/*
  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, 7, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[2, 2, -w + 1],
[4, 2, -w^2 - w + 3],
[5, 5, w^3 - 5*w + 1],
[11, 11, -w^3 + 6*w - 2],
[13, 13, -w^2 + 4],
[17, 17, 2*w^3 + w^2 - 10*w - 2],
[37, 37, w^3 + w^2 - 6*w - 3],
[37, 37, -w^3 + 6*w - 4],
[41, 41, -w^3 + 5*w - 5],
[43, 43, w^3 - 4*w + 4],
[43, 43, -2*w^3 + 11*w - 2],
[47, 47, -2*w^3 - w^2 + 12*w],
[47, 47, 2*w - 1],
[61, 61, -w^3 + w^2 + 6*w - 5],
[71, 71, w^3 + w^2 - 4*w - 3],
[71, 71, 2*w^3 + 2*w^2 - 9*w - 2],
[79, 79, w^3 + w^2 - 6*w - 5],
[81, 3, -3],
[83, 83, -3*w^3 - w^2 + 16*w + 3],
[97, 97, w^3 + w^2 - 6*w + 1],
[103, 103, -2*w^3 - 2*w^2 + 7*w - 2],
[103, 103, -3*w^3 + 14*w - 12],
[107, 107, -w^3 + 3*w + 3],
[125, 5, -2*w^3 - w^2 + 8*w - 2],
[127, 127, w^3 - w^2 - 4*w + 1],
[131, 131, 2*w^3 - 10*w + 3],
[137, 137, -5*w^3 - 2*w^2 + 26*w + 2],
[137, 137, w^3 - 7*w + 1],
[139, 139, -2*w^3 - w^2 + 8*w],
[149, 149, w^3 - 2*w^2 - 7*w + 9],
[149, 149, -2*w^3 + 11*w],
[151, 151, 2*w^2 + 2*w - 9],
[151, 151, -5*w^3 - 2*w^2 + 27*w + 1],
[151, 151, w^3 - 7*w + 3],
[151, 151, 2*w^3 + 2*w^2 - 11*w - 8],
[157, 157, w^3 - 7*w - 3],
[169, 13, 6*w^3 + 4*w^2 - 32*w - 11],
[173, 173, w^3 + w^2 - 4*w - 7],
[191, 191, -3*w^3 + 14*w - 10],
[193, 193, 2*w^3 - 12*w + 3],
[197, 197, w + 4],
[197, 197, w^3 + 2*w^2 - 6*w - 6],
[197, 197, 2*w^3 + w^2 - 10*w + 2],
[197, 197, 2*w^2 - 3],
[199, 199, w^3 + 2*w^2 - 4*w - 4],
[223, 223, -w^3 - 2*w^2 + 5*w + 7],
[229, 229, 5*w^3 + 3*w^2 - 26*w - 5],
[233, 233, 2*w + 3],
[241, 241, 2*w^3 + 2*w^2 - 10*w - 3],
[251, 251, w^3 - 2*w^2 - 5*w + 7],
[263, 263, 2*w^3 + w^2 - 12*w + 2],
[269, 269, 2*w^3 + 3*w^2 - 14*w - 4],
[271, 271, 3*w^3 + 2*w^2 - 15*w - 7],
[281, 281, 2*w^2 + w - 4],
[281, 281, w^2 - 8],
[283, 283, -3*w^3 + 18*w - 2],
[283, 283, -w^3 - w^2 + 4*w - 3],
[283, 283, -w^3 + w^2 + 6*w - 3],
[283, 283, w^3 + 2*w^2 - 5*w - 3],
[293, 293, -4*w^3 + 19*w - 16],
[307, 307, -w^3 + 2*w^2 + 6*w - 8],
[311, 311, 3*w^3 + 2*w^2 - 17*w - 9],
[311, 311, w^3 + 2*w^2 - 6*w - 8],
