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

NN = ZF.ideal([12, 6, w^2 - 2*w - 3])

primes_array = [
[2, 2, -w^2 + 3*w],\
[3, 3, -w^2 - w + 3],\
[3, 3, w + 2],\
[7, 7, 2*w^2 - 6*w - 1],\
[11, 11, -w^2 + 2*w + 4],\
[17, 17, -w^2 + 4*w - 2],\
[19, 19, w^2 - 6],\
[19, 19, -w^2 - 3*w - 1],\
[19, 19, 3*w^2 - 9*w - 1],\
[41, 41, w^2 - 8],\
[43, 43, -2*w^2 - 3*w + 4],\
[47, 47, -2*w - 3],\
[49, 7, -9*w^2 + 29*w - 3],\
[67, 67, 2*w - 3],\
[71, 71, 4*w^2 - 14*w + 5],\
[79, 79, w^2 - 3*w - 3],\
[79, 79, -w^2 + 5*w - 5],\
[79, 79, 6*w^2 - 3*w - 38],\
[97, 97, 3*w^2 - 10*w],\
[103, 103, 3*w^2 - 4*w - 24],\
[103, 103, 2*w^2 - 2*w - 11],\
[103, 103, 2*w^2 - 5*w - 2],\
[107, 107, 3*w^2 - 16],\
[109, 109, 2*w^2 - 3*w - 16],\
[113, 113, 2*w^2 - 3*w - 12],\
[121, 11, 2*w^2 - 3*w - 18],\
[125, 5, -5],\
[127, 127, 4*w^2 - 21],\
[127, 127, 2*w^2 - 9],\
[131, 131, -7*w^2 + 24*w - 6],\
[139, 139, w - 6],\
[149, 149, -w^2 + 10*w + 28],\
[151, 151, w^2 + 2*w - 4],\
[157, 157, -2*w^2 + 7*w - 4],\
[157, 157, -10*w^2 + 33*w - 6],\
[157, 157, 3*w - 8],\
[167, 167, 2*w^2 - 4*w - 5],\
[167, 167, -3*w - 4],\
[167, 167, w^2 + w - 7],\
[173, 173, 2*w^2 - 6*w - 3],\
[173, 173, w^2 - 3*w - 5],\
[173, 173, -3*w - 8],\
[179, 179, w^2 - 5*w - 1],\
[181, 181, 4*w^2 - 11*w - 4],\
[191, 191, 5*w^2 - 15*w - 1],\
[193, 193, w^2 + 3*w - 3],\
[197, 197, 7*w^2 + 13*w - 7],\
[223, 223, -5*w - 12],\
[227, 227, 3*w^2 - 5*w - 15],\
[229, 229, -2*w^2 + 13],\
[241, 241, w^2 - 3*w - 9],\
[251, 251, 9*w^2 - 28*w],\
[263, 263, -5*w^2 + 15*w + 3],\
[269, 269, 5*w^2 - 17*w + 1],\
[277, 277, 4*w^2 - 12*w - 3],\
[281, 281, -6*w^2 + 21*w - 4],\
[281, 281, -6*w - 13],\
[281, 281, 3*w^2 - 3*w - 17],\
[283, 283, -5*w^2 + 18*w - 8],\
[289, 17, 3*w^2 + 2*w - 12],\
[293, 293, -7*w^2 + 23*w - 5],\
[293, 293, 2*w^2 - 7],\
[293, 293, 4*w^2 - w - 22],\
[307, 307, -2*w^2 + 21],\
[313, 313, w^2 - 4*w - 4],\
[317, 317, 2*w^2 + 3*w - 6],\
[317, 317, 3*w^2 - 8*w - 6],\
[317, 317, 2*w^2 - w - 8],\
[331, 331, 3*w^2 - 7*w - 9],\
[337, 337, -4*w - 3],\
[349, 349, 7*w^2 - 4*w - 46],\
[353, 353, 11*w^2 - 35*w + 1],\
[359, 359, 2*w^2 - 3*w - 4],\
[359, 359, -w^2 + 7*w - 11],\
[359, 359, -2*w^2 - 8*w - 7],\
[373, 373, 5*w^2 + w - 23],\
[379, 379, 2*w^2 - 4*w - 11],\
