/*
  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, -9, 2, 11, -5, -2, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([71, 71, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 1])

primes_array = [
[13, 13, 3*w^5 - 2*w^4 - 17*w^3 + 10*w^2 + 16*w - 3],\
[13, 13, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 3],\
[13, 13, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 13*w],\
[13, 13, -w^2 + 3],\
[41, 41, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 1],\
[41, 41, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 4],\
[41, 41, w^5 - 6*w^3 + w^2 + 7*w - 2],\
[41, 41, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 20*w - 1],\
[43, 43, 3*w^5 - 2*w^4 - 17*w^3 + 11*w^2 + 16*w - 6],\
[43, 43, -2*w^5 + w^4 + 12*w^3 - 6*w^2 - 14*w + 5],\
[49, 7, 2*w^5 - w^4 - 12*w^3 + 5*w^2 + 13*w - 4],\
[64, 2, -2],\
[71, 71, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 1],\
[71, 71, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 15*w],\
[83, 83, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 26*w - 1],\
[83, 83, -4*w^5 + 2*w^4 + 23*w^3 - 9*w^2 - 23*w],\
[97, 97, -w^5 + 7*w^3 + w^2 - 11*w - 3],\
[97, 97, -2*w^5 + w^4 + 11*w^3 - 6*w^2 - 9*w + 3],\
[113, 113, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 27*w + 2],\
[113, 113, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 6],\
[113, 113, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 12*w + 3],\
[113, 113, 3*w^5 - w^4 - 18*w^3 + 4*w^2 + 20*w],\
[127, 127, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w],\
[127, 127, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 2],\
[127, 127, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 5],\
[127, 127, -8*w^5 + 5*w^4 + 46*w^3 - 23*w^2 - 45*w + 4],\
[139, 139, w^3 + w^2 - 4*w - 1],\
[139, 139, 5*w^5 - 2*w^4 - 29*w^3 + 9*w^2 + 29*w + 1],\
[139, 139, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 5],\
[139, 139, w^5 - 6*w^3 + 5*w + 2],\
[167, 167, w^4 - 4*w^2 + 1],\
[167, 167, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 21*w - 1],\
[169, 13, -3*w^5 + w^4 + 18*w^3 - 5*w^2 - 20*w],\
[181, 181, -w^4 + w^3 + 5*w^2 - 3*w - 3],\
[181, 181, -5*w^5 + 3*w^4 + 29*w^3 - 15*w^2 - 30*w + 6],\
[181, 181, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 4],\
[181, 181, w^3 + w^2 - 4*w - 2],\
[197, 197, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 2],\
[197, 197, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 26*w + 4],\
[197, 197, -2*w^5 + 2*w^4 + 12*w^3 - 11*w^2 - 12*w + 5],\
[197, 197, 2*w^5 - w^4 - 12*w^3 + 6*w^2 + 13*w - 7],\
[197, 197, w^5 - 6*w^3 - w^2 + 7*w + 1],\
[197, 197, -4*w^5 + 3*w^4 + 24*w^3 - 14*w^2 - 25*w + 2],\
[211, 211, 2*w^5 - w^4 - 13*w^3 + 5*w^2 + 18*w - 1],\
[211, 211, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 4],\
[223, 223, 4*w^5 - w^4 - 23*w^3 + 5*w^2 + 24*w - 1],\
[223, 223, -3*w^5 + w^4 + 17*w^3 - 4*w^2 - 15*w - 4],\
[223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 15*w^2 + 11*w - 5],\
[223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 5],\
[239, 239, -w^5 + w^4 + 6*w^3 - 6*w^2 - 7*w + 4],\
[239, 239, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 12*w + 2],\
[251, 251, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 4],\
