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

NN := ideal<ZF | {53, 53, 2*w^3 - 2*w^2 - 8*w - 3}>;

primesArray := [
[5, 5, -w^3 + w^2 + 5*w],
[7, 7, -w^3 + 2*w^2 + 3*w - 3],
[7, 7, w^3 - 2*w^2 - 3*w],
[13, 13, -w^2 + w + 3],
[13, 13, -w^2 + w + 2],
[16, 2, 2],
[23, 23, -w^2 + 3*w + 1],
[23, 23, -2*w^3 + 3*w^2 + 9*w - 2],
[25, 5, w^3 - 2*w^2 - 2*w + 2],
[49, 7, w^3 - w^2 - 6*w - 1],
[53, 53, 2*w^3 - 2*w^2 - 8*w - 3],
[53, 53, 2*w^3 - 2*w^2 - 8*w - 1],
[67, 67, 2*w^3 - 4*w^2 - 7*w + 2],
[67, 67, -2*w^3 + 4*w^2 + 6*w - 1],
[81, 3, -3],
[83, 83, 2*w^3 - 3*w^2 - 6*w - 1],
[83, 83, 3*w^3 - 4*w^2 - 12*w + 1],
[103, 103, 3*w^3 - 4*w^2 - 12*w - 1],
[103, 103, 2*w^3 - 3*w^2 - 6*w + 1],
[107, 107, w^3 - w^2 - 3*w - 3],
[107, 107, 2*w^3 - 2*w^2 - 9*w],
[109, 109, -3*w^3 + 5*w^2 + 10*w - 5],
[109, 109, -2*w^3 + 3*w^2 + 9*w + 1],
[109, 109, -2*w^3 + 2*w^2 + 7*w],
[109, 109, w^2 - 3*w - 4],
[121, 11, 3*w^3 - 4*w^2 - 12*w],
[121, 11, 2*w^2 - 3*w - 6],
[139, 139, w^3 - 7*w - 1],
[139, 139, -w^3 + 2*w^2 + 5*w - 4],
[139, 139, 3*w^3 - 3*w^2 - 14*w - 2],
[139, 139, -3*w^3 + 5*w^2 + 13*w - 2],
[149, 149, -w^2 + 4*w + 1],
[149, 149, -3*w^3 + 4*w^2 + 14*w - 1],
[149, 149, 2*w^2 - 3*w - 5],
[149, 149, -w^3 + 3*w^2 + 3*w - 4],
[167, 167, -w^3 - w^2 + 7*w + 4],
[167, 167, -4*w^3 + 6*w^2 + 15*w - 4],
[169, 13, w^3 - w^2 - 6*w + 4],
[173, 173, w^2 - 3*w - 5],
[173, 173, 2*w^3 - 2*w^2 - 11*w],
[179, 179, 2*w^3 - 2*w^2 - 9*w + 1],
[179, 179, w^3 - w^2 - 3*w - 4],
[179, 179, w^2 - 5],
[179, 179, w^3 - 6*w - 1],
[197, 197, w^3 + w^2 - 7*w - 6],
[197, 197, 2*w^3 - 2*w^2 - 9*w + 2],
[223, 223, -w^2 + 7],
[223, 223, 2*w^2 - w - 4],
[227, 227, 2*w^3 - 2*w^2 - 11*w - 2],
[227, 227, w^3 - w^2 - 7*w - 1],
[233, 233, w^3 - w^2 - 7*w],
[233, 233, -2*w^3 + 2*w^2 + 11*w + 1],
[257, 257, -2*w^3 + 4*w^2 + 8*w - 1],
[257, 257, 2*w^2 - 4*w - 7],
[277, 277, -4*w^3 + 6*w^2 + 15*w - 3],
[277, 277, -3*w^3 + 5*w^2 + 9*w],
[281, 281, -2*w^3 + 5*w^2 + 3*w - 5],
[281, 281, -2*w^3 + 5*w^2 + 5*w - 7],
[281, 281, 2*w^3 - 5*w^2 - 5*w + 4],
[281, 281, -4*w^3 + 7*w^2 + 15*w - 4],
[283, 283, 3*w^3 - 4*w^2 - 13*w - 3],
