/*
  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, -2, -6, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([9, 9, w^3 - w^2 - 6*w + 3])

primes_array = [
[2, 2, w + 1],\
[3, 3, -w^3 + w^2 + 5*w - 2],\
[11, 11, -w^3 + 5*w + 1],\
[11, 11, -w + 2],\
[13, 13, -w^3 + w^2 + 4*w + 1],\
[17, 17, -w^3 + w^2 + 6*w - 1],\
[17, 17, -w^2 - w + 3],\
[23, 23, w^3 - 6*w],\
[27, 3, w^3 + w^2 - 5*w - 4],\
[41, 41, -w^3 + w^2 + 5*w],\
[41, 41, 2*w^3 - 11*w - 4],\
[53, 53, 2*w^3 - 2*w^2 - 11*w + 6],\
[59, 59, 2*w^3 - 11*w - 2],\
[67, 67, w^3 - 7*w - 1],\
[71, 71, 3*w^3 - 2*w^2 - 15*w + 1],\
[79, 79, -3*w^3 + w^2 + 16*w + 3],\
[89, 89, -4*w^3 + 2*w^2 + 22*w - 3],\
[101, 101, -2*w^3 + 13*w + 6],\
[101, 101, w^3 + w^2 - 6*w - 3],\
[101, 101, -3*w^3 + 2*w^2 + 17*w - 3],\
[101, 101, w^3 - w^2 - 8*w - 3],\
[107, 107, 2*w^3 - 3*w^2 - 6*w],\
[109, 109, -w^3 + w^2 + 4*w - 3],\
[113, 113, w^3 - 8*w - 4],\
[113, 113, 7*w^3 - 4*w^2 - 39*w + 7],\
[121, 11, w^3 - w^2 - 6*w - 1],\
[127, 127, -2*w^3 + w^2 + 9*w + 3],\
[127, 127, 2*w^3 - 13*w],\
[131, 131, 6*w^3 - 3*w^2 - 34*w + 6],\
[139, 139, w^2 - 2*w - 4],\
[149, 149, 3*w^3 - w^2 - 18*w - 3],\
[151, 151, -2*w^3 + 2*w^2 + 9*w - 4],\
[163, 163, 3*w^3 - 2*w^2 - 16*w + 2],\
[163, 163, 2*w^3 - 10*w + 1],\
[167, 167, 3*w^3 - 18*w - 4],\
[173, 173, -3*w^3 + w^2 + 18*w - 3],\
[193, 193, 4*w^3 - 3*w^2 - 23*w + 9],\
[197, 197, 3*w^3 - w^2 - 17*w + 2],\
[197, 197, -3*w^3 - 2*w^2 + 15*w + 9],\
[199, 199, 2*w^3 - w^2 - 8*w - 6],\
[223, 223, -6*w^3 + 3*w^2 + 35*w - 7],\
[227, 227, 2*w^3 - 12*w - 1],\
[227, 227, -w^3 + w^2 + 7*w - 8],\
[227, 227, 3*w^3 - w^2 - 18*w - 1],\
[227, 227, -4*w^3 - 2*w^2 + 21*w + 14],\
[229, 229, -w^3 + 2*w^2 + 3*w - 5],\
[233, 233, w^3 - 3*w - 3],\
[233, 233, -3*w^3 + 3*w^2 + 15*w - 8],\
[241, 241, 4*w^2 - 7*w - 8],\
[241, 241, w^3 + w^2 - 4*w - 5],\
[251, 251, 2*w^3 - w^2 - 13*w - 1],\
[251, 251, 2*w^3 - 9*w - 6],\
[257, 257, w^3 - 8*w],\
[263, 263, -w^3 + 2*w^2 + 5*w - 7],\
[269, 269, 3*w^3 - 3*w^2 - 12*w - 1],\
[271, 271, 2*w^3 - w^2 - 12*w - 2],\
[271, 271, 9*w^3 - 5*w^2 - 50*w + 9],\
[277, 277, w^3 - 2*w^2 - 6*w + 6],\
[277, 277, w^3 + w^2 - 7*w - 2],\
[281, 281, 4*w^3 - 2*w^2 - 24*w + 5],\
