/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![-2, 4, 3, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {8, 2, w^4 - 7*w^2 - 3*w + 5}>;

primesArray := [
[2, 2, w],
[7, 7, -2*w^4 + w^3 + 12*w^2 + w - 5],
[8, 2, w^4 - 7*w^2 - 3*w + 5],
[19, 19, -2*w^4 + w^3 + 12*w^2 - 5],
[19, 19, -w^3 + w^2 + 5*w - 1],
[29, 29, 2*w^4 - 2*w^3 - 11*w^2 + 4*w + 3],
[31, 31, -2*w^4 + w^3 + 12*w^2 + 2*w - 5],
[37, 37, -2*w^4 + 13*w^2 + 5*w - 7],
[53, 53, 3*w^4 - 2*w^3 - 18*w^2 + 3*w + 9],
[59, 59, 2*w^4 - 13*w^2 - 6*w + 5],
[61, 61, -3*w^4 + 2*w^3 + 18*w^2 - 2*w - 11],
[61, 61, w^2 - w - 3],
[61, 61, 6*w^4 - 2*w^3 - 38*w^2 - 6*w + 23],
[67, 67, -w^4 + 7*w^2 + 4*w - 5],
[67, 67, -w^4 + 6*w^2 + 2*w - 1],
[71, 71, -w^2 + 3],
[73, 73, 3*w^4 - w^3 - 19*w^2 - 3*w + 9],
[79, 79, 3*w^4 - w^3 - 19*w^2 - 4*w + 11],
[83, 83, -w^4 - w^3 + 7*w^2 + 8*w - 3],
[97, 97, -2*w^4 + w^3 + 12*w^2 + 2*w - 3],
[97, 97, w^4 - w^3 - 5*w^2 + 2*w + 3],
[103, 103, -4*w^4 + w^3 + 25*w^2 + 7*w - 13],
[103, 103, 7*w^4 - 3*w^3 - 43*w^2 - 4*w + 23],
[107, 107, 4*w^4 - 2*w^3 - 24*w^2 - w + 9],
[113, 113, 5*w^4 - 3*w^3 - 30*w^2 + w + 15],
[137, 137, -3*w^4 + 2*w^3 + 17*w^2 - 7],
[139, 139, w^2 - 5],
[149, 149, 4*w^4 - w^3 - 26*w^2 - 5*w + 17],
[151, 151, -4*w^4 + 2*w^3 + 24*w^2 + w - 13],
[157, 157, -4*w^4 + w^3 + 26*w^2 + 5*w - 15],
[157, 157, 5*w^4 - 2*w^3 - 31*w^2 - 4*w + 15],
[157, 157, -3*w^4 + 2*w^3 + 19*w^2 - 2*w - 15],
[169, 13, -w^4 + 2*w^3 + 5*w^2 - 8*w - 3],
[173, 173, w^4 - w^3 - 5*w^2 + w - 1],
[179, 179, -2*w^4 + 2*w^3 + 12*w^2 - 5*w - 7],
[191, 191, -4*w^4 + 2*w^3 + 24*w^2 + 2*w - 11],
[193, 193, 2*w^4 - 2*w^3 - 11*w^2 + 5*w + 5],
[197, 197, w^4 - 6*w^2 - w + 1],
[197, 197, -w^4 + 8*w^2 + 2*w - 11],
[199, 199, -3*w^4 + w^3 + 20*w^2 + 3*w - 17],
[211, 211, 4*w^4 - 2*w^3 - 25*w^2 + 13],
[223, 223, w^4 - 6*w^2 - 3*w - 1],
[223, 223, 2*w^4 - 13*w^2 - 4*w + 9],
[223, 223, -2*w^4 + 2*w^3 + 11*w^2 - 3*w - 3],
[227, 227, 2*w^4 - 12*w^2 - 7*w + 5],
[227, 227, -w^4 + 8*w^2 + w - 7],
[229, 229, -2*w^4 - w^3 + 14*w^2 + 11*w - 11],
[229, 229, -3*w^4 + 19*w^2 + 8*w - 11],
[233, 233, -w^4 + 8*w^2 + 2*w - 7],
[233, 233, 3*w^4 - w^3 - 20*w^2 - w + 11],
[239, 239, -w^4 + 6*w^2 + 5*w - 3],
[243, 3, -3],
