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

NN = ZF.ideal([83,83,-2*w^2 + w - 2])

primes_array = [
[7, 7, 2*w^2 - w - 3],\
[8, 2, 2],\
[13, 13, -w^2 - w + 3],\
[13, 13, -w^2 + 2*w + 2],\
[13, 13, -2*w^2 + w + 2],\
[27, 3, 3],\
[29, 29, 3*w^2 - 2*w - 4],\
[29, 29, 2*w^2 + w - 4],\
[29, 29, -w^2 + 3*w + 1],\
[41, 41, w^2 - w - 5],\
[41, 41, 2*w^2 - 3*w - 4],\
[41, 41, -3*w^2 + w + 3],\
[43, 43, w^2 + 2*w - 5],\
[43, 43, 2*w^2 + w - 5],\
[43, 43, 3*w^2 - 2*w - 3],\
[71, 71, 4*w^2 - 3*w - 5],\
[71, 71, 3*w^2 - 4*w - 5],\
[71, 71, -4*w^2 + w + 5],\
[83, 83, w^2 + w - 7],\
[83, 83, w^2 - 2*w - 6],\
[83, 83, -3*w^2 - 2*w + 5],\
[97, 97, 3*w^2 + w - 7],\
[97, 97, 3*w^2 - 4*w - 7],\
[97, 97, 2*w^2 - w - 8],\
[113, 113, 3*w^2 + w - 8],\
[113, 113, 2*w^2 - 4*w - 5],\
[113, 113, 2*w^2 + 2*w - 7],\
[125, 5, -5],\
[127, 127, 2*w^2 - 9],\
[127, 127, -3*w^2 - 2*w + 6],\
[127, 127, -5*w^2 + 3*w + 7],\
[139, 139, 5*w^2 - 4*w - 6],\
[139, 139, 4*w^2 - 5*w - 7],\
[139, 139, -5*w^2 + w + 6],\
[167, 167, w^2 + w - 8],\
[167, 167, 6*w^2 - 4*w - 9],\
[167, 167, -4*w^2 - 2*w + 7],\
[181, 181, 4*w^2 + w - 9],\
[181, 181, 4*w^2 - 5*w - 9],\
[181, 181, 2*w^2 - w - 9],\
[197, 197, 2*w - 7],\
[197, 197, 3*w^2 - 5*w - 6],\
[197, 197, 2*w^2 - 2*w - 9],\
[211, 211, 3*w^2 - 5*w - 7],\
[211, 211, 2*w^2 + 3*w - 8],\
[211, 211, 4*w^2 + w - 10],\
[223, 223, 3*w^2 + 2*w - 9],\
[223, 223, 4*w^2 + w - 11],\
[223, 223, 3*w^2 + 2*w - 10],\
[239, 239, 6*w^2 - 5*w - 7],\
[239, 239, -5*w^2 + 6*w + 9],\
[239, 239, -6*w^2 + w + 7],\
[251, 251, w^2 - 3*w - 8],\
[251, 251, -3*w^2 + 2*w - 3],\
[251, 251, 2*w^2 + w - 11],\
[281, 281, -7*w^2 + 6*w + 9],\
[281, 281, 6*w^2 + w - 11],\
[281, 281, w^2 - 7*w],\
[293, 293, -7*w^2 + 2*w + 10],\
[293, 293, 5*w^2 - 7*w - 7],\
[293, 293, 2*w^2 + 5*w - 6],\
[307, 307, -3*w^2 + w - 3],\
[307, 307, 2*w^2 - 3*w - 10],\
[307, 307, w^2 + 2*w - 10],\
[337, 337, 7*w^2 - 3*w - 10],\
[337, 337, 7*w^2 - 4*w - 10],\
[337, 337, -4*w^2 - 3*w + 8],\
[349, 349, 5*w^2 + w - 15],\
[349, 349, 7*w^2 - 8*w - 11],\
[349, 349, -6*w^2 + 5*w + 2],\
[379, 379, 3*w^2 + 3*w - 11],\
[379, 379, 4*w^2 + 2*w - 11],\
[379, 379, 5*w^2 + w - 14],\
[419, 419, -5*w^2 - 3*w + 9],\
[419, 419, 2*w^2 + w - 12],\
[419, 419, w^2 - 3*w - 9],\
[421, 421, 9*w^2 - 5*w - 15],\
[421, 421, 3*w^2 + 5*w - 7],\
[421, 421, 4*w^2 - w - 15],\
[433, 433, -8*w^2 + 7*w + 10],\
[433, 433, 7*w^2 + w - 13],\
[433, 433, w^2 - 8*w],\
[449, 449, -7*w^2 + 2*w + 8],\
[449, 449, 8*w^2 - 6*w - 11],\
[449, 449, -6*w^2 - 2*w + 11],\
[461, 461, 2*w^2 + 6*w - 7],\