[281, 281, -7*w^3 + 4*w^2 + 40*w - 12],\
[289, 17, -5*w^3 + 5*w^2 + 26*w - 15],\
[293, 293, -w^3 + 2*w^2 + 4*w - 4],\
[293, 293, -9*w^3 + 5*w^2 + 51*w - 12],\
[307, 307, -2*w^3 + 2*w^2 + 12*w - 7],\
[311, 311, -4*w^3 + 24*w + 11],\
[313, 313, 2*w^3 - 9*w - 4],\
[317, 317, 2*w^3 - 4*w^2 - 2*w - 3],\
[317, 317, 2*w^3 - 14*w - 7],\
[331, 331, -3*w^3 + 19*w + 3],\
[349, 349, -w^3 - 2*w^2 + 8*w + 8],\
[349, 349, w^3 + w^2 - 5*w - 8],\
[361, 19, -w^3 + w^2 + 7*w - 6],\
[361, 19, w^3 + w^2 - 3*w - 4],\
[373, 373, 2*w^3 - w^2 - 11*w - 3],\
[379, 379, -4*w^3 + 25*w + 14],\
[383, 383, -3*w^3 + w^2 + 15*w - 2],\
[383, 383, 3*w^3 - 4*w^2 - 10*w],\
[401, 401, 4*w^3 - w^2 - 24*w - 4],\
[409, 409, w^2 + 5*w + 5],\
[419, 419, w^2 - 2*w - 6],\
[419, 419, -6*w^3 + 3*w^2 + 34*w - 8],\
[433, 433, 3*w^3 - 16*w - 6],\
[433, 433, 3*w^3 - w^2 - 16*w - 5],\
[443, 443, -w^3 + 2*w^2 + w + 3],\
[449, 449, 2*w^2 - 2*w - 5],\
[457, 457, -4*w^3 + 4*w^2 + 20*w - 9],\
[457, 457, w^2 + w - 7],\
[463, 463, w^3 - 7*w - 7],\
[463, 463, w^3 + 2*w^2 - 5*w - 9],\
[487, 487, w^3 + 2*w^2 - 5*w - 5],\
[487, 487, -w^3 + w^2 + 6*w - 7],\
[509, 509, -2*w^3 - 2*w^2 + 13*w + 10],\
[509, 509, 3*w^3 - 17*w - 3],\
[509, 509, -5*w^3 + 3*w^2 + 30*w - 11],\
[509, 509, -w^3 - 2*w^2 + 4*w + 6],\
[521, 521, -2*w^3 - w^2 + 9*w + 7],\
[541, 541, -2*w^3 - 2*w^2 + 12*w + 9],\
[541, 541, -2*w^3 + 12*w + 9],\
[547, 547, 2*w^3 + w^2 - 7*w - 3],\
[547, 547, 5*w^3 - 3*w^2 - 30*w + 3],\
[557, 557, w^3 + 2*w^2 - 5*w - 11],\
[557, 557, -w^3 + w^2 + 3*w - 4],\
[563, 563, 3*w^3 - w^2 - 15*w - 4],\
[569, 569, 2*w^2 - w - 8],\
[577, 577, 4*w^3 - 22*w - 7],\
[593, 593, -3*w^3 + w^2 + 17*w + 4],\
[607, 607, -w^3 - 3*w^2 + w + 4],\
[613, 613, 3*w^3 - 2*w^2 - 14*w],\
[613, 613, w^2 - 3*w - 5],\
[619, 619, -w^3 + 2*w^2 + 4*w - 10],\
[625, 5, -5],\
[641, 641, -3*w^3 - 3*w^2 + 12*w + 7],\
[643, 643, 2*w^3 + 2*w^2 - 14*w - 13],\
[643, 643, 3*w^3 - 19*w - 7],\
[647, 647, w^3 - 5*w - 7],\
[653, 653, -3*w^3 + 2*w^2 + 14*w - 2],\
[659, 659, -6*w^3 + 2*w^2 + 34*w - 1],\
[659, 659, -2*w^3 + w^2 + 17*w + 9],\
[683, 683, w^3 + 2*w^2 - 7*w - 7],\
[691, 691, 3*w^3 - 15*w - 7],\
