/*
  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![12, -9, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {12, 6, w^2 + w - 9}>;

primesArray := [
[2, 2, w - 2],
[3, 3, -w + 3],
[3, 3, w - 1],
[4, 2, w^2 + w - 7],
[19, 19, w + 1],
[23, 23, w^2 - 2*w - 1],
[31, 31, -2*w^2 + 19],
[31, 31, -w^2 + 5],
[31, 31, -3*w + 5],
[41, 41, w^2 + 2*w - 7],
[43, 43, w^2 - 11],
[47, 47, 3*w - 7],
[53, 53, -3*w^2 - 6*w + 11],
[59, 59, 2*w - 1],
[67, 67, 2*w^2 + w - 19],
[67, 67, 3*w^2 + 2*w - 25],
[67, 67, w - 5],
[73, 73, -4*w^2 - 3*w + 29],
[73, 73, 2*w^2 - w - 11],
[73, 73, w^2 + 2*w - 11],
[79, 79, 2*w^2 + 2*w - 11],
[83, 83, 2*w^2 - 13],
[89, 89, -2*w + 7],
[97, 97, -2*w^2 + 4*w - 1],
[101, 101, -2*w - 5],
[103, 103, 2*w^2 + w - 13],
[107, 107, -2*w^2 - 2*w + 17],
[109, 109, -4*w^2 - 2*w + 31],
[113, 113, w^2 + 2*w - 1],
[125, 5, -5],
[127, 127, 2*w^2 + 3*w - 13],
[137, 137, -2*w^2 + 7*w - 7],
[139, 139, 2*w^2 + w - 7],
[149, 149, -6*w^2 - 9*w + 31],
[157, 157, w^2 - 4*w + 1],
[163, 163, 2*w^2 + 5*w - 5],
[173, 173, 5*w^2 + 4*w - 37],
[179, 179, 2*w^2 + w - 11],
[179, 179, 3*w^2 - 2*w - 17],
[179, 179, -4*w^2 - w + 37],
[191, 191, -6*w^2 - 9*w + 29],
[193, 193, -2*w^2 - 4*w + 5],
[223, 223, -2*w^2 + 2*w + 7],
[227, 227, 6*w^2 + 4*w - 47],
[233, 233, 2*w^2 + 4*w - 13],
[233, 233, -4*w^2 - 4*w + 25],
[233, 233, 4*w + 13],
[239, 239, -4*w^2 - 8*w + 13],
[239, 239, 2*w^2 + 2*w - 5],
[239, 239, -w^2 + 8*w - 11],
[241, 241, 3*w - 1],
[251, 251, w^2 + 4*w - 19],
[263, 263, -2*w^2 + 2*w + 25],
[263, 263, -5*w^2 - 4*w + 35],
[263, 263, 3*w^2 - 25],
[269, 269, 6*w^2 + 3*w - 47],
[269, 269, -w^2 - 1],
[269, 269, 2*w^2 - 7],
[277, 277, 2*w^2 - w - 5],
[277, 277, -5*w^2 - 6*w + 31],
[277, 277, 2*w^2 - 2*w - 11],
[281, 281, 4*w^2 + 5*w - 25],
[307, 307, 4*w^2 + 2*w - 37],
[317, 317, -w - 7],
[317, 317, 3*w^2 + 2*w - 19],
[317, 317, -2*w^2 + 2*w + 5],
[331, 331, 2*w^2 + 3*w - 19],
[337, 337, w^2 - 4*w + 7],
[343, 7, -7],