[251, 251, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 28*w - 1],\
[251, 251, 6*w^5 - 4*w^4 - 35*w^3 + 19*w^2 + 37*w - 7],\
[251, 251, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 8],\
[281, 281, -6*w^5 + 4*w^4 + 35*w^3 - 18*w^2 - 35*w + 3],\
[281, 281, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 16*w - 10],\
[281, 281, -3*w^5 + w^4 + 17*w^3 - 5*w^2 - 18*w + 3],\
[281, 281, 5*w^5 - 3*w^4 - 30*w^3 + 14*w^2 + 33*w - 5],\
[293, 293, -w^5 + w^4 + 6*w^3 - 5*w^2 - 8*w + 3],\
[307, 307, 5*w^5 - 3*w^4 - 30*w^3 + 13*w^2 + 32*w],\
[307, 307, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 20*w + 5],\
[307, 307, 3*w^5 - 2*w^4 - 17*w^3 + 8*w^2 + 15*w],\
[307, 307, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 11*w - 1],\
[307, 307, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 10],\
[307, 307, -4*w^5 + 2*w^4 + 23*w^3 - 8*w^2 - 22*w - 3],\
[337, 337, -w^5 + w^4 + 5*w^3 - 4*w^2 - 3*w + 3],\
[337, 337, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 29*w],\
[337, 337, 5*w^5 - 3*w^4 - 28*w^3 + 15*w^2 + 27*w - 5],\
[337, 337, -4*w^5 + 4*w^4 + 22*w^3 - 20*w^2 - 19*w + 10],\
[349, 349, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 45*w + 9],\
[349, 349, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3],\
[419, 419, 5*w^5 - 3*w^4 - 29*w^3 + 13*w^2 + 28*w - 1],\
[419, 419, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 35*w + 10],\
[421, 421, -6*w^5 + 3*w^4 + 34*w^3 - 15*w^2 - 32*w + 3],\
[421, 421, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 28*w],\
[421, 421, w^5 - 5*w^3 - w^2 + 2*w + 3],\
[421, 421, -6*w^5 + 3*w^4 + 35*w^3 - 14*w^2 - 37*w + 2],\
[433, 433, -4*w^5 + 4*w^4 + 22*w^3 - 19*w^2 - 19*w + 6],\
[433, 433, 7*w^5 - 4*w^4 - 40*w^3 + 19*w^2 + 40*w - 6],\
[449, 449, 2*w^4 - w^3 - 10*w^2 + 4*w + 6],\
[449, 449, -4*w^5 + 4*w^4 + 23*w^3 - 20*w^2 - 21*w + 10],\
[461, 461, -6*w^5 + 3*w^4 + 36*w^3 - 15*w^2 - 40*w + 4],\
[461, 461, -7*w^5 + 5*w^4 + 40*w^3 - 23*w^2 - 39*w + 6],\
[461, 461, 4*w^5 - 2*w^4 - 22*w^3 + 9*w^2 + 19*w + 1],\
[461, 461, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 32*w - 1],\
[463, 463, 4*w^5 - 2*w^4 - 23*w^3 + 10*w^2 + 23*w],\
[463, 463, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 34*w + 7],\
[491, 491, 7*w^5 - 5*w^4 - 41*w^3 + 24*w^2 + 41*w - 8],\
[491, 491, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 15*w - 2],\
[491, 491, -w^5 + 6*w^3 + w^2 - 7*w],\
[491, 491, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 3],\
[491, 491, 5*w^5 - 3*w^4 - 29*w^3 + 16*w^2 + 30*w - 9],\
[491, 491, -3*w^5 + 2*w^4 + 18*w^3 - 8*w^2 - 20*w - 2],\
[503, 503, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 7],\
[503, 503, 6*w^5 - 3*w^4 - 35*w^3 + 14*w^2 + 37*w - 3],\
[547, 547, 2*w^5 - 12*w^3 - w^2 + 15*w + 3],\
[547, 547, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 6*w + 3],\
[547, 547, 7*w^5 - 4*w^4 - 41*w^3 + 19*w^2 + 41*w - 3],\
[547, 547, 2*w^5 - w^4 - 13*w^3 + 6*w^2 + 17*w - 3],\
[547, 547, -4*w^5 + 3*w^4 + 23*w^3 - 13*w^2 - 21*w],\
[547, 547, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 7],\
[587, 587, 2*w^4 - w^3 - 10*w^2 + 4*w + 7],\
[587, 587, 5*w^5 - 2*w^4 - 29*w^3 + 10*w^2 + 31*w - 2],\
[601, 601, w^3 + 2*w^2 - 4*w - 4],\
[601, 601, w^5 - 7*w^3 - 2*w^2 + 11*w + 5],\
[617, 617, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 9],\
