/*
  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, -3, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {32, 2, 2}>;

primesArray := [
[3, 3, w^4 - 2*w^3 - 3*w^2 + 4*w + 1],
[13, 13, w^3 - 2*w^2 - 2*w + 2],
[23, 23, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1],
[25, 5, -w^2 + 2*w + 2],
[29, 29, -w^4 + w^3 + 4*w^2 - 3*w - 1],
[31, 31, w^4 - w^3 - 5*w^2 + 2*w + 4],
[32, 2, 2],
[37, 37, w^4 - 2*w^3 - 2*w^2 + 4*w + 1],
[47, 47, w^4 - w^3 - 5*w^2 + 2*w + 3],
[47, 47, 2*w^4 - 3*w^3 - 6*w^2 + 6*w + 2],
[49, 7, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],
[53, 53, -2*w^4 + 3*w^3 + 7*w^2 - 7*w - 2],
[59, 59, -2*w^4 + 3*w^3 + 6*w^2 - 5*w - 2],
[67, 67, -w^4 + 3*w^3 + 2*w^2 - 7*w],
[67, 67, -w^4 + 3*w^3 + w^2 - 7*w + 1],
[71, 71, w^4 - 2*w^3 - 4*w^2 + 3*w + 3],
[71, 71, -w^2 + 5],
[79, 79, 2*w^4 - 3*w^3 - 5*w^2 + 3*w + 1],
[81, 3, -w^4 + 3*w^3 + 3*w^2 - 8*w - 2],
[83, 83, -3*w^4 + 3*w^3 + 10*w^2 - 2*w - 2],
[89, 89, -w^4 + 2*w^3 + 3*w^2 - 6*w - 1],
[101, 101, w^4 - w^3 - 4*w^2 + 3],
[103, 103, w^4 - w^3 - 5*w^2 + 3*w + 1],
[103, 103, 2*w^4 - 3*w^3 - 8*w^2 + 8*w + 3],
[103, 103, -w^4 + 2*w^3 + 2*w^2 - 5*w + 4],
[109, 109, -2*w^4 + 4*w^3 + 7*w^2 - 10*w - 4],
[121, 11, w^4 - 3*w^3 - 2*w^2 + 7*w + 1],
[125, 5, -2*w^4 + 2*w^3 + 7*w^2 - 3*w + 1],
[127, 127, -2*w^3 + 2*w^2 + 7*w - 4],
[149, 149, 2*w^4 - 5*w^3 - 4*w^2 + 11*w + 1],
[157, 157, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6],
[163, 163, 2*w^4 - 2*w^3 - 6*w^2 + w - 1],
[163, 163, 3*w^4 - 5*w^3 - 9*w^2 + 10*w + 4],
[163, 163, -3*w^4 + 5*w^3 + 10*w^2 - 12*w - 3],
[167, 167, w^4 - 2*w^3 - 2*w^2 + 4*w - 4],
[173, 173, 2*w^4 - 3*w^3 - 6*w^2 + 7*w],
[173, 173, 2*w^4 - 5*w^3 - 4*w^2 + 10*w],
[179, 179, -w^4 + 3*w^3 + 2*w^2 - 6*w - 2],
[181, 181, -w^4 + w^3 + 2*w^2 + w + 2],
[181, 181, -2*w^4 + 3*w^3 + 7*w^2 - 4*w - 4],
[193, 193, 2*w^4 - 4*w^3 - 5*w^2 + 8*w + 3],
[197, 197, 2*w^4 - 2*w^3 - 9*w^2 + 3*w + 6],
[199, 199, w^4 - 2*w^3 - w^2 + 3*w - 4],
[211, 211, -w^4 + 6*w^2 + w - 3],
[223, 223, 3*w^4 - 5*w^3 - 9*w^2 + 11*w + 1],
[223, 223, -w^3 + w^2 + w - 3],
[229, 229, 2*w^2 - w - 4],
[241, 241, -w^4 + 6*w^2 - 6],
[241, 241, -w^4 + 3*w^3 + 3*w^2 - 8*w],
