/*
  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 | {11,11,w^2 - 2*w - 4}>;

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 - 5*x^3 - 36*x^2 + 119*x + 353;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/9*e^3 - 1/9*e^2 - 31/9*e - 14/9, -1/9*e^3 - 2/9*e^2 + 31/9*e + 116/9, -1, e, -1/9*e^3 - 2/9*e^2 + 40/9*e + 62/9, -1/3*e^2 + e + 31/3, 1/9*e^3 - 1/9*e^2 - 31/9*e - 32/9, 1/9*e^3 + 2/9*e^2 - 31/9*e - 89/9, -2/9*e^3 + 5/9*e^2 + 44/9*e - 20/9, 1/9*e^3 + 2/9*e^2 - 31/9*e - 53/9, 1/3*e^2 - e - 31/3, 1/9*e^3 + 2/9*e^2 - 31/9*e - 53/9, -e + 3, -2/9*e^3 + 5/9*e^2 + 44/9*e - 11/9, 1/9*e^3 + 2/9*e^2 - 40/9*e - 107/9, 2, 1/3*e^3 - 31/3*e - 12, 1/9*e^3 - 4/9*e^2 - 4/9*e + 43/9, -e - 1, -2/9*e^3 + 2/9*e^2 + 44/9*e + 64/9, -1/9*e^3 + 1/9*e^2 + 40/9*e - 4/9, e + 4, -1/9*e^3 - 5/9*e^2 + 40/9*e + 119/9, 2/9*e^3 - 2/9*e^2 - 62/9*e + 8/9, -1/3*e^3 - e^2 + 34/3*e + 39, 1/9*e^3 - 1/9*e^2 - 31/9*e + 58/9, -2/9*e^3 - 7/9*e^2 + 71/9*e + 235/9, 2/9*e^3 + 4/9*e^2 - 44/9*e - 214/9, -4/9*e^3 - 2/9*e^2 + 106/9*e + 296/9, 1/9*e^3 - 1/9*e^2 - 4/9*e - 23/9, 1/9*e^3 - 1/9*e^2 - 22/9*e - 77/9, 1/3*e^3 - 4/3*e^2 - 22/3*e + 25/3, 2/9*e^3 + 4/9*e^2 - 80/9*e - 124/9, 4/9*e^3 - 10/9*e^2 - 106/9*e + 58/9, -2/9*e^3 + 8/9*e^2 + 44/9*e - 68/9, 1/3*e^3 - 25/3*e - 8, 2/9*e^3 + 4/9*e^2 - 62/9*e - 250/9, 5/9*e^3 - 8/9*e^2 - 128/9*e - 49/9, -2/9*e^3 + 2/9*e^2 + 44/9*e - 8/9, -4/9*e^3 + 4/9*e^2 + 106/9*e + 110/9, -5/9*e^3 - 4/9*e^2 + 164/9*e + 259/9, -2/9*e^3 + 8/9*e^2 + 26/9*e - 104/9, 2/9*e^3 + 16/9*e^2 - 80/9*e - 442/9, 4/9*e^3 - 1/9*e^2 - 88/9*e - 203/9, 2/9*e^3 - 8/9*e^2 - 44/9*e + 50/9, 2/9*e^3 - 5/9*e^2 - 53/9*e + 173/9, -5/9*e^3 - 4/9*e^2 + 146/9*e + 403/9, 5/9*e^3 + 7/9*e^2 - 164/9*e - 244/9, -1/3*e^3 + 1/3*e^2 + 19/3*e + 38/3, 2*e^2 - 4*e - 42, 2/3*e^3 + 1/3*e^2 - 56/3*e - 133/3, -1/9*e^3 + 1/9*e^2 + 22/9*e + 14/9, 4/9*e^3 + 2/9*e^2 - 106/9*e - 224/9, 