/*
  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![-1, -3, 6, 4, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {53,53,w^5 - w^4 - 4*w^3 + 4*w^2 + 3*w - 1}>;

primesArray := [
[13, 13, w^5 - 5*w^3 + 4*w],
[25, 5, w^5 - 5*w^3 + 6*w - 1],
[25, 5, -w^3 + w^2 + 3*w - 1],
[25, 5, w^5 - 4*w^3 - w^2 + 3*w + 2],
[27, 3, w^4 - w^3 - 4*w^2 + 2*w + 2],
[27, 3, w^4 - w^3 - 4*w^2 + 2*w + 3],
[53, 53, -w^4 + w^3 + 3*w^2 - 2*w + 1],
[53, 53, -w^4 + w^3 + 4*w^2 - 3*w - 4],
[53, 53, -w^5 + w^4 + 4*w^3 - 4*w^2 - 3*w + 1],
[53, 53, w^3 - 2*w - 2],
[53, 53, w^5 - 5*w^3 - w^2 + 5*w],
[53, 53, -w^4 + 4*w^2 + w - 4],
[64, 2, -2],
[79, 79, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 9*w + 2],
[79, 79, -w^5 - w^4 + 5*w^3 + 4*w^2 - 6*w - 1],
[79, 79, w^3 - w^2 - 4*w + 1],
[79, 79, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 9*w + 3],
[79, 79, -2*w^4 + w^3 + 7*w^2 - 3*w - 3],
[79, 79, -w^5 + 6*w^3 - w^2 - 8*w + 1],
[103, 103, 2*w^4 - 7*w^2 - w + 3],
[103, 103, w^5 - w^4 - 4*w^3 + 5*w^2 + 2*w - 4],
[103, 103, -w^5 - w^4 + 4*w^3 + 4*w^2 - 2*w - 3],
[103, 103, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 4*w - 1],
[103, 103, -w^4 + w^3 + 5*w^2 - 3*w - 4],
[103, 103, 2*w^5 - w^4 - 10*w^3 + 3*w^2 + 10*w - 1],
[131, 131, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w - 1],
[131, 131, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 4*w + 2],
[131, 131, w^5 - w^4 - 4*w^3 + 5*w^2 + 2*w - 5],
[131, 131, 2*w^4 - 7*w^2 - w + 2],
[131, 131, -2*w^5 + w^4 + 9*w^3 - 2*w^2 - 8*w],
[131, 131, 2*w^5 - w^4 - 10*w^3 + 3*w^2 + 10*w],
[157, 157, -w^5 - w^4 + 5*w^3 + 5*w^2 - 6*w - 3],
[157, 157, -2*w^3 + w^2 + 5*w - 2],
[157, 157, -w^5 - w^4 + 4*w^3 + 5*w^2 - 3*w - 3],
[157, 157, -w^5 + 5*w^3 - 7*w],
[157, 157, -2*w^5 + w^4 + 8*w^3 - 3*w^2 - 6*w + 2],
[157, 157, -w^5 + w^4 + 5*w^3 - 4*w^2 - 7*w + 2],
[181, 181, w^2 - 2*w - 2],
[181, 181, 2*w^5 - 9*w^3 + 7*w],
[181, 181, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 5*w + 4],
[181, 181, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w + 1],
[181, 181, 2*w^5 - 2*w^4 - 8*w^3 + 6*w^2 + 5*w - 2],
[181, 181, w^4 - 2*w^2 - 2],
[233, 233, 2*w^3 - w^2 - 5*w + 1],
[233, 233, -w^5 - w^4 + 6*w^3 + 3*w^2 - 8*w],
[233, 233, w^5 - w^4 - 5*w^3 + 4*w^2 + 7*w - 3],
