/*
  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([29, 11, -10, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

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

primes_array = [
[11, 11, -w - 1],\
[11, 11, w^2 - 5],\
[11, 11, -w^2 + 2*w + 4],\
[11, 11, w - 2],\
[16, 2, 2],\
[19, 19, w^2 - 2*w - 5],\
[19, 19, -w^2 + 6],\
[25, 5, -2*w^2 + 2*w + 11],\
[29, 29, w],\
[29, 29, 2*w^2 - w - 10],\
[29, 29, -2*w^2 + 3*w + 9],\
[29, 29, w - 1],\
[31, 31, w^3 - 6*w - 6],\
[31, 31, -w^3 + 3*w^2 + 3*w - 11],\
[41, 41, w^3 - 4*w^2 - 2*w + 16],\
[41, 41, w^3 - 5*w^2 - 2*w + 24],\
[59, 59, w^3 - w^2 - 5*w - 2],\
[59, 59, 2*w^2 - w - 13],\
[61, 61, w^3 - w^2 - 6*w + 3],\
[61, 61, -w^3 + 2*w^2 + 5*w - 3],\
[71, 71, 2*w^2 - w - 16],\
[71, 71, -w^3 + 3*w^2 + 4*w - 10],\
[71, 71, -w^3 + 7*w + 4],\
[71, 71, w^3 - 5*w^2 - 2*w + 23],\
[79, 79, -w^3 + 4*w^2 + 3*w - 16],\
[79, 79, w^3 + w^2 - 8*w - 10],\
[81, 3, -3],\
[131, 131, 4*w^2 - 5*w - 24],\
[131, 131, 4*w^2 - 3*w - 25],\
[139, 139, w^2 + w - 9],\
[139, 139, -w^3 + w^2 + 6*w - 5],\
[149, 149, w^3 - w^2 - 4*w + 2],\
[149, 149, w^3 - 2*w^2 - 3*w + 2],\
[151, 151, -w^3 + 7*w + 3],\
[151, 151, -w^3 + 3*w^2 + 4*w - 9],\
[179, 179, -w^3 - 2*w^2 + 10*w + 16],\
[179, 179, -w^3 + 3*w^2 + 3*w - 6],\
[179, 179, w^3 - 6*w - 1],\
[179, 179, w^3 - 5*w^2 - 3*w + 23],\
[191, 191, w^3 - 6*w^2 - w + 27],\
[191, 191, w^3 + 3*w^2 - 10*w - 21],\
[199, 199, -w^3 + 2*w^2 + 6*w - 4],\
[199, 199, -4*w^2 + 3*w + 22],\
[199, 199, 4*w^2 - 5*w - 21],\
[199, 199, 3*w^2 - 4*w - 12],\
[239, 239, -w^3 + w^2 + 4*w - 6],\
[239, 239, 2*w^3 - 5*w^2 - 9*w + 17],\
[239, 239, 2*w^3 - w^2 - 13*w - 5],\
[239, 239, w^3 - 8*w^2 + 2*w + 41],\
[241, 241, -w^3 + 5*w^2 + 4*w - 25],\
[241, 241, w^3 - 6*w - 8],\
[241, 241, -w^3 + 3*w^2 + 3*w - 13],\
[241, 241, w^3 - 5*w^2 + w + 19],\
[269, 269, w^2 - 2*w - 10],\
[269, 269, w^2 - 11],\
[271, 271, w^3 + 2*w^2 - 8*w - 18],\
[271, 271, 2*w^3 - 13*w - 10],\
[271, 271, 2*w^3 + w^2 - 15*w - 16],\
[271, 271, w^3 - 5*w^2 - w + 23],\
[289, 17, 2*w^2 - 11],\
[289, 17, -2*w^2 + 4*w + 9],\
[311, 311, w^3 - 4*w^2 - 3*w + 10],\
[311, 311, w^3 - 7*w^2 - w + 34],\
[331, 331, w^3 - w^2 - 7*w + 4],\
[331, 331, w^3 - 2*w^2 - 6*w + 3],\
[349, 349, w^3 + w^2 - 6*w - 12],\
[349, 349, -w^3 + 4*w^2 + w - 16],\
[361, 19, 4*w^2 - 4*w - 21],\