[383, 383, -w^2 + 6*w - 6],\
[389, 389, 15*w^2 - 48*w + 4],\
[409, 409, w^2 - 6*w - 18],\
[419, 419, -4*w^2 + 15*w - 6],\
[421, 421, 2*w^2 - 4*w - 17],\
[431, 431, w^2 + 3*w - 5],\
[433, 433, 5*w^2 - 7*w - 41],\
[433, 433, 2*w^2 - 2*w - 5],\
[433, 433, -4*w^2 + 15*w - 4],\
[439, 439, 5*w^2 - w - 29],\
[443, 443, -9*w^2 + 3*w + 53],\
[449, 449, 5*w^2 - 3*w - 31],\
[461, 461, w^2 - w - 13],\
[461, 461, 4*w^2 - 7*w - 16],\
[461, 461, -3*w^2 - w + 15],\
[479, 479, 2*w^2 - 5*w - 8],\
[487, 487, -11*w^2 + 34*w + 2],\
[491, 491, 3*w^2 - 6*w - 8],\
[491, 491, -6*w^2 + 21*w - 8],\
[491, 491, -8*w^2 + 26*w - 1],\
[499, 499, 2*w^2 - 8*w - 29],\
[503, 503, 3*w^2 - 7*w - 3],\
[509, 509, -w^2 - 4],\
[521, 521, 2*w^2 - w - 20],\
[541, 541, 8*w^2 - w - 44],\
[547, 547, 12*w^2 - 38*w + 3],\
[557, 557, -17*w^2 + 57*w - 11],\
[563, 563, 5*w^2 - 2*w - 32],\
[563, 563, 6*w^2 - 22*w + 9],\
[563, 563, -2*w^2 + 5*w + 20],\
[569, 569, 3*w^2 - 6*w - 28],\
[571, 571, w^2 - 4*w - 10],\
[587, 587, 6*w^2 - 7*w - 48],\
[601, 601, 7*w^2 - 21*w + 1],\
[613, 613, -6*w^2 + 9*w + 52],\
[619, 619, 4*w^2 - 10*w - 43],\
[631, 631, -2*w - 9],\
[641, 641, 2*w^2 - 6*w - 25],\
[641, 641, 11*w^2 - 6*w - 70],\
[641, 641, -w^2 - 6*w - 10],\
[643, 643, -4*w^2 + 11*w + 46],\
[643, 643, 7*w^2 - 2*w - 40],\
[643, 643, -3*w - 10],\
[653, 653, -2*w^2 - w + 20],\
[677, 677, w^2 - 3*w - 15],\
[683, 683, w^2 + 3*w - 17],\
[683, 683, 4*w^2 - 6*w - 23],\
[683, 683, 4*w^2 + 4*w - 13],\
[691, 691, 5*w^2 - 14*w - 2],\
[709, 709, -w^2 + w - 5],\
[709, 709, 2*w^2 - 6*w + 5],\
[709, 709, 4*w^2 - 5*w - 34],\
[719, 719, 2*w^2 + w - 14],\
[727, 727, -w^2 - 8*w - 14],\
[751, 751, 3*w^2 - 6*w - 14],\
[769, 769, -8*w^2 + 12*w + 69],\
[773, 773, 3*w^2 - 4*w - 12],\
[797, 797, 8*w^2 - 28*w + 7],\
[809, 809, 10*w^2 - 31*w + 2],\
[823, 823, -13*w^2 + 44*w - 8],\
[823, 823, 9*w^2 - 26*w - 6],\
[823, 823, 3*w^2 - 5*w - 9],\
[829, 829, 2*w^2 + 2*w - 11],\
[853, 853, -12*w^2 + 40*w - 9],\
[859, 859, 4*w - 15],\
[877, 877, -w^2 + 4*w - 8],\
[877, 877, 3*w^2 + w - 9],\
[877, 877, 3*w^2 - 3*w - 13],\
[911, 911, w^2 - 5*w - 13],\
[929, 929, 8*w^2 - 24*w - 1],\
[929, 929, -16*w^2 + 50*w + 1],\
[929, 929, -w^2 + 3*w - 7],\
[937, 937, -3*w^2 + 26],\