[283, 283, -w^3 + 2*w^2 + w - 4],
[313, 313, -2*w^3 + 4*w^2 + 5*w - 5],
[313, 313, 3*w^3 - 5*w^2 - 11*w],
[347, 347, -w^3 + 2*w^2 + w - 5],
[347, 347, 3*w^3 - 4*w^2 - 13*w - 4],
[353, 353, 3*w^3 - 5*w^2 - 12*w + 1],
[353, 353, -w^3 + 3*w^2 - 5],
[361, 19, 3*w^3 - 3*w^2 - 13*w - 9],
[361, 19, 2*w^3 - 2*w^2 - 7*w + 4],
[373, 373, -3*w^3 + 4*w^2 + 11*w + 1],
[373, 373, -3*w^3 + 4*w^2 + 11*w],
[383, 383, 3*w^3 - 5*w^2 - 13*w + 1],
[383, 383, 2*w^2 - 5*w - 6],
[397, 397, 4*w^3 - 5*w^2 - 19*w - 1],
[397, 397, 2*w^3 - 2*w^2 - 9*w - 7],
[431, 431, 2*w^3 - 2*w^2 - 7*w - 5],
[431, 431, 3*w^3 - 6*w^2 - 11*w + 3],
[431, 431, 3*w^3 - 3*w^2 - 13*w],
[431, 431, w^3 - 4*w^2 + w + 8],
[457, 457, 4*w^3 - 7*w^2 - 13*w],
[457, 457, -3*w^2 + 4*w + 5],
[463, 463, -3*w^3 + 5*w^2 + 11*w + 1],
[463, 463, 2*w^3 - 4*w^2 - 5*w + 6],
[487, 487, -w^3 + 4*w^2 - 7],
[487, 487, -2*w^3 + 5*w^2 + 6*w - 5],
[499, 499, -6*w^3 + 8*w^2 + 25*w - 1],
[499, 499, -w^3 + 4*w^2 - 5],
[499, 499, -2*w^3 + 5*w^2 + 6*w - 7],
[499, 499, 3*w^3 - 5*w^2 - 7*w],
[509, 509, -4*w^3 + 8*w^2 + 15*w - 8],
[509, 509, 3*w^3 - 4*w^2 - 10*w - 2],
[509, 509, -3*w^3 + 3*w^2 + 11*w + 6],
[509, 509, 4*w^3 - 5*w^2 - 16*w],
[521, 521, -4*w^3 + 5*w^2 + 15*w - 2],
[521, 521, w^3 - w^2 - 8*w + 1],
[521, 521, -4*w^3 + 7*w^2 + 12*w - 6],
[521, 521, 3*w^3 - 3*w^2 - 16*w - 1],
[523, 523, 3*w^3 - w^2 - 17*w - 8],
[523, 523, -5*w^3 + 5*w^2 + 23*w + 4],
[529, 23, 3*w^3 - 6*w^2 - 10*w + 2],
[547, 547, -2*w^3 + 5*w^2 + 6*w - 6],
[547, 547, -w^3 + 4*w^2 - 6],
[557, 557, -3*w^3 + 6*w^2 + 9*w - 10],
[557, 557, 3*w^3 - 6*w^2 - 9*w - 1],
[571, 571, 4*w^3 - 4*w^2 - 17*w - 6],
[571, 571, -2*w^3 + 6*w^2 + 3*w - 10],
[571, 571, 4*w^3 - 5*w^2 - 18*w + 3],
[571, 571, -3*w^3 + 7*w^2 + 9*w - 5],
[587, 587, 3*w^3 - 3*w^2 - 14*w],
[587, 587, w^3 - w^2 - 2*w - 4],
[593, 593, -w^3 + 3*w^2 + 2*w - 10],
[593, 593, -5*w^3 + 7*w^2 + 19*w - 4],
[613, 613, 3*w^3 - 5*w^2 - 12*w - 4],
[613, 613, -w^3 + 3*w^2 - 10],
[631, 631, 4*w^3 - 8*w^2 - 13*w + 4],
[631, 631, 3*w^3 - 4*w^2 - 15*w + 2],
[631, 631, w^3 - 9*w - 1],
[631, 631, 3*w^3 - 7*w^2 - 7*w + 9],