[347, 347, 4*w^2 + 2*w - 29],
[349, 349, 3*w^2 - 19],
[361, 19, w^2 - 2*w - 7],
[367, 367, 4*w^2 + 5*w - 19],
[367, 367, -6*w + 11],
[373, 373, -4*w^2 + w + 25],
[379, 379, -5*w^2 - 10*w + 19],
[383, 383, 4*w^2 + 6*w - 23],
[401, 401, -3*w + 11],
[401, 401, -3*w - 1],
[401, 401, -3*w - 7],
[409, 409, -w^2 - 4*w + 13],
[419, 419, 2*w^2 - w - 17],
[421, 421, -3*w^2 + 31],
[431, 431, w^2 + 4*w - 1],
[449, 449, -w^2 + 4*w + 1],
[457, 457, -w^2 + 2*w + 13],
[461, 461, 4*w^2 - 4*w - 19],
[463, 463, 2*w^2 + 4*w - 23],
[467, 467, 2*w^2 - 5*w + 5],
[479, 479, 3*w + 11],
[491, 491, 8*w^2 + 9*w - 49],
[499, 499, -6*w^2 - 6*w + 41],
[499, 499, 2*w^2 + 2*w - 23],
[499, 499, 2*w^2 + 5*w - 29],
[523, 523, -4*w^2 + w + 41],
[529, 23, 6*w^2 + 7*w - 35],
[547, 547, -2*w^2 + 6*w + 1],
[557, 557, 3*w^2 + 2*w - 13],
[563, 563, -w^2 + 4*w + 23],
[569, 569, 10*w^2 + 8*w - 73],
[587, 587, 2*w^2 - 2*w - 13],
[593, 593, -6*w^2 - 8*w + 31],
[601, 601, 4*w^2 + 5*w - 41],
[601, 601, 10*w^2 + 16*w - 49],
[601, 601, w^2 - 2*w + 5],
[613, 613, -8*w^2 - 7*w + 55],
[617, 617, -8*w^2 - 12*w + 37],
[617, 617, 2*w^2 + 3*w - 25],
[617, 617, 3*w^2 - 17],
[641, 641, 4*w^2 - 35],
[643, 643, 4*w^2 + 4*w - 37],
[659, 659, -3*w^2 - 4*w + 23],
[661, 661, -w^2 - 4*w - 5],
[673, 673, 4*w^2 + 4*w - 31],
[683, 683, 2*w^2 + 4*w - 19],
[683, 683, 2*w^2 + w - 23],
[683, 683, 3*w^2 + 6*w - 19],
[701, 701, -6*w^2 - 12*w + 23],
[701, 701, -2*w^2 + 7*w - 1],
[701, 701, 3*w^2 + 2*w - 31],
[709, 709, 5*w^2 + 4*w - 41],
[719, 719, -6*w^2 - 2*w + 55],
[719, 719, w^2 + 2*w - 17],
[719, 719, 3*w^2 + 4*w - 29],
[727, 727, -9*w^2 - 12*w + 49],
[733, 733, 13*w^2 + 20*w - 65],
[739, 739, 11*w^2 + 6*w - 91],
[743, 743, -2*w^2 - 8*w + 35],
[751, 751, 3*w^2 - 4*w - 13],
[769, 769, 2*w^2 - 1],
[797, 797, 2*w^2 - 8*w + 11],
[811, 811, 4*w^2 - w - 29],
[821, 821, 2*w^2 + 5*w - 17],
[827, 827, -3*w^2 - 6*w + 7],
[827, 827, -11*w^2 - 8*w + 83],
[827, 827, -5*w^2 - 6*w + 25],
[829, 829, -4*w^2 + 41],