5/9*e^3 - 11/9*e^2 - 146/9*e + 44/9, -2/9*e^3 + 2/9*e^2 + 62/9*e - 44/9, 1/9*e^3 + 5/9*e^2 - 67/9*e - 182/9, 2/9*e^3 + 16/9*e^2 - 71/9*e - 469/9, -4/9*e^3 + 16/9*e^2 + 97/9*e - 127/9, 1/9*e^3 + 5/9*e^2 - 13/9*e - 272/9, -4/9*e^3 + 16/9*e^2 + 70/9*e - 244/9, -1/9*e^3 - 2/9*e^2 + 49/9*e + 188/9, -2/3*e^3 + 1/3*e^2 + 50/3*e + 65/3, 2/9*e^3 + 16/9*e^2 - 98/9*e - 478/9, -1/3*e^3 - 4/3*e^2 + 43/3*e + 100/3, 4/9*e^3 + 2/9*e^2 - 142/9*e - 170/9, -2/9*e^3 + 5/9*e^2 + 35/9*e + 133/9, 2/3*e^3 + 5/3*e^2 - 77/3*e - 161/3, 2/9*e^3 - 8/9*e^2 - 62/9*e - 58/9, -8/9*e^3 - 4/9*e^2 + 257/9*e + 313/9, -1/3*e^3 + 1/3*e^2 + 25/3*e + 26/3, 1/9*e^3 - 25/9*e^2 + 23/9*e + 478/9, -1/3*e^3 - 4/3*e^2 + 34/3*e + 121/3, -4/9*e^3 - 23/9*e^2 + 187/9*e + 623/9, 2/3*e^3 + 5/3*e^2 - 74/3*e - 146/3, -4/3*e^2 + 4*e + 136/3, -1/9*e^3 + 22/9*e^2 - 14/9*e - 313/9, -1/9*e^3 - 8/9*e^2 + 58/9*e + 437/9, 1/3*e^3 - 2*e^2 - 16/3*e + 29, -1/3*e^3 + 5/3*e^2 + 25/3*e - 26/3, -5/9*e^3 + 17/9*e^2 + 110/9*e - 59/9, -5/9*e^3 + 20/9*e^2 + 110/9*e - 251/9, 5/9*e^3 - 8/9*e^2 - 128/9*e + 5/9, 4/9*e^3 - 7/9*e^2 - 70/9*e + 91/9, 4/9*e^3 - 10/9*e^2 - 70/9*e - 50/9, 1/3*e^3 + 2*e^2 - 46/3*e - 63, 1/3*e^3 - 4/3*e^2 - 25/3*e + 34/3, -2/3*e^3 - 2*e^2 + 71/3*e + 60, 1/3*e^3 + 5/3*e^2 - 34/3*e - 125/3, 3*e^2 - 6*e - 63, 4/9*e^3 - 7/9*e^2 - 70/9*e + 19/9, -8/9*e^3 + 8/9*e^2 + 194/9*e + 184/9, 5/9*e^3 - 2/9*e^2 - 146/9*e - 199/9, -1/9*e^3 - 2/9*e^2 + 31/9*e + 44/9, -8/9*e^3 + 5/9*e^2 + 203/9*e + 367/9, 4/9*e^3 - 1/9*e^2 - 151/9*e - 149/9, 2/9*e^3 + 28/9*e^2 - 116/9*e - 562/9, -1/3*e^3 - 2*e^2 + 49/3*e + 50, 1/3*e^3 + 4/3*e^2 - 40/3*e - 157/3, -2/9*e^3 - 16/9*e^2 + 62/9*e + 514/9, -7/9*e^3 - 23/9*e^2 + 226/9*e + 848/9, 2/3*e^3 - 62/3*e + 4, -2/3*e^3 - 2*e^2 + 68/3*e + 88, 7/9*e^3 + 2/9*e^2 - 208/9*e - 161/9, 4/9*e^3 + 14/9*e^2 - 151/9*e - 335/9, -1/3*e^3 + 8/3*e^2 + 1/3*e - 158/3, e^3 - e^2 - 30*e - 22, 