/*
  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([31, 31, 2*w^2 - 2*w - 9])

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^4 - 2*x^3 - 6*x^2 + 12*x - 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -1/2*e^3 + 1/2*e^2 + 7/2*e - 5/2, 3/2*e^3 - 1/2*e^2 - 19/2*e + 9/2, -e^3 - e^2 + 5*e + 7, -e^2 + 5, e^2 - 9, -e^3 - e^2 + 7*e + 3, 2*e^3 - e^2 - 12*e + 5, 1, 1/2*e^3 + 1/2*e^2 - 9/2*e + 3/2, -e^3 + 5*e - 2, 3/2*e^3 - 3/2*e^2 - 19/2*e + 11/2, -1/2*e^3 + 3/2*e^2 + 11/2*e - 17/2, e^3 - 2*e^2 - 7*e + 8, -1/2*e^3 - 1/2*e^2 + 11/2*e + 7/2, e^3 - 9*e + 6, -e^3 + 2*e^2 + 3*e - 12, -1/2*e^3 + 1/2*e^2 + 11/2*e + 13/2, e^3 + e^2 - 5*e + 5, -1/2*e^3 - 1/2*e^2 + 5/2*e - 7/2, -4*e^3 + 3*e^2 + 20*e - 19, -5*e^3 + 2*e^2 + 27*e - 12, 4*e^3 - e^2 - 28*e + 13, 2*e^3 - 10*e + 6, 3/2*e^3 + 5/2*e^2 - 11/2*e - 41/2, 2*e^3 - 4*e^2 - 14*e + 18, -5/2*e^3 + 9/2*e^2 + 27/2*e - 39/2, -3/2*e^3 + 7/2*e^2 + 17/2*e - 33/2, -1/2*e^3 + 1/2*e^2 + 5/2*e + 23/2, -e^3 + 2*e^2 + 5*e - 8, -e^2 + 9, -2*e^3 - e^2 + 8*e + 5, e^3 - 3*e^2 - 3*e + 5, e^3 + 2*e^2 - 7*e, -7/2*e^3 + 5/2*e^2 + 27/2*e - 21/2, e^3 - 5*e^2 - 9*e + 17, -3/2*e^3 - 13/2*e^2 + 23/2*e + 45/2, e^3 + e^2 - 7*e - 1, 5/2*e^3 + 11/2*e^2 - 31/2*e - 41/2, -2*e^3 + 3*e^2 + 16*e - 19, 5/2*e^3 + 5/2*e^2 - 35/2*e - 7/2, 1/2*e^3 + 1/2*e^2 + 11/2*e - 13/2, e^3 - e - 12, 7*e^3 - 2*e^2 - 41*e + 20, -4*e^3 + 3*e^2 + 18*e - 21, -1/2*e^3 + 1/2*e^2 + 19/2*e + 9/2, 4*e^3 - 3*e^2 - 20*e + 23, e^3 - 7*e^2 - 7*e + 21, -e^3 - e^2 + 3*e + 9, -7*e^3 + 4*e^2 + 35*e - 14, -13/2*e^3 + 5/2*e^2 + 75/2*e - 23/2, -2*e^3 - 7*e^2 + 16*e + 23, 9*e^3 - 4*e^2 - 51*e + 28, e^3 + 4*e^2 - 5*e - 22, -9/2*e^3 - 3/2*e^2 + 55/2*e + 17/2, 1/2*e^3 - 3/2*e^2 - 5/2*e + 31/2, -2*e^3 - 3*e^2 + 16*e + 17, -e^3 + 3*e + 2, -6*e^3 + 4*e^2 + 34*e - 12, -7/2*e^3 + 11/2*e^2 + 49/2*e - 45/2, -5/2*e^3 - 1/2*e^2 + 29/2*e + 5/2, 3/2*e^3 - 5/2*e^2 - 9/2*e - 9/2, 5/2*e^3 + 3/2*e^2 - 9/2*e - 31/2, 7*e^3 + 4*e^2 - 43*e - 4, 11/2*e^3 - 7/2*e^2 - 63/2*e + 35/2, 3*e^3 + 4*e^2 - 25*e - 18, 8*e^3 - 46*e + 2, -7/2*e^3 + 1/2*e^2 + 39/2*e - 13/2, -15/2*e^3 - 1/2*e^2 + 99/2*e - 23/2, 7*e^3 - 3*e^2 - 41*e + 11, 7*e^3 - 3*e^2 - 35*e + 19, -11/2*e^3 + 3/2*e^2 + 61/2*e - 9/2, 3*e^3 + 4*e^2 - 21*e - 10, -7/2*e^3 - 1/2*e^2 + 49/2*e + 11/2, 3*e^3 - e^2 - 13*e + 3, 4*e^3 - 7*e^2 - 24*e + 33, -3*e^3 + 4*e^2 + 19*e - 2, -e^3 - 2*e^2 + 9*e + 20, 3/2*e^3 + 3/2*e^2 - 33/2*e + 15/2, -3*e^3 + 6*e^2 + 15*e - 20, -6*e^3 + 2*e^2 + 28*e + 4, -7*e^3 + 4*e^2 + 37*e - 20, 10*e^3 - 5*e^2 - 66*e + 39, -e^3 - 6*e^2 + 3*e + 24, -5/2*e^3 + 1/2*e^2 + 41/2*e - 45/2, -5*e^3 - 4*e^2 + 31*e - 6, -6*e^3 + 3*e^2 + 34*e - 27, 5*e^3 - 25*e + 