[701, 701, w^3 + 2*w^2 - 6*w - 10],\
[701, 701, -w^3 + 3*w^2 + 6*w - 13],\
[701, 701, 5*w^3 - 2*w^2 - 30*w + 4],\
[701, 701, 4*w^3 - 2*w^2 - 21*w + 2],\
[727, 727, -2*w^3 - 3*w^2 + 9*w + 7],\
[743, 743, 3*w - 4],\
[751, 751, w^3 - 9*w - 3],\
[757, 757, w^3 - 5*w^2 + 4*w + 7],\
[757, 757, 4*w^3 - 5*w^2 - 11*w - 5],\
[761, 761, 4*w^3 - 5*w^2 - 20*w + 16],\
[769, 769, -3*w^3 + 16*w + 12],\
[773, 773, 2*w^3 + 2*w^2 - 11*w - 10],\
[773, 773, w^3 - 3*w - 5],\
[773, 773, -w^3 + 2*w^2 + 9*w + 3],\
[773, 773, 5*w^3 - w^2 - 27*w],\
[787, 787, -4*w^3 + 3*w^2 + 22*w - 4],\
[787, 787, -2*w^3 + w^2 + 14*w - 6],\
[797, 797, 2*w^3 - w^2 - 14*w + 2],\
[797, 797, -3*w^3 + 2*w^2 + 19*w - 7],\
[811, 811, 2*w^3 + w^2 - 14*w - 12],\
[811, 811, 2*w^3 + 2*w^2 - 8*w - 3],\
[829, 829, -2*w^3 + 14*w + 11],\
[857, 857, -3*w^3 + 3*w^2 + 14*w - 7],\
[859, 859, 3*w^3 - w^2 - 15*w],\
[859, 859, -11*w^3 + 7*w^2 + 60*w - 15],\
[863, 863, 2*w^2 - 2*w - 13],\
[863, 863, 2*w^3 + w^2 - 12*w - 4],\
[877, 877, -6*w^3 - 2*w^2 + 32*w + 19],\
[887, 887, 3*w^3 + w^2 - 14*w - 7],\
[887, 887, 2*w^3 - w^2 - 15*w - 5],\
[911, 911, 3*w^3 - 18*w - 2],\
[919, 919, w^3 - w^2 - 6*w - 5],\
[919, 919, -4*w^3 + w^2 + 24*w - 2],\
[941, 941, 3*w^3 - 3*w^2 - 13*w],\
[947, 947, 2*w^3 - 2*w^2 - 10*w - 1],\
[947, 947, -3*w^3 + 2*w^2 + 14*w + 6],\
[953, 953, -4*w^3 + 3*w^2 + 21*w - 3],\
[953, 953, 4*w^3 - w^2 - 23*w + 1],\
[961, 31, -2*w^3 + 3*w^2 + 12*w - 8],\
[961, 31, 3*w^2 - 4*w - 6],\
[967, 967, -5*w^3 - 3*w^2 + 23*w + 14],\
[971, 971, 5*w^3 - w^2 - 27*w - 8],\
[971, 971, -9*w^3 + 6*w^2 + 50*w - 12],\
[983, 983, -w^3 + w^2 + 10*w + 3],\
[997, 997, 2*w^3 + w^2 - 11*w - 3],\
[997, 997, w^3 + 2*w^2 - 6*w - 8],\
[997, 997, -5*w^3 + 4*w^2 + 29*w - 15]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 72*x^2 + 400
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [1/80*e^3 - 13/20*e, 0, -1/40*e^3 + 13/10*e, -1/40*e^3 + 23/10*e, -1/80*e^3 + 1/8*e^2 + 13/20*e - 9/2, 1/80*e^3 + 1/8*e^2 - 13/20*e - 9/2, -4, -1/40*e^3 + 23/10*e + 4, 1/4*e^2 - 9, -3/80*e^3 + 1/8*e^2 + 39/20*e - 9/2, 3/80*e^3 - 1/8*e^2 - 39/20*e + 9/2, 1/16*e^3 + 1/8*e^2 - 13/4*e - 9/2, 1/40*e^3 - 23/10*e + 4, 1/40*e^3 - 23/10*e - 4, 