8/9*e^3 - 14/9*e^2 - 221/9*e - 61/9, 2/3*e^3 + 11/3*e^2 - 77/3*e - 335/3, 2/9*e^3 + 1/9*e^2 - 62/9*e + 5/9, 2/9*e^3 + 19/9*e^2 - 98/9*e - 265/9, -1/9*e^3 + 13/9*e^2 - 14/9*e - 223/9, 4/3*e^2 - 8*e - 46/3, -5/9*e^3 - 19/9*e^2 + 200/9*e + 472/9, -4/9*e^3 - 17/9*e^2 + 178/9*e + 383/9, -1/3*e^3 - 2/3*e^2 + 25/3*e + 38/3, -2/3*e^3 + e^2 + 50/3*e + 30, 4/3*e^3 + e^2 - 133/3*e - 57, -14/9*e^3 + 14/9*e^2 + 362/9*e + 286/9, -1/3*e^3 + 19/3*e + 2, -5/9*e^3 + 11/9*e^2 + 146/9*e - 251/9, 2/3*e^3 - 65/3*e - 26, -11/9*e^3 - 13/9*e^2 + 332/9*e + 745/9, -11/9*e^3 - 1/9*e^2 + 332/9*e + 301/9, -2/9*e^3 - 1/9*e^2 + 71/9*e + 283/9, 1/9*e^3 - 7/9*e^2 - 31/9*e + 154/9, 11/9*e^3 - 2/9*e^2 - 287/9*e - 451/9, 8/9*e^3 + 13/9*e^2 - 293/9*e - 475/9, -2/9*e^3 + 23/9*e^2 + 26/9*e - 542/9, -2/3*e^3 - 13/3*e^2 + 95/3*e + 373/3, 1/9*e^3 - 13/9*e^2 - 31/9*e + 448/9, -4/9*e^3 - 8/9*e^2 + 142/9*e + 392/9, -10/9*e^3 - 5/9*e^2 + 274/9*e + 515/9, -10/9*e^3 - 11/9*e^2 + 301/9*e + 611/9, -8/9*e^3 - 16/9*e^2 + 311/9*e + 559/9, -8/9*e^3 - 13/9*e^2 + 266/9*e + 691/9, -7/9*e^3 + 4/9*e^2 + 208/9*e + 11/9, 1/9*e^3 - 7/9*e^2 - 40/9*e + 100/9, 1/9*e^3 - 10/9*e^2 - 13/9*e + 166/9, 11/9*e^3 + 16/9*e^2 - 332/9*e - 811/9, 4/9*e^3 + 14/9*e^2 - 151/9*e - 506/9, -1/3*e^3 + 2/3*e^2 + 7/3*e + 52/3, 2/9*e^3 - 17/9*e^2 - 44/9*e + 527/9, 11/9*e^3 - 17/9*e^2 - 305/9*e - 94/9, -7/9*e^3 - 20/9*e^2 + 271/9*e + 638/9, -2/9*e^3 + 38/9*e^2 - 19/9*e - 701/9, -2/9*e^3 + 2/9*e^2 + 62/9*e - 260/9, 2/3*e^3 + 2/3*e^2 - 62/3*e - 224/3, 4/9*e^3 + 14/9*e^2 - 124/9*e - 596/9, 1/9*e^3 + 26/9*e^2 - 103/9*e - 806/9, -2/3*e^3 - 2*e^2 + 62/3*e + 80, -5/9*e^3 - 1/9*e^2 + 101/9*e + 130/9, 4/3*e^3 - 103/3*e - 80, 2/3*e^3 - 4/3*e^2 - 44/3*e - 38/3, -4/9*e^3 + 34/9*e^2 + 70/9*e - 388/9];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11,11,w^2 - 2*w - 4}>] := 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;