/*
  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![31, 14, -14, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, 4*w^3 + 7*w^2 - 36*w - 45}>;

primesArray := [
[5, 5, 4*w^3 + 7*w^2 - 37*w - 45],
[11, 11, 2*w^3 + 3*w^2 - 18*w - 16],
[11, 11, -w + 2],
[16, 2, 2],
[19, 19, 4*w^3 + 7*w^2 - 36*w - 45],
[19, 19, -w^3 - w^2 + 9*w + 3],
[19, 19, -2*w^3 - 4*w^2 + 17*w + 23],
[19, 19, w^3 + 2*w^2 - 10*w - 10],
[29, 29, -w - 2],
[29, 29, -w^3 - 2*w^2 + 8*w + 15],
[29, 29, 2*w^3 + 3*w^2 - 18*w - 20],
[29, 29, -3*w^3 - 5*w^2 + 27*w + 28],
[31, 31, 3*w^3 + 5*w^2 - 27*w - 31],
[31, 31, -3*w^3 - 5*w^2 + 27*w + 30],
[31, 31, -2*w^3 - 3*w^2 + 18*w + 17],
[31, 31, 2*w^3 + 3*w^2 - 18*w - 18],
[71, 71, -w^3 - 2*w^2 + 10*w + 8],
[71, 71, 4*w^3 + 8*w^2 - 35*w - 50],
[71, 71, -4*w^3 - 7*w^2 + 36*w + 47],
[71, 71, -w^3 - 2*w^2 + 8*w + 17],
[79, 79, w^3 + 2*w^2 - 10*w - 14],
[79, 79, -w^3 - w^2 + 9*w + 7],
[79, 79, -2*w^3 - 4*w^2 + 17*w + 27],
[79, 79, -4*w^3 - 7*w^2 + 36*w + 41],
[81, 3, -3],
[109, 109, -2*w^3 - 3*w^2 + 19*w + 13],
[109, 109, 3*w^3 + 5*w^2 - 26*w - 26],
[109, 109, 2*w^3 + 3*w^2 - 19*w - 22],
[109, 109, 3*w^3 + 5*w^2 - 26*w - 35],
[139, 139, 4*w^3 + 7*w^2 - 36*w - 40],
[139, 139, -w^3 - w^2 + 9*w + 8],
[139, 139, -w^3 - 2*w^2 + 10*w + 15],
[139, 139, 2*w^3 + 4*w^2 - 17*w - 28],
[149, 149, 4*w^3 + 8*w^2 - 35*w - 52],
[149, 149, -3*w^3 - 5*w^2 + 26*w + 34],
[149, 149, -2*w^3 - 3*w^2 + 19*w + 14],
[149, 149, -6*w^3 - 11*w^2 + 54*w + 66],
[181, 181, w^3 + 2*w^2 - 11*w - 13],
[181, 181, -3*w^3 - 4*w^2 + 27*w + 24],
[181, 181, -7*w^3 - 12*w^2 + 63*w + 72],
[181, 181, 3*w^3 + 6*w^2 - 25*w - 39],
[191, 191, -7*w^3 - 12*w^2 + 66*w + 80],
[191, 191, 5*w^3 + 9*w^2 - 47*w - 57],
[191, 191, -2*w^3 - 2*w^2 + 18*w + 9],
[191, 191, 2*w^3 + 2*w^2 - 17*w - 13],
[239, 239, 7*w^3 + 13*w^2 - 62*w - 81],
[239, 239, -2*w^3 - 5*w^2 + 17*w + 32],
[239, 239, 6*w^3 + 11*w^2 - 55*w - 67],
[239, 239, -w^3 - 3*w^2 + 10*w + 20],
[251, 251, 7*w^3 + 13*w^2 - 63*w - 82],
[251, 251, -5*w^3 - 10*w^2 + 44*w + 61],
[251, 251, 4*w^3 + 8*w^2 - 37*w - 48],
[251, 251, -2*w^3 - 5*w^2 + 18*w + 34],
