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

NN = ZF.ideal([49,49,2*w^3 - 3*w^2 - 7*w])

primes_array = [
[5, 5, -w^3 + w^2 + 5*w],\
[7, 7, -w^3 + 2*w^2 + 3*w - 3],\
[7, 7, w^3 - 2*w^2 - 3*w],\
[13, 13, -w^2 + w + 3],\
[13, 13, -w^2 + w + 2],\
[16, 2, 2],\
[23, 23, -w^2 + 3*w + 1],\
[23, 23, -2*w^3 + 3*w^2 + 9*w - 2],\
[25, 5, w^3 - 2*w^2 - 2*w + 2],\
[49, 7, w^3 - w^2 - 6*w - 1],\
[53, 53, 2*w^3 - 2*w^2 - 8*w - 3],\
[53, 53, 2*w^3 - 2*w^2 - 8*w - 1],\
[67, 67, 2*w^3 - 4*w^2 - 7*w + 2],\
[67, 67, -2*w^3 + 4*w^2 + 6*w - 1],\
[81, 3, -3],\
[83, 83, 2*w^3 - 3*w^2 - 6*w - 1],\
[83, 83, 3*w^3 - 4*w^2 - 12*w + 1],\
[103, 103, 3*w^3 - 4*w^2 - 12*w - 1],\
[103, 103, 2*w^3 - 3*w^2 - 6*w + 1],\
[107, 107, w^3 - w^2 - 3*w - 3],\
[107, 107, 2*w^3 - 2*w^2 - 9*w],\
[109, 109, -3*w^3 + 5*w^2 + 10*w - 5],\
[109, 109, -2*w^3 + 3*w^2 + 9*w + 1],\
[109, 109, -2*w^3 + 2*w^2 + 7*w],\
[109, 109, w^2 - 3*w - 4],\
[121, 11, 3*w^3 - 4*w^2 - 12*w],\
[121, 11, 2*w^2 - 3*w - 6],\
[139, 139, w^3 - 7*w - 1],\
[139, 139, -w^3 + 2*w^2 + 5*w - 4],\
[139, 139, 3*w^3 - 3*w^2 - 14*w - 2],\
[139, 139, -3*w^3 + 5*w^2 + 13*w - 2],\
[149, 149, -w^2 + 4*w + 1],\
[149, 149, -3*w^3 + 4*w^2 + 14*w - 1],\
[149, 149, 2*w^2 - 3*w - 5],\
[149, 149, -w^3 + 3*w^2 + 3*w - 4],\
[167, 167, -w^3 - w^2 + 7*w + 4],\
[167, 167, -4*w^3 + 6*w^2 + 15*w - 4],\
[169, 13, w^3 - w^2 - 6*w + 4],\
[173, 173, w^2 - 3*w - 5],\
[173, 173, 2*w^3 - 2*w^2 - 11*w],\
[179, 179, 2*w^3 - 2*w^2 - 9*w + 1],\
[179, 179, w^3 - w^2 - 3*w - 4],\
[179, 179, w^2 - 5],\
[179, 179, w^3 - 6*w - 1],\
[197, 197, w^3 + w^2 - 7*w - 6],\
[197, 197, 2*w^3 - 2*w^2 - 9*w + 2],\
[223, 223, -w^2 + 7],\
[223, 223, 2*w^2 - w - 4],\
[227, 227, 2*w^3 - 2*w^2 - 11*w - 2],\
[227, 227, w^3 - w^2 - 7*w - 1],\
[233, 233, w^3 - w^2 - 7*w],\
[233, 233, -2*w^3 + 2*w^2 + 11*w + 1],\
[257, 257, -2*w^3 + 4*w^2 + 8*w - 1],\
[257, 257, 2*w^2 - 4*w - 7],\
[277, 277, -4*w^3 + 6*w^2 + 15*w - 3],\
[277, 277, -3*w^3 + 5*w^2 + 9*w],\
[281, 281, -2*w^3 + 5*w^2 + 3*w - 5],\
[281, 281, -2*w^3 + 5*w^2 + 5*w - 7],\
[281, 281, 2*w^3 - 5*w^2 - 5*w + 4],\
[281, 281, -4*w^3 + 7*w^2 + 15*w - 4],\
[283, 283, 3*w^3 - 4*w^2 - 13*w - 3],\
[283, 283, -w^3 + 2*w^2 + w - 4],\
[313, 313, -2*w^3 + 4*w^2 + 5*w - 5],\
