/*
  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([1, 1, 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^2 + x - 21
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, e, -e + 1, -e + 1, -e + 2, -e + 2, e + 1, e + 1, -10, -10, e + 9, 11, -2*e - 8, -2*e - 8, 2*e + 2, 2*e + 2, e + 3, e + 3, e - 8, -3*e - 6, -3*e - 6, e - 8, -2*e - 6, -2*e - 6, -2*e - 6, -2*e - 6, 2*e + 4, 2*e + 4, 2*e + 4, 2*e + 4, 6, 6, 4*e + 6, -e - 20, -e - 20, -3*e + 8, -3*e + 8, -3*e - 3, 3*e - 6, e - 5, 3*e - 6, -3*e - 3, e - 5, 4*e - 6, 4*e - 6, 8, 8, 8, 8, 2*e - 4, 2*e - 4, 6*e + 6, -2*e + 10, -2*e + 10, 6*e + 6, -e - 1, 3*e - 3, -e - 1, 3*e - 3, -30, -2*e + 12, -4*e - 14, -4*e - 14, -2*e + 12, -4*e - 14, -4*e - 14, -3*e + 2, -3*e + 2, -e + 1, -e + 1, 3*e - 16, 3*e - 16, 2*e + 14, 2*e + 14, -3*e + 5, -3*e + 5, -e - 23, -e - 23, -e - 11, -e - 11, 5*e + 2, 5*e + 2, 5*e + 14, -4*e - 22, -4*e - 22, 5*e + 14, -6*e + 8, -6*e + 8, 6, 6, 8*e + 2, 8*e + 2, 6, 6, 4*e + 16, 4*e + 16, 8, 8, -4*e + 10, -4*e + 10, 2*e - 20, 2*e - 20, -2*e + 22, -2*e + 22, -e + 22, -e + 22, e + 10, e + 10, 4*e - 6, 4*e - 6, -4*e + 14, -4*e + 14, -4*e - 26, e + 12, -e - 14, -e - 14, -4*e - 26, e + 12, e + 13, e + 13, -4*e + 28, -4*e + 28, 3*e + 40, 6*e + 12, 6*e + 12, -2*e + 16, -2*e + 16, 3*e + 14, 3*e + 14, -7*e + 19, -7*e + 19, -4*e - 10, -4*e - 10, 8*e - 2, 8*e - 2, 18, 18, 18, 18, 4*e + 28, 4*e + 28, -22, -4*e + 34, -4*e + 34, -3*e + 5, -3*e + 5, -e - 49, -e + 32, -e - 49, -e + 32, -2*e - 6, -2*e - 6, 6*e + 18, -6*e + 24, 6*e + 18, -6*e + 24, 5*e + 4, 5*e + 4, 2*e + 22, 2*e + 22]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}

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