/*
  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![25, 5, -11, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19,19,-2/5*w^3 + 2/5*w^2 + 17/5*w}>;

primesArray := [
[5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7],
[5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*w + 4],
[11, 11, w + 1],
[11, 11, 1/5*w^3 - 1/5*w^2 - 11/5*w + 2],
[16, 2, 2],
[19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1],
[19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w],
[19, 19, -2/5*w^3 + 2/5*w^2 + 17/5*w],
[19, 19, w - 1],
[29, 29, w^2 - w - 8],
[29, 29, 2/5*w^3 - 2/5*w^2 - 7/5*w + 2],
[31, 31, 1/5*w^3 - 1/5*w^2 - 1/5*w + 2],
[31, 31, 2/5*w^3 - 2/5*w^2 - 17/5*w + 3],
[61, 61, -2/5*w^3 - 3/5*w^2 + 12/5*w + 5],
[61, 61, -4/5*w^3 - 6/5*w^2 + 39/5*w + 15],
[61, 61, 3/5*w^3 + 2/5*w^2 - 18/5*w - 3],
[61, 61, 7/5*w^3 + 3/5*w^2 - 62/5*w - 13],
[71, 71, 1/5*w^3 - 6/5*w^2 - 6/5*w + 4],
[71, 71, w^2 - 8],
[79, 79, 3/5*w^3 + 12/5*w^2 - 33/5*w - 22],
[79, 79, 1/5*w^3 + 4/5*w^2 - 11/5*w - 9],
[81, 3, -3],
[101, 101, 3/5*w^3 + 7/5*w^2 - 28/5*w - 15],
[101, 101, 1/5*w^3 + 9/5*w^2 - 16/5*w - 16],
[121, 11, 3/5*w^3 - 3/5*w^2 - 18/5*w + 1],
[131, 131, 1/5*w^3 + 4/5*w^2 - 16/5*w - 8],
[131, 131, w^2 - 2*w - 2],
[149, 149, 4/5*w^3 + 1/5*w^2 - 29/5*w - 5],
[149, 149, 4/5*w^3 - 9/5*w^2 - 19/5*w + 6],
[151, 151, 2/5*w^3 + 3/5*w^2 - 17/5*w - 3],
[151, 151, -2/5*w^3 + 7/5*w^2 + 7/5*w - 8],
[151, 151, 4/5*w^3 + 1/5*w^2 - 29/5*w - 2],
[151, 151, -4/5*w^3 + 9/5*w^2 + 19/5*w - 9],
[179, 179, -3/5*w^3 + 8/5*w^2 + 18/5*w - 6],
[179, 179, 2/5*w^3 + 3/5*w^2 - 12/5*w - 6],
[211, 211, 1/5*w^3 + 4/5*w^2 - 6/5*w - 10],
[211, 211, 2/5*w^3 - 7/5*w^2 - 12/5*w + 2],
[229, 229, 6/5*w^3 - 11/5*w^2 - 41/5*w + 11],
[229, 229, 2*w^2 - 2*w - 9],
[229, 229, -4/5*w^3 - 1/5*w^2 + 19/5*w + 2],
[229, 229, -3/5*w^3 + 3/5*w^2 + 28/5*w - 3],
[239, 239, 11/5*w^3 + 4/5*w^2 - 96/5*w - 19],
[239, 239, -6/5*w^3 + 1/5*w^2 + 51/5*w + 3],
[241, 241, 1/5*w^3 + 4/5*w^2 - 1/5*w - 6],
[241, 241, 4/5*w^3 - 4/5*w^2 - 29/5*w - 1],
[251, 251, -1/5*w^3 + 6/5*w^2 + 11/5*w - 12],
[251, 251, -1/5*w^3 + 11/5*w^2 + 1/5*w - 15],
[269, 269, 2/5*w^3 + 3/5*w^2 - 17/5*w - 2],
[269, 269, -2/5*w^3 + 7/5*w^2 + 7/5*w - 9],
[271, 271, 1/5*w^3 - 1/5*w^2 + 4/5*w],