[233, 233, w^5 + w^4 - 5*w^3 - 5*w^2 + 6*w + 4],
[233, 233, -w^5 + 4*w^3 + 2*w^2 - 2*w - 3],
[233, 233, 2*w^5 - w^4 - 8*w^3 + 3*w^2 + 6*w - 1],
[311, 311, 2*w^5 - 2*w^4 - 9*w^3 + 6*w^2 + 8*w - 2],
[311, 311, w^4 + w^3 - 3*w^2 - 4*w + 1],
[311, 311, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 5*w + 2],
[311, 311, w^5 - w^4 - 4*w^3 + 4*w^2 + w - 3],
[311, 311, w^4 - 2*w^2 - w - 2],
[311, 311, -2*w^5 + w^4 + 9*w^3 - 4*w^2 - 7*w + 2],
[313, 313, -w^5 + 5*w^3 + w^2 - 4*w - 3],
[313, 313, -w^5 + 6*w^3 - 7*w + 1],
[313, 313, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[313, 313, -w^5 + w^4 + 3*w^3 - 4*w^2 + 4],
[313, 313, -w^5 + 2*w^4 + 4*w^3 - 7*w^2 - 4*w + 2],
[313, 313, -2*w^4 + w^3 + 7*w^2 - 2*w - 4],
[337, 337, 2*w^5 - w^4 - 7*w^3 + 2*w^2 + 2*w + 2],
[337, 337, w^5 - 2*w^4 - 3*w^3 + 7*w^2 + 2*w - 3],
[337, 337, 2*w^5 - w^4 - 9*w^3 + 2*w^2 + 10*w - 1],
[337, 337, -2*w^5 - w^4 + 9*w^3 + 4*w^2 - 8*w],
[337, 337, 3*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 11*w - 5],
[337, 337, w^4 - w^3 - 2*w^2 + 4*w],
[389, 389, 2*w^5 - w^4 - 9*w^3 + 4*w^2 + 7*w],
[389, 389, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 7*w + 3],
[389, 389, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 5*w],
[389, 389, -2*w^5 + 3*w^4 + 9*w^3 - 10*w^2 - 8*w + 2],
[389, 389, -3*w^5 + w^4 + 14*w^3 - 2*w^2 - 12*w],
[389, 389, -w^5 + w^4 + 5*w^3 - 3*w^2 - 3*w + 1],
[443, 443, -2*w^5 + w^4 + 8*w^3 - w^2 - 7*w - 3],
[443, 443, 2*w^5 - w^4 - 9*w^3 + 4*w^2 + 8*w],
[443, 443, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 2*w - 4],
[443, 443, 2*w^5 - w^4 - 9*w^3 + 2*w^2 + 8*w - 2],
[443, 443, -2*w^4 + 2*w^3 + 7*w^2 - 4*w - 3],
[443, 443, -3*w^5 + w^4 + 15*w^3 - 3*w^2 - 17*w + 2],
[467, 467, 2*w^5 - 2*w^4 - 9*w^3 + 6*w^2 + 10*w - 1],
[467, 467, w^5 + w^4 - 5*w^3 - 6*w^2 + 6*w + 5],
[467, 467, -w^5 - 2*w^4 + 5*w^3 + 7*w^2 - 5*w - 3],
[467, 467, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 13*w + 3],
[467, 467, 3*w^5 - 2*w^4 - 13*w^3 + 6*w^2 + 11*w - 1],
[467, 467, -2*w^5 + w^4 + 9*w^3 - w^2 - 8*w - 3],
[521, 521, w^5 + 2*w^4 - 6*w^3 - 7*w^2 + 7*w + 3],
[521, 521, 2*w^5 - 2*w^4 - 8*w^3 + 6*w^2 + 7*w - 1],
[521, 521, -w^5 + w^4 + 6*w^3 - 4*w^2 - 10*w + 3],
[521, 521, 2*w^5 - 3*w^4 - 9*w^3 + 10*w^2 + 9*w - 3],
[521, 521, -2*w^5 + 11*w^3 - 13*w + 1],