[461, 461, -8*w^2 + 2*w + 11],\
[461, 461, 6*w^2 - 8*w - 9],\
[463, 463, -7*w^2 + 8*w + 12],\
[463, 463, w^2 + 7*w - 7],\
[463, 463, -8*w^2 + w + 10],\
[491, 491, w^2 + 2*w - 11],\
[491, 491, 4*w^2 - 7*w - 7],\
[491, 491, 2*w^2 - 3*w - 11],\
[503, 503, 5*w^2 - 7*w - 10],\
[503, 503, 2*w^2 - 2*w - 11],\
[503, 503, -7*w^2 + 2*w + 7],\
[547, 547, 4*w^2 + 3*w - 15],\
[547, 547, -3*w^2 + 10*w],\
[547, 547, -7*w^2 + 4*w + 3],\
[587, 587, 6*w^2 + w - 17],\
[587, 587, 5*w^2 - 3*w - 17],\
[587, 587, w^2 - 7*w - 6],\
[601, 601, 4*w^2 + 3*w - 14],\
[601, 601, 4*w^2 + 3*w - 12],\
[601, 601, 6*w^2 + w - 16],\
[617, 617, 5*w^2 + 2*w - 14],\
[617, 617, 4*w^2 + 3*w - 13],\
[617, 617, 5*w^2 + 2*w - 15],\
[631, 631, -9*w^2 + 8*w + 11],\
[631, 631, 8*w^2 + w - 15],\
[631, 631, w^2 - 9*w],\
[643, 643, 6*w^2 - 9*w - 8],\
[643, 643, 3*w^2 + 6*w - 8],\
[643, 643, 4*w^2 - w - 16],\
[659, 659, -8*w^2 + 2*w + 9],\
[659, 659, 9*w^2 - 7*w - 12],\
[659, 659, 7*w^2 + 2*w - 13],\
[673, 673, -8*w^2 + 9*w + 14],\
[673, 673, w^2 + 8*w - 8],\
[673, 673, -9*w^2 + w + 11],\
[701, 701, 9*w^2 - 4*w - 13],\
[701, 701, -5*w^2 - 4*w + 10],\
[701, 701, -4*w^2 - 5*w + 9],\
[727, 727, -7*w^2 - 3*w + 12],\
[727, 727, w^2 + 2*w - 12],\
[727, 727, 10*w^2 - 7*w - 15],\
[743, 743, 5*w^2 - 2*w - 18],\
[743, 743, 4*w^2 - 5*w - 16],\
[743, 743, w^2 + 4*w - 14],\
[757, 757, 11*w^2 - 7*w - 18],\
[757, 757, w^2 + 3*w - 13],\
[757, 757, 3*w^2 + 5*w - 15],\
[769, 769, 2*w^2 - 11*w + 1],\
[769, 769, -10*w^2 + w + 13],\
[769, 769, 9*w^2 + 2*w - 15],\
[797, 797, -8*w^2 + 5*w + 3],\
[797, 797, -3*w^2 + 11*w],\
[797, 797, 5*w^2 + 3*w - 18],\
[811, 811, -9*w^2 + 2*w + 11],\
[811, 811, 7*w^2 - 9*w - 12],\
[811, 811, 9*w^2 - 7*w - 11],\
[827, 827, 3*w^2 + 5*w - 14],\
[827, 827, 5*w^2 + 3*w - 13],\
[827, 827, 8*w^2 - 21],\
[839, 839, -6*w^2 - 4*w + 11],\
[839, 839, w^2 - 4*w - 11],\
[839, 839, 3*w^2 + w - 16],\
[853, 853, -8*w^2 + 7*w + 3],\
[853, 853, 7*w^2 + w - 20],\
[853, 853, w^2 - 8*w - 7],\
[881, 881, -10*w^2 + 9*w + 12],\
[881, 881, 9*w^2 + w - 17],\
[881, 881, w^2 - 10*w],\
[883, 883, 7*w^2 + w - 18],\
[883, 883, -8*w^2 + 5*w + 4],\
[883, 883, 5*w^2 + 3*w - 17],\
[911, 911, 10*w^2 - 7*w - 14],\
[911, 911, 2*w^2 + w - 14],\
[911, 911, -7*w^2 - 3*w + 13],\
[937, 937, -9*w^2 + 10*w + 16],\
[937, 937, w^2 + 9*w - 9],\
[937, 937, -10*w^2 + w + 12],\
[953, 953, 9*w^2 - 7*w - 10],\
[953, 953, -10*w^2 + 2*w + 13],\
[953, 953, 8*w^2 - 10*w - 13],\
[967, 967, 2*w^2 + 2*w - 15],\
[967, 967, 11*w^2 - 7*w - 17],\
[967, 967, -7*w^2 - 4*w + 12]]
primes = [ZF.ideal(I) for I in primes_array]