[271, 271, 3/5*w^3 - 3/5*w^2 - 28/5*w + 2],
[281, 281, w^3 - 3*w^2 - 4*w + 13],
[281, 281, -3/5*w^3 + 3/5*w^2 + 28/5*w - 6],
[281, 281, -6/5*w^3 + 26/5*w^2 + 11/5*w - 16],
[281, 281, w^3 + w^2 - 8*w - 9],
[289, 17, 2/5*w^3 + 3/5*w^2 - 7/5*w - 5],
[289, 17, 4/5*w^3 - 9/5*w^2 - 29/5*w + 8],
[311, 311, -w^2 + 2*w + 11],
[311, 311, -1/5*w^3 + 6/5*w^2 + 1/5*w - 1],
[311, 311, 1/5*w^3 + 4/5*w^2 - 16/5*w + 1],
[311, 311, -1/5*w^3 - 4/5*w^2 + 11/5*w + 10],
[349, 349, -1/5*w^3 - 9/5*w^2 + 11/5*w + 16],
[349, 349, 2/5*w^3 - 12/5*w^2 - 7/5*w + 7],
[359, 359, -4/5*w^3 + 4/5*w^2 + 19/5*w - 1],
[359, 359, -w^3 + 3*w^2 + 5*w - 12],
[359, 359, w^3 - w^2 - 7*w + 2],
[359, 359, 4/5*w^3 + 6/5*w^2 - 29/5*w - 11],
[379, 379, 4/5*w^3 + 6/5*w^2 - 34/5*w - 9],
[379, 379, 3/5*w^3 - 3/5*w^2 - 28/5*w + 7],
[389, 389, w^2 - 3*w - 8],
[389, 389, -4/5*w^3 + 14/5*w^2 + 19/5*w - 16],
[401, 401, -3*w^2 + 2*w + 23],
[401, 401, 1/5*w^3 - 16/5*w^2 + 4/5*w + 11],
[409, 409, -w^3 + 2*w^2 + 3*w - 7],
[409, 409, 7/5*w^3 - 2/5*w^2 - 57/5*w - 2],
[419, 419, -1/5*w^3 + 1/5*w^2 + 16/5*w - 2],
[419, 419, 1/5*w^3 - 1/5*w^2 - 16/5*w],
[421, 421, 2/5*w^3 + 3/5*w^2 - 27/5*w - 3],
[421, 421, 1/5*w^3 + 4/5*w^2 + 4/5*w - 3],
[439, 439, -w^3 + 3*w^2 + 5*w - 16],
[439, 439, -4/5*w^3 - 6/5*w^2 + 29/5*w + 7],
[439, 439, -3/5*w^3 + 13/5*w^2 + 3/5*w - 11],
[439, 439, 4/5*w^3 + 6/5*w^2 - 39/5*w - 10],
[449, 449, 2/5*w^3 - 12/5*w^2 - 2/5*w + 5],
[449, 449, 4/5*w^3 - 24/5*w^2 - 4/5*w + 15],
[461, 461, -1/5*w^3 - 4/5*w^2 - 4/5*w + 4],
[461, 461, 2*w^2 - 2*w - 11],
[461, 461, 3/5*w^3 + 2/5*w^2 - 33/5*w - 4],
[479, 479, -w^3 + 3*w^2 + 6*w - 14],
[479, 479, -3/5*w^3 + 3/5*w^2 + 28/5*w + 3],
[499, 499, 2/5*w^3 + 8/5*w^2 - 17/5*w - 6],
[499, 499, -3/5*w^3 + 13/5*w^2 + 13/5*w - 17],
[509, 509, -w^3 + 2*w^2 + 4*w - 6],
[509, 509, 6/5*w^3 - 1/5*w^2 - 46/5*w - 4],
[521, 521, -2/5*w^3 + 2/5*w^2 + 7/5*w - 7],
[521, 521, -3/5*w^3 + 3/5*w^2 + 23/5*w - 8],
[529, 23, -7/5*w^3 + 17/5*w^2 + 42/5*w - 15],
[529, 23, 2/5*w^3 + 13/5*w^2 - 27/5*w - 21],
[541, 541, 1/5*w^3 - 11/5*w^2 - 1/5*w + 10],
[541, 541, 3/5*w^3 - 8/5*w^2 - 13/5*w + 13],
[541, 541, -2*w^2 + w + 13],
[541, 541, 3/5*w^3 + 2/5*w^2 - 23/5*w + 2],
[569, 569, w^3 - 2*w^2 - 5*w + 7],
[569, 569, -1/5*w^3 + 1/5*w^2 + 1/5*w - 6],