[617, 617, -2*w^5 + 13*w^3 - 16*w],\
[643, 643, -2*w^5 + 2*w^4 + 12*w^3 - 8*w^2 - 13*w],\
[643, 643, -3*w^5 + w^4 + 18*w^3 - 3*w^2 - 21*w - 3],\
[659, 659, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 33*w + 2],\
[659, 659, w^5 - 6*w^3 + 2*w^2 + 7*w - 4],\
[673, 673, -2*w^5 + 2*w^4 + 10*w^3 - 11*w^2 - 5*w + 6],\
[673, 673, 6*w^5 - 4*w^4 - 34*w^3 + 20*w^2 + 33*w - 9],\
[673, 673, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 16*w + 5],\
[673, 673, 7*w^5 - 4*w^4 - 40*w^3 + 20*w^2 + 38*w - 6],\
[673, 673, -3*w^5 + 2*w^4 + 16*w^3 - 11*w^2 - 12*w + 6],\
[673, 673, w^5 + w^4 - 6*w^3 - 6*w^2 + 8*w + 5],\
[701, 701, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 9],\
[701, 701, 4*w^5 - 3*w^4 - 22*w^3 + 15*w^2 + 20*w - 6],\
[727, 727, -2*w^5 + 2*w^4 + 11*w^3 - 8*w^2 - 9*w - 2],\
[727, 727, -3*w^5 + 3*w^4 + 18*w^3 - 15*w^2 - 20*w + 9],\
[729, 3, -3],\
[743, 743, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 22*w],\
[743, 743, -5*w^5 + 2*w^4 + 29*w^3 - 9*w^2 - 31*w - 1],\
[743, 743, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 31*w + 6],\
[743, 743, -3*w^5 + 3*w^4 + 17*w^3 - 14*w^2 - 17*w + 3],\
[757, 757, -6*w^5 + 3*w^4 + 35*w^3 - 13*w^2 - 37*w + 2],\
[757, 757, 3*w^5 - 3*w^4 - 17*w^3 + 13*w^2 + 16*w - 1],\
[757, 757, -8*w^5 + 5*w^4 + 47*w^3 - 25*w^2 - 48*w + 10],\
[757, 757, 4*w^5 - 2*w^4 - 25*w^3 + 10*w^2 + 29*w - 4],\
[797, 797, -6*w^5 + 4*w^4 + 35*w^3 - 19*w^2 - 34*w + 3],\
[797, 797, -w^5 + w^4 + 7*w^3 - 6*w^2 - 10*w + 3],\
[811, 811, 5*w^5 - 4*w^4 - 28*w^3 + 18*w^2 + 25*w - 4],\
[811, 811, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 17*w + 7],\
[827, 827, w^5 - w^4 - 5*w^3 + 5*w^2 + w - 4],\
[827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 3],\
[827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 2],\
[827, 827, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 4],\
[839, 839, -2*w^5 + 2*w^4 + 10*w^3 - 10*w^2 - 6*w + 7],\
[839, 839, -6*w^5 + 4*w^4 + 34*w^3 - 20*w^2 - 32*w + 5],\
[841, 29, -4*w^5 + 3*w^4 + 24*w^3 - 15*w^2 - 25*w + 7],\
[841, 29, 5*w^5 - 3*w^4 - 30*w^3 + 15*w^2 + 32*w - 6],\
[841, 29, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 33*w - 2],\
[853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 29*w + 9],\
[853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 30*w + 7],\
[881, 881, -3*w^5 + 3*w^4 + 16*w^3 - 14*w^2 - 13*w + 3],\
[881, 881, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 25*w + 2],\
[881, 881, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 9],\
[881, 881, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 34*w + 2],\
[883, 883, 3*w^5 - w^4 - 17*w^3 + 6*w^2 + 16*w - 5],\
[883, 883, 7*w^5 - 5*w^4 - 40*w^3 + 24*w^2 + 37*w - 8],\
[911, 911, w^5 - 8*w^3 + 14*w],\
[911, 911, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 17*w - 1],\
[911, 911, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 44*w + 9],\
[911, 911, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 11*w],\
[937, 937, 5*w^5 - 2*w^4 - 30*w^3 + 9*w^2 + 32*w - 1],\
[937, 937, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 15*w - 4],\
[967, 967, -w^5 + 7*w^3 - 9*w + 2],\
[967, 967, 6*w^5 - 4*w^4 - 35*w^3 + 20*w^2 + 36*w - 11]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 + 5*x^2 - 16*x - 82