[313, 313, 3*w^3 - 5*w^2 - 11*w],\
[347, 347, -w^3 + 2*w^2 + w - 5],\
[347, 347, 3*w^3 - 4*w^2 - 13*w - 4],\
[353, 353, 3*w^3 - 5*w^2 - 12*w + 1],\
[353, 353, -w^3 + 3*w^2 - 5],\
[361, 19, 3*w^3 - 3*w^2 - 13*w - 9],\
[361, 19, 2*w^3 - 2*w^2 - 7*w + 4],\
[373, 373, -3*w^3 + 4*w^2 + 11*w + 1],\
[373, 373, -3*w^3 + 4*w^2 + 11*w],\
[383, 383, 3*w^3 - 5*w^2 - 13*w + 1],\
[383, 383, 2*w^2 - 5*w - 6],\
[397, 397, 4*w^3 - 5*w^2 - 19*w - 1],\
[397, 397, 2*w^3 - 2*w^2 - 9*w - 7],\
[431, 431, 2*w^3 - 2*w^2 - 7*w - 5],\
[431, 431, 3*w^3 - 6*w^2 - 11*w + 3],\
[431, 431, 3*w^3 - 3*w^2 - 13*w],\
[431, 431, w^3 - 4*w^2 + w + 8],\
[457, 457, 4*w^3 - 7*w^2 - 13*w],\
[457, 457, -3*w^2 + 4*w + 5],\
[463, 463, -3*w^3 + 5*w^2 + 11*w + 1],\
[463, 463, 2*w^3 - 4*w^2 - 5*w + 6],\
[487, 487, -w^3 + 4*w^2 - 7],\
[487, 487, -2*w^3 + 5*w^2 + 6*w - 5],\
[499, 499, -6*w^3 + 8*w^2 + 25*w - 1],\
[499, 499, -w^3 + 4*w^2 - 5],\
[499, 499, -2*w^3 + 5*w^2 + 6*w - 7],\
[499, 499, 3*w^3 - 5*w^2 - 7*w],\
[509, 509, -4*w^3 + 8*w^2 + 15*w - 8],\
[509, 509, 3*w^3 - 4*w^2 - 10*w - 2],\
[509, 509, -3*w^3 + 3*w^2 + 11*w + 6],\
[509, 509, 4*w^3 - 5*w^2 - 16*w],\
[521, 521, -4*w^3 + 5*w^2 + 15*w - 2],\
[521, 521, w^3 - w^2 - 8*w + 1],\
[521, 521, -4*w^3 + 7*w^2 + 12*w - 6],\
[521, 521, 3*w^3 - 3*w^2 - 16*w - 1],\
[523, 523, 3*w^3 - w^2 - 17*w - 8],\
[523, 523, -5*w^3 + 5*w^2 + 23*w + 4],\
[529, 23, 3*w^3 - 6*w^2 - 10*w + 2],\
[547, 547, -2*w^3 + 5*w^2 + 6*w - 6],\
[547, 547, -w^3 + 4*w^2 - 6],\
[557, 557, -3*w^3 + 6*w^2 + 9*w - 10],\
[557, 557, 3*w^3 - 6*w^2 - 9*w - 1],\
[571, 571, 4*w^3 - 4*w^2 - 17*w - 6],\
[571, 571, -2*w^3 + 6*w^2 + 3*w - 10],\
[571, 571, 4*w^3 - 5*w^2 - 18*w + 3],\
[571, 571, -3*w^3 + 7*w^2 + 9*w - 5],\
[587, 587, 3*w^3 - 3*w^2 - 14*w],\
[587, 587, w^3 - w^2 - 2*w - 4],\
[593, 593, -w^3 + 3*w^2 + 2*w - 10],\
[593, 593, -5*w^3 + 7*w^2 + 19*w - 4],\
[613, 613, 3*w^3 - 5*w^2 - 12*w - 4],\
[613, 613, -w^3 + 3*w^2 - 10],\
[631, 631, 4*w^3 - 8*w^2 - 13*w + 4],\
[631, 631, 3*w^3 - 4*w^2 - 15*w + 2],\
[631, 631, w^3 - 9*w - 1],\
[631, 631, 3*w^3 - 7*w^2 - 7*w + 9],\
[643, 643, 3*w^3 - 3*w^2 - 16*w - 2],\
[643, 643, w^3 - w^2 - 8*w],\
[647, 647, -w^3 + 4*w^2 + 2*w - 4],\
[647, 647, -3*w^2 + 4*w + 10],\