[251, 251, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[263, 263, 3*w^4 - 2*w^3 - 18*w^2 + 2*w + 7],
[269, 269, -3*w^4 + 20*w^2 + 8*w - 11],
[271, 271, 4*w^4 - 2*w^3 - 25*w^2 - w + 13],
[283, 283, -w^2 + 3*w + 1],
[283, 283, 6*w^4 - 4*w^3 - 36*w^2 + 3*w + 19],
[283, 283, -6*w^4 + 3*w^3 + 36*w^2 + 2*w - 17],
[289, 17, -8*w^4 + 2*w^3 + 49*w^2 + 12*w - 21],
[293, 293, -3*w^4 + 2*w^3 + 17*w^2 - 2*w - 7],
[293, 293, 2*w^4 - 2*w^3 - 12*w^2 + 4*w + 7],
[307, 307, -2*w^4 + 13*w^2 + 7*w - 9],
[311, 311, -w^4 + 5*w^2 + 4*w + 3],
[311, 311, 3*w^4 - w^3 - 18*w^2 - 2*w + 7],
[317, 317, -w^4 + w^3 + 6*w^2 - 4*w - 3],
[337, 337, 4*w^4 - w^3 - 25*w^2 - 7*w + 15],
[347, 347, -2*w^4 + w^3 + 13*w^2 + w - 11],
[347, 347, -3*w^4 + w^3 + 20*w^2 + 2*w - 17],
[349, 349, -w^3 + w^2 + 3*w + 3],
[359, 359, 3*w^4 - w^3 - 18*w^2 - 3*w + 11],
[367, 367, -w^4 - w^3 + 8*w^2 + 6*w - 7],
[367, 367, 4*w^4 - 3*w^3 - 24*w^2 + 3*w + 11],
[373, 373, -w^3 + 6*w + 1],
[379, 379, 2*w^4 - 14*w^2 - 5*w + 9],
[397, 397, -2*w^4 - w^3 + 15*w^2 + 11*w - 13],
[397, 397, w^4 - 6*w^2 - 2*w - 1],
[397, 397, -w^4 - w^3 + 8*w^2 + 8*w - 5],
[401, 401, -2*w^4 + 14*w^2 + 7*w - 13],
[409, 409, -4*w^4 + 2*w^3 + 24*w^2 - 11],
[421, 421, 3*w^4 - 2*w^3 - 18*w^2 + 9],
[433, 433, -2*w^4 + 2*w^3 + 12*w^2 - 5*w - 5],
[433, 433, 6*w^4 - w^3 - 38*w^2 - 12*w + 21],
[439, 439, -5*w^4 + 2*w^3 + 30*w^2 + 3*w - 15],
[443, 443, 4*w^4 - 2*w^3 - 26*w^2 - w + 15],
[443, 443, -w^4 + w^3 + 5*w^2 - w - 5],
[443, 443, -w^4 + w^3 + 6*w^2 - 4*w - 5],
[443, 443, -w^4 + 6*w^2 + 5*w - 1],
[443, 443, w^4 + w^3 - 7*w^2 - 7*w + 1],
[461, 461, 2*w^4 - 14*w^2 - 6*w + 9],
[461, 461, -w^4 + w^3 + 4*w^2 - 2*w + 5],
[463, 463, w^4 - 7*w^2 - 2*w + 1],
[479, 479, -6*w^4 + 3*w^3 + 37*w^2 + w - 19],
[499, 499, -4*w^4 + 25*w^2 + 11*w - 13],
[499, 499, 5*w^4 - 3*w^3 - 31*w^2 + 17],
[503, 503, -4*w^4 + 3*w^3 + 23*w^2 - 3*w - 7],
[503, 503, -3*w^4 + w^3 + 18*w^2 + w - 9],
[503, 503, 2*w^4 - w^3 - 14*w^2 - w + 11],
[509, 509, 8*w^4 - 2*w^3 - 52*w^2 - 11*w + 35],
[521, 521, -2*w^4 + 2*w^3 + 13*w^2 - 5*w - 11],
[521, 521, 10*w^4 - 4*w^3 - 63*w^2 - 6*w + 39],
[529, 23, -w^4 - w^3 + 7*w^2 + 7*w - 3],
[547, 547, 9*w^4 - 5*w^3 - 55*w^2 + w + 29],
[547, 547, -w^4 + 2*w^3 + 6*w^2 - 9*w - 7],
