/*
  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([3, 8, -1, -6, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([11, 11, -w^4 + w^3 + 5*w^2 - 3*w - 4])

primes_array = [
[3, 3, w^2 - 3],\
[5, 5, w^4 - 5*w^2 + 4],\
[11, 11, -w^4 + w^3 + 5*w^2 - 3*w - 4],\
[17, 17, w^4 - 4*w^2 + 2],\
[32, 2, 2],\
[37, 37, -w^4 + 2*w^3 + 5*w^2 - 7*w - 5],\
[37, 37, -w^3 + 4*w - 1],\
[41, 41, w^3 + w^2 - 4*w - 1],\
[41, 41, -w^3 + 2*w + 2],\
[43, 43, w^2 - w - 4],\
[49, 7, w^4 - 5*w^2 + 2],\
[53, 53, w^4 - w^3 - 5*w^2 + 2*w + 5],\
[59, 59, w^4 - w^3 - 4*w^2 + 3*w - 1],\
[67, 67, -w^4 + 3*w^2 + 2*w + 2],\
[79, 79, w^3 - w^2 - 4*w + 2],\
[81, 3, -w^4 + 6*w^2 + w - 8],\
[83, 83, w^4 + w^3 - 5*w^2 - 4*w + 1],\
[89, 89, -w^4 + 5*w^2 + w - 1],\
[97, 97, -w^4 + 2*w^3 + 5*w^2 - 5*w - 5],\
[101, 101, 2*w^2 - w - 2],\
[107, 107, -w^4 + w^3 + 5*w^2 - 4*w - 8],\
[107, 107, w^4 - 2*w^3 - 4*w^2 + 6*w + 5],\
[107, 107, w^4 - w^3 - 7*w^2 + 4*w + 10],\
[109, 109, -w^4 + 5*w^2 + w - 7],\
[113, 113, w^3 - 5*w + 2],\
[121, 11, -w^4 + 2*w^3 + 5*w^2 - 7*w - 7],\
[121, 11, -w^4 + 4*w^2 + 2*w + 1],\
[127, 127, 2*w^3 + w^2 - 6*w - 4],\
[127, 127, w^2 + w - 5],\
[131, 131, -w^4 + w^3 + 5*w^2 - 2*w - 4],\
[137, 137, 2*w^4 - w^3 - 10*w^2 + 3*w + 7],\
[137, 137, w^3 + w^2 - 5*w - 1],\
[139, 139, 2*w^3 - 7*w - 1],\
[149, 149, w^3 - w^2 - 4*w + 1],\
[167, 167, w^4 - 2*w^3 - 5*w^2 + 6*w + 7],\
[167, 167, w^3 + 2*w^2 - 3*w - 7],\
[173, 173, -w^4 + w^3 + 4*w^2 - 4*w - 4],\
[173, 173, w^4 - w^3 - 5*w^2 + 3*w + 8],\
[181, 181, -w^3 + w^2 + 5*w - 2],\
[193, 193, 2*w - 1],\
[197, 197, 2*w^2 + w - 4],\
[211, 211, w^4 - 2*w^3 - 5*w^2 + 5*w + 4],\
[223, 223, w^2 - w + 1],\
[223, 223, 2*w^4 - w^3 - 9*w^2 + 3*w + 8],\
[227, 227, 2*w^3 - 6*w + 1],\
[233, 233, w^4 + 3*w^3 - 4*w^2 - 10*w - 4],\
[233, 233, -2*w^4 + 9*w^2 + w - 7],\
[239, 239, w^4 - w^3 - 6*w^2 + 3*w + 5],\
[239, 239, -2*w^3 - 2*w^2 + 7*w + 5],\
[241, 241, -w^2 - 1],\