[521, 521, 3*w^5 - w^4 - 13*w^3 + 3*w^2 + 11*w - 1],
[547, 547, w^4 + w^3 - 5*w^2 - 3*w + 1],
[547, 547, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 10*w + 6],
[547, 547, w^4 - w^3 - w^2 + 3*w - 3],
[547, 547, 2*w^4 + w^3 - 9*w^2 - 2*w + 6],
[547, 547, 3*w^5 - w^4 - 13*w^3 + 3*w^2 + 11*w - 2],
[547, 547, -w^5 + w^4 + 6*w^3 - 4*w^2 - 10*w + 2],
[571, 571, -w^5 + 4*w^3 + 3*w^2 - 3*w - 4],
[571, 571, -w^5 - w^4 + 3*w^3 + 6*w^2 - 5],
[571, 571, w^5 - 6*w^3 + 2*w^2 + 9*w - 4],
[571, 571, 2*w^5 - 9*w^3 - w^2 + 10*w + 1],
[571, 571, 2*w^5 - 11*w^3 + w^2 + 13*w - 3],
[571, 571, -w^5 + 7*w^3 - w^2 - 12*w + 2],
[599, 599, -2*w^5 + w^4 + 8*w^3 - 5*w^2 - 5*w + 5],
[599, 599, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w - 2],
[599, 599, w^4 - 2*w^3 - 5*w^2 + 7*w + 4],
[599, 599, -2*w^5 + 10*w^3 - w^2 - 11*w + 3],
[599, 599, w^5 - 3*w^4 - 5*w^3 + 11*w^2 + 7*w - 5],
[599, 599, -w^5 + w^4 + 6*w^3 - 3*w^2 - 8*w + 3],
[677, 677, -w^5 - 2*w^4 + 6*w^3 + 8*w^2 - 9*w - 4],
[677, 677, -3*w^5 + 2*w^4 + 13*w^3 - 7*w^2 - 12*w + 4],
[677, 677, w^4 + w^3 - 5*w^2 - 5*w + 4],
[677, 677, -2*w^5 + 9*w^3 - w^2 - 7*w + 1],
[677, 677, w^5 - 5*w^3 + 2*w^2 + 5*w - 5],
[677, 677, -w^5 - w^4 + 7*w^3 + 2*w^2 - 10*w + 1],
[701, 701, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 2*w - 6],
[701, 701, w^5 + w^4 - 6*w^3 - 5*w^2 + 6*w + 4],
[701, 701, -w^5 - w^4 + 3*w^3 + 4*w^2 + 2*w - 2],
[701, 701, 3*w^5 - 2*w^4 - 14*w^3 + 5*w^2 + 14*w],
[701, 701, 3*w^5 - 2*w^4 - 14*w^3 + 8*w^2 + 13*w - 5],
[701, 701, -w^5 + 2*w^4 + 3*w^3 - 7*w^2 + 2*w + 3],
[727, 727, w^5 + 2*w^4 - 5*w^3 - 8*w^2 + 6*w + 3],
[727, 727, -w^5 + 4*w^3 + 3*w^2 - 3*w - 6],
[727, 727, -w^5 - w^4 + 5*w^3 + 2*w^2 - 4*w + 2],
[727, 727, 3*w^5 - 2*w^4 - 13*w^3 + 6*w^2 + 12*w - 3],
[727, 727, -2*w^5 + 11*w^3 - w^2 - 13*w + 1],
[727, 727, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 5*w - 4],
[857, 857, -w^5 + 3*w^4 + 5*w^3 - 11*w^2 - 5*w + 5],
[857, 857, -w^5 - 2*w^4 + 4*w^3 + 9*w^2 - 3*w - 7],
[857, 857, 2*w^5 + w^4 - 9*w^3 - 4*w^2 + 6*w + 3],
[857, 857, 2*w^5 - 3*w^4 - 7*w^3 + 10*w^2 + 2*w - 5],
[857, 857, -w^5 + 6*w^3 - 2*w^2 - 8*w + 2],
[857, 857, -3*w^3 + w^2 + 9*w],
[859, 859, 2*w^4 - 2*w^3 - 5*w^2 + 5*w - 2],
[859, 859, -2*w^5 + 9*w^3 - 6*w - 2],
