/*
  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, 7, 11, -2, -7, 0, 1])
F.<w> = NumberField(g)
ZF = F.ring_of_integers()

NN = ZF.ideal([25, 5, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w])

primes_array = [
[5, 5, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 5],\
[9, 3, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 23*w + 6],\
[11, 11, w - 1],\
[25, 5, w^3 + w^2 - 4*w - 3],\
[29, 29, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w],\
[41, 41, w^4 - w^3 - 5*w^2 + 3*w + 3],\
[49, 7, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w + 1],\
[59, 59, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 24*w + 7],\
[59, 59, -w^5 + 8*w^3 + 2*w^2 - 15*w - 8],\
[61, 61, w^5 - 7*w^3 - 2*w^2 + 12*w + 4],\
[61, 61, -w^5 + 7*w^3 - 11*w - 1],\
[64, 2, 2],\
[71, 71, w^3 + w^2 - 5*w - 3],\
[71, 71, -3*w^5 + w^4 + 21*w^3 - w^2 - 33*w - 9],\
[79, 79, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 19*w - 5],\
[81, 3, 2*w^5 - w^4 - 13*w^3 + w^2 + 19*w + 8],\
[89, 89, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 7],\
[89, 89, -w^5 + 8*w^3 + w^2 - 16*w - 5],\
[89, 89, -3*w^5 + w^4 + 20*w^3 - 30*w - 11],\
[89, 89, -w^5 + 7*w^3 + 2*w^2 - 11*w - 4],\
[89, 89, -2*w^5 + 14*w^3 + 3*w^2 - 22*w - 8],\
[89, 89, 2*w^5 - w^4 - 14*w^3 + 3*w^2 + 22*w + 6],\
[101, 101, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 22*w + 6],\
[101, 101, w^2 - w - 4],\
[101, 101, -3*w^5 + w^4 + 20*w^3 - 2*w^2 - 29*w - 7],\
[109, 109, w^4 - w^3 - 5*w^2 + 4*w + 5],\
[121, 11, w^5 - 7*w^3 - 2*w^2 + 10*w + 6],\
[131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 4*w - 3],\
[131, 131, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 20*w - 6],\
[131, 131, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 11],\
[131, 131, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 2],\
[139, 139, -w^5 + 6*w^3 + w^2 - 8*w - 1],\
[151, 151, -w^3 + w^2 + 4*w - 2],\
[179, 179, w^4 - 6*w^2 + 6],\
[181, 181, w^3 + w^2 - 3*w],\
[191, 191, 3*w^5 - w^4 - 20*w^3 + 2*w^2 + 28*w + 5],\
[199, 199, -2*w^5 + w^4 + 12*w^3 - 3*w^2 - 15*w - 2],\
[199, 199, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 15],\
[211, 211, 3*w^5 - w^4 - 19*w^3 + w^2 + 26*w + 9],\
[229, 229, -w^5 + w^4 + 7*w^3 - 3*w^2 - 11*w],\
[229, 229, -w^4 + 2*w^3 + 6*w^2 - 7*w - 6],\
[239, 239, -w^5 + w^4 + 7*w^3 - 3*w^2 - 13*w - 6],\
[239, 239, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 10],\
[239, 239, w^3 - w^2 - 4*w + 1],\
[239, 239, 5*w^5 - 2*w^4 - 34*w^3 + 2*w^2 + 53*w + 17],\
[241, 241, -w^5 + 7*w^3 + w^2 - 13*w - 4],\
[241, 241, -3*w^5 + w^4 + 21*w^3 - 35*w - 10],\
[241, 241, -w^5 + 7*w^3 - 11*w],\
[251, 251, 3*w^5 - w^4 - 21*w^3 + 35*w + 11],\
[251, 251, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 3],\
[269, 269, -w^5 + 8*w^3 + 3*w^2 - 16*w - 8],\
[269, 269, -2*w^5 + w^4 + 13*w^3 - w^2 - 21*w - 9],\
[269, 269, 3*w^5 - 21*w^3 - 3*w^2 + 32*w + 12],\
