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

NN = ZF.ideal([11, 11, w^2 - w - 1])

primes_array = [
[2, 2, -w],\
[2, 2, w - 1],\
[11, 11, w^2 - w - 1],\
[17, 17, -w^2 - w + 3],\
[19, 19, w^2 - w + 1],\
[23, 23, 2*w - 3],\
[27, 3, 3],\
[29, 29, 2*w + 1],\
[31, 31, 2*w^2 - 2*w - 9],\
[37, 37, 2*w^2 - 2*w - 5],\
[41, 41, 2*w^2 - 9],\
[43, 43, w^2 + w - 5],\
[43, 43, -3*w^2 + w + 15],\
[43, 43, -2*w^2 + 2*w + 11],\
[53, 53, w^2 - w - 7],\
[61, 61, 4*w^2 - 2*w - 15],\
[67, 67, -5*w^2 + 3*w + 23],\
[73, 73, 2*w^2 - 3],\
[73, 73, -3*w^2 - w + 7],\
[73, 73, -6*w^2 + 4*w + 25],\
[79, 79, w^2 - 5*w + 1],\
[79, 79, 3*w^2 - 3*w - 11],\
[83, 83, 2*w^2 - 2*w - 3],\
[109, 109, w^2 - 3*w - 3],\
[113, 113, 3*w^2 - 3*w - 13],\
[121, 11, 3*w^2 - w - 9],\
[125, 5, -5],\
[131, 131, -6*w^2 + 2*w + 25],\
[137, 137, -w^2 + w - 3],\
[149, 149, 2*w - 7],\
[151, 151, w^2 - 3*w - 5],\
[157, 157, 3*w^2 - 3*w - 7],\
[163, 163, 2*w^2 - 13],\
[167, 167, -4*w + 5],\
[173, 173, -7*w^2 + 3*w + 27],\
[179, 179, 4*w^2 - 4*w - 11],\
[181, 181, 4*w^2 - 15],\
[181, 181, w^2 - 3*w + 5],\
[181, 181, w^2 + 3*w - 5],\
[193, 193, 2*w^2 - 4*w - 5],\
[197, 197, 2*w^2 + 2*w - 7],\
[211, 211, 4*w^2 - 2*w - 13],\
[211, 211, -7*w^2 + 5*w + 29],\
[211, 211, w^2 - w - 9],\
[223, 223, 5*w^2 - w - 23],\
[227, 227, -3*w^2 + 3*w + 17],\
[227, 227, 3*w^2 - w - 7],\
[227, 227, 3*w^2 + w - 11],\
[229, 229, 5*w^2 - w - 21],\
[233, 233, 3*w^2 - 5*w - 7],\
[239, 239, -w^2 + 7*w - 5],\
[257, 257, -4*w - 3],\
[257, 257, w^2 - 5*w - 1],\
[257, 257, -3*w^2 + w + 17],\
[263, 263, 3*w^2 - 3*w - 5],\
[271, 271, 4*w^2 - 4*w - 17],\
[271, 271, 3*w^2 - w - 3],\
[271, 271, w^2 + 3*w - 7],\
[283, 283, 6*w - 1],\
[289, 17, w^2 + 3*w - 9],\
[293, 293, 3*w^2 - w - 5],\
[307, 307, 2*w^2 + 2*w - 9],\
[307, 307, -2*w^2 + 2*w - 3],\
[307, 307, 4*w^2 - 17],\
[311, 311, w^2 + w - 11],\
[311, 311, 2*w^2 - 4*w - 7],\
[311, 311, -2*w^2 - 2*w + 13],\
[313, 313, -2*w - 7],\
[313, 313, 4*w^2 - 19],\
[313, 313, -5*w^2 - w + 13],\
[331, 331, 2*w^2 - 4*w - 11],\
[331, 331, 6*w^2 - 4*w - 21],\
[331, 331, -4*w^2 + 2*w + 21],\
[343, 7, -7],\