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, -2, e^2 + e - 18, e^2 - 20, -5*e^2 - 3*e + 94, -8*e^2 - 5*e + 140, 4*e^2 + e - 72, -4*e^2 - e + 70, e^2 - 14, 3*e^2 + 2*e - 56, 3*e^2 - 62, 2*e^2 + 2*e - 31, 1, -5*e^2 - 4*e + 84, 10*e^2 + 4*e - 186, -2, e^2 - 22, -9*e^2 - 2*e + 166, -10*e^2 - 6*e + 170, 15*e^2 + 6*e - 276, -13*e^2 - 7*e + 228, 13*e^2 + 8*e - 236, -12*e^2 - 6*e + 224, -2*e^2 + 44, -2*e^2 + 44, -2*e - 6, 5*e^2 + 4*e - 88, -3*e^2 - 2*e + 46, -13*e^2 - 6*e + 230, -2*e - 4, 9*e^2 + 4*e - 174, 12*e^2 + 5*e - 222, -8*e^2 - 6*e + 132, -6*e^2 - 6*e + 108, 8*e^2 + 6*e - 152, 6*e^2 + 4*e - 116, 4*e^2 + 4*e - 70, 3*e^2 - e - 68, 23*e^2 + 10*e - 426, -14*e^2 - 9*e + 240, -12*e^2 - 8*e + 210, 11*e^2 + 5*e - 202, -e^2 - 3*e + 10, 11*e^2 + 4*e - 202, -11*e^2 - 6*e + 208, 4*e^2 - 84, -e^2 + 2*e + 16, 12*e^2 + 8*e - 214, -6*e^2 - 2*e + 104, -2*e^2 + e + 40, 13*e^2 + 5*e - 240, -4*e^2 - 7*e + 72, 2*e^2 - 2*e - 36, -4*e^2 + 3*e + 74, e^2 - e - 12, 29*e^2 + 16*e - 520, -14*e^2 - 8*e + 240, -11*e^2 - 5*e + 206, 4*e^2 + 4*e - 74, 15*e^2 + 4*e - 272, -11*e^2 - 7*e + 198, -10*e^2 - 2*e + 182, 8*e^2 + 8*e - 136, 12*e^2 + 3*e - 216, -17*e^2 - 4*e + 320, 8*e^2 + 4*e - 144, -10*e^2 - 2*e + 196, -4*e^2 + 4*e + 70, -6*e^2 - 8*e + 106, 16*e^2 + 9*e - 290, 5*e^2 + 4*e - 72, -19*e^2 - 13*e + 332, 22*e^2 + 8*e - 404, 19*e^2 + 14*e - 342, 5*e^2 + 5*e - 94, 23*e^2 + 9*e - 424, 10*e^2 + 6*e - 182, -18*e^2 - 8*e + 320, -4*e^2 - 4*e + 94, 8*e^2 + 2*e - 170, -22*e^2 - 10*e + 402, 11*e^2 + 7*e - 190, -3*e^2 + 54, -7*e^2 - 6*e + 116, -27*e^2 - 12*e + 488, -4*e^2 - 2*e + 92, -17*e^2 - 4*e + 308, 8*e^2 + 8*e - 148, -23*e^2 - 11*e + 418, -e^2 + 8*e + 26, -19*e^2 - 3*e + 360, -18*e^2 - 9*e + 344, -8*e^2 - 4*e + 144, -18*e^2 - 10*e + 324, 35*e^2 + 20*e - 624, -7*e^2 + 112, -16*e^2 - 6*e + 288, -11*e^2 + e + 206, 5*e^2 + 3*e - 110, 6*e^2 + 7*e - 106, 2*e^2 - 2*e - 36, 20*e^2 + 9*e - 390, 18*e^2 + 10*e - 330, -2*e^2 - 4*e + 26, -20*e^2 - 12*e + 346, 11*e^2 + 6*e - 198, -23*e^2 - 10*e + 422, -5*e^2 - 2*e + 100, -22*e^2 - 12*e + 392, -6*e^2 - 4*e + 78, -10*e^2 - 4*e + 206, -30*e^2 - 16*e + 528, 19*e^2 + 6*e - 348, -33*e^2 - 17*e + 602, 6*e^2 + 4*e - 66, -7*e^2 - 4*e + 84, -e^2 + 6*e + 36, 11*e^2 + 7*e - 172, -2*e^2 - 4*e + 68, 35*e^2 + 14*e - 652, -2*e^2 - 5*e + 34, -11*e^2 - 6*e + 182, 40*e^2 + 20*e - 732, 12*e^2 + 10*e - 218, -40*e^2 - 16*e + 736, -17*e^2 - 9*e + 298, e - 16, -8*e^2 - 4*e + 146, 4*e^2 - 46, 22*e^2 + 12*e - 392, -14*e^2 - 6*e + 286, 19*e^2 + 7*e - 368, -9*e^2 - e + 184, -10*e^2 - 8*e + 158, -17*e^2 - 4*e + 308, 27*e^2 + 11*e - 486, 20*e^2 + 8*e - 350, -e^2 - 3*e - 22, 5*e^2 + 4*e - 108, 11*e^2 + 8*e - 198, -18*e^2 - 10*e + 300, -18*e^2 - 12*e + 330, -27*e^2 - 18*e + 494, 40*e^2 + 26*e - 712, -15*e^2 - 8*e + 264, -23*e^2 - 5*e + 416, -7*e^2 - 6*e + 140, -30*e^2 - 16*e + 540, -8*e^2 - 2*e + 152, 13*e^2 + 2*e - 242, -11*e^2 - 6*e + 182, -10*e^2 - 2*e + 186, 27*e^2 + 12*e - 496, -10*e^2 - 2*e + 148, -24*e^2 - 12*e + 420, 33*e^2 + 9*e - 610, -30*e^2 - 12*e + 584, 5*e^2 - 2*e - 68, 25*e^2 + 11*e - 440, -19*e^2 - 11*e + 376]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([71, 71, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 1])] = -1

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