[241, 241, 2*w^4 - 9*w^2 + 4],\
[251, 251, 2*w^3 - w^2 - 7*w + 2],\
[257, 257, 2*w^4 - w^3 - 9*w^2 + 3*w + 7],\
[263, 263, 2*w^4 - w^3 - 11*w^2 + 4*w + 10],\
[277, 277, w^4 + w^3 - 6*w^2 - 2*w + 5],\
[307, 307, 2*w^3 - w^2 - 7*w - 1],\
[307, 307, -w^4 - w^3 + 5*w^2 + 2*w - 1],\
[307, 307, w^4 - 2*w^2 - w - 2],\
[311, 311, 2*w^3 + w^2 - 8*w - 2],\
[311, 311, -w^4 + 2*w^3 + 7*w^2 - 7*w - 10],\
[317, 317, -w^4 + w^3 + 6*w^2 - 4*w - 4],\
[317, 317, w^4 + w^3 - 4*w^2 - 3*w - 2],\
[331, 331, 2*w^4 - w^3 - 8*w^2 + 2*w + 4],\
[337, 337, -w^4 + w^3 + 6*w^2 - 5*w - 8],\
[343, 7, w^4 - w^3 - 7*w^2 + 3*w + 8],\
[347, 347, w^2 + 2*w - 2],\
[349, 349, w^4 - w^3 - 3*w^2 + 2*w - 2],\
[349, 349, w^4 - 4*w^2 + w - 1],\
[359, 359, -2*w^4 + w^3 + 9*w^2 - 2*w - 4],\
[361, 19, -w^4 + 6*w^2 - 2*w - 7],\
[367, 367, w^3 + 2*w^2 - 5*w - 4],\
[367, 367, -w^4 + w^3 + 3*w^2 - w + 1],\
[373, 373, 2*w^4 - 2*w^3 - 9*w^2 + 7*w + 8],\
[373, 373, w^4 + w^3 - 3*w^2 - 4*w + 1],\
[379, 379, -3*w^3 - w^2 + 8*w + 2],\
[389, 389, 2*w^3 - w^2 - 6*w + 1],\
[401, 401, -w^4 + 3*w^2 + 2*w - 2],\
[431, 431, -w^4 - w^3 + 5*w^2 + 2*w - 2],\
[431, 431, -w^3 + w^2 + 2*w - 4],\
[433, 433, -w^4 + 6*w^2 - w - 11],\
[433, 433, -w^4 + 6*w^2 + w - 4],\
[433, 433, 3*w^4 - 2*w^3 - 13*w^2 + 6*w + 7],\
[439, 439, -2*w^4 - w^3 + 8*w^2 + 3*w - 1],\
[443, 443, 3*w^3 + w^2 - 8*w - 5],\
[443, 443, -w^4 + 5*w^2 + 2*w - 5],\
[449, 449, 3*w^4 - w^3 - 15*w^2 + 3*w + 14],\
[457, 457, -w^4 + 2*w^3 + 4*w^2 - 7*w - 5],\
[461, 461, 3*w^3 - w^2 - 11*w + 1],\
[461, 461, 2*w^4 - 8*w^2 - 3*w + 1],\
[463, 463, -2*w^4 + w^3 + 11*w^2 - w - 11],\
[463, 463, w^4 - 2*w^3 - 4*w^2 + 5*w + 1],\
[467, 467, w^4 + w^3 - 3*w^2 - 6*w - 5],\
[479, 479, -2*w^4 + 10*w^2 + w - 11],\
[491, 491, w^4 - w^3 - 4*w^2 + 2*w - 1],\
[521, 521, -w^4 + w^3 + 6*w^2 - 3*w - 4],\
[523, 523, w^4 - w^3 - 4*w^2 + 5*w + 5],\
[541, 541, -w^4 + 6*w^2 - w - 2],\
[547, 547, 2*w^3 - 2*w^2 - 5*w + 4],\