[347, 347, -2*w^2 + 8*w - 1],\
[349, 349, w^2 - 7*w + 1],\
[353, 353, -6*w^2 + 2*w + 21],\
[361, 19, -5*w^2 + 3*w + 25],\
[367, 367, 3*w^2 + w - 15],\
[373, 373, 2*w^2 - 4*w - 9],\
[383, 383, -9*w^2 + 3*w + 37],\
[401, 401, 4*w^2 - 2*w - 11],\
[409, 409, 4*w^2 - 4*w - 9],\
[431, 431, -5*w^2 + 5*w + 19],\
[439, 439, 2*w - 9],\
[443, 443, w^2 - 5*w - 3],\
[443, 443, 5*w^2 - 5*w - 23],\
[443, 443, 3*w^2 - 5*w - 17],\
[449, 449, -4*w^2 + 4*w + 1],\
[461, 461, 5*w^2 - 7*w - 9],\
[467, 467, 5*w^2 + w - 17],\
[479, 479, 2*w^2 + 4*w - 7],\
[487, 487, 5*w^2 - 5*w - 13],\
[499, 499, -w^2 + 3*w - 7],\
[509, 509, -7*w^2 + w + 27],\
[521, 521, w^2 - w - 11],\
[523, 523, 4*w^2 - 6*w - 11],\
[529, 23, 4*w^2 + 2*w - 13],\
[547, 547, -11*w^2 + 5*w + 43],\
[563, 563, 4*w^2 - 7],\
[569, 569, -6*w - 1],\
[569, 569, 4*w^2 - 5],\
[569, 569, w^2 - 5*w - 9],\
[571, 571, 5*w^2 - 3*w - 15],\
[577, 577, w^2 + 5*w - 7],\
[599, 599, 4*w^2 - 4*w - 7],\
[601, 601, -12*w^2 + 8*w + 51],\
[613, 613, 4*w^2 + 1],\
[643, 643, 6*w^2 - 2*w - 31],\
[647, 647, 2*w^2 - 2*w - 15],\
[647, 647, 4*w^2 - 4*w - 23],\
[647, 647, w^2 - 5*w - 7],\
[661, 661, -8*w^2 + 6*w + 39],\
[673, 673, 3*w^2 - 5*w - 15],\
[683, 683, -w^2 + 9*w - 7],\
[701, 701, 4*w^2 - 4*w - 5],\
[709, 709, 4*w^2 - 2*w - 7],\
[719, 719, 3*w^2 - 5*w - 13],\
[727, 727, w^2 - 7*w - 1],\
[733, 733, 2*w^2 + 4*w - 9],\
[733, 733, 3*w^2 + 3*w - 11],\
[733, 733, w^2 + w - 13],\
[743, 743, 2*w^2 - 6*w - 5],\
[751, 751, 6*w^2 - 2*w - 19],\
[761, 761, -12*w^2 + 6*w + 47],\
[761, 761, 3*w^2 - 5*w - 21],\
[761, 761, 2*w^2 + 6*w - 7],\
[769, 769, -6*w^2 + 6*w + 23],\
[773, 773, -2*w^2 + 2*w - 5],\
[773, 773, -13*w^2 + 5*w + 57],\
[773, 773, -11*w^2 + 7*w + 51],\
[787, 787, 5*w^2 - w - 13],\
[797, 797, -2*w^2 + 8*w - 11],\
[811, 811, -5*w^2 + 5*w + 1],\
[821, 821, -7*w^2 + 3*w + 35],\
[821, 821, -4*w^2 + 2*w + 23],\
[821, 821, 5*w^2 + w - 19],\
[823, 823, w^2 + 5*w - 13],\
[823, 823, -12*w^2 + 4*w + 49],\
[823, 823, -7*w^2 + w + 29],\
[829, 829, -w^2 + w - 7],\
[839, 839, 4*w^2 + 2*w - 15],\
[841, 29, 4*w^2 - 6*w - 13],\
[853, 853, 7*w^2 - 7*w - 25],\