[271, 271, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 14],\
[281, 281, -w^5 + 8*w^3 + 3*w^2 - 16*w - 11],\
[281, 281, 4*w^5 - w^4 - 29*w^3 + 49*w + 13],\
[281, 281, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 5],\
[281, 281, -5*w^5 + 3*w^4 + 33*w^3 - 8*w^2 - 50*w - 12],\
[311, 311, 2*w^5 - w^4 - 14*w^3 + w^2 + 21*w + 9],\
[311, 311, -3*w^5 + 20*w^3 + 3*w^2 - 30*w - 10],\
[311, 311, 5*w^5 - 2*w^4 - 35*w^3 + 2*w^2 + 57*w + 19],\
[331, 331, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 7],\
[349, 349, -6*w^5 + 3*w^4 + 41*w^3 - 7*w^2 - 64*w - 17],\
[349, 349, 4*w^5 - w^4 - 29*w^3 - w^2 + 50*w + 17],\
[349, 349, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 17],\
[359, 359, w^5 - 6*w^3 - w^2 + 5*w + 3],\
[379, 379, -w^5 + w^4 + 8*w^3 - 3*w^2 - 17*w - 3],\
[379, 379, 2*w^5 - 14*w^3 - 4*w^2 + 22*w + 11],\
[379, 379, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 13],\
[379, 379, 4*w^5 - 2*w^4 - 27*w^3 + 3*w^2 + 43*w + 14],\
[389, 389, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 44*w + 13],\
[401, 401, -2*w^5 + 2*w^4 + 13*w^3 - 6*w^2 - 20*w - 6],\
[409, 409, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w],\
[419, 419, -5*w^5 + w^4 + 34*w^3 + 2*w^2 - 53*w - 19],\
[419, 419, 2*w^5 - w^4 - 14*w^3 + 4*w^2 + 23*w + 3],\
[421, 421, -w^5 - w^4 + 8*w^3 + 6*w^2 - 13*w - 7],\
[421, 421, 2*w^5 - 2*w^4 - 13*w^3 + 8*w^2 + 18*w + 2],\
[421, 421, 4*w^5 - w^4 - 26*w^3 + 37*w + 11],\
[431, 431, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 9],\
[431, 431, w^5 - 8*w^3 - w^2 + 14*w + 2],\
[449, 449, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 38*w + 9],\
[449, 449, -3*w^5 + w^4 + 20*w^3 - 31*w - 14],\
[461, 461, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 23*w + 4],\
[461, 461, -4*w^5 + 2*w^4 + 26*w^3 - 5*w^2 - 36*w - 9],\
[461, 461, -4*w^5 + 2*w^4 + 27*w^3 - 3*w^2 - 42*w - 15],\
[491, 491, -3*w^5 + w^4 + 20*w^3 - 29*w - 10],\
[491, 491, -5*w^5 + w^4 + 35*w^3 + 2*w^2 - 56*w - 18],\
[491, 491, 2*w^5 - 14*w^3 - 4*w^2 + 24*w + 13],\
[491, 491, -3*w^5 + w^4 + 21*w^3 - 2*w^2 - 33*w - 10],\
[499, 499, 6*w^5 - 3*w^4 - 40*w^3 + 6*w^2 + 61*w + 18],\
[499, 499, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 17],\
[499, 499, -w^5 - w^4 + 8*w^3 + 5*w^2 - 14*w - 4],\
[499, 499, -3*w^5 + w^4 + 19*w^3 - 2*w^2 - 26*w - 8],\
[509, 509, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 11],\
[509, 509, -2*w^5 + 16*w^3 + 2*w^2 - 30*w - 9],\
[521, 521, 3*w^5 - 21*w^3 - 3*w^2 + 34*w + 10],\
[521, 521, -3*w^5 + w^4 + 19*w^3 - 26*w - 9],\
[521, 521, 4*w^5 - w^4 - 27*w^3 + 40*w + 15],\
[521, 521, -3*w^5 + 22*w^3 + 3*w^2 - 37*w - 11],\
[541, 541, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 18],\
[569, 569, -6*w^5 + 2*w^4 + 40*w^3 - 2*w^2 - 60*w - 19],\
[569, 569, -w^5 + w^4 + 7*w^3 - 2*w^2 - 12*w - 6],\
[571, 571, 4*w^5 - w^4 - 28*w^3 + 43*w + 14],\
[571, 571, -4*w^5 + 2*w^4 + 28*w^3 - 4*w^2 - 45*w - 14],\