[311, 311, 5*w^3 + 9*w^2 - 44*w - 50],
[311, 311, -4*w^3 - 7*w^2 + 37*w + 48],
[311, 311, 2*w^3 + 4*w^2 - 19*w - 19],
[311, 311, w^3 + w^2 - 8*w - 11],
[331, 331, 4*w^3 + 6*w^2 - 36*w - 37],
[331, 331, -6*w^3 - 10*w^2 + 54*w + 59],
[331, 331, -2*w - 1],
[331, 331, -2*w^3 - 4*w^2 + 16*w + 27],
[349, 349, -3*w^3 - 5*w^2 + 28*w + 27],
[349, 349, 4*w^3 + 8*w^2 - 35*w - 53],
[349, 349, -3*w^3 - 6*w^2 + 28*w + 40],
[349, 349, 6*w^3 + 11*w^2 - 54*w - 65],
[359, 359, -w^3 - 3*w^2 + 8*w + 19],
[359, 359, -6*w^3 - 11*w^2 + 53*w + 69],
[359, 359, 5*w^3 + 9*w^2 - 46*w - 54],
[359, 359, w^2 - w - 8],
[401, 401, 5*w^3 + 8*w^2 - 45*w - 50],
[401, 401, w^3 + 2*w^2 - 7*w - 15],
[401, 401, w^3 + 2*w^2 - 7*w - 11],
[401, 401, 5*w^3 + 8*w^2 - 45*w - 46],
[409, 409, -9*w^3 - 16*w^2 + 81*w + 97],
[409, 409, w^3 + w^2 - 11*w - 10],
[409, 409, 6*w^3 + 10*w^2 - 53*w - 66],
[409, 409, -3*w^3 - 6*w^2 + 29*w + 38],
[421, 421, -3*w^3 - 4*w^2 + 26*w + 23],
[421, 421, -4*w^3 - 6*w^2 + 35*w + 39],
[421, 421, 8*w^3 + 14*w^2 - 71*w - 86],
[421, 421, -7*w^3 - 12*w^2 + 62*w + 70],
[439, 439, w^3 + w^2 - 10*w - 9],
[439, 439, -4*w^3 - 7*w^2 + 35*w + 47],
[439, 439, 3*w^3 + 5*w^2 - 28*w - 26],
[439, 439, 2*w^3 + 3*w^2 - 17*w - 14],
[479, 479, 3*w^3 + 6*w^2 - 29*w - 33],
[479, 479, 5*w^3 + 10*w^2 - 43*w - 59],
[479, 479, w^3 - 9*w + 6],
[479, 479, 9*w^3 + 16*w^2 - 81*w - 102],
[521, 521, 4*w^3 + 7*w^2 - 38*w - 44],
[521, 521, 3*w^3 + 4*w^2 - 26*w - 25],
[521, 521, 5*w^3 + 8*w^2 - 44*w - 46],
[521, 521, -8*w^3 - 14*w^2 + 71*w + 84],
[569, 569, 21*w^3 + 37*w^2 - 191*w - 236],
[569, 569, -13*w^3 - 23*w^2 + 120*w + 149],
[569, 569, 10*w^3 + 17*w^2 - 92*w - 111],
[569, 569, -18*w^3 - 32*w^2 + 164*w + 201],
[631, 631, -2*w^3 - 3*w^2 + 20*w + 25],
[631, 631, 7*w^3 + 12*w^2 - 64*w - 80],
[631, 631, -9*w^3 - 17*w^2 + 80*w + 106],
[631, 631, 19*w^3 + 33*w^2 - 175*w - 215],
[641, 641, 4*w^3 + 6*w^2 - 35*w - 35],
[641, 641, 4*w^3 + 6*w^2 - 35*w - 38],
[641, 641, 7*w^3 + 12*w^2 - 62*w - 71],
[641, 641, 7*w^3 + 12*w^2 - 62*w - 74],
[659, 659, 6*w^3 + 11*w^2 - 54*w - 63],
[659, 659, 4*w^3 + 6*w^2 - 37*w - 38],