[673, 673, w^3 + 2*w^2 - 9*w - 8],\
[673, 673, -w^3 + 4*w^2 + 3*w - 7],\
[683, 683, 3*w^3 - 3*w^2 - 11*w - 2],\
[683, 683, 4*w^3 - 4*w^2 - 17*w - 5],\
[709, 709, -5*w^3 + 7*w^2 + 21*w + 1],\
[709, 709, -3*w^3 + 2*w^2 + 16*w + 4],\
[709, 709, 4*w^3 - 4*w^2 - 19*w - 2],\
[709, 709, -2*w^3 + 4*w^2 + 3*w - 4],\
[787, 787, 2*w^2 - w - 9],\
[787, 787, w^3 + w^2 - 7*w - 2],\
[811, 811, 4*w^3 - 7*w^2 - 11*w],\
[811, 811, 3*w^3 - 4*w^2 - 9*w - 1],\
[811, 811, -6*w^3 + 9*w^2 + 23*w - 5],\
[811, 811, 5*w^3 - 6*w^2 - 21*w - 2],\
[821, 821, -3*w^3 + 6*w^2 + 11*w - 2],\
[821, 821, 2*w^3 - 13*w - 5],\
[821, 821, -w^3 + 4*w^2 - w - 9],\
[821, 821, -w^3 + 3*w^2 + 5*w - 6],\
[841, 29, 2*w^3 - 2*w^2 - 12*w - 1],\
[857, 857, w^3 - 4*w^2 + 3*w + 6],\
[857, 857, -5*w^3 + 8*w^2 + 21*w - 3],\
[863, 863, -w^3 + 4*w^2 + w - 7],\
[863, 863, -w^3 + 4*w^2 + w - 6],\
[877, 877, -w^3 - w^2 + 8*w + 2],\
[877, 877, -w^3 + 3*w^2 + 4*w - 8],\
[883, 883, 4*w^3 - 7*w^2 - 14*w + 1],\
[883, 883, 3*w^3 - 6*w^2 - 8*w + 7],\
[929, 929, 3*w^3 - 5*w^2 - 12*w - 3],\
[929, 929, -w^3 + 3*w^2 - 9],\
[929, 929, 3*w - 5],\
[929, 929, 3*w^3 - 3*w^2 - 15*w + 2],\
[937, 937, w^3 - 2*w^2 - 6*w + 6],\
[937, 937, 2*w^3 - w^2 - 12*w],\
[941, 941, -w^3 + 4*w^2 + 2*w - 7],\
[941, 941, -4*w^3 + 6*w^2 + 13*w - 3],\
[941, 941, -5*w^3 + 7*w^2 + 19*w + 2],\
[941, 941, 3*w^2 - 4*w - 7],\
[953, 953, 3*w^3 - 5*w^2 - 12*w - 2],\
[953, 953, -w^3 + 3*w^2 - 8],\
[961, 31, 4*w^3 - 6*w^2 - 13*w - 2],\
[961, 31, 5*w^3 - 7*w^2 - 19*w + 3],\
[977, 977, -4*w^3 + 7*w^2 + 19*w - 6],\
[977, 977, -3*w^3 + 7*w^2 + 10*w - 6]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^6 - 34*x^4 + 240*x^2 - 32
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [-1/40*e^4 + 9/20*e^2 + 6/5, 0, e, 1/20*e^4 - 7/5*e^2 + 33/5, 1/40*e^5 - 7/10*e^3 + 33/10*e, 1/40*e^4 - 19/20*e^2 + 24/5, -1/20*e^4 + 9/10*e^2 + 2/5, -1/10*e^5 + 33/10*e^3 - 106/5*e, -1/16*e^5 + 17/8*e^3 - 15*e, 13/80*e^5 - 207/40*e^3 + 327/10*e, 1/10*e^5 - 71/20*e^3 + 277/10*e, 1/80*e^5 - 9/40*e^3 + 2/5*e, -1/20*e^4 + 9/10*e^2 + 12/5, 1/20*e^5 - 7/5*e^3 + 23/5*e, -1/40*e^5 + 7/10*e^3 - 53/10*e, -3/20*e^4 + 47/10*e^2 - 84/5, 1/8*e^5 - 17/4*e^3 + 33*e, 1/20*e^4 - 9/10*e^2 + 8/5, -3/20*e^5 + 