[571, 571, w^2 + w - 9],
[571, 571, -2/5*w^3 + 12/5*w^2 + 7/5*w - 12],
[571, 571, 1/5*w^3 + 9/5*w^2 - 11/5*w - 11],
[571, 571, -2/5*w^3 + 7/5*w^2 + 17/5*w - 4],
[601, 601, -3/5*w^3 + 13/5*w^2 + 18/5*w - 11],
[601, 601, -4/5*w^3 + 4/5*w^2 + 34/5*w + 1],
[631, 631, 2/5*w^3 + 18/5*w^2 - 27/5*w - 30],
[631, 631, 3/5*w^3 - 23/5*w^2 - 3/5*w + 15],
[661, 661, -2*w^3 + 17*w + 12],
[661, 661, -9/5*w^3 + 4/5*w^2 + 74/5*w + 1],
[691, 691, -7/5*w^3 - 18/5*w^2 + 72/5*w + 35],
[691, 691, 6/5*w^3 - 31/5*w^2 - 6/5*w + 19],
[701, 701, -1/5*w^3 + 6/5*w^2 + 16/5*w - 6],
[701, 701, 2/5*w^3 - 7/5*w^2 - 22/5*w + 8],
[709, 709, -3/5*w^3 + 13/5*w^2 + 13/5*w - 18],
[709, 709, 1/5*w^3 + 14/5*w^2 - 11/5*w - 20],
[709, 709, -2/5*w^3 - 8/5*w^2 + 17/5*w + 5],
[709, 709, 1/5*w^3 + 4/5*w^2 - 21/5*w - 10],
[719, 719, -3/5*w^3 + 8/5*w^2 + 13/5*w - 12],
[719, 719, 3/5*w^3 + 2/5*w^2 - 23/5*w + 1],
[719, 719, -7/5*w^3 + 17/5*w^2 + 32/5*w - 14],
[719, 719, 7/5*w^3 + 3/5*w^2 - 52/5*w - 8],
[739, 739, 1/5*w^3 - 1/5*w^2 - 11/5*w - 5],
[739, 739, w - 6],
[761, 761, -4/5*w^3 + 9/5*w^2 + 19/5*w - 11],
[761, 761, 4/5*w^3 + 1/5*w^2 - 29/5*w],
[769, 769, 3/5*w^3 - 13/5*w^2 - 13/5*w + 8],
[769, 769, -2/5*w^3 - 8/5*w^2 + 17/5*w + 15],
[769, 769, 7/5*w^3 - 17/5*w^2 - 37/5*w + 18],
[769, 769, -6/5*w^3 - 4/5*w^2 + 41/5*w + 5],
[809, 809, -1/5*w^3 + 6/5*w^2 - 4/5*w - 14],
[809, 809, -2/5*w^3 - 3/5*w^2 + 22/5*w - 4],
[809, 809, -6/5*w^3 + 6/5*w^2 + 41/5*w - 5],
[809, 809, w^3 - w^2 - 5*w + 4],
[821, 821, -3/5*w^3 - 17/5*w^2 + 33/5*w + 30],
[821, 821, -4/5*w^3 + 24/5*w^2 + 9/5*w - 15],
[829, 829, 7/5*w^3 - 12/5*w^2 - 52/5*w + 10],
[829, 829, -1/5*w^3 + 11/5*w^2 - 4/5*w - 19],
[829, 829, -13/5*w^3 + 3/5*w^2 + 108/5*w + 10],
[829, 829, -w^3 + 4*w^2 + 5*w - 23],
[839, 839, 4/5*w^3 - 9/5*w^2 - 24/5*w + 4],
[839, 839, 3/5*w^3 + 2/5*w^2 - 18/5*w - 8],
[841, 29, -w^3 + w^2 + 6*w - 4],
[859, 859, 6/5*w^3 - 11/5*w^2 - 36/5*w + 9],
[859, 859, w^3 - 6*w - 3],
[881, 881, 4/5*w^3 - 4/5*w^2 - 9/5*w - 1],
[881, 881, 9/5*w^3 - 24/5*w^2 - 19/5*w + 14],
[881, 881, -w^3 + 6*w + 2],
[881, 881, 1/5*w^3 - 1/5*w^2 - 1/5*w - 6],
[911, 911, w^3 - w^2 - 5*w + 1],
[911, 911, 6/5*w^3 - 6/5*w^2 - 41/5*w + 2],
[919, 919, -7/5*w^3 + 12/5*w^2 + 42/5*w - 7],
