],\
[683, 683, 3*w^3 + 2*w^2 - 16*w - 10],\
[691, 691, -2*w^3 - 2*w^2 + 12*w + 15],\
[691, 691, 3*w^3 - 2*w^2 - 15*w - 2],\
[691, 691, w^3 - 2*w^2 - 6*w + 6],\
[691, 691, -4*w^3 + 3*w^2 + 15*w - 2],\
[719, 719, 2*w^3 - 7*w - 8],\
[719, 719, w^2 + 2*w - 5],\
[727, 727, -w^3 + 3*w^2 + 4*w - 8],\
[727, 727, 2*w^3 - 9*w - 9],\
[733, 733, w^3 - 8*w - 2],\
[739, 739, -w^3 - 2*w^2 + 4*w + 10],\
[743, 743, w^3 - w^2 - 2*w + 8],\
[743, 743, 4*w^3 - w^2 - 17*w - 4],\
[751, 751, -2*w^3 + 2*w^2 + 6*w - 5],\
[751, 751, w^3 - 5*w - 7],\
[769, 769, 3*w^3 - 2*w^2 - 10*w],\
[787, 787, -2*w^3 + 3*w^2 + 8*w],\
[787, 787, 3*w^3 - w^2 - 10*w - 4],\
[809, 809, 3*w^3 + w^2 - 16*w - 6],\
[811, 811, 3*w^3 - 14*w - 2],\
[811, 811, -w^3 + 3*w^2 + 5*w - 5],\
[811, 811, w^3 - 2*w - 6],\
[811, 811, 3*w^3 - w^2 - 11*w - 3],\
[823, 823, w^3 + 2*w^2 - 3*w - 8],\
[827, 827, -w^3 + 3*w^2 + 3*w - 7],\
[829, 829, w^3 + w^2 - 3*w - 7],\
[853, 853, 3*w^3 + w^2 - 14*w - 6],\
[857, 857, 3*w^3 + w^2 - 13*w - 5],\
[857, 857, 3*w^3 - 3*w^2 - 11*w - 1],\
[859, 859, 4*w^3 - 2*w^2 - 18*w - 3],\
[859, 859, 3*w^3 - 11*w - 3],\
[863, 863, 3*w^3 + w^2 - 17*w - 9],\
[877, 877, -w^3 - 2*w^2 + 9*w + 1],\
[877, 877, 5*w^3 + w^2 - 23*w - 11],\
[911, 911, w^2 - 4*w - 6],\
[911, 911, -2*w^3 + 2*w^2 + 10*w + 3],\
[919, 919, -4*w^3 + 3*w^2 + 18*w - 1],\
[929, 929, 2*w^3 - 13*w - 12],\
[929, 929, -w^3 + 2*w^2 + 7*w - 5],\
[953, 953, -3*w^3 + 3*w^2 + 13*w - 1],\
[967, 967, -3*w^3 + 4*w^2 + 13*w - 7],\
[971, 971, 2*w^2 - 3*w - 12],\
[971, 971, 3*w^3 - w^2 - 16*w - 3]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - x^3 - 11*x^2 + 9*x + 10
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 1, 1, -1/2*e^3 + 11/2*e - 1, 1/2*e^3 - 7/2*e - 1, -2, 1/2*e^3 - e^2 - 9/2*e + 5, 2*e + 2, -1/2*e^3 - e^2 + 9/2*e + 3, 1/2*e^3 - e^2 - 9/2*e + 9, -1/2*e^3 + 11/2*e - 1, 1/2*e^3 - e^2 - 13/2*e + 9, 0, 2, 2, -e^3 + e^2 + 6*e - 8, e^3 - 11*e + 4, -2*e^2 + 12, -2*e^2 + 14, -1/2*e^3 + e^2 + 13/2*e - 13, -1/2*e^3 + 3*e^2 + 5/2*e - 19, -2*e^3 + 16*e - 2, 2*e^2 - 2*e - 12, 1/2*e^3 + e^2 - 5/2*e - 3, -e^3 + 2*e^2 + 11*e - 8, -e^3 - 2*e^2 + 9*e + 8, e^2 - 3*e - 2, -3/2*e^3 + e^2 + 31/2*e - 5, -e^3 + 7*e + 8, 1/2*e^3 - 15/2*e + 3, 2*e^3 - 18*e + 4, -1/2*e^3 + e^2 + 9/2*e - 1, 1/2*e^3 - 11/2*e - 1, -e^3 + 5*e + 4, 2*e^3 - 18*e + 8, -e^3 - 2*e^2 + 11*e + 8, -e^3 + 2*e^2 + 9*e - 2, -e^3 - 2*e^2 + 7*e + 8, -3*e^2 - e + 20, e^3 - 11*e + 4, 2*e^3 - 18*e + 2, e^3 + 2*e^2 - 9*e - 18, -e^3 + 11*e + 2, -e^3 + 2*e^2 + 13*e - 16, -e^3 + 7*e + 8, -1/2*e^3 - e^2 - 3/2*e + 13, e^3 - 2*e^2 - 9*e + 10, e^3 - 3*e^2 - 8*e + 24, -1/2*e^3 + 3/2*e - 7, -1/2*e^3 + 19/2*e - 5, 2*e^3 - 16*e + 4, -2*e^2 + 2*e + 20, e^3 - 13*e + 12, -e^3 + 2*e^2 + 9*e - 12, 2*e^3 - 2*e^2 - 12*e + 18, -e^2 - 3*e - 6, 1/2*e^3 + 3*e^2 - 5/2*e - 15, 1/2*e^3 - 2*e^2 - 11/2*e + 15, -e^3 + e^2 + 14*e - 8, -e^3 - 4*e^2 + 7*e + 28, 1/2*e^3 + 4*e^2 - 15/2*e - 17, e^3 + 2*e^2 - 13*e - 8, -3*e^3 + 25*e - 6, -e^3 + 5*e + 4, 2*e^3 - 4*e^2 - 22*e + 14, -1/2*e^3 - e^2 + 5/2*e + 9, -e^3 + 5*e + 12, e^2 + 3*e - 16, -e^3 + 4*e^2 + 5*e - 26, 3/2*e^3 - e^2 - 31/2*e + 7, 1/2*e^3 - 3/2*e + 7, -2*e^3 + 18*e - 16, e^3 + 2*e^2 - 7*e, 3*e^2 + e - 22, -e^3 + 4*e^2 + 7*e - 22, 2*e^3 - 14*e - 6, -5/2*e^3 + 3*e^2 + 53/2*e - 21, -8*e - 10, 3*e^3 - e^2 - 32*e + 10, -1/2*e^3 - e^2 + 21/2*e - 5, -2*e^3 + 2*e^2 + 18*e - 10, -e^3 + 2*e^2 + 7*e, -2*e^2 - 6*e + 20, e^3 - 6*e^2 - 9*e + 34, e^3 + 2*e^2 - 7*e - 10, e^3 + 2*e^2 - 13*e + 8, 2*e^3 + 2*e^2 - 24*e + 2, e^3 + 2*e^2 - 5*e - 6, e^3 - 2*e^2 - 13*e + 18, 1/2*e^3 - 11/2*e - 17, 2*e^3 - 28*e + 10, e^3 + 2*e^2 - 13*e - 30, 1/2*e^3 + e^2 - 21/2*e - 23, 3/2*e^3 - 25/2*e - 3, 7/2*e^3 - 3*e^2 - 63/2*e + 15, 3/2*e^3 + 3*e^2 - 23/2*e - 15, -e^3 - e^2 + 22*e + 12, 7/2*e^3 - 5*e^2 - 75/2*e + 19, e^3 - 5*e - 16, 5/2*e^3 - 35/2*e + 3, 2*e^2 - 10*e - 8, -e^3 - 2*e^2 + 13*e + 8, -3*e^3 - 2*e^2 + 33*e - 2, -5/2*e^3 + 47/2*e + 13, e^3 + 6*e^2 - 13*e - 36, e^3 - 2*e^2 - 9*e + 24, e^3 - 11*e + 10, 3*e^3 + e^2 - 30*e, -2*e^2 - 12*e + 12, -e^3 + 2*e^2 + e - 2, 1/2*e^3 - 3*e^2 - 5/2*e + 5, 2*e^3 - 20*e - 8, 2*e^3 + 4*e^2 - 16*e - 26, 3*e^3 - 4*e^2 - 21*e + 24, -3/2*e^3 - e^2 + 31/2*e - 27, e^3 - 2*e^2 - 5*e + 28, -e^3 + 9*e + 4, -3*e^3 + 27*e + 4, -1/2*e^3 - 2*e^2 - 9/2*e + 37, 5/2*e^3 - 5*e^2 - 29/2*e + 37, 3/2*e^3 + 2*e^2 - 37/2*e - 23, -2*e^3 + 2*e^2 + 18*e - 18, 2*e^3 - 4*e^2 - 16*e + 10, -e^3 + 19*e + 10, e^3 + 6*e^2 - 9*e - 22, -2*e^3 + 2*e^2 + 24*e - 12, -3/2*e^3 - e^2 + 27/2*e + 19, -1/2*e^3 - 2*e^2 - 1/2*e + 5, 4*e^3 - 32*e + 4, -e^3 + 2*e^2 + 17*e - 26, -2*e^3 + 5*e^2 + 21*e - 38, -3*e^3 - 4*e^2 + 23*e + 32, -e^3 + 3*e, 4*e^3 - 2*e^2 - 34*e - 12, 4*e^3 - 4*e^2 - 36*e + 28, -3*e^3 + 4*e^2 + 29*e - 20, 2*e^3 - 4*e^2 - 14*e + 32, -3/2*e^3 + 6*e^2 + 37/2*e - 33, 1/2*e^3 + e^2 - 13/2*e + 17, -7/2*e^3 + 61/2*e - 33, -e^3 - 2*e^2 + 7*e + 4, -2*e^3 + 28*e - 2, -3/2*e^3 - 3*e^2 + 39/2*e - 5, e^3 - 4*e^2 - 15*e + 44, -e^3 + 8*e^2 + 11*e - 32, 2*e^2 + 4*e - 12, -5/2*e^3 + 2*e^2 + 75/2*e - 15, -2*e^3 + 22*e - 40, e^3 - 4*e^2 - 7*e + 34, -3*e^3 + 2*e^2 + 27*e - 12, -16*e - 2, 5*e^3 - 4*e^2 - 41*e + 32, 2*e^3 - 3*e^2 - 11*e + 22, e^3 + 3*e^2 - 2*e, 2*e^3 - 8*e^2 - 10*e + 50, -e^3 - 4*e^2 + 19*e + 20, -3*e^3 + 4*e^2 + 33*e - 36, e^3 + 2*e^2 - 13*e + 18, 2*e^2 + 12*e - 18, -2*e^3 + 6*e^2 + 22*e - 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([5, 5, w - 1])] = -1
AL_eigenvalues[ZF.ideal([7, 7, -w^2 + w + 2])] = -1

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