[379, 379, w^3 - 3*w^2 - 3*w + 15],\
[379, 379, 2*w^3 - 2*w^2 - 12*w + 1],\
[379, 379, -2*w^3 + 4*w^2 + 10*w - 11],\
[379, 379, w^3 - 6*w - 10],\
[401, 401, -2*w^3 + 7*w^2 + 8*w - 30],\
[401, 401, -w^3 + 2*w^2 + 6*w - 2],\
[419, 419, 5*w^2 - 4*w - 26],\
[419, 419, 5*w^2 - 6*w - 25],\
[421, 421, 3*w^2 - w - 20],\
[421, 421, -2*w^3 + 2*w^2 + 10*w + 3],\
[431, 431, w^2 - 3*w - 5],\
[431, 431, w^2 + w - 7],\
[449, 449, 2*w^3 - 13*w - 7],\
[449, 449, -2*w^3 + 6*w^2 + 7*w - 18],\
[461, 461, w^3 - w^2 - 8*w + 3],\
[461, 461, -w^3 + 2*w^2 + 7*w - 5],\
[491, 491, w^3 - 3*w^2 - 6*w + 15],\
[491, 491, w^3 - 9*w - 7],\
[499, 499, 4*w^2 - 6*w - 23],\
[499, 499, -4*w^2 + 2*w + 25],\
[509, 509, 2*w^3 - 3*w^2 - 11*w + 6],\
[521, 521, -w^3 + 3*w^2 + 2*w - 12],\
[521, 521, -w^3 + 4*w^2 + 2*w - 19],\
[529, 23, w^3 - 7*w^2 - w + 35],\
[529, 23, -w^3 + 3*w^2 + 4*w - 5],\
[541, 541, 2*w^3 - w^2 - 12*w - 5],\
[541, 541, w^3 - 2*w^2 - 7*w + 9],\
[569, 569, w^3 - 8*w - 2],\
[569, 569, -w^3 + 3*w^2 + 5*w - 9],\
[599, 599, -w^3 - 4*w^2 + 10*w + 25],\
[599, 599, -w^3 + 7*w^2 - 36],\
[601, 601, 5*w^2 - 6*w - 26],\
[601, 601, -2*w^3 + 8*w^2 + 6*w - 33],\
[601, 601, -2*w^3 - 2*w^2 + 16*w + 21],\
[601, 601, 5*w^2 - 4*w - 27],\
[619, 619, w^3 + 4*w^2 - 11*w - 26],\
[619, 619, w^3 - 3*w^2 - 5*w + 18],\
[641, 641, -w^3 + 3*w^2 + 6*w - 14],\
[641, 641, -w^3 + 9*w + 6],\
[659, 659, -w^3 - 4*w^2 + 10*w + 28],\
[659, 659, -w^3 + 6*w^2 - w - 27],\
[659, 659, -w^3 - 3*w^2 + 8*w + 23],\
[659, 659, -w^3 + 7*w^2 - w - 33],\
[691, 691, w^3 - w^2 - 8*w + 2],\
[691, 691, 2*w^3 - 2*w^2 - 11*w + 1],\
[701, 701, -w^3 - w^2 + 7*w + 15],\
[701, 701, -w^3 + 4*w^2 + 2*w - 20],\
[709, 709, -2*w^3 - w^2 + 15*w + 15],\
[709, 709, 2*w^3 - 7*w^2 - 7*w + 27],\
[719, 719, -w^3 - 4*w^2 + 13*w + 26],\
[719, 719, w^3 - 7*w^2 - 2*w + 34],\
[739, 739, 2*w^3 + 2*w^2 - 15*w - 24],\
[739, 739, -2*w^3 + 8*w^2 + 5*w - 35],\
[751, 751, -w^3 - 3*w^2 + 9*w + 25],\
[751, 751, -w^3 + 6*w^2 - 30],\
[761, 761, 3*w^3 - 2*w^2 - 16*w - 8],\
[761, 761, -3*w^3 + 7*w^2 + 11*w - 23],\
[769, 769, -2*w^3 + 4*w^2 + 11*w - 10],\
[769, 769, w^3 + 2*w^2 - 9*w - 12],\
[809, 809, -3*w^3 + 3*w^2 + 16*w + 4],\
[809, 809, w^3 + 5*w^2 - 12*w - 30],\
[811, 811, -w^3 - 4*w^2 + 10*w + 27],\
[811, 811, -2*w^3 + 5*w^2 + 9*w - 14],\
[811, 811, -2*w^3 + w^2 + 13*w + 2],\