[941, 941, w^2 - 5*w - 7],\
[947, 947, 4*w^2 - 12*w - 5],\
[947, 947, 10*w^2 - 30*w - 3],\
[947, 947, -5*w^2 + 17*w - 7],\
[953, 953, 2*w^2 - 9*w - 30],\
[967, 967, 5*w - 12],\
[971, 971, 5*w^2 - 8*w - 26],\
[977, 977, 15*w^2 - 48*w + 2],\
[983, 983, -18*w^2 + 59*w - 6],\
[991, 991, 6*w^2 - 23*w + 12],\
[997, 997, 4*w^2 + w - 16]]
primes = [ZF.ideal(I) for I in primes_array]

heckePol = x^4 - 2*x^3 - 7*x^2 + 9*x + 12
K.<e> = NumberField(heckePol)

hecke_eigenvalues_array = [0, -1, e, e^3 - e^2 - 4*e, -e^3 + e^2 + 5*e, -e^2 + 2*e + 6, e^3 - e^2 - 5*e, 3*e^2 - 2*e - 12, -e^3 + 6*e + 4, e^3 + 2*e^2 - 9*e - 6, e^3 - 7*e, -e^3 + 5*e + 12, e^3 + 4*e^2 - 10*e - 18, 5*e^2 - 6*e - 20, -e^3 - 3*e^2 + 9*e + 12, -e^3 + 6*e, -e^3 - 3*e^2 + 9*e + 20, -2*e^3 - e^2 + 11*e + 12, -3*e^2 + 3*e + 10, -2*e^3 - 2*e^2 + 12*e + 8, -4*e^3 - 2*e^2 + 22*e + 24, -e^2 - 5*e + 12, -e^3 + 4*e^2 - 12, -3*e^3 + 18*e + 6, -4*e^2 + 18, 3*e^3 - 3*e^2 - 12*e - 2, e^3 - e^2 - 3*e - 6, -2*e^3 + 4*e^2 + 9*e - 12, -e^3 - 2*e^2 + 9*e + 4, e^3 + 3*e^2 - 3*e - 24, e^3 - 3*e^2 + 12, 2*e^3 + 3*e^2 - 12*e - 18, -4*e^3 - e^2 + 21*e + 20, -4*e^3 + e^2 + 23*e + 6, -3*e^3 + 2*e^2 + 18*e + 2, -e^3 - 7*e^2 + 16*e + 30, e^3 - 4*e^2 - 6*e + 24, e^3 - e^2, 2*e^3 - 4*e^2 - 15*e + 12, -e^3 - e^2 + 15*e + 6, 4*e^3 - 3*e^2 - 24*e + 6, -3*e^3 - 3*e^2 + 17*e + 30, -e^3 + 7*e^2 - 4*e - 36, -3*e^3 + 3*e^2 + 15*e + 6, -3*e^2 - 2*e + 24, 3*e^3 + 2*e^2 - 19*e - 18, -2*e^3 - 3*e^2 + 15*e + 18, 3*e^3 - 4*e^2 - 15*e + 12, e^3 - e^2 - 11*e, -4*e^3 - 6*e^2 + 31*e + 30, -2*e^3 + 3*e^2 + 9*e - 18, 2*e^3 - 5*e^2 - 3*e + 24, 10*e^2 - 9*e - 36, -3*e^3 + 3*e^2 + 10*e + 6, -2*e^2 - 2*e + 18, -4*e^2 + 12*e + 18, -2*e^3 + 3*e^2 + 3*e - 6, -4*e^3 + 7*e^2 + 19*e - 6, e^3 - 4*e^2 - 3*e + 16, 3*e^3 - e^2 - 21*e + 2, 2*e^3 - 2*e^2 - 3*e + 6, e^3 - 8*e + 6, 5*e^3 - 9*e^2 - 18*e + 18, e^3 - 2*e^2 - 6*e + 4, 7*e^3 - 3*e^2 - 36*e - 10, 8*e^3 - 7*e^2 - 42*e + 6, -2*e^3 + 7*e^2 + 5*e - 18, -4*e^3 - 8*e^2 + 30*e + 42, 5*e^3 + 2*e^2 - 30*e - 20, -e^3 + 2*e^2 - 2, 6*e^2 - 6*e - 38, -2*e^3 + 5*e^2 + 13*e - 18, e^3 + 3*e^2 - 6*e, 5*e^3 + 2*e^2 - 37*e - 12, 7*e^3 - 3*e^2 - 42*e, -2*e^3 - 4*e^2 + 18*e + 18, -4*e^2 + 15*e + 16, e^3 - 4*e^2, -e^3 - 2*e^2 + 6*e + 6, e^3 - 4*e^2 - 12*e + 