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

hecke_eigenvalues_array = [e, -3/2*e + 1/2, -1/2*e - 5/2, 2*e - 2, -1/2*e - 5/2, -2*e + 3, -2*e + 5, -e - 3, 3/2*e - 5/2, -e + 9, e - 7, e - 7, e - 5, 7/2*e - 9/2, 3, -2*e + 6, 1/2*e + 13/2, -e - 2, 2, -5/2*e + 3/2, 1, -1/2*e - 1/2, e + 8, -3*e - 1, -1/2*e + 31/2, -3*e + 15, -e - 1, e - 5, e - 3, -3/2*e - 7/2, 6*e - 2, 3*e - 7, -2*e - 8, -3*e, -5*e + 3, 2*e - 12, 4, -4*e + 9, 2*e + 2, 3*e - 6, 3*e + 10, e - 15, 2*e + 18, 7/2*e - 1/2, e - 1, 2*e - 9, -9*e + 9, -4*e + 10, e + 11, 3*e + 11, -2*e + 10, 2*e - 22, 7*e - 9, -6*e + 13, -2*e + 22, 17/2*e - 23/2, -8*e + 18, -6*e + 2, 4*e + 16, 3*e - 17, -3*e - 10, -13/2*e + 23/2, 4*e - 11, 8*e - 2, 4*e - 22, -15/2*e + 17/2, -e - 23, -e - 11, 2*e + 6, -5*e + 21, -e + 19, 6*e + 4, 3*e - 13, -15/2*e + 17/2, 3/2*e - 45/2, -3*e + 34, e - 37, 1/2*e + 57/2, -5*e + 11, -6*e - 10, 7*e + 9, 6*e - 24, -6*e + 6, 6*e - 8, 4*e + 8, -30, 8*e - 12, -9*e + 1, -3/2*e + 9/2, -4*e + 4, 4*e + 22, -10*e - 2, -3*e - 17, 4*e - 32, -7*e + 27, -8*e - 6, 6*e + 5, -8*e - 3, 6*e + 8, 8*e - 8, 4*e - 18, -2*e + 30, 7*e - 1, -5*e - 14, 4*e - 4, e - 21, 1/2*e - 43/2, -2*e - 22, -11*e + 9, -13/2*e + 15/2, 11/2*e - 13/2, 13*e - 5, 10*e - 10, 15/2*e - 21/2, 3*e + 5, 2*e - 12, 2*e - 12, -4*e + 36, 9*e - 13, 3*e + 35, -15/2*e + 33/2, -7*e - 21, e - 3, 5, 10*e - 8, 12*e - 24, -10*e - 12, -e - 27, 15/2*e + 15/2, -9*e - 4, 3*e + 37, 13/2*e - 23/2, 3*e - 45, -6*e - 2, -8*e + 14, 6*e - 16, -9*e + 9, 19, 8*e - 4, -5/2*e - 17/2, -12*e + 6, 4*e - 40, -4*e + 40, -1/2*e - 17/2, e, 23/2*e - 21/2, 7*e + 5, 9*e - 11, -6*e, 2*e + 18, -11*e - 1, -6*e + 28, 5*e - 19, 6*e + 14, -9*e + 13, e + 15, -6*e + 30, -e - 23, -7*e + 25, 13/2*e - 43/2, -e + 3, 3*e - 29, -9*e - 15, 11*e + 5, 8*e - 12, -9*e + 1, -16*e + 30, 7*e - 31, -11*e - 10]
hecke_eigenvalues = {}
for i in range(len(hecke_eigenvalues_array)):
    hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i]

AL_eigenvalues = {}
AL_eigenvalues[ZF.ideal([83,83,-2*w^2 + w - 2])] = -1

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