/*
  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, -2, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {31, 31, -w^3 + 2*w^2 + 3*w - 2}>;

primesArray := [
[5, 5, -w^3 + 2*w^2 + 3*w],
[7, 7, w - 1],
[11, 11, -w^3 + 2*w^2 + 4*w],
[13, 13, -2*w^3 + 3*w^2 + 10*w - 2],
[16, 2, 2],
[17, 17, -w^3 + 2*w^2 + 5*w - 3],
[17, 17, -w^3 + w^2 + 5*w],
[17, 17, -w^2 + 2*w + 1],
[19, 19, w^2 - w - 2],
[29, 29, w^3 - 2*w^2 - 5*w],
[29, 29, w^2 - w - 3],
[31, 31, -w^3 + 2*w^2 + 3*w - 2],
[43, 43, 2*w^3 - 3*w^2 - 11*w],
[47, 47, w^3 - 7*w - 4],
[53, 53, -2*w^3 + 3*w^2 + 9*w - 1],
[61, 61, -w - 3],
[73, 73, -w^3 + 2*w^2 + 3*w - 3],
[73, 73, w^3 - w^2 - 7*w - 1],
[81, 3, -3],
[83, 83, -2*w^3 + 3*w^2 + 9*w + 1],
[83, 83, -w^3 + 3*w^2 + 2*w - 3],
[89, 89, w - 4],
[103, 103, w^3 - 2*w^2 - 6*w + 3],
[113, 113, 2*w^3 - w^2 - 14*w - 6],
[121, 11, -w^2 + 2*w + 7],
[125, 5, -3*w^3 + 4*w^2 + 15*w + 1],
[139, 139, w^3 - 2*w^2 - 5*w - 3],
[139, 139, w^3 - 5*w - 5],
[151, 151, w^3 - 9*w - 7],
[157, 157, -3*w^3 + 6*w^2 + 13*w - 6],
[163, 163, w^3 - 3*w^2 - 2*w + 9],
[167, 167, -w^3 + 7*w + 3],
[167, 167, -3*w^3 + 4*w^2 + 15*w - 1],
[173, 173, 2*w^3 - 3*w^2 - 7*w - 2],
[181, 181, -w^3 + 3*w^2 + 4*w - 5],
[191, 191, w^3 - w^2 - 4*w - 4],
[193, 193, -w^3 + w^2 + 7*w + 6],
[193, 193, 2*w^3 - 2*w^2 - 13*w - 5],
[199, 199, -2*w^3 + 5*w^2 + 5*w - 5],
[211, 211, w^3 - w^2 - 8*w - 3],
[211, 211, -2*w^2 + 3*w + 8],
[227, 227, -w^3 + 3*w^2 + 4*w - 4],
[227, 227, w^2 - 3*w - 6],
[229, 229, w^2 - 5],
[229, 229, 3*w^3 - 4*w^2 - 17*w - 2],
[233, 233, 3*w^3 - 6*w^2 - 13*w + 3],
[233, 233, 3*w^3 - 4*w^2 - 16*w + 1],
[241, 241, -2*w^3 + 4*w^2 + 8*w + 1],
[251, 251, 2*w^3 - 2*w^2 - 9*w - 3],
[277, 277, 3*w^3 - 4*w^2 - 17*w - 1],
[281, 281, -2*w^2 + 5*w + 1],
[283, 283, -4*w^3 + 6*w^2 + 19*w - 2],
[293, 293, -3*w^3 + 6*w^2 + 12*w - 2],
[307, 307, w^3 - w^2 - 4*w - 5],
[307, 307, w^3 - w^2 - 8*w - 2],
[311, 311, 2*w^2 - 3*w - 7],
[313, 313, -2*w^3 + w^2 + 15*w + 3],
[317, 317, -w^3 + 3*w^2 + w - 5],
[331, 331, 2*w^3 - 3*w^2 - 12*w],