[811, 811, -w^3 + 7*w^2 - w - 32],\
[821, 821, 2*w^3 - 8*w^2 - 7*w + 35],\
[821, 821, -2*w^3 - 2*w^2 + 17*w + 22],\
[829, 829, -3*w^3 + 20*w + 13],\
[829, 829, -w^3 + 7*w^2 - w - 36],\
[829, 829, w^3 + 4*w^2 - 10*w - 31],\
[829, 829, 2*w^3 - 3*w^2 - 11*w + 14],\
[859, 859, -w^3 + 7*w^2 - 3*w - 31],\
[859, 859, 7*w^2 - 9*w - 39],\
[859, 859, 7*w^2 - 5*w - 41],\
[859, 859, 3*w^3 - 4*w^2 - 17*w + 2],\
[929, 929, w^3 + 5*w^2 - 14*w - 31],\
[929, 929, -2*w^3 + 4*w^2 + 10*w - 5],\
[929, 929, w^3 - 9*w^2 + 4*w + 44],\
[929, 929, -w^3 + 8*w^2 + w - 39],\
[941, 941, -w^3 + 7*w^2 - 34],\
[941, 941, w^3 + 4*w^2 - 11*w - 28],\
[961, 31, -5*w^2 + 5*w + 28],\
[971, 971, -3*w^3 + w^2 + 19*w + 14],\
[971, 971, -3*w^3 + 8*w^2 + 12*w - 31],\
[991, 991, -w^3 - 5*w^2 + 11*w + 35],\
[991, 991, w^3 - 8*w^2 + 2*w + 40]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 + 2*x^2 - 25*x - 59
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1, e, -1, 1/9*e^2 - 2/3*e - 31/9, 1/3*e^2 - e - 25/3, 1/3*e^2 - 31/3, 1, -7/9*e^2 + 2/3*e + 109/9, 2/9*e^2 - 1/3*e - 89/9, -1/9*e^2 - 4/3*e + 4/9, 2/9*e^2 + 2/3*e - 80/9, -2/3*e^2 + 29/3, -4/9*e^2 + 5/3*e + 70/9, -2/9*e^2 + 1/3*e + 89/9, -1/3*e^2 + e + 34/3, 5/9*e^2 - 4/3*e - 92/9, 1/3*e^2 - 13/3, -2/9*e^2 - 2/3*e + 26/9, 7/9*e^2 - 2/3*e - 163/9, -11/9*e^2 + 1/3*e + 161/9, 8/9*e^2 - 4/3*e - 131/9, 2/3*e^2 - e - 65/3, -e + 1, -5/9*e^2 + 1/3*e + 128/9, -10/9*e^2 + 5/3*e + 229/9, 1/3*e^2 + e - 7/3, e - 3, -1/3*e^2 + 2*e + 7/3, 4/3*e^2 - 2*e - 76/3, 5/9*e^2 - 7/3*e - 65/9, -2/9*e^2 + 10/3*e + 26/9, -14/9*e^2 - 2/3*e + 236/9, 16/9*e^2 - 5/3*e - 199/9, -e^2 + 16, 1/9*e^2 - 2/3*e - 4/9, 1/3*e^2 - 2*e - 28/3, -1/3*e^2 + e + 13/3, 10/9*e^2 + 7/3*e - 193/9, -7/9*e^2 - 1/3*e + 73/9, -10/9*e^2 - 4/3*e + 157/9, -2/9*e^2 + 4/3*e + 8/9, 2/3*e^2 - e - 44/3, 7/9*e^2 - 2/3*e - 181/9, 14/9*e^2 - 7/3*e - 236/9, 2/9*e^2 - 1/3*e - 188/9, -5/3*e^2 + 2*e + 86/3, 1/9*e^2 + 10/3*e - 4/9, 17/9*e^2 - 10/3*e - 410/9, 2*e^2 - 2*e - 34, 2*e^2 + e - 38, -13/9*e^2 + 2/3*e + 178/9, -13/9*e^2 - 1/3*e + 169/9, -5/9*e^2 + 7/3*e + 56/9, -1/3*e^2 + 2*e + 22/3, 4/9*e^2 - 2/3*e + 2/9, 13/9*e^2 - 8/3*e - 277/9, 5/9*e^2 - 7/3*e - 164/9, -17/9*e^2 + 10/3*e + 437/9, -e - 2, 1/9*e^2 + 7/3*e - 121/9, 1/9*e^2 - 5/3*e - 157/9, -17/9*e^2 - 2/3*e + 221/9, -2/9*e^2 + 16/3*e + 62/9, 8/3*e^2 - 4*e - 182/3, 1/9*e^2 + 7/3*e - 103/9, -e^2 - 3*e + 20, 23/9*e^2 - 16/3*e - 533/9, 7/9*e^2 + 10/3*e - 271/9, -3*e^2 + 7*e + 65, -8/3*e^2 + e + 164/3, 2/9*e^2 + 2/3*e + 1/9, -31/9*e^2 + 5/3*e + 475/9, -7/3*e^2 + 2*e + 145/3, 1/9*e^2 + 4/3*e - 121/9, 5/9*e^2 + 2/3*e - 335/9, -7/3*e^2 + 3*e + 157/3, 2*e^2 - 6*e - 46, 11/9*e^2 - 1/3*e - 26/9, -11/9*e^2 - 5/3*e + 71/9, -4/3*e^2 - e + 70/3, -7/9*e^2 + 5/3*e - 26/9, -2*e^2 + 26, -11/3*e^2 + 182/3, -8/9*e^2 + 4/3*e - 94/9, 16/9*e^2 - 2/3*e - 73/9, 4/9*e^2 + 19/3*e - 196/9, 1/3*e^2 + 2*e + 50/3, -16/9*e^2 - 1/3*e + 280/9, 26/9*e^2 - 1/3*e - 536/9, e^2 + e + 1, 5/3*e^2 - 4*e - 179/3, 31/9*e^2 - 14/3*e - 646/9, 25/9*e^2 - 17/3*e - 451/9, 19/9*e^2 - 14/3*e - 436/9, 16/9*e^2 - 11/3*e - 325/9, 11/9*e^2 - 7/3*e - 197/9, 26/9*e^2 + 2/3*e - 383/9, -16/9*e^2 + 14/3*e + 307/9, -8/9*e^2 - 2/3*e + 383/9, -e^2 - e + 25, 2/9*e^2 + 8/3*e - 71/9, -5/3*e^2 + 2*e + 134/3, 17/9*e^2 - 13/3*e - 293/9, 35/9*e^2 - 22/3*e - 779/9, 7/9*e^2 - 14/3*e - 1/9, -20/9*e^2 + 19/3*e + 413/9, 5/9*e^2 + 2/3*e - 290/9, e^2 - e + 1, -7/3*e^2 + 3*e + 133/3, -8/9*e^2 - 5/3*e + 113/9, -4*e^2 + 8*e + 81, e^2 - 3*e - 5, -1/9*e^2 - 4/3*e - 185/9, -25/9*e^2 + 23/3*e + 640/9, 41/9*e^2 - 16/3*e - 830/9, -5/9*e^2 - 5/3*e + 101/9, -10/3*e^2 + 4*e + 169/3, 7/3*e^2 - 5*e - 103/3, -34/9*e^2 + 17/3*e + 568/9, 8/9*e^2 + 11/3*e - 347/9, 10/3*e^2 + e - 208/3, 4*e^2 - 2*e - 76, -2*e^2 + 6*e + 37, -4/9*e^2 - 13/3*e + 295/9, 10/9*e^2 + 4/3*e - 328/9, 7/9*e^2 - 5/3*e - 19/9, -1/3*e^2 - 5*e + 31/3, -3*e^2 - 4*e + 47, 5/9*e^2 + 14/3*e + 106/9, 8/9*e^2 + 14/3*e + 58/9, -38/9*e^2 - 2/3*e + 620/9, 38/9*e^2 - 19/3*e - 629/9, e^2 - 3*e - 16, -29/9*e^2 - 2/3*e + 611/9, -19/9*e^2 - 16/3*e + 427/9, 5/9*e^2 + 8/3*e - 317/9, -4/3*e^2 + 2*e + 61/3, 40/9*e^2 - 14/3*e - 853/9, -4/3*e^2 + 4*e + 58/3, -29/9*e^2 + 19/3*e + 530/9, -34/9*e^2 + 14/3*e + 721/9, 44/9*e^2 - 10/3*e - 815/9, -8/3*e^2 + 2*e + 140/3, -20/9*e^2 + 22/3*e + 440/9, 44/9*e^2 - 7/3*e - 716/9, 11/9*e^2 - 16/3*e - 215/9, 4*e^2 - e - 61, 16/9*e^2 - 29/3*e - 388/9, 29/9*e^2 + 2/3*e - 539/9, 25/9*e^2 - 14/3*e - 361/9, -2/3*e^2 - 6*e + 11/3, 22/9*e^2 - 8/3*e - 691/9, -14/3*e^2 + 3*e + 281/3, -28/9*e^2 + 17/3*e + 787/9]
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 - 1])] = -1

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