[919, 919, -6/5*w^3 + 1/5*w^2 + 36/5*w + 5],
[929, 929, -2/5*w^3 - 8/5*w^2 + 17/5*w + 16],
[929, 929, -3/5*w^3 + 13/5*w^2 + 13/5*w - 7],
[961, 31, w^3 - w^2 - 6*w + 2],
[971, 971, 8/5*w^3 - 13/5*w^2 - 48/5*w + 6],
[971, 971, -2/5*w^3 - 3/5*w^2 + 22/5*w],
[971, 971, 7/5*w^3 - 7/5*w^2 - 57/5*w + 7],
[971, 971, -1/5*w^3 + 6/5*w^2 - 4/5*w - 10],
[991, 991, 2/5*w^3 + 8/5*w^2 - 22/5*w - 7],
[991, 991, -2/5*w^3 + 12/5*w^2 + 2/5*w - 15]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 7*x^5 + 6*x^4 - 42*x^3 - 75*x^2 - 13*x + 15;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-5/19*e^5 - 12/19*e^4 + 48/19*e^3 + 69/19*e^2 - 83/19*e - 51/19, e, -9/19*e^5 - 14/19*e^4 + 94/19*e^3 + 52/19*e^2 - 119/19*e - 12/19, 26/19*e^5 + 89/19*e^4 - 147/19*e^3 - 507/19*e^2 - 222/19*e + 60/19, e^5 + 3*e^4 - 7*e^3 - 17*e^2 - e + 2, -31/19*e^5 - 120/19*e^4 + 157/19*e^3 + 747/19*e^2 + 272/19*e - 244/19, -10/19*e^5 - 24/19*e^4 + 77/19*e^3 + 100/19*e^2 - 33/19*e - 7/19, 1, -20/19*e^5 - 67/19*e^4 + 116/19*e^3 + 371/19*e^2 + 143/19*e + 5/19, 8/19*e^5 + 4/19*e^4 - 92/19*e^3 + 53/19*e^2 + 148/19*e - 78/19, -31/19*e^5 - 120/19*e^4 + 138/19*e^3 + 709/19*e^2 + 424/19*e - 168/19, 22/19*e^5 + 68/19*e^4 - 158/19*e^3 - 391/19*e^2 + 65/19*e + 137/19, 16/19*e^5 + 46/19*e^4 - 127/19*e^3 - 274/19*e^2 + 125/19*e + 167/19, -24/19*e^5 - 107/19*e^4 + 67/19*e^3 + 658/19*e^2 + 487/19*e - 241/19, 18/19*e^5 + 66/19*e^4 - 93/19*e^3 - 389/19*e^2 - 180/19*e - 52/19, -14/19*e^5 - 45/19*e^4 + 85/19*e^3 + 235/19*e^2 + 7/19*e - 25/19, -61/19*e^5 - 211/19*e^4 + 350/19*e^3 + 1237/19*e^2 + 458/19*e - 322/19, 41/19*e^5 + 125/19*e^4 - 291/19*e^3 - 733/19*e^2 - 30/19*e + 270/19, -9/19*e^5 - 14/19*e^4 + 113/19*e^3 + 71/19*e^2 - 290/19*e - 126/19, 31/19*e^5 + 158/19*e^4 - 43/19*e^3 - 994/19*e^2 - 842/19*e + 92/19, 59/19*e^5 + 229/19*e^4 - 289/19*e^3 - 1426/19*e^2 - 666/19*e + 446/19, 36/19*e^5 + 113/19*e^4 - 224/19*e^3 - 588/19*e^2 - 151/19*e - 104/19, 7/19*e^5 - 6/19*e^4 - 109/19*e^3 + 120/19*e^2 + 253/19*e - 168/19, e^5 + 6*e^4 - 40*e^2 - 35*e, -33/19*e^5 - 159/19*e^4 + 66/19*e^3 + 976/19*e^2 + 805/19*e - 196/19, -44/19*e^5 - 193/19*e^4 + 164/19*e^3 + 1238/19*e^2 + 763/19*e - 312/19, 7/19*e^5 + 32/19*e^4 - 71/19*e^3 - 298/19*e^2 + 253/19*e + 345/19, 21/19*e^5 + 39/19*e^4 - 232/19*e^3 - 229/19*e^2 + 493/19*e + 237/19, -13/19*e^5 - 16/19*e^4 + 159/19*e^3 + 54/19*e^2 - 307/19*e - 144/19, 117/19*e^5 + 429/19*e^4 - 652/19*e^3 - 2633/19*e^2 - 866/19*e + 650/19, -34/19*e^5 - 131/19*e^4 + 163/19*e^3 + 853/19*e^2 + 454/19*e - 457/19, -2/19*e^5 + 37/19*e^4 + 118/19*e^3 - 303/19*e^2 - 436/19*e + 29/19, -44/19*e^5 - 136/19*e^4 + 316/19*e^3 + 744/19*e^2 - 168/19*e - 46/19, 46/19*e^5 + 175/19*e^4 - 206/19*e^3 - 1049/19*e^2 - 745/19*e + 264/19, -63/19*e^5 - 250/19*e^4 + 240/19*e^3 + 1428/19*e^2 + 1143/19*e - 198/19, 41/19*e^5 + 106/19*e^4 - 348/19*e^3 - 543/19*e^2 + 388/19*e - 34/19, 91/19*e^5 + 359/19*e^4 - 429/19*e^3 - 2240/19*e^2 - 1176/19*e + 590/19, 85/19*e^5 + 280/19*e^4 - 531/19*e^3 - 1648/19*e^2 - 451/19*e + 506/19, -67/19*e^5 - 214/19*e^4 + 419/19*e^3 + 1164/19*e^2 + 328/19*e + 50/19, -108/19*e^5 - 415/19*e^4 + 539/19*e^3 + 2524/19*e^2 + 1080/19*e - 448/19, 16/19*e^5 + 84/19*e^4 - 13/19*e^3 - 521/19*e^2 - 521/19*e + 224/19, 12/19*e^5 - 32/19*e^4 - 233/19*e^3 + 336/19*e^2 + 640/19*e - 60/19, 3*e^5 + 10*e^4 - 19*e^3 - 58*e^2 - 8*e + 9, 100/19*e^5 + 335/19*e^4 - 599/19*e^3 - 1950/19*e^2 - 696/19*e + 545/19, e^5 - 16*e^3 + 4*e^2 + 45*e - 4, 106/19*e^5 + 357/19*e^4 - 668/19*e^3 - 2124/19*e^2 - 262/19*e + 933/19, -132/19*e^5 - 408/19*e^4 + 910/19*e^3 + 2308/19*e^2 + 47/19*e - 480/19, 62/19*e^5 + 221/19*e^4 - 352/19*e^3 - 1323/19*e^2 - 316/19*e + 621/19, -68/19*e^5 - 205/19*e^4 + 459/19*e^3 + 1174/19*e^2 + 243/19*e - 420/19, -25/19*e^5 - 136/19*e^4 + 31/19*e^3 + 877/19*e^2 + 573/19*e - 160/19, -30/19*e^5 - 167/19*e^4 + 60/19*e^3 + 1174/19*e^2 + 661/19*e - 439/19, -12/19*e^5 - 63/19*e^4 + 43/19*e^3 + 500/19*e^2 + 234/19*e - 339/19, -25/19*e^5 - 3/19*e^4 + 354/19*e^3 - 149/19*e^2 - 814/19*e + 87/19, -122/19*e^5 - 460/19*e^4 + 662/19*e^3 + 2892/19*e^2 + 897/19*e - 1005/19, -35/19*e^5 - 122/19*e^4 + 184/19*e^3 + 673/19*e^2 + 369/19*e - 15/19, -140/19*e^5 - 469/19*e^4 + 869/19*e^3 + 2749/19*e^2 + 602/19*e - 649/19, 154/19*e^5 + 552/19*e^4 - 859/19*e^3 - 3250/19*e^2 - 1084/19*e + 389/19, 87/19*e^5 + 395/19*e^4 - 307/19*e^3 - 2599/19*e^2 - 1497/19*e + 762/19, -22/19*e^5 - 49/19*e^4 + 196/19*e^3 + 315/19*e^2 - 122/19*e - 384/19, 86/19*e^5 + 328/19*e^4 - 381/19*e^3 - 1886/19*e^2 - 1164/19*e + 