[257, 257, -2*w^3 + 2*w^2 + 5*w],
[269, 269, 2*w^3 - w^2 - 5*w],
[269, 269, 3*w^4 - 6*w^3 - 7*w^2 + 12*w],
[277, 277, -w^4 + 3*w^3 + w^2 - 7*w],
[277, 277, -w^3 - w^2 + 5*w + 4],
[283, 283, -3*w^4 + 5*w^3 + 8*w^2 - 8*w - 1],
[283, 283, 3*w^4 - 4*w^3 - 9*w^2 + 7*w + 1],
[283, 283, 3*w^4 - 5*w^3 - 9*w^2 + 8*w + 4],
[293, 293, 2*w^4 - 4*w^3 - 7*w^2 + 11*w + 2],
[307, 307, -w^3 + 3*w^2 - 5],
[311, 311, w^2 - 2*w - 4],
[317, 317, w^4 - w^3 - 2*w^2 + w - 4],
[317, 317, -w^3 + 6*w],
[331, 331, -w^4 + w^3 + 2*w^2 - 2*w + 3],
[343, 7, -2*w^3 + w^2 + 7*w - 1],
[349, 349, 3*w^4 - 5*w^3 - 8*w^2 + 10*w],
[349, 349, w^4 + w^3 - 6*w^2 - 4*w + 4],
[353, 353, w^3 - 3*w - 4],
[359, 359, 2*w^4 - 2*w^3 - 8*w^2 + w + 4],
[367, 367, -5*w^4 + 8*w^3 + 17*w^2 - 18*w - 8],
[367, 367, -w^4 + 4*w^3 - 9*w + 1],
[373, 373, 3*w^4 - 6*w^3 - 8*w^2 + 12*w + 1],
[379, 379, 4*w^4 - 7*w^3 - 12*w^2 + 15*w + 6],
[379, 379, -w^4 - w^3 + 6*w^2 + 5*w],
[379, 379, -3*w^4 + 4*w^3 + 9*w^2 - 6*w - 1],
[421, 421, -3*w^4 + 4*w^3 + 11*w^2 - 7*w - 3],
[433, 433, 2*w^4 - 5*w^3 - 5*w^2 + 10*w],
[439, 439, -4*w^4 + 7*w^3 + 11*w^2 - 15*w - 1],
[457, 457, -2*w^4 + 4*w^3 + 6*w^2 - 7*w - 2],
[461, 461, w^3 - 2*w - 3],
[463, 463, -3*w^4 + 5*w^3 + 10*w^2 - 11*w - 2],
[463, 463, -2*w^4 + 2*w^3 + 8*w^2 - 2*w - 7],
[467, 467, w^4 - 2*w^3 - 4*w^2 + 4*w + 6],
[487, 487, w^2 - 3*w - 3],
[487, 487, w^2 + w - 5],
[491, 491, -3*w^4 + 6*w^3 + 8*w^2 - 12*w],
[491, 491, -w^4 + 3*w^3 + 3*w^2 - 9*w - 1],
[503, 503, -w^4 + 7*w^2 - 7],
[503, 503, -w^4 + 2*w^3 + 3*w^2 - 5*w + 3],
[509, 509, -2*w^4 + 4*w^3 + 7*w^2 - 7*w - 3],
[521, 521, -w^4 + w^3 + 6*w^2 - 2*w - 5],
[541, 541, -w^4 + 2*w^3 + 3*w^2 - 4*w - 5],
[541, 541, -3*w^4 + 5*w^3 + 10*w^2 - 10*w - 2],
[541, 541, -w^4 + 3*w^2 + 5*w - 1],
[547, 547, -3*w^4 + 5*w^3 + 10*w^2 - 9*w - 4],
[557, 557, 5*w^4 - 5*w^3 - 18*w^2 + 6*w + 3],
[557, 557, w^4 - 2*w^2 - 2*w - 2],
[563, 563, -2*w^4 + 4*w^3 + 5*w^2 - 6*w],
[563, 563, 3*w^4 - 4*w^3 - 10*w^2 + 8*w + 1],
[587, 587, -w^4 + 5*w^2 + 2*w - 6],
[599, 599, 3*w^4 - 4*w^3 - 11*w^2 + 9*w + 3],
[599, 599, 3*w^3 - 3*w^2 - 9*w + 1],
[601, 601, -w^4 + 3*w^3 + w^2 - 8*w],
[607, 607, 4*w^4 - 8*w^3 - 12*w^2 + 17*w + 6],
