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

NN = ZF.ideal([19,19,-w^3 - w^2 + 5*w + 3])

primes_array = [
[5, 5, -2*w^3 - 2*w^2 + 13*w + 8],\
[11, 11, 2*w^3 + 3*w^2 - 12*w - 11],\
[11, 11, -w^3 - 2*w^2 + 6*w + 9],\
[11, 11, w^3 + w^2 - 7*w - 4],\
[11, 11, w - 1],\
[16, 2, 2],\
[19, 19, -w^2 + 4],\
[19, 19, 2*w^3 + 2*w^2 - 13*w - 6],\
[19, 19, 3*w^3 + 4*w^2 - 18*w - 16],\
[19, 19, -w^3 - w^2 + 5*w + 3],\
[49, 7, 2*w^3 + 3*w^2 - 13*w - 15],\
[59, 59, -3*w^3 - 4*w^2 + 19*w + 14],\
[59, 59, 3*w^3 + 4*w^2 - 18*w - 17],\
[59, 59, 2*w^3 + 3*w^2 - 11*w - 13],\
[59, 59, -w^3 - 2*w^2 + 7*w + 9],\
[71, 71, 3*w^3 + 3*w^2 - 17*w - 10],\
[71, 71, -2*w^3 - w^2 + 14*w + 2],\
[71, 71, 5*w^3 + 7*w^2 - 30*w - 25],\
[71, 71, -3*w^3 - 4*w^2 + 16*w + 16],\
[81, 3, -3],\
[89, 89, -2*w^3 - 2*w^2 + 13*w + 4],\
[89, 89, 3*w^3 + 4*w^2 - 17*w - 16],\
[89, 89, 3*w^3 + 4*w^2 - 18*w - 18],\
[89, 89, -4*w^3 - 5*w^2 + 25*w + 18],\
[139, 139, -w^3 + 6*w + 1],\
[139, 139, -3*w^3 - 3*w^2 + 19*w + 8],\
[139, 139, 4*w^3 + 5*w^2 - 24*w - 21],\
[139, 139, -2*w^3 - 2*w^2 + 11*w + 5],\
[151, 151, -6*w^3 - 7*w^2 + 37*w + 27],\
[151, 151, 3*w^3 + 5*w^2 - 19*w - 17],\
[151, 151, -6*w^3 - 8*w^2 + 37*w + 31],\
[151, 151, 4*w^3 + 6*w^2 - 25*w - 26],\
[191, 191, 5*w^3 + 7*w^2 - 31*w - 26],\
[191, 191, -2*w^3 - 3*w^2 + 10*w + 13],\
[191, 191, -5*w^3 - 7*w^2 + 31*w + 29],\
[191, 191, 2*w^3 + 4*w^2 - 13*w - 17],\
[199, 199, 2*w^3 + 4*w^2 - 11*w - 20],\
[199, 199, -2*w^3 - 2*w^2 + 11*w + 3],\
[199, 199, 3*w^3 + 5*w^2 - 17*w - 17],\
[199, 199, 3*w^3 + 3*w^2 - 19*w - 6],\
[211, 211, 3*w^3 + 5*w^2 - 18*w - 19],\
[211, 211, -3*w^3 - 5*w^2 + 18*w + 21],\
[211, 211, 2*w^3 + 4*w^2 - 12*w - 17],\
[211, 211, -4*w^3 - 6*w^2 + 24*w + 23],\
[229, 229, 2*w^3 + 4*w^2 - 13*w - 19],\
[229, 229, -2*w^3 - 3*w^2 + 10*w + 15],\
[229, 229, -5*w^3 - 7*w^2 + 31*w + 24],\
[229, 229, 2*w^3 + 2*w^2 - 14*w - 3],\
[269, 269, 2*w^3 + w^2 - 11*w - 1],\
[269, 269, -6*w^3 - 7*w^2 + 37*w + 24],\
[269, 269, 5*w^3 + 6*w^2 - 29*w - 24],\
[269, 269, 3*w^3 + 2*w^2 - 19*w - 7],\
[281, 281, -4*w^3 - 6*w^2 + 25*w + 25],\
[281, 281, -5*w^3 - 7*w^2 + 31*w + 27],\