1/4*e^2 - 9, 3/40*e^3 - 39/10*e, -1/80*e^3 - 3/8*e^2 + 13/20*e + 27/2, 12, 6, -3/80*e^3 - 1/8*e^2 + 39/20*e + 9/2, -1/16*e^3 + 1/8*e^2 + 13/4*e - 9/2, 1/40*e^3 - 13/10*e, -7/80*e^3 - 1/8*e^2 + 91/20*e + 9/2, 1/20*e^3 - 23/5*e + 4, -11/80*e^3 - 1/8*e^2 + 143/20*e + 9/2, 7/80*e^3 - 3/8*e^2 - 91/20*e + 27/2, -1/40*e^3 + 1/2*e^2 + 13/10*e - 18, -1/40*e^3 + 23/10*e + 8, -3/40*e^3 + 69/10*e, -1/4*e^2 + 9, -3/80*e^3 - 3/8*e^2 + 39/20*e + 27/2, 1/40*e^3 - 1/4*e^2 - 13/10*e + 9, -3/40*e^3 + 1/4*e^2 + 39/10*e - 9, -1/40*e^3 + 23/10*e - 12, -1/20*e^3 + 23/5*e + 4, 1/20*e^3 - 23/5*e - 10, 1/20*e^3 - 23/5*e - 2, 1/20*e^3 - 23/5*e + 12, -1/20*e^3 + 23/5*e - 4, 1/20*e^3 - 23/5*e + 12, 1/20*e^3 - 23/5*e - 4, 1/10*e^3 - 46/5*e - 4, -1/20*e^3 + 23/5*e, -1/40*e^3 + 23/10*e - 12, 1/20*e^3 - 23/5*e + 16, -17/80*e^3 - 1/8*e^2 + 221/20*e + 9/2, -12, -1/80*e^3 - 7/8*e^2 + 13/20*e + 63/2, 7/80*e^3 + 1/8*e^2 - 91/20*e - 9/2, -1/10*e^3 + 46/5*e - 4, -1/20*e^3 + 23/5*e + 8, -1/40*e^3 + 23/10*e - 4, 22, -1/20*e^3 - 3/4*e^2 + 13/5*e + 27, 11/80*e^3 - 3/8*e^2 - 143/20*e + 27/2, -1/10*e^3 + 1/4*e^2 + 26/5*e - 9, -7/40*e^3 + 91/10*e, -1/80*e^3 - 3/8*e^2 + 13/20*e + 27/2, 1/20*e^3 - 23/5*e + 12, -1/20*e^3 + 23/5*e + 10, 1/10*e^3 - 46/5*e + 10, 3/80*e^3 - 3/8*e^2 - 39/20*e + 27/2, -11/80*e^3 - 3/8*e^2 + 143/20*e + 27/2, 1/20*e^3 - 23/5*e + 22, -1/10*e^3 + 46/5*e + 4, 1/20*e^3 + 1/4*e^2 - 13/5*e - 9, -11/80*e^3 + 1/8*e^2 + 143/20*e - 9/2, -1/10*e^3 + 46/5*e + 2, 1/20*e^3 - 23/5*e + 4, 3/40*e^3 - 69/10*e + 8, 9/80*e^3 - 3/8*e^2 - 117/20*e + 27/2, -1/10*e^3 + 46/5*e + 4, -28, -1/20*e^3 + 23/5*e + 4, 3/16*e^3 + 3/8*e^2 - 39/4*e - 27/2, 3/40*e^3 + 1/4*e^2 - 39/10*e - 9, 1/4*e^2 - 9, -1/40*e^3 + 13/10*e, -16, 3/16*e^3 + 3/8*e^2 - 39/4*e - 27/2, e^2 - 36, 1/20*e^3 - 23/5*e - 16, -7/80*e^3 + 1/8*e^2 + 91/20*e - 9/2, -3/80*e^3 - 9/8*e^2 + 39/20*e + 81/2, 3/40*e^3 - 69/10*e + 12, -17/80*e^3 + 3/8*e^2 + 221/20*e - 27/2, -13/80*e^3 + 3/8*e^2 + 169/20*e - 27/2, 1/10*e^3 - 46/5*e + 6, 1/20*e^3 - 23/5*e + 12, 3/40*e^3 + 3/4*e^2 - 39/10*e - 27, 3/40*e^3 - 1/4*e^2 - 39/10*e + 9, 1/40*e^3 - 23/10*e + 24, 1/16*e^3 + 1/8*e^2 - 13/4*e - 9/2, -1/20*e^3 + 23/5*e - 2, -1/20*e^3 + 