[607, 607, w^4 + 2*w^3 - 7*w^2 - 6*w + 3],
[613, 613, -3*w^4 + 4*w^3 + 9*w^2 - 7*w - 2],
[617, 617, 2*w^3 - 4*w^2 - 5*w + 2],
[619, 619, -w^4 + w^3 + 2*w^2 + 3*w + 2],
[641, 641, -2*w^4 + 4*w^3 + 6*w^2 - 11*w - 3],
[641, 641, -2*w^4 + 5*w^3 + 5*w^2 - 11*w - 2],
[643, 643, -3*w^4 + 6*w^3 + 7*w^2 - 10*w + 3],
[653, 653, -4*w^4 + 6*w^3 + 15*w^2 - 15*w - 6],
[659, 659, 2*w^4 - w^3 - 9*w^2 + w + 5],
[659, 659, -w^4 + w^3 + w^2 + 2*w + 2],
[659, 659, -2*w^4 + 4*w^3 + 5*w^2 - 9*w - 3],
[673, 673, 2*w^4 + w^3 - 10*w^2 - 7*w + 3],
[677, 677, -3*w^4 + w^3 + 14*w^2 + 3*w - 5],
[677, 677, -3*w^4 + 4*w^3 + 8*w^2 - 4*w - 1],
[677, 677, -2*w^4 + 2*w^3 + 6*w^2 + w + 1],
[683, 683, -4*w^4 + 7*w^3 + 12*w^2 - 15*w],
[691, 691, -3*w^4 + 4*w^3 + 9*w^2 - 4*w + 1],
[691, 691, 3*w^4 - 5*w^3 - 8*w^2 + 9*w + 1],
[709, 709, -w^4 + 4*w^3 - 8*w - 1],
[733, 733, 4*w^4 - 6*w^3 - 13*w^2 + 10*w + 2],
[743, 743, 2*w^4 - 5*w^3 - 4*w^2 + 14*w - 3],
[751, 751, -w^4 - 2*w^3 + 7*w^2 + 6*w - 4],
[757, 757, 3*w^3 - 2*w^2 - 9*w + 1],
[761, 761, 4*w^4 - 5*w^3 - 14*w^2 + 9*w + 3],
[761, 761, 5*w^4 - 8*w^3 - 16*w^2 + 15*w + 8],
[761, 761, w^4 - 2*w^3 - 2*w^2 + 2*w - 3],
[769, 769, 3*w^3 - 4*w^2 - 8*w + 7],
[769, 769, 3*w^4 - 5*w^3 - 11*w^2 + 12*w + 4],
[769, 769, -2*w^3 + 3*w^2 + 6*w - 1],
[773, 773, 3*w^4 - 3*w^3 - 12*w^2 + 7*w + 8],
[773, 773, -2*w^4 + 6*w^3 + 3*w^2 - 14*w + 2],
[773, 773, -2*w^4 + w^3 + 9*w^2 - w - 6],
[787, 787, w^4 - 4*w^2 - 2*w + 6],
[809, 809, -2*w^4 + 2*w^3 + 8*w^2 - w - 6],
[809, 809, -2*w^4 + 3*w^3 + 9*w^2 - 5*w - 8],
[821, 821, -2*w^4 + 9*w^2 + 3*w + 1],
[821, 821, w^4 - 4*w^2 - 3*w + 4],
[829, 829, -w^4 + 3*w^3 - 4*w + 4],
[853, 853, -3*w^4 + 5*w^3 + 8*w^2 - 9*w - 2],
[859, 859, 2*w^4 - 2*w^3 - 6*w^2 - 3],
[863, 863, 5*w^4 - 9*w^3 - 14*w^2 + 17*w + 3],
[863, 863, 2*w^4 - 4*w^3 - 7*w^2 + 12*w + 6],
[863, 863, 2*w^4 - 2*w^3 - 11*w^2 + 6*w + 9],
[877, 877, -2*w^4 + 3*w^3 + 8*w^2 - 7*w - 2],
[877, 877, 3*w^4 - 6*w^3 - 6*w^2 + 13*w - 1],
[877, 877, w^4 + w^3 - 4*w^2 - 8*w],
[881, 881, -w^4 - w^3 + 7*w^2 + 2*w - 4],
[883, 883, 4*w^4 - 4*w^3 - 16*w^2 + 7*w + 8],
[887, 887, -2*w^4 + 3*w^3 + 10*w^2 - 8*w - 9],
[929, 929, -w^3 + 2*w^2 - 4],
[929, 929, -3*w^4 + 6*w^3 + 9*w^2 - 14*w],