[337, 337, w^3 - w^2 - 8*w - 1],
[343, 7, -w^3 + 6*w + 8],
[347, 347, 2*w^3 - 4*w^2 - 7*w + 5],
[349, 349, 2*w^3 - 4*w^2 - 9*w],
[353, 353, -3*w^3 + 4*w^2 + 16*w + 5],
[353, 353, -3*w^3 + 4*w^2 + 14*w],
[353, 353, -w^3 + 3*w^2 + 4*w - 8],
[353, 353, -w^3 + 2*w^2 + 8*w - 3],
[359, 359, -4*w^3 + 7*w^2 + 19*w - 3],
[367, 367, -w^3 + 2*w^2 + 6*w - 5],
[373, 373, 5*w^3 - 8*w^2 - 26*w + 3],
[379, 379, -3*w^3 + 5*w^2 + 17*w - 4],
[379, 379, -w^3 + 2*w^2 + 4*w - 6],
[383, 383, w^2 - 6],
[389, 389, w^3 - 2*w^2 - 3*w - 4],
[389, 389, w^2 + w - 4],
[397, 397, -4*w^3 + 5*w^2 + 23*w + 5],
[397, 397, -3*w^3 + 5*w^2 + 12*w - 2],
[419, 419, -3*w^3 + 4*w^2 + 17*w + 5],
[419, 419, -3*w^3 + 5*w^2 + 15*w],
[419, 419, 3*w^3 - 5*w^2 - 14*w - 1],
[419, 419, 2*w^3 - 3*w^2 - 10*w - 6],
[431, 431, 2*w^3 - 2*w^2 - 11*w],
[433, 433, w^3 - 8*w - 1],
[433, 433, 2*w^2 - 2*w - 5],
[439, 439, 2*w^3 - 2*w^2 - 9*w - 6],
[443, 443, 2*w^3 - 5*w^2 - 7*w],
[443, 443, 2*w^3 - 5*w^2 - 9*w + 4],
[457, 457, -2*w^3 + 5*w^2 + 7*w - 4],
[457, 457, -w^3 + 2*w^2 - 3],
[461, 461, -4*w^3 + 7*w^2 + 17*w - 7],
[479, 479, 3*w^3 - 5*w^2 - 12*w + 6],
[487, 487, -2*w^3 + 5*w^2 + 9*w - 9],
[487, 487, w^3 - 10*w - 8],
[499, 499, 2*w^3 - 2*w^2 - 12*w + 1],
[503, 503, w^2 - 7],
[503, 503, w^3 + w^2 - 7*w - 8],
[529, 23, -3*w^3 + 5*w^2 + 12*w + 1],
[529, 23, 3*w^3 - 4*w^2 - 14*w - 1],
[541, 541, w^3 - 4*w^2 - 2*w + 11],
[563, 563, -2*w^3 + w^2 + 15*w + 6],
[569, 569, w^3 - 2*w^2 - 4*w - 4],
[601, 601, 3*w^3 - 5*w^2 - 17*w + 1],
[607, 607, -2*w^3 + 5*w^2 + 7*w - 5],
[607, 607, -w^3 + 3*w^2 + 6*w - 5],
[613, 613, -4*w^3 + 5*w^2 + 21*w + 1],
[613, 613, -5*w^3 + 8*w^2 + 23*w - 4],
[619, 619, w^3 + w^2 - 9*w - 5],
[619, 619, -3*w^3 + 4*w^2 + 14*w + 2],
[641, 641, -6*w^3 + 8*w^2 + 31*w + 3],
[643, 643, 3*w^3 - 4*w^2 - 17*w - 6],
[643, 643, -2*w^3 + 5*w^2 + 7*w - 6],
[647, 647, 3*w^3 - 4*w^2 - 13*w - 10],
[653, 653, -3*w^3 + 5*w^2 + 14*w + 2],
[653, 653, w^3 - 2*w^2 - w - 3],
[661, 661, 5*w^3 - 9*w^2 - 24*w + 5],
[683, 683, 2*w^3 - 3*w^2 - 13*w - 2],