[863, 863, 7*w^2 - 5*w - 35],\
[877, 877, -6*w - 5],\
[877, 877, w^2 + 5*w - 11],\
[877, 877, 5*w^2 - 3*w - 13],\
[881, 881, w^2 - 3*w - 13],\
[887, 887, -9*w^2 + 5*w + 33],\
[907, 907, w^2 - 9*w + 1],\
[911, 911, -6*w^2 + 6*w + 25],\
[919, 919, 5*w^2 - 3*w - 27],\
[929, 929, 2*w - 11],\
[937, 937, -7*w^2 + w + 31],\
[941, 941, -6*w^2 + 4*w + 31],\
[941, 941, -8*w^2 + 2*w + 27],\
[941, 941, 3*w^2 + 5*w - 9],\
[947, 947, -10*w + 7],\
[947, 947, 2*w^2 + 8*w - 7],\
[947, 947, 2*w^2 + 4*w - 11],\
[961, 31, -14*w^2 + 8*w + 55],\
[967, 967, -w^2 - w - 7],\
[971, 971, -10*w^2 + 8*w + 39],\
[977, 977, -10*w^2 + 6*w + 37],\
[983, 983, 3*w^2 + 3*w - 13],\
[983, 983, -4*w - 11],\
[983, 983, -10*w^2 + 4*w + 37],\
[997, 997, w^2 + 9*w - 7],\
[997, 997, 5*w^2 - 7*w - 27],\
[997, 997, 6*w^2 - 8*w - 11]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^5 - x^4 - 10*x^3 + 8*x^2 + 20*x - 4
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -1/2*e^3 - 1/2*e^2 + 3*e + 2, 1, -e^4 + 7*e^2 - 4, e^2 - e - 6, -e^4 - e^3 + 6*e^2 + 6*e, e^3 + e^2 - 8*e - 4, -e^3 + 5*e + 2, e^4 + e^3 - 7*e^2 - 5*e + 2, -e^4 + 8*e^2 - e - 10, e^4 + 2*e^3 - 7*e^2 - 12*e + 6, 2*e^2 - 2*e - 8, e^4 - e^3 - 9*e^2 + 7*e + 10, -e^3 - e^2 + 4*e + 8, e^4 + 2*e^3 - 8*e^2 - 13*e + 10, -e^4 + 9*e^2 - 2*e - 12, 2*e^3 - e^2 - 13*e + 2, -e^4 - 2*e^3 + 7*e^2 + 10*e - 4, -2*e^3 + 14*e + 6, e^4 - 7*e^2 + 4*e + 4, -e^4 - e^3 + 8*e^2 + 8*e - 8, -e^4 + e^3 + 8*e^2 - 10*e - 8, e^4 + e^3 - 5*e^2 - 3*e - 10, e^4 - e^3 - 6*e^2 + 4*e - 2, e^4 + e^3 - 10*e^2 - 6*e + 14, 2*e^2 - 2*e - 10, e^4 - 9*e^2 - 4*e + 18, -2*e^3 + 10*e - 4, e^4 + 2*e^3 - 11*e^2 - 14*e + 24, -2*e^4 + 2*e^3 + 18*e^2 - 14*e - 22, 2*e^3 + 2*e^2 - 12*e - 4, e^4 - e^3 - 8*e^2 + 6*e + 18, e^4 - e^3 - 4*e^2 + 6*e - 12, e^3 - 3*e^2 - 8*e + 12, -e^4 - 2*e^3 + 3*e^2 + 8*e + 18, -2*e^4 - 2*e^3 + 14*e^2 + 14*e - 12, -e^4 + e^3 + 8*e^2 - 6*e + 2, -e^4 - 4*e^3 + 5*e^2 + 26*e + 8, e^4 - 2*e^3 - 8*e^2 + 15*e + 2, e^3 + 2*e^2 - 3*e - 2, 2*e^4 + 2*e^3 - 12*e^2 - 8*e - 6, 3*e^4 + 3*e^3 - 23*e^2 - 17*e + 26, -3*e^4 + e^3 + 22*e^2 - 6*e - 12, -2*e^3 - e^2 + 15*e + 10, 2*e^4 - 2*e^3 - 16*e^2 + 16*e + 12, -2*e^4 + 18*e^2 - 12, -2*e^4 - 4*e^3 + 10*e^2 + 24*e + 16, 3*e^2 - 3*e - 10, -e^4 + 5*e^2 + 6*e + 8, -2*e^4 - 2*e^3 + 16*e^2 + 12*e - 10, -2*e^3 - 6*e^2 + 16*e + 20, -e^4 - 2*e^3 + 8*e^2 + 9*e - 6, -e^4 + e^3 + 2*e^2 - 8*e + 18, -e^4 + 2*e^3 + 7*e^2 - 18*e - 4, -2*e^4 - e^3 + 13*e^2 + 12*e, -e^4 + e^3 + 9*e^2 - 3*e - 26, 2*e^4 - 4*e^3 - 14*e^2 + 28*e + 20, 2*e^3 + 4*e^2 - 10*e - 28, 2*e^3 + 4*e^2 - 10*e - 16, -2*e^4 + 3*e^3 + 16*e^2 - 25*e - 18, -e^4 - 2*e^3 + 7*e^2 + 12*e + 10, 2*e^4 - 14*e^2 + 8*e + 8, -4*e^4 - 3*e^3 + 29*e^2 + 16*e - 16, e^4 + e^3 - 12*e^2 - 4*e + 20, 2*e^4 + 2*e^3 - 14*e^2 - 6*e, -e^4 + e^3 + 11*e^2 - 13*e - 14, 6*e^3 + 4*e^2 - 34*e - 24, 4*e^4 - 32*e^2 + 4*e + 26, -4*e^3 - 2*e^2 + 22*e + 14, -3*e^4 + e^3 + 20*e^2 - 8*e - 6, 2*e^4 - 4*e^3 - 20*e^2 + 22*e + 32, 4*e^3 - 2*e^2 - 18*e + 16, 5*e^3 + e^2 - 24*e, 4*e^4 + 4*e^3 - 26*e^2 - 18*e, 8*e + 4, -e^4 - 5*e^3 + 8*e^2 + 28*e - 10, 2*e^4 + 3*e^3 - 14*e^2 - 19*e + 14, -e^4 - 2*e^3 + 9*e^2 + 12*e - 20, -2*e^4 + 2*e^3 + 14*e^2 - 14*e - 8, -e^4 + 2*e^3 + 9*e^2 - 16*e - 32, 2*e^4 + 4*e^3 - 14*e^2 - 32*e + 16, 3*e^4 - 23*e^2 + 10*e + 16, -e^4 + 2*e^3 + 6*e^2 - 13*e + 2, 2*e^4 - 14*e^2 - 4, 4*e^3 + 4*e^2 - 32*e - 8, -6*e^3 - 2*e^2 + 32*e + 8, 2*e^3 - 2*e^2 - 12*e + 4, -2*e^3 + 10*e - 12, 3*e^4 - 29*e^2 - 4*e + 40, -3*e^4 - e^3 + 26*e^2 + 8*e - 26, -2*e^4 + 12*e^2 + 2*e - 4, e^4 + e^3 - 4*e^2 - 12*e - 16, -4*e^4 + 4*e^3 + 36*e^2 - 20*e - 40, -2*e^4 - 2*e^3 + 18*e^2 + 6*e - 24, 2*e^4 + e^3 - 16*e^2 - 15*e + 30, 4*e^4 + e^3 - 30*e^2 + 5*e + 26, 4*e^4 + 2*e^3 - 30*e^2 - 12*e + 12, -e^4 + 5*e^3 + 18*e^2 - 36*e - 38, 2*e^4 - 2*e^3 - 20*e^2 + 24*e + 28, -e^4 + e^3 + 9*e^2 - 15*e - 18, 3*e^4 - 19*e^2 + 2*e + 12, 5*e^4 + 2*e^3 - 35*e^2 - 6*e + 4, -2*e^3 - 2*e^2 + 8*e + 18, e^4 + 3*e^3 - 2*e^2 - 20*e - 12, -e^4 - 2*e^3 + 7*e^2 + 18*e - 4, 2*e^4 - 15*e^2 + e + 30, -e^4 - 2*e^3 + 2*e^2 + 15*e + 14, 3*e^4 - 4*e^3 - 23*e^2 + 20*e + 26, -2*e^4 - 3*e^3 + 15*e^2 + 12*e - 24, 2*e^4 - 22*e^2 + 4*e + 32, -2*e^4 - 4*e^3 + 12*e^2 + 22*e - 16, -2*e^4 - 3*e^3 + 15*e^2 + 12*e - 20, 3*e^4 + 2*e^3 - 25*e^2 - 4*e + 