[313, 313, -4*w^3 - 2*w^2 + 17*w - 6],
[317, 317, 2*w^3 - 9*w + 2],
[317, 317, 3*w^2 + 2*w - 10],
[331, 331, 5*w^3 + 4*w^2 - 26*w - 12],
[337, 337, 2*w^3 + w^2 - 8*w + 6],
[347, 347, 2*w^3 - 9*w + 4],
[353, 353, w^3 - 5*w - 3],
[359, 359, -w^3 + w + 3],
[367, 367, -2*w^3 + 12*w - 7],
[367, 367, 2*w^3 + 4*w^2 - 6*w - 7],
[379, 379, -w^3 - 2*w^2 + w + 3],
[379, 379, -3*w^2 - 2*w + 12],
[397, 397, -2*w^3 - w^2 + 6*w + 2],
[397, 397, -w^3 + 3*w - 7],
[397, 397, -2*w^3 - 2*w^2 + 9*w + 6],
[397, 397, w^3 + 2*w^2 - 5*w - 5],
[401, 401, 3*w^3 + 3*w^2 - 16*w - 13],
[401, 401, w^3 + 2*w^2 - 6*w - 4],
[431, 431, 3*w^3 + 2*w^2 - 14*w - 4],
[431, 431, 2*w^3 + w^2 - 10*w + 4],
[433, 433, -2*w^3 + 2*w^2 + 13*w - 12],
[433, 433, -w^3 + 4*w^2 + 8*w - 18],
[439, 439, 2*w^3 + 2*w^2 - 8*w + 3],
[439, 439, -w^3 + 3*w^2 + 6*w - 11],
[443, 443, -7*w^3 - 2*w^2 + 39*w + 1],
[457, 457, w^3 + 2*w^2 - 3*w - 7],
[461, 461, w^2 - 2*w - 4],
[461, 461, 4*w^3 + 2*w^2 - 20*w - 5],
[463, 463, 3*w^3 + 2*w^2 - 13*w - 3],
[467, 467, -3*w^3 + 15*w - 11],
[467, 467, -4*w^3 + 18*w + 1],
[479, 479, 2*w^2 + 2*w - 11],
[487, 487, -3*w^3 - 2*w^2 + 19*w + 1],
[487, 487, 4*w^3 + 2*w^2 - 19*w],
[491, 491, 3*w^3 + 2*w^2 - 18*w],
[499, 499, -w^3 + 3*w - 5],
[499, 499, -6*w^3 - 2*w^2 + 31*w - 2],
[523, 523, 2*w^2 + 3*w - 10],
[541, 541, 2*w^3 + 3*w^2 - 8*w - 6],
[547, 547, -3*w^3 - w^2 + 16*w - 3],
[563, 563, -w^3 + 5*w - 7],
[563, 563, 2*w^2 + w - 2],
[571, 571, 3*w^3 + 2*w^2 - 12*w + 6],
[577, 577, -2*w^3 + w^2 + 8*w - 6],
[577, 577, -2*w^3 - 2*w^2 + 12*w + 11],
[587, 587, -5*w^3 - w^2 + 26*w - 5],
[593, 593, -w^3 + 8*w - 4],
[599, 599, -w^3 + w^2 + 4*w - 9],
[601, 601, 3*w^3 + w^2 - 14*w + 1],
[607, 607, 2*w^2 + 2*w - 3],
[607, 607, 4*w^3 - 24*w + 1],
[613, 613, -3*w^3 - 3*w^2 + 16*w + 9],
[617, 617, -2*w^3 + 13*w - 6],
[619, 619, -4*w^3 + 20*w - 1],
[631, 631, -5*w^3 - 2*w^2 + 29*w + 5],
[643, 643, -w^3 - 2*w^2 + 6*w + 10],
[647, 647, -5*w^3 - 4*w^2 + 20*w - 4],
[647, 647, -3*w^3 - 3*w^2 + 18*w + 7],
[653, 653, 4*w^3 - 20*w + 3],