[659, 659, 4*w^3 + 8*w^2 - 35*w - 55],
[659, 659, 3*w^3 + 6*w^2 - 28*w - 42],
[661, 661, -27*w^3 - 48*w^2 + 247*w + 306],
[661, 661, 14*w^3 + 24*w^2 - 129*w - 158],
[661, 661, 11*w^3 + 19*w^2 - 102*w - 126],
[661, 661, 2*w^3 + 3*w^2 - 19*w - 24],
[691, 691, 6*w^3 + 12*w^2 - 53*w - 73],
[691, 691, 8*w^3 + 15*w^2 - 72*w - 95],
[691, 691, -5*w^3 - 10*w^2 + 46*w + 60],
[691, 691, -3*w^3 - 7*w^2 + 27*w + 47],
[739, 739, -2*w^3 - 5*w^2 + 19*w + 34],
[739, 739, 7*w^3 + 13*w^2 - 64*w - 78],
[739, 739, -3*w^3 - 7*w^2 + 26*w + 43],
[739, 739, 8*w^3 + 15*w^2 - 71*w - 95],
[751, 751, -4*w^3 - 9*w^2 + 37*w + 59],
[751, 751, -5*w^3 - 9*w^2 + 43*w + 48],
[751, 751, -6*w^3 - 10*w^2 + 55*w + 68],
[751, 751, -10*w^3 - 17*w^2 + 93*w + 114],
[769, 769, 6*w^3 + 11*w^2 - 52*w - 69],
[769, 769, w^3 + w^2 - 7*w - 5],
[769, 769, 2*w^3 + 2*w^2 - 19*w - 11],
[769, 769, 7*w^3 + 12*w^2 - 64*w - 72],
[809, 809, -3*w^3 - 4*w^2 + 29*w + 17],
[809, 809, -5*w^3 - 8*w^2 + 43*w + 54],
[809, 809, w^3 - 9*w + 9],
[809, 809, 10*w^3 + 18*w^2 - 90*w - 109],
[881, 881, -10*w^3 - 17*w^2 + 92*w + 113],
[881, 881, -6*w^3 - 10*w^2 + 56*w + 69],
[881, 881, -31*w^3 - 55*w^2 + 283*w + 350],
[881, 881, -11*w^3 - 19*w^2 + 101*w + 125],
[911, 911, 5*w^3 + 8*w^2 - 44*w - 49],
[911, 911, -6*w^3 - 10*w^2 + 53*w + 60],
[911, 911, 5*w^3 + 8*w^2 - 44*w - 48],
[911, 911, 6*w^3 + 10*w^2 - 53*w - 61],
[919, 919, 4*w^3 + 7*w^2 - 37*w - 38],
[919, 919, 5*w^3 + 9*w^2 - 44*w - 60],
[919, 919, 8*w^3 + 14*w^2 - 73*w - 85],
[919, 919, -7*w^3 - 13*w^2 + 61*w + 81],
[971, 971, 28*w^3 + 49*w^2 - 255*w - 312],
[971, 971, 16*w^3 + 28*w^2 - 145*w - 176],
[971, 971, -25*w^3 - 44*w^2 + 227*w + 278],
[971, 971, 15*w^3 + 26*w^2 - 136*w - 165],
[991, 991, 5*w^3 + 10*w^2 - 42*w - 64],
[991, 991, -4*w^3 - 5*w^2 + 36*w + 29],
[991, 991, -11*w^3 - 19*w^2 + 99*w + 115],