42, e^4 - e^3 - 12*e^2 + 2*e + 22, -2*e^4 - 6*e^3 + 6*e^2 + 38*e + 40, -5*e^3 + 4*e^2 + 21*e - 22, 2*e^4 + 5*e^3 - 16*e^2 - 27*e - 2, 2*e^4 - 3*e^3 - 17*e^2 + 12*e + 16, -2*e^4 - 9*e^3 + 13*e^2 + 56*e - 12, e^4 - e^3 - 4*e^2 + 2*e + 10, -3*e^4 - 6*e^3 + 14*e^2 + 45*e + 18, 3*e^3 + 8*e^2 - 23*e - 30, 2*e^4 + 8*e^3 - 17*e^2 - 53*e + 18, 2*e^4 + 2*e^3 - 13*e^2 + e - 10, e^4 + 4*e^3 - 8*e^2 - 35*e + 10, -3*e^4 - 2*e^3 + 23*e^2 - 28, 5*e^4 - 4*e^3 - 41*e^2 + 36*e + 46, 3*e^4 + 2*e^3 - 16*e^2 - 19*e - 26, 3*e^4 + 5*e^3 - 28*e^2 - 18*e + 42, -e^4 + 2*e^3 + 5*e^2 - 20*e + 16, 4*e^4 + 3*e^3 - 36*e^2 - 11*e + 46, -3*e^4 - 3*e^3 + 20*e^2 + 28*e - 12, -e^4 + e^2 + 4*e + 18, 10*e^3 + 6*e^2 - 56*e - 24, 2*e^4 + 5*e^3 - 12*e^2 - 23*e - 18, 4*e^4 + 4*e^3 - 28*e^2 - 32*e + 18, -3*e^4 + 2*e^3 + 25*e^2 - 10*e - 24, -2*e^3 - 2*e^2 + 20*e - 4, -e^3 - 9*e^2 + 12*e + 36, -2*e^4 + 8*e^3 + 12*e^2 - 54*e - 16, -4*e^4 - 9*e^3 + 28*e^2 + 49*e - 2, -2*e^4 - 7*e^3 + 19*e^2 + 48*e - 24, -3*e^4 - 4*e^3 + 25*e^2 + 28*e - 36, -2*e^4 - 4*e^3 + 16*e^2 + 22*e - 6, -2*e^4 - 6*e^3 + 20*e^2 + 40*e - 32, e^4 + 8*e^3 - 9*e^2 - 50*e + 12, -e^4 - 10*e^3 - 3*e^2 + 60*e + 36, 2*e^4 + 2*e^3 - 12*e^2 - 20*e - 10, 3*e^4 + 2*e^3 - 20*e^2 - 7*e + 6, -e^4 - 7*e^3 + 10*e^2 + 48*e - 16, 3*e^4 + e^3 - 18*e^2 + 8*e - 4, -2*e^4 - 4*e^3 + 13*e^2 + 21*e - 18, -2*e^4 + 6*e^3 + 11*e^2 - 31*e + 14, -3*e^4 - e^3 + 16*e^2 + 10*e + 6, 5*e^3 - 2*e^2 - 27*e + 22, -2*e^4 - 6*e^3 + 8*e^2 + 40*e - 6, 5*e^4 + 6*e^3 - 44*e^2 - 29*e + 38, 3*e^4 + 8*e^3 - 13*e^2 - 44*e - 28, -7*e^4 - 3*e^3 + 54*e^2 + 10*e - 60, -2*e^4 - 4*e^3 + 18*e^2 + 40*e - 24, -4*e^4 - 4*e^3 + 32*e^2 + 32*e - 44, -2*e^4 + 3*e^3 + 22*e^2 - 7*e - 54, 2*e^4 - 4*e^3 - 25*e^2 + 39*e + 42, -7*e^4 - 7*e^3 + 53*e^2 + 27*e - 30, e^4 - 9*e^2 + 6*e, 2*e^4 - 8*e^3 - 22*e^2 + 56*e + 28, 6*e^4 - 40*e^2 + 6*e, -6*e^4 - 4*e^3 + 47*e^2 + 23*e - 54, 5*e^4 + 2*e^3 - 34*e^2 - 7*e + 6, e^4 + 8*e^3 - 9*e^2 - 52*e + 18, -e^4 + 6*e^3 + 9*e^2 - 32*e - 28]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([11, 11, w^2 - w - 1])] = -1

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