8, 27/2*e^3 - 9/2*e^2 - 161/2*e + 87/2, -3/2*e^3 - 13/2*e^2 + 1/2*e + 55/2, -3/2*e^3 + 15/2*e^2 + 7/2*e - 67/2, 13/2*e^3 + 11/2*e^2 - 69/2*e - 35/2, -5/2*e^3 - 1/2*e^2 + 17/2*e - 3/2, -4*e^3 + 8*e^2 + 20*e - 36, -5*e^3 + 7*e^2 + 23*e - 39, -4*e^3 + 2*e^2 + 26*e - 2, 4*e^3 + 6*e^2 - 18*e - 24, 5/2*e^3 + 9/2*e^2 - 19/2*e - 35/2, 2*e^3 + 2*e^2 - 18*e + 18, -8*e^3 + 8*e^2 + 58*e - 50, 6*e^2 + 2*e - 26, -3*e^3 + 7*e^2 + 23*e - 9, 17/2*e^3 - 17/2*e^2 - 99/2*e + 67/2, -e^3 + 8*e^2 + 5*e - 24, 9*e^3 - 10*e^2 - 59*e + 42, -15/2*e^3 + 7/2*e^2 + 105/2*e - 77/2, 7*e^3 - 41*e - 8, -3*e^3 + 9*e^2 + 19*e - 19, -4*e^3 + 2*e^2 + 20*e - 14, -11/2*e^3 - 1/2*e^2 + 87/2*e - 35/2, -1/2*e^3 - 9/2*e^2 + 11/2*e + 51/2, e^3 - 2*e^2 - 11*e, -1/2*e^3 + 9/2*e^2 + 9/2*e - 85/2, 4*e^3 - e^2 - 14*e - 7, 29/2*e^3 - 19/2*e^2 - 163/2*e + 117/2, 23/2*e^3 - 5/2*e^2 - 137/2*e + 63/2, 4*e^3 - 6*e^2 - 26*e + 38, -3/2*e^3 + 19/2*e^2 + 17/2*e - 21/2, -4*e^3 + 20*e + 8, -15/2*e^3 + 7/2*e^2 + 63/2*e - 27/2, -1/2*e^3 + 1/2*e^2 - 15/2*e + 11/2, -19/2*e^3 + 23/2*e^2 + 93/2*e - 97/2, 2*e^3 - 3*e^2 - 18*e + 11, 5*e^3 + 5*e^2 - 25*e - 9, -9/2*e^3 - 11/2*e^2 + 81/2*e + 39/2, -9/2*e^3 + 9/2*e^2 + 53/2*e - 41/2, 7*e^3 - 8*e^2 - 57*e + 48, 2*e^3 + 6*e - 6, 6*e^3 - 12*e^2 - 40*e + 32, -9*e^3 - 5*e^2 + 45*e + 31, -5*e^3 - 3*e^2 + 29*e + 1, -5*e^3 + 2*e^2 + 33*e - 2, -3*e^3 - 2*e^2 + 19*e, 2*e^3 - 6*e^2 - 32*e + 36, -e^3 - e^2 - 3*e + 19, -5*e^3 + 33*e - 14, 7/2*e^3 + 9/2*e^2 - 41/2*e + 17/2, 1/2*e^3 - 7/2*e^2 - 9/2*e + 79/2, -13/2*e^3 + 23/2*e^2 + 73/2*e - 75/2, 4*e^3 - 3*e^2 - 14*e + 17, 5/2*e^3 - 5/2*e^2 - 21/2*e + 17/2, -7*e^3 + 9*e^2 + 29*e - 43, 6*e^3 - 6*e^2 - 32*e + 38, -2*e^3 + 7*e^2 - 2*e - 37, -4*e^3 + e^2 + 12*e - 9, 4*e^3 - 3*e^2 - 26*e - 21, -12*e^3 - e^2 + 58*e + 13, 6*e^3 - 10*e^2 - 30*e + 32, 4*e^3 - 4*e^2 - 24*e + 34, 29/2*e^3 - 13/2*e^2 - 153/2*e + 65/2, -3*e^3 - 5*e^2 + 11*e + 25, 6*e^3 - 7*e^2 - 36*e + 57, -e^3 - 9*e^2 + 3*e + 19, -5*e^3 + 9*e^2 + 41*e - 35, 3*e^3 - 4*e^2 - 13*e + 4, 7*e^3 - 8*e^2 - 45*e + 16, -13/2*e^3 - 23/2*e^2 + 81/2*e + 55/2, -15/2*e^3 - 3/2*e^2 + 111/2*e - 1/2, -9/2*e^3 - 3/2*e^2 + 39/2*e + 25/2, 4*e^3 + 6*e^2 - 28*e - 42, 9/2*e^3 - 3/2*e^2 - 33/2*e - 45/2, 12*e^3 - 4*e^2 - 80*e + 34, 11/2*e^3 - 9/2*e^2 - 69/2*e + 15/2, -9*e^3 + 8*e^2 + 51*e - 50, 2*e^3 - 5*e^2 - 4*e - 11, -9/2*e^3 + 7/2*e^2 + 63/2*e - 49/2, -13*e^3 + 5*e^2 + 73*e - 33, -13/2*e^3 + 1/2*e^2 + 71/2*e - 39/2, 3*e^3 - 12*e^2 - 27*e + 58, -e^3 - 2*e^2 - 3*e + 20, -14*e^3 + 4*e^2 + 92*e - 64]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([31, 31, 2*w^2 - 2*w - 9])] = -1

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