6, 4*e^3 - 4*e^2 - 18*e - 12, 5*e^3 + 4*e^2 - 30*e - 38, e^3 - 5*e^2, -2*e^3 + 3*e^2 - 14, 2*e^3 - 10*e^2 - 6*e + 42, -2*e^3 + 4*e^2 + 5*e - 18, 4*e^3 + 2*e^2 - 24*e - 32, 5*e^3 + 2*e^2 - 40*e - 12, -e^3 + e^2 + 6*e + 6, -5*e^3 + 5*e^2 + 18*e - 6, 3*e^3 - 5*e^2 - 18*e - 6, -4*e^3 + 4*e^2 + 19*e - 6, 5*e^3 - 28*e - 24, -8*e^3 + 13*e^2 + 30*e - 32, -e^2 - e + 24, -3*e^3 + e^2 + 22*e + 12, e^3 + e^2 - 2*e + 12, -e^3 - 2*e^2 + 8*e + 12, 4*e^3 - e^2 - 21*e + 12, e^3 - 5*e^2 - 13*e + 18, -5*e^3 + 9*e^2 + 18*e - 18, 6*e^3 - 8*e^2 - 27*e + 10, 3*e^3 - 10*e^2 - 15*e + 24, -2*e^3 + 9*e^2 + 9*e - 30, 6*e^3 - 6*e^2 - 35*e + 24, -2*e^3 - e^2 + 19*e, -6*e^3 - e^2 + 34*e + 36, -5*e^3 - 2*e^2 + 30*e + 30, -8*e^3 - 6*e^2 + 42*e + 68, 3*e^3 + 6*e^2 - 24*e - 12, 4*e^3 - 2*e^2 - 21*e + 10, 5*e^3 - e^2 - 42*e + 2, -e^3 + 13*e^2 - 6*e - 52, 6*e^3 - 3*e^2 - 22*e - 24, e^3 + e^2 - 8*e - 6, 2*e^3 - 2*e^2 - 10*e + 18, 4*e^3 + 9*e^2 - 30*e - 54, -3*e^3 + 7*e^2 + 4*e - 12, 3*e^3 + 4*e^2 - 21*e, -e^3 + 7*e^2 + 3*e - 8, -2*e^3 + 2*e^2 + e + 18, e^3 - 6*e^2 - e + 30, -9*e^3 + 2*e^2 + 42*e + 36, e^3 + e^2 - 12, -6*e^3 + 10*e^2 + 19*e - 24, 4*e^3 + 3*e^2 - 18*e - 12, -5*e^3 + 12*e^2 + 26*e - 30, 10*e^3 - 9*e^2 - 45*e + 10, -4*e^3 + 7*e^2 + 16*e - 30, -6*e^3 - 11*e^2 + 46*e + 48, -6*e^3 + 25*e + 12, -4*e^3 + 2*e^2 + 32*e, -3*e^3 + 13*e^2 + 3*e - 58, e^3 - 2*e^2 - 3*e - 6, 8*e^3 + 3*e^2 - 44*e - 30, -10*e^3 + 16*e^2 + 36*e - 42, 6*e^3 + 2*e^2 - 33*e - 28, 6*e^3 + 6*e^2 - 47*e - 36, 9*e^3 - 11*e^2 - 33*e + 12, 3*e^3 - 4*e^2 - 15*e + 14, 3*e^3 - 2*e^2 - 6*e - 18, e^3 + 15*e^2 - 12*e - 60, 9*e^2 + 10*e - 42, 5*e^2 - 2*e - 30, -3*e^3 - 8*e^2 + 24*e + 74, 14*e^3 - 10*e^2 - 69*e + 12, -e^3 - 2*e^2 + 14*e + 18, 11*e^3 - 6*e^2 - 62*e + 6, -3*e^3 - 7*e^2 + 15*e + 66, -9*e^3 - 5*e^2 + 63*e + 42, e^3 - 19*e^2 + 10*e + 54, 8*e^3 - 39*e - 24, -2*e^3 + 14*e^2 - e - 48, 2*e^3 + 4*e^2 - 18*e - 12, -2*e^3 - 9*e^2 + 15*e + 54, -3*e^3 - 7*e^2 + 15*e + 68, -3*e^2 + 4*e - 12, 2*e^3 - 11*e^2 - 12*e + 30, -e^3 + 2*e^2 - 2*e, 2*e^3 - 14*e^2 - 12*e + 56, 2*e^3 - 13*e^2 + 16*e + 66]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

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

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