23/5*e + 20, -23/80*e^3 + 1/8*e^2 + 299/20*e - 9/2, -1/20*e^3 + 23/5*e - 20, 1/80*e^3 + 1/8*e^2 - 13/20*e - 9/2, 26, 1/20*e^3 - 13/5*e, -36, -22, -17/80*e^3 + 1/8*e^2 + 221/20*e - 9/2, 1/20*e^3 + e^2 - 13/5*e - 36, -19/80*e^3 - 1/8*e^2 + 247/20*e + 9/2, -7/80*e^3 + 5/8*e^2 + 91/20*e - 45/2, -17/80*e^3 + 1/8*e^2 + 221/20*e - 9/2, -1/5*e^3 - 1/2*e^2 + 52/5*e + 18, -1/80*e^3 - 11/8*e^2 + 13/20*e + 99/2, -9/80*e^3 - 9/8*e^2 + 117/20*e + 81/2, -1/10*e^3 + 1/2*e^2 + 26/5*e - 18, -1/10*e^3 + 46/5*e + 8, 14, 3/4*e^2 - 27, -1/10*e^3 + 46/5*e + 20, -1/40*e^3 + 23/10*e - 24, -1/80*e^3 + 3/8*e^2 + 13/20*e - 27/2, 1/20*e^3 - 23/5*e + 8, 1/10*e^3 - 46/5*e + 20, 1/5*e^3 - 3/4*e^2 - 52/5*e + 27, 7/40*e^3 - 1/4*e^2 - 91/10*e + 9, 27/80*e^3 + 1/8*e^2 - 351/20*e - 9/2, 22, 3/20*e^3 - 69/5*e, -3/80*e^3 - 5/8*e^2 + 39/20*e + 45/2, -3/40*e^3 + 39/10*e, -1/40*e^3 + 23/10*e - 20, -1/20*e^3 + 23/5*e - 20, -9/80*e^3 - 5/8*e^2 + 117/20*e + 45/2, -2, -13/80*e^3 + 9/8*e^2 + 169/20*e - 81/2, -1/10*e^3 + 46/5*e - 8, 1/80*e^3 - 3/8*e^2 - 13/20*e + 27/2, -22, 9/80*e^3 + 5/8*e^2 - 117/20*e - 45/2, -1/10*e^3 + 46/5*e + 10, 1/40*e^3 + 1/2*e^2 - 13/10*e - 18, -3/40*e^3 + 69/10*e + 28, -1/10*e^3 + 46/5*e - 2, -1/10*e^3 + 46/5*e + 18, -9/40*e^3 + 117/10*e, -1/8*e^3 + 23/2*e - 24, -4, -11/80*e^3 - 9/8*e^2 + 143/20*e + 81/2, 17/40*e^3 + 1/4*e^2 - 221/10*e - 9, -3/40*e^3 + 39/10*e, -9/40*e^3 - 1/2*e^2 + 117/10*e + 18, -1/8*e^3 + 23/2*e - 32, -3/80*e^3 + 3/8*e^2 + 39/20*e - 27/2, 13/40*e^3 + 1/4*e^2 - 169/10*e - 9, -1/20*e^3 + 23/5*e - 20, 1/40*e^3 - 23/10*e + 4, 1/5*e^3 - 92/5*e + 8, 1/40*e^3 - 23/10*e + 52, 7/80*e^3 + 3/8*e^2 - 91/20*e - 27/2, 1/40*e^3 - 13/10*e, 3/20*e^3 + 5/4*e^2 - 39/5*e - 45, 11/80*e^3 - 3/8*e^2 - 143/20*e + 27/2, 1/5*e^3 - 92/5*e - 14, -11/80*e^3 - 7/8*e^2 + 143/20*e + 63/2, 13/80*e^3 - 5/8*e^2 - 169/20*e + 45/2, 3/40*e^3 + 3/2*e^2 - 39/10*e - 54, -1/4*e^3 + 13*e, -1/20*e^3 - e^2 + 13/5*e + 36, 1/20*e^3 - 23/5*e + 4, 3/20*e^3 - 69/5*e + 4, -1/80*e^3 - 9/8*e^2 + 13/20*e + 81/2, -1/20*e^3 + 23/5*e - 16]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([3, 3, -w^3 + w^2 + 5*w - 2])] = 1

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