[599, 599, -w^5 + w^4 + 7*w^3 - 2*w^2 - 13*w - 7],\
[601, 601, 4*w^5 - w^4 - 26*w^3 + 38*w + 13],\
[601, 601, 4*w^5 - 2*w^4 - 28*w^3 + 5*w^2 + 44*w + 11],\
[601, 601, 3*w^5 - w^4 - 21*w^3 + 2*w^2 + 35*w + 9],\
[619, 619, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 13],\
[619, 619, -4*w^5 + w^4 + 26*w^3 - 36*w - 12],\
[619, 619, w^4 - w^3 - 6*w^2 + 4*w + 4],\
[631, 631, -3*w^5 + w^4 + 20*w^3 - 30*w - 9],\
[631, 631, -5*w^5 + 2*w^4 + 35*w^3 - 3*w^2 - 57*w - 15],\
[641, 641, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 16],\
[641, 641, -5*w^5 + 2*w^4 + 34*w^3 - 2*w^2 - 52*w - 18],\
[659, 659, 3*w^5 - w^4 - 20*w^3 + w^2 + 32*w + 9],\
[659, 659, -4*w^5 + w^4 + 27*w^3 - w^2 - 40*w - 11],\
[661, 661, 4*w^5 - w^4 - 26*w^3 + 38*w + 12],\
[661, 661, -w^5 + 6*w^3 - 8*w + 1],\
[691, 691, 3*w^5 - 20*w^3 - 3*w^2 + 29*w + 9],\
[691, 691, w^5 + w^4 - 8*w^3 - 7*w^2 + 14*w + 9],\
[691, 691, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w + 2],\
[701, 701, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 9],\
[701, 701, 5*w^5 - 2*w^4 - 33*w^3 + 3*w^2 + 48*w + 14],\
[701, 701, w^4 + 3*w^3 - 3*w^2 - 10*w - 1],\
[701, 701, -2*w^5 + w^4 + 14*w^3 - 3*w^2 - 24*w - 3],\
[709, 709, -2*w^5 + 2*w^4 + 14*w^3 - 7*w^2 - 22*w - 5],\
[709, 709, -3*w^5 + 21*w^3 + 4*w^2 - 34*w - 11],\
[719, 719, 5*w^5 - 2*w^4 - 35*w^3 + 3*w^2 + 58*w + 17],\
[719, 719, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 37*w + 11],\
[719, 719, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 12],\
[719, 719, 2*w^5 - 16*w^3 - 3*w^2 + 30*w + 10],\
[739, 739, -5*w^5 + w^4 + 34*w^3 + w^2 - 51*w - 17],\
[751, 751, -w^5 - w^4 + 9*w^3 + 7*w^2 - 18*w - 10],\
[761, 761, 4*w^5 - 2*w^4 - 28*w^3 + 3*w^2 + 46*w + 16],\
[761, 761, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 12],\
[769, 769, -4*w^5 + w^4 + 28*w^3 - 44*w - 12],\
[769, 769, -w^4 + w^3 + 3*w^2 - 3*w + 1],\
[809, 809, 2*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 20*w + 1],\
[811, 811, -2*w^5 + 15*w^3 + 2*w^2 - 24*w - 8],\
[811, 811, -w^5 + 9*w^3 + w^2 - 18*w - 6],\
[821, 821, -w^5 + 8*w^3 + 3*w^2 - 17*w - 9],\
[821, 821, w^5 + w^4 - 7*w^3 - 6*w^2 + 11*w + 8],\
[821, 821, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 46*w + 11],\
[821, 821, 5*w^5 - 2*w^4 - 34*w^3 + 4*w^2 + 51*w + 13],\
[829, 829, -2*w^5 + w^4 + 15*w^3 - w^2 - 28*w - 10],\
[829, 829, -6*w^5 + 3*w^4 + 42*w^3 - 7*w^2 - 68*w - 21],\
[829, 829, 4*w^5 - 2*w^4 - 26*w^3 + 5*w^2 + 38*w + 12],\
[829, 829, w^5 - 7*w^3 - 3*w^2 + 11*w + 6],\
[839, 839, 3*w^5 - 2*w^4 - 20*w^3 + 5*w^2 + 32*w + 11],\
[839, 839, -w^4 + w^3 + 6*w^2 - 3*w - 4],\
[841, 29, -3*w^5 + w^4 + 22*w^3 - 38*w - 11],\
[911, 911, 3*w^5 - w^4 - 21*w^3 + 36*w + 12],\
[919, 919, -w^5 - w^4 + 7*w^3 + 6*w^2 - 12*w - 5],\
[941, 941, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w],\
[941, 941, 4*w^5 - w^4 - 29*w^3 + 48*w + 15],\
[971, 971, -4*w^5 + w^4 + 27*w^3 + w^2 - 41*w - 13],\