[281, 281, w^2 + 2*w - 7],\
[281, 281, 3*w^3 + 5*w^2 - 19*w - 18],\
[331, 331, -2*w^3 - 3*w^2 + 12*w + 7],\
[331, 331, w^3 + w^2 - 7*w - 8],\
[331, 331, w^3 + 2*w^2 - 6*w - 13],\
[331, 331, w - 5],\
[349, 349, 2*w^3 + 5*w^2 - 16*w - 12],\
[349, 349, 2*w^3 + 4*w^2 - 15*w - 10],\
[349, 349, 8*w^3 + 9*w^2 - 50*w - 32],\
[349, 349, 2*w^3 + 4*w^2 - 15*w - 9],\
[401, 401, 4*w^3 + 3*w^2 - 21*w - 12],\
[401, 401, 3*w^3 + 2*w^2 - 16*w - 9],\
[401, 401, -3*w^3 - 6*w^2 + 20*w + 19],\
[401, 401, -4*w^3 - 3*w^2 + 27*w + 9],\
[409, 409, -5*w^3 - 6*w^2 + 31*w + 20],\
[409, 409, 3*w^3 + 4*w^2 - 20*w - 12],\
[409, 409, -3*w^3 - 5*w^2 + 17*w + 23],\
[409, 409, 4*w^3 + 5*w^2 - 23*w - 21],\
[419, 419, -4*w^3 - 4*w^2 + 21*w + 18],\
[419, 419, -4*w^3 - 4*w^2 + 20*w + 19],\
[419, 419, -3*w^3 - 4*w^2 + 16*w + 18],\
[419, 419, -2*w^3 - w^2 + 8*w + 10],\
[421, 421, -6*w^3 - 7*w^2 + 36*w + 26],\
[421, 421, 5*w^3 + 5*w^2 - 31*w - 17],\
[421, 421, 3*w^3 + 2*w^2 - 18*w - 6],\
[421, 421, 4*w^3 + 4*w^2 - 23*w - 14],\
[431, 431, 6*w^3 + 7*w^2 - 36*w - 25],\
[431, 431, -3*w^3 - 2*w^2 + 18*w + 5],\
[431, 431, 4*w^3 + 4*w^2 - 23*w - 15],\
[431, 431, -5*w^3 - 5*w^2 + 31*w + 18],\
[439, 439, w^3 + w^2 - 4*w - 1],\
[439, 439, w^3 + 3*w^2 - 6*w - 10],\
[439, 439, 3*w^3 + 3*w^2 - 20*w - 7],\
[439, 439, 5*w^3 + 7*w^2 - 30*w - 30],\
[479, 479, -4*w^3 - 5*w^2 + 25*w + 15],\
[479, 479, -5*w^3 - 6*w^2 + 32*w + 21],\
[479, 479, 5*w^3 + 7*w^2 - 29*w - 28],\
[479, 479, 2*w^2 + w - 9],\
[491, 491, 7*w^3 + 8*w^2 - 43*w - 31],\
[491, 491, 6*w^3 + 6*w^2 - 37*w - 24],\
[491, 491, 4*w^3 + 3*w^2 - 24*w - 6],\
[491, 491, 5*w^3 + 5*w^2 - 29*w - 21],\
[509, 509, -4*w^3 - 5*w^2 + 25*w + 14],\
[509, 509, -3*w^3 - 4*w^2 + 17*w + 20],\
[509, 509, -6*w^3 - 7*w^2 + 38*w + 25],\
[509, 509, -6*w^3 - 8*w^2 + 35*w + 31],\
[541, 541, 9*w^3 + 12*w^2 - 55*w - 46],\
[541, 541, w^3 + 4*w^2 - 7*w - 17],\
[541, 541, -4*w^3 - 5*w^2 + 21*w + 19],\
[541, 541, -4*w^3 - 3*w^2 + 27*w + 8],\
[571, 571, -3*w^3 - 6*w^2 + 18*w + 23],\
[571, 571, 6*w^3 + 9*w^2 - 36*w - 37],\
[571, 571, 7*w^3 + 10*w^2 - 42*w - 36],\
[571, 571, 2*w^3 + 5*w^2 - 12*w - 24],\
[619, 619, -9*w^3 - 11*w^2 + 55*w + 39],\