[859, 859, -w^4 - w^3 + 2*w^2 + 3*w + 4],
[859, 859, -w^5 - 2*w^4 + 6*w^3 + 9*w^2 - 7*w - 7],
[859, 859, -2*w^5 - w^4 + 10*w^3 + 4*w^2 - 9*w - 4],
[859, 859, -w^5 - w^4 + 4*w^3 + 4*w^2 - 3*w + 1],
[883, 883, 2*w^5 + w^4 - 10*w^3 - 4*w^2 + 11*w],
[883, 883, -2*w^5 + w^4 + 9*w^3 - 3*w^2 - 10*w + 3],
[883, 883, 3*w^5 - w^4 - 13*w^3 + 2*w^2 + 12*w + 1],
[883, 883, 3*w^5 - 2*w^4 - 12*w^3 + 6*w^2 + 8*w - 3],
[883, 883, w^5 - 6*w^3 + w^2 + 10*w - 3],
[883, 883, -2*w^5 + 11*w^3 - w^2 - 14*w + 3],
[911, 911, -3*w^5 + 15*w^3 - 15*w + 1],
[911, 911, -2*w^5 + 2*w^4 + 8*w^3 - 8*w^2 - 6*w + 3],
[911, 911, -2*w^5 + 11*w^3 - 13*w + 2],
[911, 911, -2*w^5 + 10*w^3 + 2*w^2 - 10*w - 1],
[911, 911, 2*w^4 - 2*w^3 - 8*w^2 + 6*w + 7],
[911, 911, -3*w^4 + 12*w^2 - 7],
[937, 937, 3*w^4 - w^3 - 10*w^2 + 2*w + 4],
[937, 937, -w^5 + 2*w^4 + 5*w^3 - 9*w^2 - 6*w + 5],
[937, 937, -w^5 + 2*w^4 + 2*w^3 - 6*w^2 + 3*w + 3],
[937, 937, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 12*w - 3],
[937, 937, -2*w^5 + 11*w^3 + w^2 - 14*w],
[937, 937, 2*w^5 + w^4 - 9*w^3 - 5*w^2 + 7*w + 5],
[961, 31, -3*w^5 + w^4 + 12*w^3 - 3*w^2 - 8*w + 2],
[961, 31, -2*w^5 + 2*w^4 + 10*w^3 - 7*w^2 - 13*w + 3],
[961, 31, -w^5 + w^4 + 7*w^3 - 5*w^2 - 10*w + 4]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 - 5*x^2 - 11*x + 42;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/3*e^2 - 1/3*e - 3, 1/3*e^2 - 4/3*e - 8, e - 5, -e + 2, 2/3*e^2 - 8/3*e - 6, -1/3*e^2 + 1/3*e + 7, e^2 - 3*e - 9, -1, -2/3*e^2 - 1/3*e + 12, -2*e + 4, -2/3*e^2 + 8/3*e + 10, 2*e - 1, 4/3*e^2 - 10/3*e - 12, -e^2 + e + 17, -1/3*e^2 + 7/3*e + 7, 2/3*e^2 + 1/3*e - 11, 4, -4/3*e^2 + 7/3*e + 15, -e^2 + 3*e + 13, 4/3*e^2 - 13/3*e - 14, -1/3*e^2 + 10/3*e - 2, 2/3*e^2 - 11/3*e - 11, -2/3*e^2 - 1/3*e + 22, -e^2 + 3*e + 13, -4/3*e^2 + 4/3*e + 24, e^2 - 24, 1/3*e^2 + 2/3*e - 4, -1/3*e^2 - 2/3*e + 6, -e^2 + 3*e + 7, -1/3*e^2 - 2/3*e + 23, 1/3*e^2 + 8/3*e - 8, e^2 - e - 9, -2*e^2 + 6*e + 16, -4/3*e^2 + 10/3*e + 20, 4, -4/3*e^2 + 1/3*e + 22, 2*e^2 - 6*e - 18, 2*e + 4, 4/3*e^2 - 4/3*e - 12, -5/3*e^2 + 5/3*e + 27, -e^2 + 3*e + 17, 1/3*e^2 - 10/3*e + 3, -e^2 + 3*e - 5, 4/3*e^2 - 13/3*e - 15, 2/3*e^2 - 2/3*e - 14, 1/3*e^2 - 4/3*e - 9, -e + 18, 6, 8/3*e^2 - 23/3*e - 16, -e^2 + 5*e + 3, -e^2 + e + 17, -7/3*e^2 + 4/3*e + 35, -4/3*e^2 + 4/3*e + 10, 4/3*e^2 + 2/3*e - 26, -11/3*e^2 + 29/3*e + 31, -1/3*e^2 - 2/3*e + 15, 5/3*e^2 - 11/3*e - 33, -2/3*e^2 + 2/3*e - 4, 1/3*e^2 + 2/3*e + 5, e^2 - 3*e - 13, e^2 - e - 19, 2*e + 4, 6*e - 10, 4/3*e^2 - 7/3*e - 34, -2*e^2 + 8*e + 16, 6*e - 10, 2/3*e^2 - 8/3*e - 22, 3*e^2 - 9*e - 27, 1/3*e^2 + 2/3*e - 14, -4/3*e^2 + 10/3*e + 24, 2*e^2 + e - 37, -2/3*e^2 + 11/3*e + 9, 7*e - 11, -2/3*e^2 + 2/3*e + 8, 2*e^2 - 7*e - 12, -4/3*e^2 + 16/3*e + 14, -1/3*e^2 + 22/3*e - 1, -1/3*e^2 - 5/3*e + 5, -4/3*e^2 - 5/3*e + 20, -e^2 + 3*e + 35, 8/3*e^2 - 8/3*e - 34, -4*e - 8, -2/3*e^2 + 5/3*e + 20, -4/3*e^2 - 2/3*e + 8, -2/3*e^2 + 17/3*e + 3, 5/3*e^2 - 20/3*e + 2, 2/3*e^2 - 14/3*e - 14, 10/3*e^2 - 19/3*e - 38, e + 14, 4/3*e^2 - 4/3*e - 14, 5/3*e^2 - 14/3*e - 36, -2/3*e^2 + 2/3*e + 32, -4/3*e^2 - 2/3*e + 8, -1/3*e^2 - 2/3*e + 34, -4*e - 8, 2/3*e^2 + 7/3*e - 27, -5/3*e^2 - 4/3*e + 41, -28, 2/3*e^2 - 23/3*e + 2, e^2 + 2*e - 26, -28, -7/3*e^2 + 1/3*e + 49, -1/3*e^2 - 14/3*e + 32, 5/3*e^2 - 5/3*e - 37, -e^2 + 6*e, -2/3*e^2 - 4/3*e + 6, -2/3*e^2 + 14/3*e + 2, 5/3*e^2 - 17/3*e - 23, -2*e^2 + 3*e + 42, 4*e - 20, -1/3*e^2 + 1/3*e - 13, -2/3*e^2 - 4/3*e + 38, -4/3*e^2 + 7/3*e + 5, 7/3*e^2 - 22/3*e + 1, -2/3*e^2 - 7/3*e + 6, 5/3*e^2 - 32/3*e - 26, 1/3*e^2 - 1/3*e + 25, 2*e^2 - 2*e - 42, 1/3*e^2 - 4/3*e - 14, 7/3*e^2 - 25/3*e - 31, 2/3*e^2 - 11/3*e - 16, 2*e^2 - 4*e - 34, 2*e^2 - 5*e - 22, -2/3*e^2 - 7/3*e + 24, -7/3*e^2 - 5/3*e + 69, -11/3*e^2 + 8/3*e + 56, -2/3*e^2 - 16/3*e + 10, 3*e^2 - 8*e - 10, 2*e^2 - 4*e - 16, -1/3*e^2 + 7/3*e - 11, -4/3*e^2 + 4/3*e + 26, 6*e - 18, e^2 + 6*e - 9, -8/3*e^2 + 20/3*e + 52, -4*e, -10*e + 4, -11/3*e^2 + 5/3*e + 69, 4/3*e^2 - 25/3*e - 21, -4*e^2 + 8*e + 48, -8*e + 4, -2*e^2 + 7*e + 10, 4/3*e^2 - 13/3*e + 23, 1/3*e^2 + 26/3*e - 6, -1/3*e^2 - 5/3*e + 27, 4/3*e^2 + 17/3*e - 22, -4/3*e^2 + 4/3*e + 40, 5/3*e^2 - 11/3*e - 43, -11/3*e^2 - 4/3*e + 68, -2/3*e^2 + 11/3*e + 1, 2/3*e^2 - 14/3*e + 28, -11/3*e^2 + 14/3*e + 37, -2*e + 14, -e^2 + 3*e + 47, 4/3*e^2 - 7/3*e - 4, 3*e^2 - 10*e - 50, -1/3*e^2 + 7/3*e + 27, -14/3*e^2 + 44/3*e + 36, 4/3*e^2 - 28/3*e - 30, -e^2 - 3*e + 9];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {53,53,w^5 - w^4 - 4*w^3 + 4*w^2 + 3*w - 1}>] := 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;