[829, 829, 4*w^2 + 8*w - 23],
[829, 829, 8*w - 19],
[839, 839, 2*w^2 + 6*w - 7],
[853, 853, 10*w^2 + 6*w - 77],
[859, 859, -5*w^2 - 2*w + 47],
[863, 863, 12*w^2 + 19*w - 59],
[877, 877, 8*w^2 + 6*w - 61],
[907, 907, -4*w - 1],
[907, 907, 9*w^2 + 4*w - 71],
[907, 907, 2*w^2 + 7*w - 17],
[919, 919, -10*w^2 - 5*w + 79],
[929, 929, 2*w^2 + 7*w - 11],
[937, 937, -4*w^2 + 6*w + 11],
[941, 941, 3*w^2 - 11],
[947, 947, 6*w - 5],
[967, 967, -2*w^2 + 5*w + 5],
[971, 971, 5*w^2 - 2*w - 29],
[991, 991, 4*w^2 + 2*w - 25],
[997, 997, 3*w^2 - 2*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 - x^4 - 8*x^3 + 6*x^2 + 13*x - 3;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1, 1/2*e^4 - 3*e^2 + 5/2, 1, -e^2 + 5, e^3 - 2*e^2 - 5*e + 6, -3/2*e^4 + 9*e^2 - e - 5/2, -1/2*e^4 + e^3 + 3*e^2 - 5*e - 5/2, 1/2*e^4 - 2*e^3 - 4*e^2 + 11*e + 13/2, -e^3 + 3*e, -1/2*e^4 - e^3 + 2*e^2 + 4*e + 7/2, -1/2*e^4 + e^3 - 4*e + 15/2, 1/2*e^4 + 2*e^3 - 2*e^2 - 10*e - 9/2, 3/2*e^4 - e^3 - 8*e^2 + 3*e + 9/2, -e^4 - e^3 + 8*e^2 + 7*e - 13, -e^4 + e^3 + 8*e^2 - 9*e - 7, 1/2*e^4 - 2*e^2 + 4*e - 5/2, -e^4 - 2*e^3 + 7*e^2 + 10*e - 4, 3/2*e^4 - 3*e^3 - 11*e^2 + 14*e + 25/2, 2*e^3 + 2*e^2 - 14*e - 4, 3*e^3 - 13*e + 2, 1/2*e^4 - 5*e^2 - 3*e + 15/2, -2*e^3 + e^2 + 14*e - 3, -2*e^4 + 14*e^2 - 10, -3*e^3 + 2*e^2 + 13*e - 6, -2*e^4 + 10*e^2 - 2*e + 2, 1/2*e^4 - e^3 - e^2 + 8*e + 3/2, -1/2*e^4 + e^3 - 3*e + 25/2, -3/2*e^4 + 4*e^3 + 11*e^2 - 21*e - 21/2, -e^2 + 4*e + 3, 3/2*e^4 - e^3 - 4*e^2 + 3*e - 23/2, 3/2*e^4 + e^3 - 9*e^2 - 3*e + 15/2, e^4 - 2*e^3 - 7*e^2 + 6*e + 14, -3/2*e^4 + e^3 + 12*e^2 - 4*e - 27/2, 1/2*e^4 - 4*e^3 - 5*e^2 + 22*e + 25/2, 2*e^4 + 3*e^3 - 12*e^2 - 19*e + 14, e^2 - 6*e + 3, -e^3 - 2*e^2 + 7*e, -e^4 - e^3 + 6*e^2 + 3*e - 3, -3*e^4 - 4*e^3 + 22*e^2 + 16*e - 27, 3*e^4 - e^3 - 16*e^2 - e + 3, -2*e^3 - e^2 + 10*e + 11, 2*e^4 - 13*e^2 + 4*e + 23, 2*e^4 - 6*e^3 - 15*e^2 + 26*e + 21, 5/2*e^4 - 2*e^3 - 20*e^2 + 12*e + 27/2, e^4 - 2*e^3 - 