[557, 557, -5*w^4 + w^3 + 32*w^2 + 6*w - 17],
[563, 563, 4*w^4 - w^3 - 26*w^2 - 4*w + 19],
[587, 587, -2*w^3 + 3*w^2 + 10*w - 5],
[587, 587, 7*w^4 - w^3 - 45*w^2 - 15*w + 25],
[593, 593, -9*w^4 + 3*w^3 + 57*w^2 + 10*w - 33],
[599, 599, w^4 + w^3 - 6*w^2 - 7*w + 3],
[601, 601, w^4 - 2*w^3 - 4*w^2 + 9*w - 1],
[601, 601, 3*w^4 - 3*w^3 - 16*w^2 + 5*w + 3],
[607, 607, 6*w^4 - 2*w^3 - 38*w^2 - 5*w + 23],
[607, 607, 4*w^4 - 2*w^3 - 24*w^2 - 3*w + 11],
[613, 613, -w^4 + w^3 + 7*w^2 - 3*w - 5],
[641, 641, 2*w^4 - 2*w^3 - 11*w^2 + 4*w + 7],
[641, 641, -w^3 + 6*w - 1],
[647, 647, 7*w^4 - 3*w^3 - 45*w^2 - 2*w + 31],
[659, 659, -3*w^4 + w^3 + 19*w^2 + w - 9],
[661, 661, -2*w^4 + 15*w^2 + 5*w - 13],
[683, 683, -w^4 + 8*w^2 + 2*w - 9],
[683, 683, -4*w^4 + 3*w^3 + 24*w^2 - 4*w - 17],
[683, 683, 6*w^4 - 3*w^3 - 36*w^2 - w + 15],
[691, 691, -3*w^4 + w^3 + 18*w^2 + 4*w - 11],
[709, 709, w^4 - w^3 - 4*w^2 + 4*w - 3],
[733, 733, -5*w^4 + 2*w^3 + 30*w^2 + 2*w - 15],
[743, 743, -9*w^4 + 5*w^3 + 55*w^2 - 31],
[751, 751, -4*w^4 + 2*w^3 + 23*w^2 + 2*w - 9],
[751, 751, 5*w^4 - w^3 - 32*w^2 - 8*w + 19],
[761, 761, w^4 + w^3 - 7*w^2 - 6*w + 3],
[761, 761, -6*w^4 + w^3 + 38*w^2 + 11*w - 19],
[787, 787, -w^3 + w^2 + 3*w - 5],
[787, 787, -6*w^4 + 3*w^3 + 35*w^2 + 3*w - 13],
[809, 809, -7*w^4 + 5*w^3 + 42*w^2 - 6*w - 23],
[821, 821, 3*w^4 - 2*w^3 - 18*w^2 + 2*w + 5],
[827, 827, 7*w^4 - 3*w^3 - 43*w^2 - 3*w + 23],
[829, 829, 4*w^4 - w^3 - 26*w^2 - 7*w + 19],
[829, 829, 5*w^4 - 2*w^3 - 30*w^2 - 4*w + 17],
[841, 29, w^4 - w^3 - 5*w^2 - 1],
[841, 29, 6*w^4 - 2*w^3 - 38*w^2 - 6*w + 19],
[853, 853, 3*w^4 - 20*w^2 - 7*w + 13],
[857, 857, 2*w^3 - 3*w^2 - 8*w + 3],
[859, 859, -5*w^4 + 2*w^3 + 33*w^2 + 2*w - 27],
[863, 863, -3*w^4 + 2*w^3 + 17*w^2 - 3],
[881, 881, -5*w^4 + w^3 + 32*w^2 + 10*w - 21],
[883, 883, w^2 - 2*w - 5],
[887, 887, 3*w^4 + w^3 - 21*w^2 - 12*w + 13],
[911, 911, -4*w^4 + w^3 + 26*w^2 + 7*w - 17],
[911, 911, 6*w^4 - 2*w^3 - 37*w^2 - 5*w + 19],
[919, 919, 14*w^4 - 3*w^3 - 86*w^2 - 26*w + 37],
[919, 919, -7*w^4 + 4*w^3 + 42*w^2 - 21],
[937, 937, 3*w^4 - 2*w^3 - 19*w^2 + 4*w + 13],
[947, 947, 6*w^4 - 3*w^3 - 36*w^2 - 2*w + 19],
[953, 953, w^4 - 2*w^3 - 3*w^2 + 6*w - 5],
[953, 953, -9*w^4 + 3*w^3 + 56*w^2 + 10*w - 31],