[643, 643, 3*w^3 - 3*w^2 - 16*w - 2],
[643, 643, w^3 - w^2 - 8*w],
[647, 647, -w^3 + 4*w^2 + 2*w - 4],
[647, 647, -3*w^2 + 4*w + 10],
[673, 673, w^3 + 2*w^2 - 9*w - 8],
[673, 673, -w^3 + 4*w^2 + 3*w - 7],
[683, 683, 3*w^3 - 3*w^2 - 11*w - 2],
[683, 683, 4*w^3 - 4*w^2 - 17*w - 5],
[709, 709, -5*w^3 + 7*w^2 + 21*w + 1],
[709, 709, -3*w^3 + 2*w^2 + 16*w + 4],
[709, 709, 4*w^3 - 4*w^2 - 19*w - 2],
[709, 709, -2*w^3 + 4*w^2 + 3*w - 4],
[787, 787, 2*w^2 - w - 9],
[787, 787, w^3 + w^2 - 7*w - 2],
[811, 811, 4*w^3 - 7*w^2 - 11*w],
[811, 811, 3*w^3 - 4*w^2 - 9*w - 1],
[811, 811, -6*w^3 + 9*w^2 + 23*w - 5],
[811, 811, 5*w^3 - 6*w^2 - 21*w - 2],
[821, 821, -3*w^3 + 6*w^2 + 11*w - 2],
[821, 821, 2*w^3 - 13*w - 5],
[821, 821, -w^3 + 4*w^2 - w - 9],
[821, 821, -w^3 + 3*w^2 + 5*w - 6],
[841, 29, 2*w^3 - 2*w^2 - 12*w - 1],
[857, 857, w^3 - 4*w^2 + 3*w + 6],
[857, 857, -5*w^3 + 8*w^2 + 21*w - 3],
[863, 863, -w^3 + 4*w^2 + w - 7],
[863, 863, -w^3 + 4*w^2 + w - 6],
[877, 877, -w^3 - w^2 + 8*w + 2],
[877, 877, -w^3 + 3*w^2 + 4*w - 8],
[883, 883, 4*w^3 - 7*w^2 - 14*w + 1],
[883, 883, 3*w^3 - 6*w^2 - 8*w + 7],
[929, 929, 3*w^3 - 5*w^2 - 12*w - 3],
[929, 929, -w^3 + 3*w^2 - 9],
[929, 929, 3*w - 5],
[929, 929, 3*w^3 - 3*w^2 - 15*w + 2],
[937, 937, w^3 - 2*w^2 - 6*w + 6],
[937, 937, 2*w^3 - w^2 - 12*w],
[941, 941, -w^3 + 4*w^2 + 2*w - 7],
[941, 941, -4*w^3 + 6*w^2 + 13*w - 3],
[941, 941, -5*w^3 + 7*w^2 + 19*w + 2],
[941, 941, 3*w^2 - 4*w - 7],
[953, 953, 3*w^3 - 5*w^2 - 12*w - 2],
[953, 953, -w^3 + 3*w^2 - 8],
[961, 31, 4*w^3 - 6*w^2 - 13*w - 2],
[961, 31, 5*w^3 - 7*w^2 - 19*w + 3],
[977, 977, -4*w^3 + 7*w^2 + 19*w - 6],
[977, 977, -3*w^3 + 7*w^2 + 10*w - 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 2*x^3 - 10*x^2 + 4*x + 12;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/2*e^3 + 3/2*e^2 + 3*e - 1, -1/2*e^2 + e + 2, -1/2*e^2 + 2*e + 2, 1/2*e^3 - e^2 - 4*e + 2, -e + 5, e^2 - e - 6, 1/2*e^3 - 2*e^2 - 3*e + 6, -1/2*e^3 + 2*e^2 + 2*e - 4, -e^3 + 5/2*e^2 + 5*e - 7, -1, 3/2*e^2 - 3*e - 9, -e^3 + 2*e^2 + 6*e - 4, 3/2*e^3 - 5*e^2 - 9*e + 14, 1/2*e^3 - e^2 - 8*e + 4, 1/2*e^3 - 1/2*e^2 - 5*e - 3, -1/2*e^3 + 2*e^2 + 3*e - 6, e^3 - 5*e^2 - e + 20, 1/2*e^3 - 3*e^2 - 2*e + 8, -2*e + 12, -e^3 + 7/2*e^2 + 3*e - 9, -1/2*e^3 + 6*e + 2, 3/2*e^3 - 4*e^2 - 7*e + 8, -1/2*e^3 + 3/2*e^2 + 2*e - 1, 1/2*e^3 - e^2 - 6*e + 2, 1/2*e^2 - 3*e - 4, 1/2*e^3 - 3/2*e^2 - 2*e + 8, 2*e^3 - 13/2*e^2 - 11*e + 17, e^3 - 5/2*e^2 - 10*e + 2, -1/2*e^3 - e^2 + 8*e + 2, -3/2*e^3 + 7/2*e^2 + 8*e + 2, e^2 - 6*e - 12, -3*e^2 + 5*e + 12, e^3 - 5/2*e^2 - 8*e + 18, 2*e^3 - 4*e^2 - 16*e + 12, 3/2*e^3 - 9/2*e^2 - 5*e + 9, 3/2*e^3 - 3*e^2 - 9*e + 6, e^3 - 4*e^2 + 2*e + 14, e^3 - 5/2*e^2 - 6*e + 3, -3/2*e^3 + 6*e^2 + 5*e - 18, -3*e^2 + 10*e + 12, 1/2*e^3 - 9/2*e^2 + e + 12, 2*e^3 - 5*e^2 - 14*e + 12, e^3 - 2*e^2 - 13*e + 6, 1/2*e^2 + 2*e + 6, -e^3 + e^2 + 7*e + 12, -1/2*e^3 + 5/2*e^2 - 6*e - 10, -3/2*e^3 + 7*e^2 + 5*e - 22, -1/2*e^3 + 10*e + 6, 2*e^3 - 13/2*e^2 - 15*e + 12, 2*e^3 - 11/2*e^2 - 17*e + 21, -1/2*e^3 + e^2 + 5*e + 6, -1/2*e^3 + 5/2*e^2 + 8*e - 15, e^2 + 2*e - 6, -3/2*e^3 + 8*e^2 - 28, -1/2*e^3 + 3*e^2 - e - 4, -e^3 + 2*e^2 + 10*e - 12, -e^3 + 1/2*e^2 + 17*e, -1/2*e^3 - 3*e^2 + 8*e + 18, 4*e^2 - 10*e - 18, -5/2*e^3 + 8*e^2 + 12*e - 22, e^3 - 3*e^2 - 7*e + 2, 1/2*e^3 - e^2 + e + 2, -1/2*e^3 + 5/2*e^2 - 2*e + 2, 2*e^3 - 6*e^2 - 9*e + 24, -1/2*e^3 - 5/2*e^2 + 7*e + 18, -3/2*e^3 + 1/2*e^2 + 16*e - 3, -3/2*e^3 + 2*e^2 + 7*e + 24, 2*e^3 - 3*e^2 - 11*e - 4, -3/2*e^3 + 15/2*e^2 + e - 22, 2*e^3 - 4*e^2 - 15*e + 14, 5/2*e^3 - 17/2*e^2 - 16*e + 20, -9/2*e^2 + 4*e + 36, -e^3 + 7/2*e^2 + 5*e - 15, -3/2*e^3 + 13/2*e^2 + 5*e - 1, -1/2*e^3 + 7/2*e^2 - 5*e - 22, 3/2*e^3 - 9/2*e^2 - e + 6, -2*e^3 + 27*e, -9/2*e^3 + 13*e^2 + 26*e - 30, -e^3 + 15/2*e^2 - 8*e - 33, -1/2*e^3 - 1/2*e^2 + 13*e + 2, 7/2*e^3 - 8*e^2 - 34*e + 14, -2*e^3 + 8*e^2 + 4*e - 28, -1/2*e^3 + 1/2*e^2 + 11*e + 11, 4*e^3 - 9*e^2 - 29*e + 14, -3/2*e^3 + 3*e^2 + 6*e + 20, 3/2*e^3 - 2*e^2 - 15*e - 4, 2*e^3 - 6*e^2 - 14*e + 26, -3*e^3 + 10*e^2 + 16*e - 28, 4*e^3 - 21/2*e^2 - 26*e + 17, 3/2*e^3 - 1/2*e^2 - 17*e + 9, -3*e^3 + 7*e^2 + 20*e - 24, -5/2*e^3 + 11/2*e^2 + 20*e + 3, -5/2*e^3 + 19/2*e^2 + 9*e - 21, e^3 + 5/2*e^2 - 9*e - 27, 9/2*e^3 - 9*e^2 - 32*e + 12, -7/2*e^3 + 9*e^2 + 26*e - 12, -3*e^3 + 15/2*e^2 + 20*e - 3, -2*e^3 + 4*e^2 + 25*e - 4, -3*e^3 + 4*e^2 + 23*e + 8, 5/2*e^3 - 7*e^2 - 15*e + 8, -5/2*e^3 + 19/2*e^2 + 9*e - 28, -e^3 + 5*e^2 + 10*e - 34, e^3 + e^2 - 12*e, 1/2*e^3 - e^2 + 3*e - 18, 5/2*e^3 - 13/2*e^2 - 9*e - 4, e^3 - 7*e^2 - 2*e + 26, -e^3 + 5/2*e^2 + 7*e - 19, 2*e^3 - 9/2*e^2 - 18*e + 5, 2*e^3 - 11/2*e^2 - 10*e + 9, -3*e^3 + 25/2*e^2 + 12*e - 42, -2*e^3 - 1/2*e^2 + 27*e + 18, -3/2*e^3 + 2*e^2 + 13*e + 12, 1/2*e^3 - 6*e^2 + 8*e + 8, -3/2*e^3 + 2*e^2 + 16*e - 4, -3*e^3 + 9/2*e^2 + 23*e + 17, 7/2*e^3 - 25/2*e^2 - 13*e + 35, -2*e^3 + 13/2*e^2 + 19*e - 31, -2*e^2 + 7*e - 4, -e^3 + 11/2*e^2 + e - 10, -3/2*e^3 + 6*e^2 + e - 28, -3*e^3 + 19/2*e^2 + 12*e - 36, 5/2*e^3 - 13*e^2 - 8*e + 30, 1/2*e^3 + 2*e^2 - 18*e - 10, 2*e^3 - 8*e^2 + e + 20, 4*e^3 - 11*e^2 - 32*e + 30, -e^3 + 2*e^2 + 6*e - 18, 4*e^3 - 9*e^2 - 27*e + 8, -3*e^3 + 6*e^2 + 30*e - 10, -7/2*e^3 + 9*e^2 + 26*e - 40, -1/2*e^3 + 3/2*e^2 - 6*e - 10, -2*e^3 + 5*e^2 + 15*e - 22, 3*e^3 - 4*e^2 - 23*e - 10, -4*e^3 + 10*e^2 + 31*e - 22, 4*e^3 - 31/2*e^2 - 9*e + 56, 11/2*e^3 - 13*e^2 - 44*e + 14, 38, 1/2*e^3 + 1/2*e^2 - e - 27, 2*e^3 - 10*e^2 + 3*e + 30, -e^3 + 1/2*e^2 + 10*e - 12, 7/2*e^3 - 3/2*e^2 - 42*e - 6, 3/2*e^3 - 5/2*e^2 - 18*e - 13, 5*e^2 - 5*e - 48, 2*e^3 - 15*e - 24, -4*e^2 - e + 42, -3/2*e^3 + 15/2*e^2 + e - 6, -3/2*e^3 + 13/2*e^2 - 4*e - 34, -e^3 + 23/2*e^2 - 5*e - 46, -1/2*e^3 + 11/2*e^2 - 15*e - 7, 2*e^3 - 5*e^2 - 10*e - 10, -1/2*e^3 + e^2 + 16*e - 12, -5/2*e^3 + 11/2*e^2 + 31*e + 6, -7/2*e^3 + 8*e^2 + 30*e - 24, -5/2*e^2 + 5*e + 45, -6*e^3 + 17*e^2 + 32*e - 46, 3*e^2 + 4*e - 22, -2*e^3 + 13*e^2 + 8*e - 60, -5*e^3 + 17*e^2 + 36*e - 48, -2*e^3 + 8*e^2 + 5*e - 24, -e^3 + 3/2*e^2 + 19*e - 9, 1/2*e^3 - e^2 - 3*e - 18, 1/2*e^3 - 2*e^2 - 6*e - 18, 1/2*e^3 + 7/2*e^2 - 17*e - 7, 7/2*e^3 - 13*e^2 - 17*e + 20, e^3 + 3*e^2 - 32*e - 18, 5/2*e^3 - 17/2*e^2 - 9*e + 15];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {53, 53, 2*w^3 - 2*w^2 - 8*w - 3}>] := 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;