8*e^2 + 14*e + 21, e^4 - 2*e^3 - 6*e^2 + 6*e + 3, -5/2*e^4 + 12*e^2 + 3*e + 15/2, e^4 + 3*e^3 - 17*e - 15, -2*e^4 + 2*e^3 + 12*e^2 - 8*e - 12, e^4 + 6*e^3 - 4*e^2 - 32*e - 1, -2*e^4 + 2*e^3 + 11*e^2 - 6*e - 9, 1/2*e^4 - 9*e^2 - 4*e + 33/2, -2*e^4 - 3*e^3 + 6*e^2 + 21*e + 18, e^4 + e^3 - e - 21, 3/2*e^4 - e^3 - 7*e^2 + 4*e + 9/2, 1/2*e^4 - e^3 + 10*e - 15/2, e^4 + 2*e^3 - 9*e^2 - 12*e + 24, e^4 - 2*e^3 - 5*e^2 + 6*e + 14, 3*e^4 - 2*e^3 - 20*e^2 + 14*e + 11, -e^4 - 2*e^3 + 10*e^2 + 6*e - 7, -7/2*e^4 + 3*e^3 + 22*e^2 - 17*e - 45/2, -2*e^4 + 3*e^3 + 14*e^2 - 15*e - 4, -3/2*e^4 + 5*e^3 + 6*e^2 - 25*e + 3/2, -4*e^4 - 2*e^3 + 29*e^2 + 4*e - 21, -3*e^4 + e^3 + 22*e^2 - 7*e - 27, 2*e^4 - 6*e^3 - 14*e^2 + 32*e + 14, -1/2*e^4 + 2*e^3 + 6*e^2 - 6*e - 23/2, -1/2*e^4 + 5*e^3 - e^2 - 21*e + 43/2, -1/2*e^4 - e^3 + 8*e^2 + 9*e - 63/2, 3/2*e^4 - 3*e^3 - 11*e^2 + 25*e + 43/2, -e^4 + 6*e^3 + 4*e^2 - 26*e + 11, -3/2*e^4 + 2*e^2 + 2*e + 19/2, -e^4 - e^3 + 12*e^2 + 9*e - 31, -1/2*e^4 - 7*e^3 - 2*e^2 + 38*e + 43/2, 3/2*e^4 - 3*e^3 - 11*e^2 + 14*e + 13/2, -2*e^4 + 10*e^2 - 6*e + 6, 3*e^4 - e^3 - 16*e^2 + 7*e + 9, -5*e^4 - 5*e^3 + 38*e^2 + 23*e - 45, -3*e^4 + 26*e^2 - 4*e - 33, e^4 + 9*e^3 - 49*e - 19, 2*e^4 - e^3 - 8*e^2 - 3*e - 18, -e^4 - 6*e^3 + 10*e^2 + 26*e - 7, 1/2*e^4 - 3*e^2 - 3/2, -e^4 + 2*e^3 + 4*e^2 - 8*e - 3, 4*e^4 - 2*e^3 - 32*e^2 + 6*e + 26, 6*e^4 - 42*e^2 + 42, 5/2*e^4 + 2*e^3 - 20*e^2 - 10*e + 43/2, -4*e^4 - 2*e^3 + 29*e^2 + 20*e - 39, -2*e^4 - 8*e^3 + 16*e^2 + 36*e - 18, -1/2*e^4 + 2*e^2 - 5*e + 39/2, -3/2*e^4 + e^3 + 12*e^2 - 8*e - 23/2, 3*e^4 - 22*e^2 + 6*e + 41, 3*e^4 - 2*e^3 - 6*e^2 + 8*e - 31, -2*e^4 + 4*e^3 + 14*e^2 - 12*e - 16, -e^4 + 6*e^3 + 16*e^2 - 30*e - 37, 2*e^3 + 5*e^2 - 6*e - 13, -13/2*e^4 - e^3 + 38*e^2 - 9/2, -4*e^3 + 2*e^2 + 10*e - 12, -4*e^3 - 4*e^2 + 16*e + 18, -3/2*e^4 + 9*e^3 + 11*e^2 - 37*e - 27/2, -1/2*e^4 - e^3 + 9*e^2 + 11*e - 57/2, 3*e^4 + 6*e^3 - 10*e^2 - 34*e - 19, -6*e^4 + 6*e^3 + 39*e^2 - 24*e - 37, 1/2*e^4 + 4*e^3 - 11*e^2 - 28*e + 49/2, 3*e^4 + 9*e^3 - 22*e^2 - 43*e + 23, -2*e^4 - 2*e^3 + 18*e^2 + 20*e - 24, 1/2*e^4 - 3*e^3 - 2*e^2 + 16*e - 27/2, -9/2*e^4 + 7*e^3 + 31*e^2 - 43*e - 81/2, -5*e^4 + 2*e^3 + 34*e^2 - 8*e - 45, 2*e^4 - 22*e^2 - 8*e + 32, 2*e^4 + 4*e^3 - 9*e^2 - 26*e - 15, 8*e^4 + e^3 - 52*e^2 + 5*e + 32, 6*e^4 + 9*e^3 - 44*e^2 - 37*e + 44, 4*e^4 + 5*e^3 - 28*e^2 - 15*e + 30, 2*e^4 + 6*e^3 - 8*e^2 - 22*e - 30, 5/2*e^4 + 4*e^3 - 11*e^2 - 25*e + 27/2, -11/2*e^4 - e^3 + 33*e^2 + 15*e - 63/2, -e^4 + 2*e^3 + 6*e^2 - 2*e - 15, -3*e^4 - 3*e^3 + 14*e^2 + 9*e + 9, 4*e^4 + 3*e^3 - 30*e^2 - 29*e + 50, -3*e^4 - 12*e^3 + 22*e^2 + 56*e - 15, -5*e^4 - 7*e^3 + 36*e^2 + 35*e - 39, 13/2*e^4 - e^3 - 46*e^2 + 12*e + 81/2, -2*e^2 - 12*e + 14, 6*e^4 + 7*e^3 - 42*e^2 - 35*e + 38, -3/2*e^4 - 2*e^3 + 8*e^2 + 9*e - 11/2, -10*e^3 - 6*e^2 + 62*e + 18, 8*e^4 - 51*e^2 + 4*e + 47, 6*e^4 - 2*e^3 - 26*e^2 + 8*e - 16, -9*e^3 - 6*e^2 + 47*e + 6, -5*e^4 - e^3 + 28*e^2 - 3*e - 7, 2*e^4 - 2*e^3 - 10*e^2 - 2*e - 6, 4*e^3 - 3*e^2 - 12*e - 9, -5/2*e^4 + 6*e^3 + 22*e^2 - 38*e - 39/2, -2*e^4 + 2*e^3 + 24*e^2 - 6*e - 54, -e^4 + e^3 + 2*e^2 - 7*e - 25, -3*e^4 - 4*e^3 + 20*e^2 + 18*e - 1, -7*e^4 + 3*e^3 + 38*e^2 - 9*e + 5, 2*e^4 + 6*e^3 - 24*e^2 - 30*e + 54, -3/2*e^4 - 5*e^3 + 17*e^2 + 23*e - 95/2, -9/2*e^4 + 6*e^3 + 21*e^2 - 19*e + 25/2, 11/2*e^4 - 3*e^3 - 36*e^2 + 23*e + 93/2, -9/2*e^4 + 4*e^3 + 31*e^2 - 6*e - 77/2, 3*e^4 + 5*e^3 - 18*e^2 - 31*e - 7, -e^4 + 2*e^3 + 10*e^2 - 28*e - 19, -2*e^3 + 10*e^2 + 6*e - 10, 11/2*e^4 - 8*e^3 - 27*e^2 + 38*e + 7/2, e^4 + 7*e^3 - 39*e - 27, -4*e^4 + 8*e^3 + 26*e^2 - 40*e - 4, 2*e^4 - 12*e^3 - 16*e^2 + 50*e + 30, -15/2*e^4 + e^3 + 49*e^2 - 4*e - 93/2, -9*e^4 + 2*e^3 + 52*e^2 - 6*e - 31, 9/2*e^4 + 2*e^3 - 32*e^2 + 2*e + 63/2, -2*e^4 - 2*e^3 + 8*e^2 + 26*e + 2, -11/2*e^4 + e^3 + 33*e^2 + 6*e - 17/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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