[991, 991, 10*w^3 + 17*w^2 - 90*w - 108]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 28*x^4 + 207*x^2 - 162;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/27*e^5 + 19/27*e^3 - 10/3*e, 2/27*e^5 - 29/27*e^3 + 1/3*e, -1/9*e^4 + 10/9*e^2 + 1, 1, 4/9*e^4 - 58/9*e^2 + 4, -1/27*e^5 + 19/27*e^3 - 10/3*e, -1/3*e^3 + 13/3*e, -5/9*e^4 + 77/9*e^2 - 14, -1/9*e^4 + 19/9*e^2 - 12, -2/9*e^5 + 35/9*e^3 - 35/3*e, 1/27*e^5 - 10/27*e^3 - 3*e, -2/27*e^5 + 20/27*e^3 + 5*e, 5/27*e^5 - 95/27*e^3 + 41/3*e, 1/9*e^5 - 13/9*e^3 - 5/3*e, -2/27*e^5 + 29/27*e^3 - 4/3*e, -2/9*e^4 + 11/9*e^2 + 14, -8/9*e^4 + 125/9*e^2 - 22, 11/9*e^4 - 164/9*e^2 + 22, -2/3*e^4 + 29/3*e^2 - 2, 4/9*e^4 - 76/9*e^2 + 20, 2/9*e^5 - 35/9*e^3 + 29/3*e, 7/9*e^4 - 97/9*e^2 + 4, -2/9*e^5 + 23/9*e^3 + 23/3*e, -2/9*e^4 + 29/9*e^2 - 8, -1/27*e^5 - 8/27*e^3 + 29/3*e, e^4 - 15*e^2 + 10, -5/27*e^5 + 104/27*e^3 - 15*e, 5/9*e^4 - 59/9*e^2 - 12, 1/3*e^5 - 19/3*e^3 + 26*e, 4/3*e^3 - 52/3*e, -e^4 + 15*e^2 - 28, -1/9*e^4 + 1/9*e^2 + 2, -2/9*e^4 + 20/9*e^2 - 6, 5/9*e^4 - 59/9*e^2 - 14, 5/27*e^5 - 68/27*e^3 - 1/3*e, -1/27*e^5 - 17/27*e^3 + 12*e, -2/27*e^5 + 92/27*e^3 - 89/3*e, 7/27*e^5 - 115/27*e^3 + 23/3*e, 7/27*e^5 - 142/27*e^3 + 80/3*e, -4/9*e^5 + 58/9*e^3 - 3*e, -4/27*e^5 + 103/27*e^3 - 58/3*e, 1/27*e^5 - 1/27*e^3 - 37/3*e, -1/27*e^5 + 10/27*e^3 + 4*e, 2/3*e^3 - 23/3*e, -10/9*e^4 + 136/9*e^2 + 8, -4/9*e^5 + 70/9*e^3 - 64/3*e, 2/9*e^5 - 17/9*e^3 - 37/3*e, -11/9*e^4 + 137/9*e^2 + 6, -5/9*e^4 + 86/9*e^2 - 34, -2*e^4 + 27*e^2 + 2, -2/9*e^4 + 47/9*e^2 - 24, -1/3*e^4 + 10/3*e^2 + 12, 2/3*e^4 - 32/3*e^2 + 20, 2*e^2 - 4, 14/9*e^4 - 194/9*e^2 + 4, 4/3*e^4 - 64/3*e^2 + 36, 1/27*e^5 + 53/27*e^3 - 103/3*e, -16/27*e^5 + 259/27*e^3 - 68/3*e, 4/27*e^5 - 103/27*e^3 + 82/3*e, 1/9*e^5 - 34/9*e^3 + 80/3*e, 5/27*e^5 - 95/27*e^3 + 14/3*e, -22/9*e^4 + 346/9*e^2 - 66, 2/3*e^4 - 20/3*e^2 - 16, 10/27*e^5 - 190/27*e^3 + 88/3*e, 1/3*e^5 - 7/3*e^3 - 28*e, 11/9*e^4 - 137/9*e^2 - 6, 22/27*e^5 - 400/27*e^3 + 140/3*e, 10/9*e^4 - 154/9*e^2 + 12, -5/27*e^5 + 95/27*e^3 - 56/3*e, 1/27*e^5 + 35/27*e^3 - 74/3*e, -2/9*e^5 + 23/9*e^3 + 17/3*e, -17/27*e^5 + 323/27*e^3 - 122/3*e, 1/27*e^5 - 64/27*e^3 + 19*e, 2/27*e^5 + 7/27*e^3 - 19*e, 4/9*e^4 - 76/9*e^2 + 12, 2/9*e^4 - 38/9*e^2 - 2, -2/27*e^5 + 83/27*e^3 - 73/3*e, 5/9*e^5 - 53/9*e^3 - 68/3*e, -2/27*e^5 + 65/27*e^3 - 71/3*e, -8/27*e^5 + 89/27*e^3 + 