/*
  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![29, 11, -10, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, -2*w^2 + 2*w + 11}>;

primesArray := [
[11, 11, -w - 1],
[11, 11, w^2 - 5],
[11, 11, -w^2 + 2*w + 4],
[11, 11, w - 2],
[16, 2, 2],
[19, 19, w^2 - 2*w - 5],
[19, 19, -w^2 + 6],
[25, 5, -2*w^2 + 2*w + 11],
[29, 29, w],
[29, 29, 2*w^2 - w - 10],
[29, 29, -2*w^2 + 3*w + 9],
[29, 29, w - 1],
[31, 31, w^3 - 6*w - 6],
[31, 31, -w^3 + 3*w^2 + 3*w - 11],
[41, 41, w^3 - 4*w^2 - 2*w + 16],
[41, 41, w^3 - 5*w^2 - 2*w + 24],
[59, 59, w^3 - w^2 - 5*w - 2],
[59, 59, 2*w^2 - w - 13],
[61, 61, w^3 - w^2 - 6*w + 3],
[61, 61, -w^3 + 2*w^2 + 5*w - 3],
[71, 71, 2*w^2 - w - 16],
[71, 71, -w^3 + 3*w^2 + 4*w - 10],
[71, 71, -w^3 + 7*w + 4],
[71, 71, w^3 - 5*w^2 - 2*w + 23],
[79, 79, -w^3 + 4*w^2 + 3*w - 16],
[79, 79, w^3 + w^2 - 8*w - 10],
[81, 3, -3],
[131, 131, 4*w^2 - 5*w - 24],
[131, 131, 4*w^2 - 3*w - 25],
[139, 139, w^2 + w - 9],
[139, 139, -w^3 + w^2 + 6*w - 5],
[149, 149, w^3 - w^2 - 4*w + 2],
[149, 149, w^3 - 2*w^2 - 3*w + 2],
[151, 151, -w^3 + 7*w + 3],
[151, 151, -w^3 + 3*w^2 + 4*w - 9],
[179, 179, -w^3 - 2*w^2 + 10*w + 16],
[179, 179, -w^3 + 3*w^2 + 3*w - 6],
[179, 179, w^3 - 6*w - 1],
[179, 179, w^3 - 5*w^2 - 3*w + 23],
[191, 191, w^3 - 6*w^2 - w + 27],
[191, 191, w^3 + 3*w^2 - 10*w - 21],
[199, 199, -w^3 + 2*w^2 + 6*w - 4],
[199, 199, -4*w^2 + 3*w + 22],
[199, 199, 4*w^2 - 5*w - 21],
[199, 199, 3*w^2 - 4*w - 12],
[239, 239, -w^3 + w^2 + 4*w - 6],
[239, 239, 2*w^3 - 5*w^2 - 9*w + 17],
[239, 239, 2*w^3 - w^2 - 13*w - 5],
[239, 239, w^3 - 8*w^2 + 2*w + 41],
[241, 241, -w^3 + 5*w^2 + 4*w - 25],
[241, 241, w^3 - 6*w - 8],
[241, 241, -w^3 + 3*w^2 + 3*w - 13],
[241, 241, w^3 - 5*w^2 + w + 19],
[269, 269, w^2 - 2*w - 10],
[269, 269, w^2 - 11],
[271, 271, w^3 + 2*w^2 - 8*w - 18],
[271, 271, 2*w^3 - 13*w - 10],
[271, 271, 2*w^3 + w^2 - 15*w - 16],
[271, 271, w^3 - 5*w^2 - w + 23],
[289, 17, 2*w^2 - 11],
[289, 17, -2*w^2 + 4*w + 9],
[311, 311, w^3 - 4*w^2 - 3*w + 10],
[311, 311, w^3 - 7*w^2 - w + 34],
[331, 331, w^3 - w^2 - 7*w + 4],
[331, 331, w^3 - 2*w^2 - 6*w + 3],
[349, 349, w^3 + w^2 - 6*w - 12],
[349, 349, -w^3 + 4*w^2 + w - 16],
[361, 19, 4*w^2 - 4*w - 21],
[379, 379, w^3 - 3*w^2 - 3*w + 15],