[547, 547, -2*w^4 + w^3 + 7*w^2 - w - 4],\
[557, 557, -3*w^4 + 3*w^3 + 15*w^2 - 10*w - 16],\
[557, 557, -2*w^4 + w^3 + 10*w^2 - 2*w - 8],\
[563, 563, 2*w^3 - w^2 - 3*w - 2],\
[569, 569, 3*w^4 - 2*w^3 - 12*w^2 + 5*w + 5],\
[569, 569, w^4 - 7*w^2 + w + 13],\
[569, 569, -w^4 + 6*w^2 - 4],\
[577, 577, w^3 + 2*w^2 - 3*w - 8],\
[577, 577, 2*w^4 - 7*w^2 - 1],\
[587, 587, 2*w^4 - w^3 - 11*w^2 + 5*w + 14],\
[593, 593, -2*w^3 - 2*w^2 + 8*w + 5],\
[599, 599, -2*w^3 + w^2 + 6*w - 2],\
[601, 601, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 7],\
[613, 613, 3*w^4 - 3*w^3 - 13*w^2 + 10*w + 10],\
[613, 613, w^4 - w^3 - 3*w^2 - w + 1],\
[625, 5, -w^4 + 4*w^3 + 5*w^2 - 14*w - 7],\
[641, 641, -2*w^4 - w^3 + 8*w^2 + 5*w - 4],\
[641, 641, -3*w^4 + w^3 + 13*w^2 - 4*w - 5],\
[643, 643, -w^4 - 2*w^3 + 6*w^2 + 5*w - 5],\
[643, 643, -2*w^4 - w^3 + 10*w^2 + 4*w - 10],\
[647, 647, -2*w^4 + 3*w^3 + 11*w^2 - 9*w - 10],\
[653, 653, -2*w^4 + w^3 + 6*w^2 + 1],\
[653, 653, -3*w^3 - 2*w^2 + 12*w + 7],\
[659, 659, w^4 - 3*w^3 - 7*w^2 + 10*w + 8],\
[661, 661, 3*w^3 + w^2 - 9*w - 2],\
[661, 661, 2*w^2 - 3*w - 5],\
[661, 661, 2*w^4 - w^3 - 7*w^2 + 3*w + 1],\
[683, 683, w^4 - 4*w^2 - 3*w + 2],\
[701, 701, -2*w^4 + 8*w^2 + w - 4],\
[733, 733, -2*w^4 + w^3 + 11*w^2 - 5*w - 13],\
[733, 733, -3*w^4 + w^3 + 13*w^2 - w - 7],\
[743, 743, -2*w^4 + 2*w^3 + 10*w^2 - 9*w - 8],\
[751, 751, 2*w^4 - 9*w^2 + w + 5],\
[769, 769, -w^4 - 2*w^3 + 3*w^2 + 7*w + 5],\
[787, 787, 2*w^4 - w^3 - 10*w^2 + 4*w + 4],\
[809, 809, 2*w^4 - 10*w^2 + w + 8],\
[809, 809, -w^4 + 4*w^3 + 5*w^2 - 14*w - 5],\
[821, 821, -w^4 + 2*w^3 + 6*w^2 - 9*w - 10],\
[823, 823, -w^3 + 2*w^2 + 4*w - 4],\
[829, 829, -w^4 + 2*w^3 + 8*w^2 - 7*w - 10],\
[839, 839, w^4 + w^3 - 4*w^2 - 4*w + 4],\
[841, 29, -3*w^3 - 2*w^2 + 10*w + 7],\
[857, 857, 3*w^4 - 13*w^2 + 7],\
[863, 863, w^4 + 2*w^3 - 2*w^2 - 8*w - 2],\
[911, 911, w^4 - 4*w^3 - 4*w^2 + 14*w + 4],\