[991, 991, w^5 - w^4 - 8*w^3 + 3*w^2 + 16*w + 2],\
[991, 991, -4*w^5 + 3*w^4 + 27*w^3 - 9*w^2 - 42*w - 10]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^3 - 2*x^2 - 7*x + 5
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, e, e + 1, e^2 - 2*e - 4, e + 1, -e^2 + e + 8, e^2 + e - 11, -e^2 - e + 9, -e^2 + 4*e - 1, -3*e^2 + 5*e + 13, -e^2 + 5, -3*e + 7, -e^2 + 10, -e^2 + 5*e, e^2 - 3*e - 5, e^2 - 3*e + 6, -e^2 + 2*e + 9, -4*e^2 + 5*e + 15, 2*e^2 - 6*e - 11, 3*e^2 - e - 19, -2*e - 2, 2*e^2 - 2*e - 7, e^2 - 2*e - 3, e^2 - 3*e + 2, -e^2 + e + 11, -e + 7, 3*e^2 - 5*e - 13, 2*e^2 - 5*e - 13, -e^2 + 1, 2*e^2 + e - 19, 5*e + 5, -3*e^2 + 7*e + 15, 2*e^2 - 3*e - 13, -e^2 - 3*e + 1, -2*e^2 + 5*e + 5, -2*e^2 + 14, -3*e^2 + 6*e + 25, -8, e + 5, 2*e^2 - 8*e - 3, -e^2 + 7*e + 1, 7*e + 1, e^2 + 3*e + 2, -2*e^2 - 6*e + 23, -e^2 + 7*e - 4, 3*e^2 - 12*e - 19, -2*e^2 - 2*e + 10, -2*e^2 + 3*e + 9, -e^2 - 2*e + 17, -9, e^2 + 4*e - 12, e^2 + 4*e - 9, e^2 + e, 2*e^2 - 5*e - 6, 5*e^2 - 8*e - 16, 4*e^2 - 7*e - 17, e^2 - 6*e - 10, 2*e^2 - 11*e - 13, -e^2 - 3*e + 1, e^2 - 7*e + 7, -5*e^2 + 9*e + 20, 19, 2*e^2 - 9*e + 3, -e^2 + 4*e + 13, -7*e - 9, -e^2 + 3*e - 8, 8*e + 6, 1, 2*e^2 - 6*e - 9, -e^2 - 3*e, -e^2 - 6*e + 25, 5*e^2 - 9*e - 17, e^2 - 7*e + 2, 3*e^2 - 6, 4*e^2 - 3*e - 25, 3*e^2 - 9*e - 22, 2*e^2 + 9*e - 22, 4*e^2 - 8*e - 35, e^2 + 9*e - 10, -3*e^2 + 9*e + 15, -3*e^2 - 3*e + 21, -2*e^2 + 9*e + 32, -4*e^2 + 7*e + 20, 2*e^2 - 3*e - 23, 6*e - 6, -7*e^2 + 9*e + 37, -3*e^2 + 9*e + 12, -3*e + 12, 3*e^2 - 8*e - 5, -e^2 + 8*e - 10, -2*e^2 - 4*e + 24, 3*e^2 - 2*e - 4, -3*e^2 + 10*e + 26, 2*e^2 + e - 19, -3*e^2 - 6*e + 33, 2*e^2 - 5*e - 1, -3*e^2 - 2*e + 43, -5*e^2 + 12*e + 17, -3*e^2 - 3*e + 27, -4*e^2 + 9*e + 21, -e^2 + 6*e + 4, -6*e^2 + 6*e + 24, 4*e^2 - 5*e - 2, 7*e^2 - 12*e - 32, 5*e^2 - 9*e - 26, -2*e^2 + e + 13, -5*e^2 + 9*e + 36, -5*e^2 + 4*e + 29, -2*e^2 + 3*e + 30, e^2 - 4*e + 27, -4*e^2 + 10*e + 4, -e^2 + e + 9, 4*e^2 + 5*e - 37, -3*e^2 + e + 19, -3*e^2 + 39, 3*e^2 + e - 17, -e^2 + 13*e + 11, -e^2 + 5*e - 31, -8*e^2 + 2*e + 50, 5*e^2 - 30, 3*e^2 + 5*e - 14, -4*e^2 + 15*e + 9, -7*e^2 + 12*e + 37, 2*e^2 + 3*e - 35, -2*e^2 + 9*e + 17, -e^2 + 6*e - 11, 8*e^2 - 10*e - 22, 3*e^2 - 11*e + 11, e^2 - 15*e + 2, 2*e^2 - 7*e - 15, -e^2 + 6*e - 2, 3*e^2 - 17*e - 20, -6*e^2 + 11*e + 31, -4*e^2 + 9, -2*e - 5, -4*e^2 + 12*e + 1, e^2 + 3*e + 21, 7*e^2 - 15*e - 20, -e^2 + 3*e + 31, -2*e^2 - 7*e + 11, -e^2 + 8, -e^2 + 4*e + 26, -e^2 - 5*e + 8, -4*e^2 - 7*e + 36, -e^2 - 2*e + 44, -2*e^2 - e + 44, -e^2 - 7*e + 28, 3*e^2 - 8*e - 27, -5*e^2 + 7*e + 23, 4*e^2 - 6*e - 13, -7*e^2 + 4*e + 35, -3*e^2 + 10*e + 33, -2*e^2 - 8*e + 51, -6*e^2 + 24*e + 35, -3*e^2 + e - 8, 8*e^2 - 3*e - 47, 6*e^2 - 10*e - 7, -3*e^2 + 4*e - 27, -2*e^2 - e - 21]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([5, 5, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 5])] = 1

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