[379, 379, 2*w^3 - 2*w^2 - 12*w + 1],
[379, 379, -2*w^3 + 4*w^2 + 10*w - 11],
[379, 379, w^3 - 6*w - 10],
[401, 401, -2*w^3 + 7*w^2 + 8*w - 30],
[401, 401, -w^3 + 2*w^2 + 6*w - 2],
[419, 419, 5*w^2 - 4*w - 26],
[419, 419, 5*w^2 - 6*w - 25],
[421, 421, 3*w^2 - w - 20],
[421, 421, -2*w^3 + 2*w^2 + 10*w + 3],
[431, 431, w^2 - 3*w - 5],
[431, 431, w^2 + w - 7],
[449, 449, 2*w^3 - 13*w - 7],
[449, 449, -2*w^3 + 6*w^2 + 7*w - 18],
[461, 461, w^3 - w^2 - 8*w + 3],
[461, 461, -w^3 + 2*w^2 + 7*w - 5],
[491, 491, w^3 - 3*w^2 - 6*w + 15],
[491, 491, w^3 - 9*w - 7],
[499, 499, 4*w^2 - 6*w - 23],
[499, 499, -4*w^2 + 2*w + 25],
[509, 509, 2*w^3 - 3*w^2 - 11*w + 6],
[521, 521, -w^3 + 3*w^2 + 2*w - 12],
[521, 521, -w^3 + 4*w^2 + 2*w - 19],
[529, 23, w^3 - 7*w^2 - w + 35],
[529, 23, -w^3 + 3*w^2 + 4*w - 5],
[541, 541, 2*w^3 - w^2 - 12*w - 5],
[541, 541, w^3 - 2*w^2 - 7*w + 9],
[569, 569, w^3 - 8*w - 2],
[569, 569, -w^3 + 3*w^2 + 5*w - 9],
[599, 599, -w^3 - 4*w^2 + 10*w + 25],
[599, 599, -w^3 + 7*w^2 - 36],
[601, 601, 5*w^2 - 6*w - 26],
[601, 601, -2*w^3 + 8*w^2 + 6*w - 33],
[601, 601, -2*w^3 - 2*w^2 + 16*w + 21],
[601, 601, 5*w^2 - 4*w - 27],
[619, 619, w^3 + 4*w^2 - 11*w - 26],
[619, 619, w^3 - 3*w^2 - 5*w + 18],
[641, 641, -w^3 + 3*w^2 + 6*w - 14],
[641, 641, -w^3 + 9*w + 6],
[659, 659, -w^3 - 4*w^2 + 10*w + 28],
[659, 659, -w^3 + 6*w^2 - w - 27],
[659, 659, -w^3 - 3*w^2 + 8*w + 23],
[659, 659, -w^3 + 7*w^2 - w - 33],
[691, 691, w^3 - w^2 - 8*w + 2],
[691, 691, 2*w^3 - 2*w^2 - 11*w + 1],
[701, 701, -w^3 - w^2 + 7*w + 15],
[701, 701, -w^3 + 4*w^2 + 2*w - 20],
[709, 709, -2*w^3 - w^2 + 15*w + 15],
[709, 709, 2*w^3 - 7*w^2 - 7*w + 27],
[719, 719, -w^3 - 4*w^2 + 13*w + 26],
[719, 719, w^3 - 7*w^2 - 2*w + 34],
[739, 739, 2*w^3 + 2*w^2 - 15*w - 24],
[739, 739, -2*w^3 + 8*w^2 + 5*w - 35],
[751, 751, -w^3 - 3*w^2 + 9*w + 25],
[751, 751, -w^3 + 6*w^2 - 30],
[761, 761, 3*w^3 - 2*w^2 - 16*w - 8],
[761, 761, -3*w^3 + 7*w^2 + 11*w - 23],
[769, 769, -2*w^3 + 4*w^2 + 11*w - 10],
[769, 769, w^3 + 2*w^2 - 9*w - 12],
[809, 809, -3*w^3 + 3*w^2 + 16*w + 4],
[809, 809, w^3 + 5*w^2 - 12*w - 30],
[811, 811, -w^3 - 4*w^2 + 10*w + 27],
[811, 811, -2*w^3 + 5*w^2 + 9*w - 14],
[811, 811, -2*w^3 + w^2 + 13*w + 2],