26/5*e^3 - 179/5*e, -3/20*e^5 + 26/5*e^3 - 194/5*e, 3/20*e^5 - 47/10*e^3 + 134/5*e, -7/80*e^5 + 123/40*e^3 - 119/5*e, -1/8*e^4 + 13/4*e^2 - 6, 1/8*e^4 - 13/4*e^2 + 16, -1/4*e^3 + 7/2*e, -19/80*e^5 + 311/40*e^3 - 248/5*e, 1/16*e^5 - 17/8*e^3 + 16*e, 1/20*e^4 - 9/10*e^2 + 48/5, -e^2 + 8, 1/5*e^5 - 71/10*e^3 + 262/5*e, -e^2 + 22, -1/40*e^4 + 9/20*e^2 + 26/5, 3/10*e^5 - 203/20*e^3 + 671/10*e, 17/80*e^5 - 273/40*e^3 + 199/5*e, -1/4*e^4 + 7*e^2 - 23, -1/10*e^4 + 14/5*e^2 - 66/5, -1/4*e^5 + 8*e^3 - 48*e, 9/40*e^4 - 101/20*e^2 + 56/5, -17/80*e^5 + 283/40*e^3 - 503/10*e, 7/80*e^5 - 113/40*e^3 + 153/10*e, -3/10*e^5 + 47/5*e^3 - 283/5*e, 3/20*e^4 - 37/10*e^2 + 104/5, 1/5*e^5 - 71/10*e^3 + 252/5*e, -1/10*e^4 + 14/5*e^2 - 6/5, -17/80*e^5 + 303/40*e^3 - 583/10*e, -1/5*e^4 + 23/5*e^2 - 12/5, 1/10*e^4 - 14/5*e^2 + 66/5, -1/10*e^4 + 19/5*e^2 - 36/5, 9/40*e^5 - 141/20*e^3 + 216/5*e, 1/20*e^4 - 19/10*e^2 + 38/5, -1/2*e^2 - 7, -9/40*e^4 + 121/20*e^2 - 106/5, -1/40*e^4 + 9/20*e^2 - 64/5, -1/4*e^4 + 7*e^2 - 27, 29/80*e^5 - 491/40*e^3 + 891/10*e, 1/10*e^4 - 4/5*e^2 - 94/5, 3/20*e^4 - 26/5*e^2 + 79/5, -1/10*e^4 + 13/10*e^2 + 39/5, 5/16*e^5 - 85/8*e^3 + 74*e, -7/40*e^5 + 59/10*e^3 - 431/10*e, 7/40*e^5 - 113/20*e^3 + 148/5*e, 3/40*e^5 - 47/20*e^3 + 97/5*e, 19/80*e^5 - 311/40*e^3 + 258/5*e, -1/2*e^2 + 3, 1/10*e^4 - 4/5*e^2 - 94/5, 1/4*e^4 - 15/2*e^2 + 14, -23/80*e^5 + 397/40*e^3 - 717/10*e, 1/10*e^4 - 4/5*e^2 - 64/5, 23/80*e^5 - 397/40*e^3 + 737/10*e, 3/10*e^4 - 47/5*e^2 + 158/5, -1/10*e^4 + 13/10*e^2 + 9/5, 7/80*e^5 - 123/40*e^3 + 124/5*e, -e^3 + 21*e, -1/4*e^5 + 9*e^3 - 71*e, -3/8*e^5 + 13*e^3 - 183/2*e, -3/40*e^5 + 31/10*e^3 - 329/10*e, 1/5*e^5 - 38/5*e^3 + 307/5*e, -7/20*e^4 + 83/10*e^2 - 76/5, -11/40*e^5 + 179/20*e^3 - 259/5*e, -1/10*e^4 + 9/5*e^2 - 106/5, 1/16*e^5 - 23/8*e^3 + 51/2*e, -3/80*e^5 + 17/40*e^3 + 83/10*e, 1/20*e^4 + 11/10*e^2 - 162/5, 1/20*e^4 + 11/10*e^2 - 92/5, 1/4*e^4 - 15/2*e^2 + 42, 1/10*e^4 - 4/5*e^2 - 164/5, 13/40*e^5 - 227/20*e^3 + 407/5*e, 7/20*e^4 - 83/10*e^2 + 186/5, 3/10*e^4 - 42/5*e^2 + 108/5, -3/8*e^5 + 47/4*e^3 - 67*e, -11/40*e^4 + 119/20*e^2 - 64/5, 9/40*e^5 - 39/5*e^3 + 617/10*e, 1/4*e^4 - 7*e^2 + 11, 13/40*e^4 - 137/20*e^2 + 32/5, 5/2*e^2 - 23, -3/80*e^5 + 67/40*e^3 - 56/5*e, 5/2*e^2 - 37, 5/2*e^2 - 23, -1/5*e^4 + 28/5*e^2 - 32/5, -3/20*e^4 + 57/10*e^2 - 94/5, -13/80*e^5 + 227/40*e^3 - 447/10*e, -3/20*e^5 + 21/5*e^3 - 89/5*e, 3/8*e^5 - 49/4*e^3 + 76*e, 3/80*e^5 - 37/40*e^3 + 37/10*e, -11/80*e^5 + 229/40*e^3 - 509/10*e, 1/4*e^4 - 9/2*e^2 + 4, -2/5*e^4 + 41/5*e^2 - 14/5, -11/40*e^5 + 179/20*e^3 - 259/5*e, -13/40*e^5 + 247/20*e^3 - 512/5*e, 1/20*e^4 + 1/10*e^2 - 62/5, -3/10*e^4 + 32/5*e^2 + 22/5, 3/40*e^4 - 47/20*e^2 - 138/5, -1/10*e^4 + 43/10*e^2 - 131/5, -9/80*e^5 + 161/40*e^3 - 128/5*e, 3/20*e^5 - 119/20*e^3 + 493/10*e, 2*e, 3/10*e^5 - 99/10*e^3 + 338/5*e, 7/20*e^4 - 103/10*e^2 + 226/5, 1/20*e^4 - 39/10*e^2 + 108/5, 1/10*e^4 - 14/5*e^2 + 66/5, 1/20*e^4 - 29/10*e^2 - 12/5, 11/20*e^5 - 92/5*e^3 + 608/5*e, 2*e^2 - 40, 1/40*e^4 - 9/20*e^2 + 34/5, 11/80*e^5 - 199/40*e^3 + 172/5*e, -1/2*e^5 + 17*e^3 - 113*e, 1/10*e^4 - 9/5*e^2 + 156/5, 1/10*e^5 - 51/20*e^3 + 37/10*e, -1/5*e^4 + 21/10*e^2 + 233/5, 3/4*e^3 - 41/2*e, -1/40*e^4 + 49/20*e^2 - 194/5, 1/10*e^4 + 1/5*e^2 - 174/5, 1/5*e^5 - 61/10*e^3 + 137/5*e, 1/5*e^4 - 23/5*e^2 - 48/5, -9/40*e^5 + 151/20*e^3 - 286/5*e, -1/20*e^4 + 19/10*e^2 - 228/5, 1/5*e^4 - 23/5*e^2 - 118/5, 61/80*e^5 - 969/40*e^3 + 747/5*e, 1/10*e^4 - 13/10*e^2 + 91/5, -3/40*e^5 + 21/10*e^3 - 69/10*e, -3/10*e^4 + 49/10*e^2 + 167/5, -29/80*e^5 + 451/40*e^3 - 671/10*e, -13/40*e^5 + 58/5*e^3 - 869/10*e, -27/80*e^5 + 463/40*e^3 - 419/5*e, 9/40*e^5 - 131/20*e^3 + 156/5*e, -7/40*e^5 + 133/20*e^3 - 293/5*e, -1/40*e^4 + 9/20*e^2 + 26/5, 1/2*e^4 - 25/2*e^2 + 29, -3/8*e^5 + 51/4*e^3 - 90*e, -1/10*e^4 + 9/5*e^2 - 156/5, 1/5*e^4 - 28/5*e^2 + 92/5, 3/10*e^4 - 27/5*e^2 - 102/5, -17/80*e^5 + 243/40*e^3 - 283/10*e, 1/5*e^4 - 28/5*e^2 + 22/5, -41/80*e^5 + 699/40*e^3 - 1259/10*e, 9/16*e^5 - 151/8*e^3 + 279/2*e, 1/10*e^4 - 33/10*e^2 - 69/5, -19/40*e^4 + 191/20*e^2 + 14/5, -39/80*e^5 + 611/40*e^3 - 428/5*e, -9/40*e^4 + 61/20*e^2 + 54/5, 1/8*e^4 - 9/4*e^2 - 16, -1/8*e^5 + 4*e^3 - 65/2*e, -5/16*e^5 + 89/8*e^3 - 87*e, -3/40*e^4 + 67/20*e^2 - 172/5, 33/80*e^5 - 537/40*e^3 + 381/5*e, 13/20*e^4 - 81/5*e^2 + 199/5]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([7,7,-w^3 + 2*w^2 + 3*w - 3])] = 1

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