[929, 929, -4*w^4 + 7*w^3 + 11*w^2 - 14*w - 4],
[937, 937, 2*w^4 - 2*w^3 - 5*w^2 - w - 2],
[941, 941, -3*w^4 + 3*w^3 + 13*w^2 - 8*w - 8],
[941, 941, w^4 - w^3 - 3*w^2 + 3*w - 3],
[947, 947, -3*w^4 + 6*w^3 + 10*w^2 - 16*w - 3],
[961, 31, -w^4 + 5*w^3 - 14*w],
[961, 31, -w^4 - w^3 + 8*w^2 + 2*w - 4],
[971, 971, -2*w^4 + 3*w^3 + 4*w^2 - 3*w + 2],
[983, 983, 4*w^4 - 7*w^3 - 11*w^2 + 16*w],
[983, 983, -3*w^4 + 2*w^3 + 12*w^2],
[997, 997, 3*w^4 - 4*w^3 - 8*w^2 + 5*w - 2],
[997, 997, w^4 - 3*w^3 - w^2 + 3*w - 2],
[997, 997, -4*w^4 + 5*w^3 + 13*w^2 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 14*x^4 + 46*x^2 - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/4*e^5 + 5/2*e^3 - 7/2*e, 1/2*e^4 - 5*e^2 + 7, -e^3 + 7*e, -1/2*e^5 + 7*e^3 - 23*e, 1/2*e^5 - 7*e^3 + 23*e, 1, -1/2*e^5 + 7*e^3 - 21*e, 1/4*e^5 - 5/2*e^3 + 7/2*e, 1/2*e^4 - 3*e^2 - 1, -1/4*e^5 + 7/2*e^3 - 21/2*e, -2*e, -2, -1/2*e^4 + 2*e^2 + 9, 1/2*e^5 - 6*e^3 + 16*e, e^4 - 8*e^2 + 6, 3/4*e^5 - 19/2*e^3 + 49/2*e, -e^4 + 10*e^2 - 10, 1/2*e^5 - 7*e^3 + 20*e, 1/2*e^4 - 5*e^2 + 5, -3*e^2 + 14, -1/2*e^4 + 5*e^2 - 5, -2*e^2 + 12, -1/2*e^4 + 5*e^2 - 3, 1/2*e^5 - 5*e^3 + 9*e, e^5 - 14*e^3 + 42*e, -e^4 + 8*e^2 + 2, 2*e^3 - 16*e, -1/4*e^5 + 1/2*e^3 + 21/2*e, 3/4*e^5 - 19/2*e^3 + 57/2*e, 1/2*e^4 - 5*e^2 + 17, 1/2*e^4 - 4*e^2 + 7, e^4 - 11*e^2 + 22, -1/2*e^5 + 6*e^3 - 20*e, 3/2*e^4 - 17*e^2 + 25, -5/4*e^5 + 33/2*e^3 - 111/2*e, 4*e^2 - 14, -e^4 + 9*e^2 - 18, -1/2*e^4 + e^2 + 15, -e^4 + 8*e^2, -3/2*e^5 + 20*e^3 - 58*e, -1/2*e^5 + 5*e^3 - 11*e, 1/2*e^4 - e^2 - 5, 7/4*e^5 - 43/2*e^3 + 115/2*e, -5/4*e^5 + 37/2*e^3 - 119/2*e, -1/2*e^5 + 7*e^3 - 21*e, -e^4 + 12*e^2 - 16, -e^4 + 13*e^2 - 20, -3/4*e^5 + 19/2*e^3 - 47/2*e, e^2 - 6, 2*e^2 - 10, 2*e^2 - 10, 1/2*e^5 - 9*e^3 + 39*e, -e^4 + 6*e^2 + 4, 1/2*e^4 - 4*e^2 + 3, 1/2*e^4 - 4*e^2 + 3, -e^5 + 16*e^3 - 53*e, -1/2*e^5 + 7*e^3 - 19*e, e^5 - 14*e^3 + 45*e, -5/4*e^5 + 29/2*e^3 - 71/2*e, 1/2*e^5 - 7*e^3 + 17*e, e^4 - 8*e^2 + 4, -2*e^5 + 26*e^3 - 69*e, 2*e^5 - 28*e^3 + 84*e, -1/2*e^4 + 5*e^2 - 5, -3/2*e^4 + 7*e^2 + 33, 3/2*e^5 - 21*e^3 + 56*e, -2*e^4 + 18*e^2 - 36, 7/4*e^5 - 47/2*e^3 + 133/2*e, e^4 - 14*e^2 + 26, -e^5 + 18*e^3 - 78*e, e^5 - 12*e^3 + 31*e, -3/4*e^5 + 15/2*e^3 - 7/2*e, e^2 - 8, -1/2*e^4 + e^2 + 7, -5/4*e^5 + 33/2*e^3 - 105/2*e, -1/2*e^4 + 3*e^2 - 7, 5/4*e^5 - 31/2*e^3 + 85/2*e, 1/2*e^5 - 11*e^3 + 55*e, 5/2*e^4 - 27*e^2 + 39, 9/4*e^5 - 57/2*e^3 + 147/2*e, -1/4*e^5 + 7/2*e^3 - 29/2*e, 1/2*e^5 - e^3 - 21*e, -e^5 + 16*e^3 - 64*e, -7*e^3 + 49*e, -5/4*e^5 + 41/2*e^3 - 173/2*e, 11/4*e^5 - 71/2*e^3 + 197/2*e, e^4 - 2*e^2 - 30, -2*e^3 + 20*e, 5/2*e^5 - 32*e^3 + 80*e, 2*e^4 - 24*e^2 + 42, -1/2*e^4 + e^2 + 3, -9/4*e^5 + 57/2*e^3 - 171/2*e, 3/4*e^5 - 19/2*e^3 + 39/2*e, -1/2*e^5 + e^3 + 19*e, -1/2*e^4 + e^2 + 19, -7/4*e^5 + 45/2*e^3 - 131/2*e, e^5 - 7*e^3 - 3*e, -3/2*e^5 + 20*e^3 - 68*e, -3/2*e^5 + 19*e^3 - 49*e, -20, 3/2*e^5 - 16*e^3 + 38*e, 10*e^2 - 48, e^4 - 4*e^2 - 18, e^4 - 14*e^2 + 24, -3/4*e^5 + 15/2*e^3 - 43/2*e, -3/4*e^5 + 17/2*e^3 - 43/2*e, 2*e^4 - 15*e^2 - 2, 3*e^4 - 24*e^2 + 10, -2*e^5 + 22*e^3 - 51*e, 4*e^3 - 40*e, -e^4 + 9*e^2 - 34, 5*e^2 - 40, e^5 - 11*e^3 + 23*e, -3/2*e^4 + 14*e^2 - 3, -1/2*e^5 + 5*e^3 - 15*e, -e^4 + 8*e^2, -5/4*e^5 + 45/2*e^3 - 175/2*e, e^4 - 9*e^2 + 14, 1/4*e^5 - 9/2*e^3 + 21/2*e, 10, -e^4 + 10*e^2, 4*e^3 - 20*e, 0, -2*e^4 + 16*e^2 + 16, -2*e^4 + 16*e^2 + 22, 5/2*e^5 - 33*e^3 + 94*e, -e^4 + 12*e^2 - 42, e^5 - 18*e^3 + 83*e, -3*e^4 + 23*e^2 - 8, 5/2*e^4 - 22*e^2 + 17, 2*e^5 - 27*e^3 + 97*e, -8*e^2 + 34, 3*e^4 - 28*e^2 + 20, -1/2*e^4 + 11*e^2 - 49, 3*e^5 - 38*e^3 + 99*e, 4*e^4 - 34*e^2 + 20, -3/2*e^4 + 16*e^2 - 23, 9/4*e^5 - 53/2*e^3 + 107/2*e, 4*e, -3/4*e^5 + 39/2*e^3 - 197/2*e, -3*e^4 + 30*e^2 - 36, -e^5 + 14*e^3 - 63*e, 4*e^3 - 34*e, -3/2*e^4 + 11*e^2 - 13, -e^5 + 12*e^3 - 30*e, -2*e^4 + 18*e^2 - 14, e^5 - 6*e^3 - 18*e, 18*e, e^4 - 13*e^2 + 28, -3*e^4 + 26*e^2 - 4, 3/2*e^4 - 17*e^2 + 1, -7/2*e^5 + 45*e^3 - 134*e, 5*e^2 - 18, -3*e^4 + 20*e^2 + 14, 3/2*e^4 - 15*e^2 + 19, -2*e^4 + 22*e^2 - 14, 5/2*e^5 - 37*e^3 + 141*e, 3*e^5 - 35*e^3 + 91*e, -5/2*e^4 + 27*e^2 - 39, 3*e^5 - 42*e^3 + 129*e, -7/2*e^4 + 36*e^2 - 41, -3/4*e^5 + 23/2*e^3 - 81/2*e, 2*e^2 - 40, 10*e^2 - 30, -3/4*e^5 + 19/2*e^3 - 41/2*e, 9/4*e^5 - 57/2*e^3 + 135/2*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {32, 2, 2}>] := -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;