[937, 937, w^4 - w^2 - 4],\
[941, 941, 2*w^4 - 2*w^3 - 7*w^2 + 4*w + 1],\
[947, 947, -2*w^3 - 3*w^2 + 6*w + 10],\
[947, 947, -3*w^4 + 3*w^3 + 14*w^2 - 10*w - 13],\
[953, 953, 2*w^4 - w^3 - 11*w^2 + 6*w + 10],\
[961, 31, -2*w^4 + w^3 + 8*w^2 - 5*w - 1],\
[977, 977, w^3 + 4*w^2 - 5*w - 11],\
[977, 977, -3*w^4 + w^3 + 14*w^2 - 2*w - 11],\
[991, 991, 2*w^4 - 9*w^2 - 3*w + 4],\
[991, 991, -2*w^4 + 8*w^2 + 4*w - 1],\
[997, 997, -w^4 + 4*w^3 + 6*w^2 - 14*w - 8]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^8 - 22*x^6 + 149*x^4 - 332*x^2 + 64
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -1/8*e^7 + 9/4*e^5 - 81/8*e^3 + 19/2*e, 1, 1/2*e^6 - 17/2*e^4 + 32*e^2 - 6, -1/2*e^4 + 11/2*e^2 - 7, 1/8*e^7 - 7/4*e^5 + 21/8*e^3 + 27/2*e, -1/8*e^7 + 7/4*e^5 - 13/8*e^3 - 41/2*e, -1/8*e^7 + 7/4*e^5 - 21/8*e^3 - 27/2*e, 2, -1/4*e^7 + 9/2*e^5 - 81/4*e^3 + 17*e, 1/8*e^7 - 7/4*e^5 + 13/8*e^3 + 45/2*e, e^2 - 2, -2*e^2 + 12, e^3 - 9*e, -1/2*e^6 + 17/2*e^4 - 34*e^2 + 16, 1/8*e^7 - 7/4*e^5 + 13/8*e^3 + 41/2*e, -e^3 + 7*e, -3/8*e^7 + 25/4*e^5 - 175/8*e^3 - 3/2*e, -2*e^2 + 10, -1/8*e^7 + 7/4*e^5 - 13/8*e^3 - 49/2*e, 1/4*e^7 - 9/2*e^5 + 81/4*e^3 - 19*e, -1/2*e^5 + 15/2*e^3 - 25*e, -e^6 + 17*e^4 - 66*e^2 + 28, 1/8*e^7 - 7/4*e^5 + 21/8*e^3 + 23/2*e, 3/8*e^7 - 25/4*e^5 + 183/8*e^3 - 7/2*e, 1/8*e^7 - 7/4*e^5 + 13/8*e^3 + 41/2*e, 1/8*e^7 - 7/4*e^5 + 13/8*e^3 + 41/2*e, 1/2*e^7 - 9*e^5 + 83/2*e^3 - 46*e, -1/2*e^6 + 17/2*e^4 - 30*e^2, -e^6 + 17*e^4 - 64*e^2 + 20, -1/8*e^7 + 7/4*e^5 - 29/8*e^3 - 7/2*e, 1/8*e^7 - 7/4*e^5 + 13/8*e^3 + 49/2*e, e^6 - 17*e^4 + 66*e^2 - 20, 1/2*e^6 - 15/2*e^4 + 21*e^2 + 6, -1/4*e^7 + 9/2*e^5 - 85/4*e^3 + 24*e, 3/2*e^6 - 51/2*e^4 + 96*e^2 - 24, -3/8*e^7 + 23/4*e^5 - 115/8*e^3 - 45/2*e, e^6 - 17*e^4 + 66*e^2 - 26, 3/8*e^7 - 25/4*e^5 + 175/8*e^3 - 1/2*e, 5/8*e^7 - 43/4*e^5 + 329/8*e^3 - 15/2*e, -2, -3*e^2 + 20, 2*e^6 - 34*e^4 + 128*e^2 - 32, -e^6 + 17*e^4 - 62*e^2, e^4 - 13*e^2 + 