[619, 619, -5*w^3 - 4*w^2 + 32*w + 14],\
[619, 619, 7*w^3 + 9*w^2 - 42*w - 36],\
[619, 619, 4*w^3 + 6*w^2 - 26*w - 27],\
[631, 631, 2*w^3 + 5*w^2 - 12*w - 21],\
[631, 631, -7*w^3 - 10*w^2 + 42*w + 39],\
[631, 631, 4*w^3 + 4*w^2 - 27*w - 13],\
[631, 631, w^3 + w^2 - 3*w - 4],\
[641, 641, -9*w^3 - 10*w^2 + 56*w + 36],\
[641, 641, -5*w^3 - 9*w^2 + 30*w + 32],\
[641, 641, -3*w^3 - 6*w^2 + 19*w + 21],\
[641, 641, 4*w^3 + 3*w^2 - 22*w - 10],\
[701, 701, -2*w^3 - 5*w^2 + 14*w + 20],\
[701, 701, -6*w^3 - 7*w^2 + 36*w + 34],\
[701, 701, 4*w^3 + 6*w^2 - 23*w - 17],\
[701, 701, -9*w^3 - 12*w^2 + 56*w + 46],\
[719, 719, 3*w^3 + 5*w^2 - 17*w - 25],\
[719, 719, -3*w^3 - 2*w^2 + 17*w + 3],\
[719, 719, 7*w^3 + 8*w^2 - 43*w - 26],\
[719, 719, -6*w^3 - 7*w^2 + 35*w + 29],\
[751, 751, 2*w^3 + 2*w^2 - 15*w - 4],\
[751, 751, 5*w^3 + 8*w^2 - 30*w - 34],\
[751, 751, w^3 + w^2 - 9*w - 5],\
[751, 751, -4*w^3 - 7*w^2 + 24*w + 26],\
[769, 769, 6*w^3 + 8*w^2 - 37*w - 27],\
[769, 769, -w^3 - 3*w^2 + 7*w + 16],\
[769, 769, 2*w^3 + w^2 - 14*w - 5],\
[769, 769, -3*w^3 - 4*w^2 + 16*w + 19],\
[821, 821, 6*w^3 + 6*w^2 - 37*w - 23],\
[821, 821, 5*w^3 + 5*w^2 - 29*w - 20],\
[821, 821, -7*w^3 - 8*w^2 + 42*w + 27],\
[821, 821, 4*w^3 + 3*w^2 - 24*w - 7],\
[829, 829, 7*w^3 + 8*w^2 - 44*w - 28],\
[829, 829, 7*w^3 + 9*w^2 - 41*w - 35],\
[829, 829, -2*w^3 - w^2 + 10*w + 2],\
[829, 829, 2*w^3 - 13*w + 2],\
[839, 839, 4*w^3 + 8*w^2 - 25*w - 26],\
[839, 839, -3*w^3 - 3*w^2 + 15*w + 16],\
[839, 839, -w^3 - 4*w^2 + 8*w + 10],\
[839, 839, -3*w^3 - 7*w^2 + 18*w + 26],\
[841, 29, -5*w^3 - 5*w^2 + 30*w + 16],\
[841, 29, -5*w^3 - 5*w^2 + 30*w + 19],\
[859, 859, 5*w^3 + 6*w^2 - 32*w - 20],\
[859, 859, -4*w^3 - 5*w^2 + 23*w + 23],\
[859, 859, -5*w^3 - 6*w^2 + 31*w + 18],\
[859, 859, -2*w^3 - w^2 + 13*w + 6],\
[911, 911, 9*w^3 + 12*w^2 - 56*w - 48],\
[911, 911, 6*w^3 + 5*w^2 - 38*w - 14],\
[911, 911, 7*w^3 + 8*w^2 - 40*w - 28],\
[911, 911, 10*w^3 + 12*w^2 - 61*w - 46],\
[929, 929, -7*w^3 - 10*w^2 + 39*w + 39],\
[929, 929, -10*w^3 - 13*w^2 + 60*w + 45],\
[929, 929, -6*w^3 - 7*w^2 + 32*w + 29],\
[929, 929, -3*w^3 - w^2 + 12*w + 12],\