[811, 811, -w^3 + 7*w^2 - w - 32],
[821, 821, 2*w^3 - 8*w^2 - 7*w + 35],
[821, 821, -2*w^3 - 2*w^2 + 17*w + 22],
[829, 829, -3*w^3 + 20*w + 13],
[829, 829, -w^3 + 7*w^2 - w - 36],
[829, 829, w^3 + 4*w^2 - 10*w - 31],
[829, 829, 2*w^3 - 3*w^2 - 11*w + 14],
[859, 859, -w^3 + 7*w^2 - 3*w - 31],
[859, 859, 7*w^2 - 9*w - 39],
[859, 859, 7*w^2 - 5*w - 41],
[859, 859, 3*w^3 - 4*w^2 - 17*w + 2],
[929, 929, w^3 + 5*w^2 - 14*w - 31],
[929, 929, -2*w^3 + 4*w^2 + 10*w - 5],
[929, 929, w^3 - 9*w^2 + 4*w + 44],
[929, 929, -w^3 + 8*w^2 + w - 39],
[941, 941, -w^3 + 7*w^2 - 34],
[941, 941, w^3 + 4*w^2 - 11*w - 28],
[961, 31, -5*w^2 + 5*w + 28],
[971, 971, -3*w^3 + w^2 + 19*w + 14],
[971, 971, -3*w^3 + 8*w^2 + 12*w - 31],
[991, 991, -w^3 - 5*w^2 + 11*w + 35],
[991, 991, w^3 - 8*w^2 + 2*w + 40]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 44*x^2 + 32*x + 292;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/28*e^3 + 1/14*e^2 - 13/14*e + 2/7, 1/28*e^3 + 1/14*e^2 - 13/14*e + 2/7, 1/14*e^3 + 1/7*e^2 - 20/7*e - 10/7, -1/14*e^3 - 1/7*e^2 + 13/7*e - 4/7, -1/56*e^3 - 1/28*e^2 + 27/28*e - 37/14, 1/56*e^3 + 1/28*e^2 - 27/28*e - 47/14, 1, -3/56*e^3 - 3/28*e^2 + 25/28*e + 29/14, 1/4*e^2 - 19/2, -1/14*e^3 - 11/28*e^2 + 13/7*e + 41/14, -5/56*e^3 - 5/28*e^2 + 79/28*e + 39/14, -1/2*e^2 - e + 8, 1/14*e^3 + 9/14*e^2 - 6/7*e - 108/7, -3/28*e^3 - 5/7*e^2 + 39/14*e + 99/7, 1/28*e^3 + 4/7*e^2 - 13/14*e - 75/7, -3/56*e^3 + 1/7*e^2 + 25/28*e - 52/7, -9/56*e^3 - 4/7*e^2 + 131/28*e + 40/7, -1/2*e^2 + e + 13, 3/14*e^3 + 13/14*e^2 - 46/7*e - 93/7, 1/28*e^3 - 3/7*e^2 - 13/14*e + 44/7, -3/14*e^3 - 3/7*e^2 + 46/7*e + 23/7, -1/7*e^3 - 2/7*e^2 + 19/7*e + 13/7, 5/28*e^3 + 6/7*e^2 - 65/14*e - 130/7, -3/14*e^3 - 5/28*e^2 + 53/7*e + 39/14, -1/7*e^3 - 15/28*e^2 + 12/7*e + 173/14, 5/28*e^3 + 5/14*e^2 - 65/14*e - 60/7, 1/7*e^3 + 11/14*e^2 - 40/7*e - 90/7, -1/7*e^3 - 11/14*e^2 + 40/7*e + 104/7, 1/56*e^3 - 13/28*e^2 - 27/28*e + 135/14, 1/8*e^3 + 3/4*e^2 - 11/4*e - 29/2, -1/28*e^3 - 9/28*e^2 - 29/14*e + 143/14, -5/28*e^3 - 3/28*e^2 + 107/14*e + 29/14, 9/28*e^3 + 8/7*e^2 - 131/14*e - 73/7, 3/28*e^3 - 2/7*e^2 - 25/14*e + 111/7, -2/7*e^3 - 1/14*e^2 + 52/7*e - 23/7, 13/56*e^3 + 13/28*e^2 - 183/28*e - 163/14, 11/56*e^3 + 11/28*e^2 - 129/28*e - 153/14, -3/7*e^3 - 19/14*e^2 + 78/7*e + 151/7, -3/28*e^3 - 3/14*e^2 + 53/14*e - 48/7, -1/28*e^3 - 1/14*e^2 - 1/14*e - 58/7, 5/56*e^3 - 23/28*e^2 - 135/28*e + 129/14, -1/4*e^3 - 3/4*e^2 + 15/2*e + 21/2, -3/28*e^3 + 1/28*e^2 + 25/14*e - 47/14, 11/56*e^3 + 39/28*e^2 - 73/28*e - 517/14, 9/56*e^3 + 11/7*e^2 - 131/28*e - 208/7, 1/8*e^3 + 3/4*e^2 - 19/4*e - 21/2, -1/8*e^3 - 3/4*e^2 + 19/4*e + 33/2, -13/56*e^3 - 12/7*e^2 + 183/28*e + 232/7, -1/4*e^3 - 1/2*e^2 + 9/2*e + 11, 3/28*e^3 - 2/7*e^2 - 53/14*e + 83/7, 5/28*e^3 + 6/7*e^2 - 51/14*e - 81/7, -11/28*e^3 - 11/14*e^2 + 171/14*e + 97/7, 1/4*e^3 + e^2 - 11/2*e - 20, 5/28*e^3 - 1/7*e^2 - 79/14*e + 24/7, 1/14*e^3 + 8/7*e^2 - 13/7*e - 199/7, 1/28*e^3 + 15/14*e^2 + 15/14*e - 166/7, -3/28*e^3 - 17/14*e^2 + 11/14*e + 162/7, -3/14*e^3 - 10/7*e^2 + 39/7*e + 149/7, 5/56*e^3 - 9/28*e^2 - 23/28*e + 227/14, 19/56*e^3 + 33/28*e^2 - 289/28*e - 151/14, 3/28*e^3 + 3/14*e^2 - 39/14*e - 8/7, 3/28*e^3 + 3/14*e^2 - 39/14*e - 8/7, 5/14*e^3 + 12/7*e^2 - 65/7*e - 106/7, 1/14*e^3 - 6/7*e^2 - 13/7*e + 242/7, 11/56*e^3 + 1/7*e^2 - 185/28*e - 94/7, 9/56*e^3 + 4/7*e^2 - 75/28*e - 166/7, -22, 29/56*e^3 + 16/7*e^2 - 391/28*e - 286/7, -39/56*e^3 - 23/14*e^2 + 493/28*e + 143/7, -37/56*e^3 - 15/14*e^2 + 495/28*e + 61/7, 1/8*e^3 - e^2 - 11/4*e + 22, 3/14*e^3 + 13/14*e^2 - 53/7*e - 198/7, -1/14*e^3 - 9/14*e^2 + 27/7*e - 4/7, -1/56*e^3 - 2/7*e^2 + 27/28*e - 134/7, 5/56*e^3 + 3/7*e^2 - 79/28*e - 226/7, 1/7*e^3 + 11/14*e^2 - 19/7*e + 50/7, 1/14*e^3 - 5/14*e^2 - 20/7*e + 214/7, 13/28*e^3 + 10/7*e^2 - 239/14*e - 37/7, -1/28*e^3 - 4/7*e^2 + 83/14*e + 187/7, -3/56*e^3 + 23/14*e^2 + 81/28*e - 185/7, -25/56*e^3 - 37/14*e^2 + 283/28*e + 409/7, 3/28*e^3 + 12/7*e^2 + 17/14*e - 260/7, -1/28*e^3 - 11/7*e^2 - 43/14*e + 222/7, 1/4*e^3 - 23/2*e - 5, 1/28*e^3 + 4/7*e^2 + 57/14*e - 159/7, -1/14*e^3 - 1/7*e^2 + 27/7*e + 108/7, 1/14*e^3 + 1/7*e^2 - 27/7*e + 88/7, 1/7*e^3 + 2/7*e^2 - 26/7*e - 202/7, 3/14*e^3 + 3/7*e^2 - 18/7*e + 5/7, 3/7*e^3 + 6/7*e^2 - 99/7*e - 25/7, -1/4*e^3 + 1/4*e^2 + 15/2*e - 51/2, -11/28*e^3 - 43/28*e^2 + 129/14*e + 145/14, -1/4*e^3 - e^2 + 17/2*e + 3, 1/28*e^3 + 4/7*e^2 - 41/14*e - 173/7, -9/28*e^3 - 25/28*e^2 + 159/14*e + 27/14, -1/28*e^3 + 5/28*e^2 - 29/14*e - 207/14, 15/56*e^3 - 5/7*e^2 - 293/28*e - 34/7, 3/8*e^3 + 2*e^2 - 25/4*e - 62, 3/14*e^3 + 10/7*e^2 - 11/7*e - 331/7, 1/14*e^3 + 23/14*e^2 - 27/7*e - 402/7, -1/2*e^3 - 5/2*e^2 + 15*e + 20, 3/14*e^3 - 4/7*e^2 - 67/7*e - 23/7, 23/56*e^3 + 11/7*e^2 - 397/28*e - 166/7, -3/56*e^3 - 6/7*e^2 + 137/28*e + 130/7, 5/14*e^3 - 11/14*e^2 - 65/7*e + 174/7, 11/14*e^3 + 43/14*e^2 - 143/7*e - 348/7, 1/2*e^3 + 5/4*e^2 - 15*e - 7/2, 1/2*e^3 + 7/2*e^2 - 13*e - 71, -3/14*e^3 - 41/14*e^2 + 39/7*e + 373/7, 2/7*e^3 + 9/28*e^2 - 38/7*e + 165/14, -3/14*e^3 - 3/7*e^2 + 60/7*e + 163/7, -3*e + 19, 5/28*e^3 + 19/14*e^2 + 5/14*e - 200/7, 1/4*e^3 - 1/2*e^2 - 23/2*e + 14, 4/7*e^3 + 22/7*e^2 - 90/7*e - 360/7, 1/7*e^3 - 12/7*e^2 - 40/7*e + 316/7, 3/28*e^3 + 31/14*e^2 - 25/14*e - 379/7, -11/28*e^3 - 39/14*e^2 + 129/14*e + 307/7, -5/28*e^3 - 17/28*e^2 + 107/14*e + 15/14, 3/28*e^3 + 13/28*e^2 - 81/14*e - 219/14, 5/28*e^3 - 9/14*e^2 - 79/14*e + 52/7, 11/28*e^3 + 25/14*e^2 - 129/14*e - 286/7, 2/7*e^3 - 3/7*e^2 - 101/7*e + 156/7, 1/14*e^3 + 8/7*e^2 + 36/7*e - 122/7, -1/56*e^3 - 43/28*e^2 - 29/28*e + 341/14, 17/56*e^3 + 59/28*e^2 - 179/28*e - 673/14, -3/14*e^3 - 47/28*e^2 + 11/7*e + 249/14, -1/7*e^3 + 27/28*e^2 + 54/7*e - 541/14, 1/14*e^3 - 5/14*e^2 - 34/7*e - 10/7, 3/14*e^3 - 1/14*e^2 - 88/7*e + 33/7, -1/7*e^3 + 3/14*e^2 + 75/7*e - 71/7, 1/2*e^2 + 3*e - 22, 2/7*e^3 + 29/14*e^2 - 66/7*e - 320/7, -2/7*e^3 - 29/14*e^2 + 66/7*e + 222/7, 3/56*e^3 + 3/28*e^2 - 249/28*e - 113/14, -5/14*e^3 - 12/7*e^2 + 23/7*e + 246/7, -1/2*e^3 + 19*e - 6, -27/56*e^3 - 27/28*e^2 + 561/28*e + 37/14, 5/7*e^3 + 61/28*e^2 - 144/7*e - 319/14, 19/28*e^3 + 31/28*e^2 - 233/14*e - 85/14, 23/28*e^3 + 53/28*e^2 - 313/14*e - 279/14, 5/14*e^3 - 1/28*e^2 - 51/7*e + 243/14, -2/7*e^3 - 65/28*e^2 + 80/7*e + 675/14, -17/56*e^3 - 59/28*e^2 + 235/28*e + 869/14, 9/56*e^3 + 51/28*e^2 - 131/28*e - 185/14, 1/2*e^3 + 11/4*e^2 - 17*e - 89/2, -1/4*e^3 - 2*e^2 + 11/2*e + 15, 3/28*e^3 + 12/7*e^2 - 25/14*e - 407/7, -5/4*e^3 - 5/2*e^2 + 65/2*e + 25, -3/28*e^3 - 5/7*e^2 - 17/14*e + 29/7, -1/4*e^3 + 21/2*e - 15, -13/14*e^3 - 33/14*e^2 + 162/7*e + 102/7, -6/7*e^3 - 17/14*e^2 + 163/7*e - 62/7];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {25, 5, -2*w^2 + 2*w + 11}>] := -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;