23/3*e, 11/27*e^5 - 110/27*e^3 - 17*e, 1/3*e^4 - 19/3*e^2 + 4, -2/3*e^3 + 8/3*e, 2/9*e^4 - 56/9*e^2 + 26, -14/27*e^5 + 176/27*e^3 + 14/3*e, -17/27*e^5 + 233/27*e^3 + 2/3*e, -e^4 + 13*e^2 + 10, 1/9*e^4 - 1/9*e^2 - 42, 8/9*e^5 - 143/9*e^3 + 50*e, -7/9*e^5 + 124/9*e^3 - 44*e, 10/27*e^5 - 118/27*e^3 - 19/3*e, -11/27*e^5 + 92/27*e^3 + 80/3*e, -7/3*e^4 + 97/3*e^2 - 12, 2*e^4 - 26*e^2 - 6, -10/3*e^3 + 136/3*e, 7/27*e^5 - 196/27*e^3 + 161/3*e, 8/9*e^4 - 125/9*e^2 - 8, 32/9*e^4 - 455/9*e^2 + 26, 5/3*e^4 - 62/3*e^2 - 6, 1/9*e^4 - 46/9*e^2 + 24, 14/27*e^5 - 275/27*e^3 + 42*e, -2/27*e^5 - 52/27*e^3 + 125/3*e, 4/27*e^5 - 67/27*e^3 + 10*e, -26/27*e^5 + 413/27*e^3 - 80/3*e, -11/3*e^3 + 167/3*e, -7/27*e^5 + 133/27*e^3 - 34/3*e, 2/9*e^4 - 20/9*e^2 - 28, 10/9*e^4 - 154/9*e^2 + 32, -e^4 + 18*e^2 - 44, -4/9*e^4 + 49/9*e^2 - 28, -2/3*e^4 + 35/3*e^2 - 20, -16/9*e^4 + 223/9*e^2 - 10, 2/9*e^4 + 25/9*e^2 - 36, -23/9*e^4 + 338/9*e^2 - 52, 5/9*e^4 - 50/9*e^2 - 4, 14/9*e^4 - 185/9*e^2 - 2, 7/3*e^4 - 97/3*e^2 + 10, -5/27*e^5 + 122/27*e^3 - 53/3*e, 19/27*e^5 - 235/27*e^3 - 22/3*e, 1/9*e^4 + 17/9*e^2 - 14, -5/9*e^4 + 104/9*e^2 - 44, 20/9*e^4 - 317/9*e^2 + 74, 26/9*e^4 - 341/9*e^2 - 14, 14/9*e^4 - 221/9*e^2 + 52, 11/3*e^4 - 161/3*e^2 + 52, -28/9*e^4 + 424/9*e^2 - 50, -2/3*e^5 + 9*e^3 - 13/3*e, 4/27*e^5 - 130/27*e^3 + 142/3*e, 4*e^3 - 58*e, -10/9*e^4 + 190/9*e^2 - 34, 2/9*e^4 - 56/9*e^2 - 2, -10/3*e^3 + 100/3*e, -1/3*e^4 + 10/3*e^2 + 38, 20/9*e^4 - 281/9*e^2 + 38, -e^2 + 24, 1/3*e^4 - 28/3*e^2 + 44, 13/3*e^3 - 157/3*e, -11/3*e^3 + 137/3*e, -4/27*e^5 + 103/27*e^3 - 55/3*e, 1/3*e^5 - 29/3*e^3 + 172/3*e, 1/27*e^5 - 37/27*e^3 + 8*e, -4/9*e^4 + 76/9*e^2 - 64, 2/27*e^5 + 25/27*e^3 - 53/3*e, 1/3*e^4 - 7/3*e^2 - 46, -e^5 + 40/3*e^3 + 8/3*e, 2/27*e^5 - 11/27*e^3 - 4/3*e, 20/27*e^5 - 362/27*e^3 + 53*e, -5/27*e^5 + 158/27*e^3 - 48*e, 5/9*e^5 - 119/9*e^3 + 206/3*e, 29/27*e^5 - 389/27*e^3 - 16/3*e, 23/27*e^5 - 293/27*e^3 - 38/3*e, -8/27*e^5 + 215/27*e^3 - 63*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, 4*w^3 + 7*w^2 - 36*w - 45}>] := -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;