[683, 683, 4*w^3 - 7*w^2 - 18*w + 1],
[719, 719, -w^3 + 2*w^2 + 3*w - 7],
[719, 719, 3*w^3 - 5*w^2 - 16*w - 1],
[727, 727, -2*w^3 + 5*w^2 + 8*w - 7],
[727, 727, 2*w^3 - 5*w^2 - 7*w + 13],
[733, 733, -2*w^2 + 7],
[733, 733, -3*w^3 + 5*w^2 + 11*w + 2],
[739, 739, -6*w^3 + 9*w^2 + 29*w - 3],
[739, 739, -2*w^3 + 3*w^2 + 7*w + 5],
[751, 751, 4*w^3 - 5*w^2 - 23*w - 3],
[757, 757, -w^3 + 10*w + 4],
[757, 757, 2*w^3 - w^2 - 14*w - 4],
[769, 769, -w^3 + 4*w^2 + 4*w - 8],
[769, 769, 4*w^3 - 6*w^2 - 21*w - 2],
[787, 787, 3*w^3 - 3*w^2 - 19*w - 6],
[787, 787, -2*w^2 + 6*w + 7],
[839, 839, -3*w^3 + 3*w^2 + 19*w + 5],
[841, 29, 3*w^3 - 4*w^2 - 13*w - 4],
[853, 853, -3*w^3 + 5*w^2 + 14*w + 3],
[853, 853, 2*w^2 - w - 9],
[853, 853, 3*w^3 - 5*w^2 - 11*w + 1],
[853, 853, w - 6],
[857, 857, 2*w^3 - 2*w^2 - 9*w - 7],
[857, 857, -4*w^3 + 8*w^2 + 13*w - 1],
[859, 859, 2*w^3 - 4*w^2 - 13*w + 2],
[859, 859, w^3 - w^2 - 9*w - 2],
[863, 863, -2*w^3 - w^2 + 17*w + 11],
[877, 877, 5*w^3 - 6*w^2 - 28*w - 9],
[877, 877, -4*w^3 + 5*w^2 + 20*w + 6],
[881, 881, 4*w^3 - 5*w^2 - 18*w - 8],
[911, 911, -w^3 + 4*w^2 - w - 8],
[911, 911, 2*w^3 - 3*w^2 - 13*w - 1],
[919, 919, w^3 - w^2 - 7*w + 5],
[929, 929, 2*w^3 - 3*w^2 - 13*w + 1],
[929, 929, -2*w^3 + 5*w^2 + 6*w - 11],
[941, 941, -3*w^3 + 7*w^2 + 12*w - 5],
[953, 953, -2*w^3 + 5*w^2 + 5*w - 7],
[971, 971, w^3 - w^2 - 3*w - 5],
[971, 971, 3*w^3 - 7*w^2 - 12*w + 11],
[977, 977, 3*w^3 - 5*w^2 - 11*w],
[997, 997, -2*w^3 + 3*w^2 + 6*w - 5],
[997, 997, -4*w^3 + 7*w^2 + 16*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - x^3 - 15*x^2 + 20*x + 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/3*e^3 - 4*e + 2/3, 1/3*e^3 - 4*e + 8/3, -2, 1/3*e^3 - 5*e + 11/3, 2, 1/6*e^3 + 1/2*e^2 - 3/2*e - 11/3, 1/3*e^3 - 6*e + 14/3, 1/3*e^3 - 5*e + 2/3, 1/3*e^3 - 5*e + 20/3, -e + 2, -1, -1/3*e^3 + 4*e - 20/3, e^2 - 4, -e^2 - e + 12, -1/3*e^3 + 2*e - 2/3, -1/6*e^3 - 1/2*e^2 + 9/2*e + 17/3, 1/6*e^3 - 1/2*e^2 - 9/2*e + 43/3, 1/6*e^3 + 1/2*e^2 - 5/2*e - 5/3, 1/6*e^3 - 3/2*e^2 - 9/2*e + 49/3, -1/6*e^3 + 3/2*e^2 + 1/2*e - 49/3, -1/6*e^3 - 1/2*e^2 + 11/2*e + 11/3, 2/3*e^3 - 8*e - 2/3, -1/3*e^3 + 7*e - 8/3, -1/2*e^3 + 1/2*e^2 + 17/2*e - 5, -2/3*e^3 + e^2 + 12*e - 22/3, -1/6*e^3 - 1/2*e^2 + 7/2*e + 29/3, -e^3 + 13*e - 14, e^2 - 8, 5/6*e^3 + 1/2*e^2 - 19/2*e + 29/3, 3/2*e^3 + 1/2*e^2 - 43/2*e + 9, 1/3*e^3 - 4*e - 16/3, 2/3*e^3 - 11*e - 2/3, -5/6*e^3 - 1/2*e^2 + 19/2*e - 5/3, -4/3*e^3 + e^2 + 22*e - 26/3, 1/3*e^3 + 2*e^2 - 4*e - 40/3, -1/2*e^3 - 7/2*e^2 + 11/2*e + 25, -2/3*e^3 - e^2 + 6*e + 14/3, -2*e^2 - 4*e + 26, -2/3*e^3 + 2*e^2 + 10*e - 40/3, -e^3 + 12*e - 8, -4/3*e^3 - e^2 + 19*e - 2/3, -e^2 + e + 6, -e^3 - e^2 + 10*e - 6, -1/6*e^3 - 1/2*e^2 - 5/2*e + 35/3, 2/3*e^3 + e^2 - 6*e - 14/3, -2/3*e^3 - e^2 + 9*e - 28/3, -2/3*e^3 + 13*e - 16/3, -11/6*e^3 + 3/2*e^2 + 55/2*e - 59/3, 8/3*e^3 - 38*e + 58/3, 4/3*e^3 + e^2 - 19*e - 4/3, 1/3*e^3 + e^2 - 2*e - 52/3, 4/3*e^3 - 24*e + 62/3, -e^3 - e^2 + 14*e - 4, -e^3 + 14*e + 14, -5/6*e^3 - 7/2*e^2 + 19/2*e + 73/3, -e^3 - e^2 + 12*e - 10, e^3 + 2*e^2 - 16*e - 14, -2*e^3 - 2*e^2 + 28*e - 8, -4/3*e^3 + 18*e + 22/3, 2*e^3 - e^2 - 31*e + 14, 2/3*e^3 + 2*e^2 - 4*e - 38/3, -2*e^3 + 2*e^2 + 28*e - 14, -2/3*e^3 + e^2 + 11*e - 40/3, -e^3 - 2*e^2 + 7*e + 20, 4/3*e^3 - 3*e^2 - 19*e + 92/3, 3/2*e^3 + 1/2*e^2 - 45/2*e + 9, 5/3*e^3 - e^2 - 28*e + 70/3, -8/3*e^3 + 2*e^2 + 37*e - 70/3, -3/2*e^3 + 1/2*e^2 + 47/2*e - 15, 2*e^3 + e^2 - 29*e + 6, 5/3*e^3 + 2*e^2 - 23*e + 22/3, 2*e^2 + 3*e - 2, 1/2*e^3 - 3/2*e^2 - 11/2*e + 15, 5/2*e^3 + 1/2*e^2 - 65/2*e + 17, -13/6*e^3 - 1/2*e^2 + 55/2*e - 37/3, 5/3*e^3 - 2*e^2 - 21*e + 58/3, 5/6*e^3 - 7/2*e^2 - 17/2*e + 101/3, 3*e^2 + 4*e - 36, -7/6*e^3 - 3/2*e^2 + 27/2*e - 19/3, -2/3*e^3 + 8*e - 10/3, 3/2*e^3 - 3/2*e^2 - 47/2*e + 1, -4/3*e^3 + 18*e - 62/3, e^2 + e - 8, -7/3*e^3 - 2*e^2 + 24*e + 28/3, -2/3*e^3 - 3*e^2 + 9*e + 2/3, 5/6*e^3 + 5/2*e^2 - 23/2*e - 73/3, -1/3*e^3 - 3*e - 2/3, 17/6*e^3 - 1/2*e^2 - 81/2*e + 95/3, -e^3 + 15*e - 26, -7/6*e^3 - 1/2*e^2 + 21/2*e + 11/3, e^2 - 2*e - 6, e^3 + 2*e^2 - 6*e - 8, -1/3*e^3 + 2*e^2 + 5*e - 2/3, -2*e^3 - e^2 + 24*e + 4, 11/6*e^3 + 5/2*e^2 - 33/2*e - 43/3, e^2 + e + 12, 2*e^3 - 2*e^2 - 23*e + 24, -7/3*e^3 + 2*e^2 + 35*e - 92/3, -7/6*e^3 + 1/2*e^2 + 27/2*e - 7/3, 2/3*e^3 - e^2 - 6*e + 118/3, -1/3*e^3 - 38/3, 2/3*e^3 - 2*e^2 - 2*e + 88/3, -5/2*e^3 + 1/2*e^2 + 67/2*e - 29, e^3 - e^2 - 16*e - 6, 1/3*e^3 - e^2 - 3*e - 28/3, 4/3*e^3 + e^2 - 10*e + 2/3, -e^3 - 4*e^2 + 9*e + 22, -11/6*e^3 + 3/2*e^2 + 67/2*e - 77/3, -17/6*e^3 + 1/2*e^2 + 91/2*e - 95/3, -8/3*e^3 - 3*e^2 + 33*e + 2/3, -2/3*e^3 - 2*e^2 + 14*e + 56/3, -8/3*e^3 + 37*e - 40/3, -e^3 + 19*e + 8, e^2 - e + 20, -5/3*e^3 + 22*e - 34/3, 4/3*e^3 - 13*e + 62/3, -7/3*e^3 - 2*e^2 + 28*e + 28/3, -5*e^3 + 61*e - 30, -1/2*e^3 - 5/2*e^2 + 17/2*e - 3, -1/3*e^3 - 2*e^2 + 4*e + 52/3, 5/6*e^3 + 3/2*e^2 - 31/2*e + 5/3, -4*e^3 + 3*e^2 + 53*e - 36, -5/3*e^3 + 2*e^2 + 26*e - 130/3, -1/3*e^3 + e^2 + 5*e + 34/3, -5/3*e^3 + 20*e + 20/3, 5/6*e^3 + 3/2*e^2 - 29/2*e + 23/3, -1/6*e^3 - 1/2*e^2 - 7/2*e - 19/3, -2/3*e^3 + 16*e - 10/3, 1/2*e^3 - 3/2*e^2 + 1/2*e + 23, -4/3*e^3 + e^2 + 15*e - 74/3, -1/6*e^3 - 1/2*e^2 + 25/2*e + 71/3, 13/3*e^3 + 2*e^2 - 65*e + 86/3, -1/6*e^3 + 7/2*e^2 + 9/2*e - 115/3, -1/2*e^3 - 1/2*e^2 + 37/2*e + 3, -1/3*e^3 + 5*e^2 + 7*e - 122/3, -1/2*e^3 - 1/2*e^2 + 13/2*e - 17, -2*e^3 + 3*e^2 + 33*e - 26, -11/6*e^3 - 9/2*e^2 + 39/2*e + 127/3, 5/6*e^3 - 9/2*e^2 - 19/2*e + 89/3, 17/6*e^3 + 1/2*e^2 - 73/2*e + 125/3, -5/3*e^3 - 3*e^2 + 18*e + 20/3, -2/3*e^3 - e^2 + 7*e + 2/3, 2*e^3 + 5*e^2 - 23*e - 22, -1/3*e^3 - 4*e^2 + 4*e + 94/3, 7/3*e^3 - 2*e^2 - 37*e + 92/3, 3/2*e^3 - 1/2*e^2 - 57/2*e + 19, 17/6*e^3 + 1/2*e^2 - 69/2*e + 89/3, 4/3*e^3 - 21*e - 10/3, -2*e^3 + e^2 + 17*e - 16, 4/3*e^3 - 13*e + 26/3, 2/3*e^3 + 3*e^2 - 86/3, 7/6*e^3 + 3/2*e^2 - 7/2*e - 59/3, -4/3*e^3 - e^2 + 21*e - 26/3, 10/3*e^3 + 4*e^2 - 38*e - 52/3, 1/6*e^3 + 9/2*e^2 - 7/2*e - 71/3, 5/3*e^3 - e^2 - 15*e + 88/3, e^3 - 4*e^2 - 16*e + 26];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {31, 31, -w^3 + 2*w^2 + 3*w - 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;