[653, 653, w^3 - 8*w + 2],
[659, 659, -9*w^3 - 4*w^2 + 48*w + 4],
[659, 659, -5*w^3 - 2*w^2 + 25*w - 1],
[661, 661, 4*w^3 + 2*w^2 - 21*w],
[673, 673, -3*w^3 + w^2 + 18*w - 9],
[683, 683, 2*w^3 + 2*w^2 - 7*w - 4],
[691, 691, -5*w^3 - 3*w^2 + 28*w + 7],
[691, 691, 3*w^3 - 15*w + 5],
[701, 701, -2*w^3 + 2*w^2 + 11*w - 10],
[701, 701, -w^3 + 8*w - 10],
[709, 709, -4*w^3 - w^2 + 20*w - 4],
[709, 709, w^3 + 4*w^2 - w - 13],
[719, 719, -2*w^3 + 12*w - 9],
[739, 739, -6*w^3 - 4*w^2 + 29*w + 6],
[739, 739, -w^3 + w^2 + 8*w - 7],
[769, 769, -w^3 + w^2 + 8*w - 5],
[769, 769, 3*w^3 - 2*w^2 - 15*w + 19],
[773, 773, -4*w^3 + 22*w - 9],
[797, 797, -2*w^3 + 13*w - 2],
[827, 827, -w^3 + 6*w - 8],
[827, 827, -4*w^2 - 3*w + 14],
[839, 839, w^2 + 4*w - 6],
[857, 857, 7*w^3 + 2*w^2 - 37*w + 1],
[863, 863, 3*w^3 - 14*w + 2],
[881, 881, -3*w^3 + 2*w^2 + 15*w - 13],
[883, 883, 2*w^3 - w^2 - 10*w + 14],
[887, 887, -5*w^3 - 4*w^2 + 25*w + 11],
[887, 887, 3*w^3 - 14*w + 8],
[907, 907, 4*w^3 - 20*w + 13],
[907, 907, 2*w^2 + 4*w - 5],
[907, 907, -3*w^3 + 18*w - 4],
[907, 907, w - 6],
[947, 947, 3*w^3 + 3*w^2 - 12*w - 5],
[947, 947, 2*w^3 + 3*w^2 - 10*w - 10],
[961, 31, 2*w^3 - w^2 - 12*w + 4],
[961, 31, -w^3 + 2*w^2 + 5*w - 5],
[971, 971, -4*w^3 - w^2 + 20*w - 2],
[991, 991, -2*w^3 + 3*w^2 + 14*w - 18],
[997, 997, 7*w^3 + 2*w^2 - 36*w + 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 3*x^3 - 11*x^2 + 31*x - 2;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, -1/4*e^3 + 1/2*e^2 + 9/4*e - 5/2, e - 1, -1/4*e^3 - 1/2*e^2 + 13/4*e + 3/2, -1/4*e^3 - 1/2*e^2 + 13/4*e + 11/2, -3/4*e^3 + 3/2*e^2 + 31/4*e - 17/2, 1/4*e^3 + 1/2*e^2 - 17/4*e - 1/2, 1/4*e^3 + 1/2*e^2 - 9/4*e - 13/2, e^3 - e^2 - 10*e + 10, -1/2*e^3 + 7/2*e + 1, 1/2*e^3 - e^2 - 7/2*e + 4, 1/2*e^3 - e^2 - 9/2*e + 9, e^3 - e^2 - 11*e + 9, 1/2*e^3 - e^2 - 13/2*e + 11, e^3 - 2*e^2 - 10*e + 15, 1/2*e^3 - e^2 - 17/2*e + 5, 3/4*e^3 - 1/2*e^2 - 23/4*e - 5/2, -1/2*e^3 + e^2 + 11/2*e - 2, 3/4*e^3 + 1/2*e^2 - 43/4*e - 5/2, 1/2*e^3 - 15/2*e - 1, e^2 + e - 6, 1/2*e^3 - 11/2*e + 9, 7/4*e^3 - 1/2*e^2 - 87/4*e + 25/2, -3/2*e^3 + 33/2*e + 1, e^3 - 2*e^2 - 12*e + 13, -e^3 + 2*e^2 + 13*e - 20, -3/4*e^3 + 3/2*e^2 + 47/4*e - 17/2, e^3 - 11*e + 6, 1/4*e^3 - 3/2*e^2 - 33/4*e + 43/2, -5/4*e^3 + 5/2*e^2 + 57/4*e - 39/2, -1/2*e^3 + 11/2*e - 5, -3/2*e^3 + e^2 + 37/2*e - 18, -3/2*e^3 + 3*e^2 + 39/2*e - 25, e^2 - 2*e + 1, -3/4*e^3 + 3/2*e^2 + 35/4*e - 11/2, 3/4*e^3 + 1/2*e^2 - 35/4*e - 9/2, -5/4*e^3 - 1/2*e^2 + 49/4*e + 19/2, -3/2*e^3 + 2*e^2 + 25/2*e - 21, 2*e^3 - 2*e^2 - 22*e + 24, -e^3 + 9*e + 6, e^3 + e^2 - 15*e - 5, 3/4*e^3 - 5/2*e^2 - 11/4*e + 41/2, -3/4*e^3 - 1/2*e^2 + 15/4*e + 27/2, -1/2*e^3 - 2*e^2 + 13/2*e + 8, 3/2*e^3 - 3*e^2 - 37/2*e + 12, -3/4*e^3 + 5/2*e^2 + 23/4*e - 31/2, 3/4*e^3 + 3/2*e^2 - 27/4*e - 15/2, 1/4*e^3 + 1/2*e^2 - 13/4*e - 3/2, 3/2*e^3 + e^2 - 27/2*e - 13, e^3 - 3*e^2 - 13*e + 23, -1/4*e^3 - 7/2*e^2 + 13/4*e + 57/2, -1/2*e^3 - 2*e^2 + 17/2*e + 22, -5/4*e^3 - 5/2*e^2 + 49/4*e + 63/2, e^3 + 3*e^2 - 15*e - 19, -2*e^3 + 2*e^2 + 23*e - 15, 1/2*e^3 - 2*e^2 - 21/2*e + 28, e^3 - 12*e + 11, -e^2 + 3*e - 10, 3/4*e^3 - 1/2*e^2 - 31/4*e + 23/2, -2*e^3 + 4*e^2 + 24*e - 30, -e^3 + 2*e^2 + 10*e - 31, e^2 + 6*e - 23, 1/4*e^3 + 3/2*e^2 - 13/4*e - 37/2, 1/4*e^3 - 5/2*e^2 - 9/4*e + 49/2, -2*e^3 + 4*e^2 + 16*e - 28, -28, 1/4*e^3 - 5/2*e^2 - 17/4*e + 45/2, -e^3 + 3*e^2 + 10*e - 20, -7/4*e^3 - 1/2*e^2 + 83/4*e - 13/2, -1/2*e^3 + e^2 + 3/2*e - 10, e^3 - 7*e - 10, 3/2*e^3 - 2*e^2 - 41/2*e + 37, 2*e^3 - 3*e^2 - 25*e + 26, -1/2*e^3 + e^2 + 17/2*e + 7, 5/4*e^3 - 5/2*e^2 - 49/4*e + 43/2, e^2 + 6*e - 9, -5/4*e^3 - 1/2*e^2 + 53/4*e + 9/2, 3*e^3 - 2*e^2 - 37*e + 34, 1/4*e^3 - 5/2*e^2 - 9/4*e + 25/2, -7/4*e^3 + 7/2*e^2 + 67/4*e - 61/2, -e^3 + e^2 + 12*e - 24, -e^3 - e^2 + 10*e + 28, 2*e^3 - 2*e^2 - 24*e + 26, -3/4*e^3 + 3/2*e^2 + 35/4*e - 35/2, e^2 - 8*e - 9, 3/2*e^3 - 2*e^2 - 43/2*e + 34, -3/2*e^3 - 2*e^2 + 23/2*e + 32, -1/4*e^3 + 1/2*e^2 + 5/4*e - 3/2, -e^3 + 2*e^2 + 11*e - 22, 4*e - 6, 3/2*e^3 - e^2 - 45/2*e + 14, 5/2*e^3 - 4*e^2 - 55/2*e + 41, -e^3 + 6*e + 3, e^3 - 2*e^2 - 14*e - 5, -3/2*e^3 + 3*e^2 + 35/2*e - 15, -2*e^3 + 28*e - 10, -3/2*e^3 + 3*e^2 + 33/2*e - 6, 1/2*e^3 - 3*e^2 - 1/2*e + 19, 1/2*e^3 - 2*e^2 - 23/2*e + 17, 7/2*e^3 - 5*e^2 - 71/2*e + 37, -3/4*e^3 + 3/2*e^2 + 67/4*e - 27/2, -5/2*e^3 + 3*e^2 + 65/2*e - 17, 1/2*e^3 - e^2 - 25/2*e - 3, 3/2*e^3 - 2*e^2 - 31/2*e + 40, -e^3 + 2*e^2 + 7*e + 4, -15/4*e^3 + 5/2*e^2 + 171/4*e - 59/2, e^3 - 11*e - 4, 2*e - 14, -5/4*e^3 + 7/2*e^2 + 89/4*e - 49/2, 1/2*e^3 - 5*e^2 - 9/2*e + 37, -2*e^3 + 4*e^2 + 30*e - 46, -1/2*e^3 - 2*e^2 + 19/2*e + 33, -5/2*e^3 + 2*e^2 + 45/2*e - 10, -7/4*e^3 - 11/2*e^2 + 103/4*e + 83/2, -7/4*e^3 - 1/2*e^2 + 71/4*e + 49/2, 1/2*e^3 - 2*e^2 - 11/2*e + 43, 5/2*e^3 - 2*e^2 - 67/2*e + 41, -9/2*e^3 + 3*e^2 + 111/2*e - 42, e^3 - 17*e + 24, e^2 - 11*e - 2, -3/4*e^3 + 5/2*e^2 + 63/4*e - 83/2, -5/4*e^3 + 1/2*e^2 + 45/4*e + 11/2, 2*e^2 - 6*e - 40, 3*e^3 - 4*e^2 - 32*e + 33, 3/4*e^3 - 5/2*e^2 - 39/4*e + 47/2, -11/4*e^3 - 5/2*e^2 + 139/4*e + 21/2, -3/2*e^3 + 2*e^2 + 43/2*e - 6, -1/2*e^3 + 2*e^2 + 11/2*e - 27, e^3 + e^2 - 10*e - 32, 11/4*e^3 - 3/2*e^2 - 107/4*e + 3/2, 5/4*e^3 - 1/2*e^2 - 69/4*e + 9/2, 5/4*e^3 - 9/2*e^2 - 69/4*e + 65/2, -2*e^3 + e^2 + 20*e + 19, -1/2*e^3 - 3*e^2 + 11/2*e - 2, 2*e^3 + e^2 - 25*e - 2, -5/2*e^3 - e^2 + 57/2*e + 7, 2*e^3 - e^2 - 28*e + 5, 11/4*e^3 - 7/2*e^2 - 131/4*e + 75/2, 13/4*e^3 - 15/2*e^2 - 129/4*e + 89/2, 15/4*e^3 - 7/2*e^2 - 175/4*e + 55/2, e^3 + 2*e^2 - 9*e - 38, 3/2*e^3 - 5*e^2 - 39/2*e + 47, -e^3 + 9*e^2 + 6*e - 58, -2*e^3 + 4*e^2 + 28*e - 36, 2*e^3 - 3*e^2 - 25*e - 2, -2*e^3 + 2*e^2 + 20*e - 10, 3/2*e^3 - e^2 - 31/2*e - 1, 2*e^3 - 6*e^2 - 28*e + 48, -5/2*e^3 + 3*e^2 + 65/2*e - 53, 4*e^3 - 3*e^2 - 42*e + 5, -e^2 + 9*e - 8, e^3 - e^2 - 16*e + 40, -3/2*e^3 + 2*e^2 + 15/2*e - 32, 7/2*e^3 - 8*e^2 - 73/2*e + 45, 7/2*e^3 - 3*e^2 - 63/2*e + 23, e^3 - 4*e^2 - 9*e + 22, 9/4*e^3 - 7/2*e^2 - 77/4*e + 81/2, -7/2*e^3 + 65/2*e + 7, -1/2*e^3 - 4*e^2 + 15/2*e + 13, -3/4*e^3 - 3/2*e^2 + 23/4*e + 105/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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