20, -2*e^6 + 34*e^4 - 126*e^2 + 18, e^6 - 16*e^4 + 51*e^2 + 18, 1/2*e^5 - 21/2*e^3 + 50*e, e^6 - 17*e^4 + 68*e^2 - 32, -e^4 + 13*e^2 - 30, 5/8*e^7 - 43/4*e^5 + 337/8*e^3 - 43/2*e, e^6 - 17*e^4 + 62*e^2 - 4, e^4 - 11*e^2 + 18, -1/4*e^7 + 9/2*e^5 - 85/4*e^3 + 28*e, 7/8*e^7 - 65/4*e^5 + 619/8*e^3 - 165/2*e, -e^4 + 12*e^2 - 28, 3/4*e^7 - 27/2*e^5 + 239/4*e^3 - 45*e, 1/4*e^7 - 7/2*e^5 + 13/4*e^3 + 37*e, -1/4*e^7 + 9/2*e^5 - 77/4*e^3 + 8*e, e^6 - 17*e^4 + 60*e^2 + 8, 5/8*e^7 - 43/4*e^5 + 345/8*e^3 - 37/2*e, -1/2*e^6 + 19/2*e^4 - 43*e^2 + 22, -e^4 + 14*e^2 - 36, -e^6 + 17*e^4 - 63*e^2 + 10, 1/4*e^7 - 11/2*e^5 + 145/4*e^3 - 70*e, -1/2*e^7 + 17/2*e^5 - 31*e^3 - 5*e, 5/8*e^7 - 39/4*e^5 + 209/8*e^3 + 71/2*e, -2*e^6 + 33*e^4 - 114*e^2 - 2, e^6 - 17*e^4 + 64*e^2 - 8, -e^6 + 17*e^4 - 66*e^2 + 2, e^4 - 7*e^2 - 16, 1/4*e^7 - 7/2*e^5 + 9/4*e^3 + 48*e, -3/8*e^7 + 29/4*e^5 - 303/8*e^3 + 99/2*e, -1/8*e^7 + 3/4*e^5 + 107/8*e^3 - 121/2*e, -3/2*e^6 + 49/2*e^4 - 85*e^2 + 12, -e^4 + 14*e^2 - 18, 5/8*e^7 - 45/4*e^5 + 405/8*e^3 - 79/2*e, 5/4*e^7 - 43/2*e^5 + 349/4*e^3 - 54*e, -1/4*e^7 + 4*e^5 - 43/4*e^3 - 20*e, e^6 - 18*e^4 + 73*e^2 - 22, -3/8*e^7 + 25/4*e^5 - 167/8*e^3 - 23/2*e, -1/2*e^6 + 17/2*e^4 - 30*e^2 - 14, -3/4*e^7 + 25/2*e^5 - 179/4*e^3 + 8*e, 1/4*e^7 - 9/2*e^5 + 81/4*e^3 - 15*e, -1/4*e^7 + 9/2*e^5 - 85/4*e^3 + 33*e, 5/2*e^6 - 83/2*e^4 + 151*e^2 - 22, 5/8*e^7 - 39/4*e^5 + 209/8*e^3 + 85/2*e, -3*e^6 + 51*e^4 - 192*e^2 + 46, -e^6 + 18*e^4 - 76*e^2 + 46, -e^6 + 17*e^4 - 62*e^2, -5/4*e^7 + 21*e^5 - 307/4*e^3 + 8*e, 1/2*e^7 - 9*e^5 + 77/2*e^3 - 23*e, -1/4*e^7 + 7/2*e^5 - 25/4*e^3 - 22*e, -3/4*e^7 + 29/2*e^5 - 311/4*e^3 + 107*e, 9/8*e^7 - 79/4*e^5 + 677/8*e^3 - 131/2*e, -1/4*e^7 + 5*e^5 - 107/4*e^3 + 33*e, -5/8*e^7 + 47/4*e^5 - 465/8*e^3 + 135/2*e, e^4 - 13*e^2 + 20, e^6 - 16*e^4 + 47*e^2 + 36, -3/8*e^7 + 25/4*e^5 - 159/8*e^3 - 53/2*e, -2*e^6 + 34*e^4 - 130*e^2 + 46, 2*e^6 - 32*e^4 + 104*e^2 + 20, -e^6 + 16*e^4 - 53*e^2 + 18, 5/8*e^7 - 45/4*e^5 + 437/8*e^3 - 163/2*e, 7/8*e^7 - 61/4*e^5 + 507/8*e^3 - 83/2*e, 5/2*e^6 - 83/2*e^4 + 153*e^2 - 46, 7/8*e^7 - 65/4*e^5 + 627/8*e^3 - 167/2*e, -3*e^6 + 50*e^4 - 181*e^2 + 12, -2*e^6 + 34*e^4 - 130*e^2 + 42, 3*e^6 - 50*e^4 + 180*e^2 - 8, -3*e^6 + 51*e^4 - 194*e^2 + 58, -7/8*e^7 + 57/4*e^5 - 355/8*e^3 - 61/2*e, 3/8*e^7 - 25/4*e^5 + 175/8*e^3 - 5/2*e, -7/8*e^7 + 63/4*e^5 - 567/8*e^3 + 145/2*e, -9/8*e^7 + 73/4*e^5 - 481/8*e^3 - 43/2*e, 3/8*e^7 - 33/4*e^5 + 439/8*e^3 - 223/2*e, -e^6 + 17*e^4 - 68*e^2 + 36, 1/2*e^7 - 10*e^5 + 119/2*e^3 - 119*e, e^6 - 17*e^4 + 64*e^2 - 40, -3/8*e^7 + 25/4*e^5 - 183/8*e^3 + 31/2*e, -2*e^6 + 34*e^4 - 124*e^2 + 6, -3*e^6 + 50*e^4 - 183*e^2 + 36, 5*e^2 - 18, -3/2*e^6 + 51/2*e^4 - 96*e^2 + 14, -3/8*e^7 + 21/4*e^5 - 47/8*e^3 - 97/2*e, -1/2*e^7 + 17/2*e^5 - 31*e^3 - e, -3/2*e^6 + 51/2*e^4 - 94*e^2 + 22, -2*e^6 + 34*e^4 - 126*e^2 - 2, 5/2*e^6 - 83/2*e^4 + 151*e^2 - 18, e^4 - 18*e^2 + 56, e^7 - 17*e^5 + 62*e^3 + 2*e, -e^6 + 16*e^4 - 52*e^2 + 2, -e^7 + 18*e^5 - 83*e^3 + 93*e, 3/8*e^7 - 25/4*e^5 + 183/8*e^3 - 19/2*e, -e^6 + 17*e^4 - 64*e^2 - 22, 1/2*e^6 - 15/2*e^4 + 23*e^2 + 6, -2*e^6 + 34*e^4 - 134*e^2 + 56, -2*e^6 + 33*e^4 - 119*e^2 + 38, -3/4*e^7 + 23/2*e^5 - 115/4*e^3 - 62*e, -1/8*e^7 + 11/4*e^5 - 133/8*e^3 + 9/2*e, 2*e^6 - 34*e^4 + 130*e^2 - 30, -3/2*e^7 + 27*e^5 - 239/2*e^3 + 100*e, -3*e^6 + 51*e^4 - 188*e^2 + 32, 9/8*e^7 - 83/4*e^5 + 789/8*e^3 - 219/2*e, 5/8*e^7 - 47/4*e^5 + 481/8*e^3 - 197/2*e, -2*e^6 + 34*e^4 - 130*e^2 + 36, -e^7 + 17*e^5 - 67*e^3 + 43*e, e^4 - 18*e^2 + 26, 2*e^6 - 34*e^4 + 126*e^2 - 22, 9/8*e^7 - 87/4*e^5 + 917/8*e^3 - 299/2*e, -3*e^6 + 50*e^4 - 179*e^2 + 10, -2*e^4 + 26*e^2 - 32, -1/2*e^6 + 21/2*e^4 - 56*e^2 + 48, 1/8*e^7 - 13/4*e^5 + 225/8*e^3 - 161/2*e]
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^4 + w^3 + 5*w^2 - 3*w - 4])] = -1

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