[977, 977, -5*w^4 + w^3 + 33*w^2 + 7*w - 23],
[977, 977, -5*w^4 + 2*w^3 + 31*w^2 + 2*w - 13],
[997, 997, 6*w^4 - w^3 - 38*w^2 - 13*w + 21],
[997, 997, 4*w^4 - 2*w^3 - 24*w^2 - 4*w + 13]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + x^3 - 5*x^2 - x + 3;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e - 1, -1, -e^3 - e^2 + 6*e - 1, -e^2 - e + 2, -e^3 - 2*e^2 + 3, e^3 + 2*e^2 - 4*e - 4, e^3 + 4*e^2 - 3*e - 10, -3*e^3 - 4*e^2 + 11*e + 3, 2*e^3 + 6*e^2 - 3*e - 12, e^3 - 2*e^2 - 4*e + 8, 2*e^3 - 12*e + 5, 3*e^3 + 4*e^2 - 9*e - 4, -5*e^3 - 9*e^2 + 17*e + 5, -3*e^3 - 7*e^2 + 8*e + 8, 3*e^3 + 9*e^2 - 7*e - 15, 2*e^3 + 3*e^2 - 5*e - 4, -3*e^3 - 3*e^2 + 19*e + 2, 2*e^3 + 4*e^2 - 2*e - 9, 3*e^3 + 3*e^2 - 14*e - 4, -4*e^3 - 6*e^2 + 12*e + 14, 5*e^3 + 11*e^2 - 12*e - 16, -3*e^3 - 5*e^2 + 7*e + 8, -e^3 - 6*e^2 + 6*e + 18, 3*e^3 + 4*e^2 - 14*e - 6, -5*e^2 - 9*e + 12, 2*e^3 - 3*e^2 - 13*e + 5, -2*e^3 + e^2 + 2*e - 18, 2*e^3 + 7*e^2 + 4*e - 16, e^3 + 2*e^2 + 4*e - 10, -3*e^2 - 2*e - 1, -4*e^2 - 4*e + 5, e^3 + 4*e^2 - 3*e - 19, 4*e^3 + 14*e^2 - 7*e - 30, -7*e^3 - 6*e^2 + 26*e + 3, 2*e^3 - 3*e^2 - 8*e + 9, e^3 - e^2 - 9*e - 7, e^3 + 3*e^2 - 9*e - 18, -4*e^3 - 9*e^2 + 14*e + 12, -3*e^3 - 7*e^2 + 7*e + 5, 2*e^3 + 12*e^2 + 4*e - 25, 5*e^3 + 16*e^2 - 11*e - 31, -3*e^3 - 10*e^2 + 9*e + 17, 3*e^3 + 2*e^2 - 20*e - 13, -e - 12, -5*e^3 - 11*e^2 + 12*e + 12, 2*e^2 - 2*e - 10, 4*e^3 + 6*e^2 - 12*e - 16, 3*e^3 - 6*e + 9, 5*e^3 + 3*e^2 - 15*e, -6*e^3 - 5*e^2 + 26*e, -4*e^3 - 5*e^2 + 19*e - 2, -e^3 - 7*e^2 - 10*e + 12, 11*e^3 + 13*e^2 - 35*e - 9, 4*e^3 + 8*e^2 - 15*e - 9, -9*e^3 - 10*e^2 + 37*e + 5, -e^3 + e^2 + 8*e - 13, 4*e^3 - 20*e - 1, e^3 - 8*e^2 - 17*e + 17, -4*e^2 - 3*e + 11, -e^3 + 2*e^2 + 13*e + 3, -e^3 - 10*e^2 - 3*e + 15, -9*e^3 - 5*e^2 + 36*e + 2, 10*e^3 + 10*e^2 - 37*e - 9, e^2 + 6*e + 15, 4*e^3 + 14*e^2 - 13*e - 30, -2*e^3 + e^2 - 22, 11*e^3 + 19*e^2 - 27*e - 30, -3*e^3 - e^2 + 11*e + 18, 4*e^3 - 18*e + 5, -8*e^3 - 15*e^2 + 33*e + 9, -3*e^3 + 7*e - 7, -4*e^3 - 11*e^2 + 7*e + 32, 2*e^3 - 2*e^2 - 22*e + 5, -8*e^3 - 6*e^2 + 30*e - 1, -8*e^3 - 8*e^2 + 27*e + 8, e^3 - 2*e^2 - 12*e + 20, -3*e^3 + 2*e^2 + 9*e - 25, -3*e^3 + e^2 + 7*e - 6, -3*e^3 - 5*e^2 + 5*e + 23, -7*e^3 - 11*e^2 + 29*e + 17, -3*e^3 - 9*e^2 - 7*e + 20, 10*e^3 + 11*e^2 - 26*e - 1, -9*e^3 - 9*e^2 + 52*e + 5, -4*e^3 - 2*e^2 + 30*e + 3, e^3 - 8*e^2 - 19*e + 24, -3*e^3 - 14*e^2 - 6*e + 30, 3*e^3 + 10*e^2 - 4*e - 3, -7*e^2 - 2*e + 36, -11*e^3 - 11*e^2 + 32*e - 3, -8*e^3 - e^2 + 38*e - 12, -4*e^3 - 11*e^2 + 6*e + 11, 10*e^3 + 12*e^2 - 29*e, -5*e^3 + 3*e^2 + 18*e - 25, -2*e^3 - 2*e^2 + 15*e + 2, -3*e^3 + 3*e^2 + 14*e - 33, -2*e^3 - 2*e^2 + 11*e + 3, 11*e^3 + 13*e^2 - 38*e - 3, 2*e^3 + 3*e^2 + e - 15, 9*e^3 + 8*e^2 - 35*e - 12, -3*e^3 + 2*e^2 + 9*e, -8*e^3 - 15*e^2 + 38*e + 17, -2*e^3 - e^2 + 8*e - 19, 13*e^3 + 18*e^2 - 47*e - 16, 6*e^3 + 8*e^2 - 20*e - 39, -15*e^3 - 18*e^2 + 45*e + 15, 4*e^3 + 5*e^2 - 10*e + 24, -6*e^2 - 2*e + 9, 9*e^3 + 2*e^2 - 48*e, 5*e^3 + 13*e^2 - 16*e - 18, -14*e^3 - 19*e^2 + 38*e + 17, -9*e^3 - 18*e^2 + 21*e + 26, 6*e^2 + 9*e - 40, 2*e^3 + 7*e^2 + 3*e + 2, 6*e^3 + 5*e^2 - 28*e - 13, 11*e^3 + 12*e^2 - 48*e - 6, 2*e^3 + 9*e^2 - 6*e - 15, 5*e^3 + 4*e^2 - 21*e + 9, -6*e^3 - 19*e^2 + 5*e + 24, -13*e^3 - 11*e^2 + 41*e - 4, 5*e^3 - e^2 - 17*e + 6, e^3 + e^2 - 14*e - 30, 3*e^3 + 7*e^2 - 3*e + 9, -2*e^3 - e^2 + 15*e - 4, 4*e^3 + 10*e^2 - 10*e - 16, 8*e^3 + 15*e^2 - 31*e - 31, 8*e^3 + e^2 - 45*e - 3, 5*e^3 + 8*e^2 - 10*e - 31, 4*e^3 + 7*e^2 - 28*e - 31, -14*e^3 - 19*e^2 + 43*e + 30, 8*e^3 + 16*e^2 - 26*e - 24, -19*e^3 - 23*e^2 + 69*e + 20, -3*e^3 - 12*e^2 - 2*e + 35, -13*e^3 - 17*e^2 + 46*e - 12, -11*e^3 - 10*e^2 + 39*e + 3, 2*e^3 - 2*e^2 - 12*e + 18, -6*e^3 - 6*e^2 + 15*e + 2, 3*e^3 + 16*e^2 + 13*e - 37, 7*e^3 + 2*e^2 - 45*e + 2, 9*e^3 + 2*e^2 - 25*e + 23, 2*e^3 - 18*e - 31, -12*e^3 - 22*e^2 + 42*e + 39, 14*e^3 + 19*e^2 - 52*e - 10, -4*e^3 + 2*e^2 + 25*e - 21, -8*e^3 - 12*e^2 + 22*e - 3, -7*e^3 - 9*e^2 + 29*e + 20, -11*e^3 - 25*e^2 + 41*e + 36, 2*e^3 - 8*e^2 - 13*e + 33, -13*e^3 - 7*e^2 + 49*e - 9, 2*e^3 + 18*e^2 + 7*e - 31, 6*e^3 + 10*e^2 - 20*e - 13, 21*e^3 + 32*e^2 - 68*e - 22, -3*e^3 + 7*e^2 + 38*e - 12, 7*e^3 + 8*e^2 - 24*e + 9, -10*e^3 - 30*e^2 + 11*e + 57, e^3 - 10*e, 10*e^3 + 3*e^2 - 57*e - 3, -10*e^3 - 12*e^2 + 37*e + 2, -5*e^2 - 21*e + 5];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {8, 2, w^4 - 7*w^2 - 3*w + 5}>] := 1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;