[961, 31, -5*w^3 - 5*w^2 + 30*w + 17],\
[961, 31, -5*w^3 - 5*w^2 + 30*w + 18],\
[991, 991, 10*w^3 + 12*w^2 - 61*w - 45],\
[991, 991, -6*w^3 - 9*w^2 + 38*w + 38],\
[991, 991, 6*w^3 + 8*w^2 - 37*w - 38],\
[991, 991, -8*w^3 - 11*w^2 + 49*w + 43]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^2 - 4*x + 1
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [e, 2, -2*e + 3, e, -4, e - 4, -2*e + 8, -2*e + 5, e + 2, -1, -4, 4*e - 2, 2*e + 2, -e + 2, -4*e + 8, -4*e + 14, 2*e + 2, e - 2, 2*e + 2, -4*e + 4, -2*e + 5, 8*e - 18, 7, -6*e + 19, -6*e + 18, 2*e - 10, -8*e + 7, -5*e + 16, 2*e - 16, -10*e + 14, 6*e - 12, -21, -6*e + 3, -9*e + 12, -7*e + 26, 4*e - 5, 2*e + 14, -6*e + 9, 10*e - 26, 8*e - 7, 14*e - 32, e, -2*e + 6, 8*e - 20, 8*e - 11, 8*e - 26, 4*e - 6, 2*e + 7, -4*e - 4, -6*e + 18, 12*e - 24, -2*e + 28, -15*e + 30, 12, 3*e, 6*e - 24, 12, -2*e - 2, -6*e + 12, 10*e - 35, 14*e - 38, -10*e + 16, -6*e + 11, 8*e - 38, 10*e - 15, 4*e + 6, -4, -6*e + 5, -8*e + 38, -11*e + 26, -6*e + 4, -16*e + 36, 2*e - 24, 22, 4, -8*e + 38, 14*e - 18, -14*e + 38, -12*e + 28, -e - 6, -10*e + 36, 10*e - 25, -17*e + 26, 14*e - 36, e - 12, 8*e + 10, 6*e - 16, -14*e + 18, 17*e - 24, 4*e - 16, 28, -8*e + 26, 4*e + 26, 2*e, -4*e + 6, -4*e, 2*e + 14, -8*e + 16, -24, -8*e - 2, -12*e + 42, 10*e - 35, 12*e - 30, -6, 6*e - 8, -8*e + 8, -6*e + 22, -21*e + 40, 6*e - 38, -6*e + 40, 18*e - 47, -9*e + 28, 8*e - 20, -22, 2*e - 20, 3*e - 28, -23*e + 42, -12*e + 20, 8*e - 2, -4, 17*e - 32, -16*e + 28, 32, -4*e - 8, -4*e + 54, -8, 16*e - 16, -5*e + 2, -2*e - 10, -8*e - 7, -4*e - 18, -18*e + 16, -8*e + 18, 22*e - 51, -2*e - 18, 12*e - 46, 13*e + 2, 3*e - 38, -11*e + 56, -5*e + 32, -16*e + 56, 4*e - 26, 11*e - 10, 4*e - 50, 16*e - 13, 6*e - 35, 9*e - 26, 2*e - 18, 18*e - 44, -17, -15*e + 22, -6*e - 23, 3*e + 16, 8*e - 15, 22*e - 50, -14*e - 2, -18*e + 30, -8*e + 19, 6*e - 26, -9*e - 8, 16*e - 4, -2*e + 32, -6*e - 20, 2*e - 9, 9*e - 32, 24*e - 50, 6*e - 11, -9*e - 2]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

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

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