159/19, -54/19*e^5 - 160/19*e^4 + 393/19*e^3 + 825/19*e^2 - 144/19*e + 270/19, 43/19*e^5 + 69/19*e^4 - 447/19*e^3 - 221/19*e^2 + 748/19*e - 139/19, -1/19*e^5 + 9/19*e^4 + 40/19*e^3 - 47/19*e^2 - 180/19*e + 62/19, -32/19*e^5 - 111/19*e^4 + 197/19*e^3 + 643/19*e^2 + 111/19*e + 84/19, -36/19*e^5 - 132/19*e^4 + 205/19*e^3 + 778/19*e^2 + 189/19*e - 48/19, 86/19*e^5 + 214/19*e^4 - 647/19*e^3 - 955/19*e^2 + 14/19*e - 126/19, 75/19*e^5 + 218/19*e^4 - 473/19*e^3 - 1035/19*e^2 - 389/19*e - 375/19, -124/19*e^5 - 442/19*e^4 + 685/19*e^3 + 2608/19*e^2 + 917/19*e - 748/19, -33/19*e^5 - 7/19*e^4 + 408/19*e^3 - 316/19*e^2 - 658/19*e + 545/19, -193/19*e^5 - 733/19*e^4 + 975/19*e^3 + 4476/19*e^2 + 1949/19*e - 1182/19, 115/19*e^5 + 352/19*e^4 - 819/19*e^3 - 2024/19*e^2 + 142/19*e + 660/19, -33/19*e^5 - 121/19*e^4 + 142/19*e^3 + 615/19*e^2 + 539/19*e + 279/19, -63/19*e^5 - 212/19*e^4 + 449/19*e^3 + 1409/19*e^2 - 168/19*e - 654/19, 150/19*e^5 + 569/19*e^4 - 699/19*e^3 - 3324/19*e^2 - 1804/19*e + 542/19, -9/19*e^5 + 81/19*e^4 + 303/19*e^3 - 746/19*e^2 - 1031/19*e + 368/19, 26/19*e^5 - 6/19*e^4 - 356/19*e^3 + 291/19*e^2 + 690/19*e - 225/19, -94/19*e^5 - 237/19*e^4 + 777/19*e^3 + 1301/19*e^2 - 371/19*e - 537/19, 10/19*e^5 + 81/19*e^4 + 56/19*e^3 - 594/19*e^2 - 689/19*e + 140/19, 37/19*e^5 + 161/19*e^4 - 55/19*e^3 - 940/19*e^2 - 1396/19*e + 5/19, -2*e^5 - 6*e^4 + 14*e^3 + 34*e^2 + 2*e - 4, 65/19*e^5 + 232/19*e^4 - 415/19*e^3 - 1448/19*e^2 - 118/19*e + 188/19, -66/19*e^5 - 242/19*e^4 + 379/19*e^3 + 1496/19*e^2 + 356/19*e - 715/19, -140/19*e^5 - 412/19*e^4 + 983/19*e^3 + 2255/19*e^2 + 203/19*e - 307/19, 2/19*e^5 + 77/19*e^4 + 148/19*e^3 - 590/19*e^2 - 628/19*e + 180/19, -3/19*e^5 - 68/19*e^4 - 89/19*e^3 + 505/19*e^2 + 277/19*e - 327/19, 5*e^5 + 16*e^4 - 34*e^3 - 93*e^2 + 2*e + 27, 31/19*e^5 + 120/19*e^4 - 214/19*e^3 - 804/19*e^2 + 336/19*e + 624/19, 21/19*e^5 + 58/19*e^4 - 232/19*e^3 - 419/19*e^2 + 512/19*e + 180/19, -49/19*e^5 - 148/19*e^4 + 383/19*e^3 + 889/19*e^2 - 232/19*e - 249/19, 129/19*e^5 + 492/19*e^4 - 657/19*e^3 - 3000/19*e^2 - 1157/19*e + 723/19, 2/19*e^5 + 77/19*e^4 + 129/19*e^3 - 609/19*e^2 - 400/19*e + 218/19, 23/19*e^5 + 40/19*e^4 - 274/19*e^3 - 344/19*e^2 + 511/19*e + 569/19, -53/19*e^5 - 226/19*e^4 + 201/19*e^3 + 1461/19*e^2 + 986/19*e - 381/19, -3*e^5 - 9*e^4 + 17*e^3 + 39*e^2 + 21*e + 30, -117/19*e^5 - 391/19*e^4 + 728/19*e^3 + 2253/19*e^2 + 372/19*e - 42/19, 42/19*e^5 + 135/19*e^4 - 350/19*e^3 - 933/19*e^2 + 435/19*e + 702/19, 5/19*e^5 - 102/19*e^4 - 276/19*e^3 + 976/19*e^2 + 1185/19*e - 595/19, -193/19*e^5 - 581/19*e^4 + 1355/19*e^3 + 3298/19*e^2 + 220/19*e - 745/19, 72/19*e^5 + 245/19*e^4 - 429/19*e^3 - 1385/19*e^2 - 245/19*e + 248/19, 61/19*e^5 + 211/19*e^4 - 293/19*e^3 - 1199/19*e^2 - 952/19*e + 284/19, 36/19*e^5 + 94/19*e^4 - 319/19*e^3 - 493/19*e^2 + 495/19*e + 371/19, -77/19*e^5 - 295/19*e^4 + 401/19*e^3 + 1853/19*e^2 + 789/19*e - 280/19, 213/19*e^5 + 686/19*e^4 - 1357/19*e^3 - 3916/19*e^2 - 971/19*e + 360/19, 5*e^4 + 11*e^3 - 43*e^2 - 44*e + 30, 111/19*e^5 + 369/19*e^4 - 697/19*e^3 - 2155/19*e^2 - 483/19*e + 167/19, 25/19*e^5 - 16/19*e^4 - 354/19*e^3 + 415/19*e^2 + 890/19*e - 391/19, 45/19*e^5 + 165/19*e^4 - 261/19*e^3 - 1096/19*e^2 - 298/19*e + 269/19, 46/19*e^5 + 213/19*e^4 - 149/19*e^3 - 1391/19*e^2 - 916/19*e + 74/19, -59/19*e^5 - 229/19*e^4 + 308/19*e^3 + 1559/19*e^2 + 590/19*e - 826/19, 13/19*e^5 + 168/19*e^4 + 259/19*e^3 - 1137/19*e^2 - 1669/19*e - 103/19, -97/19*e^5 - 381/19*e^4 + 460/19*e^3 + 2338/19*e^2 + 1236/19*e - 370/19, 136/19*e^5 + 448/19*e^4 - 861/19*e^3 - 2557/19*e^2 - 467/19*e + 365/19, -195/19*e^5 - 734/19*e^4 + 1036/19*e^3 + 4515/19*e^2 + 1494/19*e - 1666/19, 129/19*e^5 + 454/19*e^4 - 676/19*e^3 - 2544/19*e^2 - 1480/19*e - 37/19, -86/19*e^5 - 233/19*e^4 + 666/19*e^3 + 1259/19*e^2 - 280/19*e - 634/19, 137/19*e^5 + 439/19*e^4 - 825/19*e^3 - 2358/19*e^2 - 895/19*e + 113/19, -83/19*e^5 - 184/19*e^4 + 698/19*e^3 + 735/19*e^2 - 557/19*e + 54/19, -16/19*e^5 + 106/19*e^4 + 507/19*e^3 - 923/19*e^2 - 1987/19*e - 72/19, 7/19*e^5 + 108/19*e^4 + 138/19*e^3 - 830/19*e^2 - 849/19*e + 668/19, 113/19*e^5 + 294/19*e^4 - 929/19*e^3 - 1624/19*e^2 + 656/19*e + 575/19, 60/19*e^5 + 220/19*e^4 - 405/19*e^3 - 1493/19*e^2 - 11/19*e + 707/19, 141/19*e^5 + 479/19*e^4 - 890/19*e^3 - 2930/19*e^2 - 612/19*e + 815/19, -18/19*e^5 - 28/19*e^4 + 226/19*e^3 + 199/19*e^2 - 428/19*e - 651/19, 104/19*e^5 + 356/19*e^4 - 626/19*e^3 - 2180/19*e^2 - 622/19*e + 810/19, 143/19*e^5 + 385/19*e^4 - 1065/19*e^3 - 1981/19*e^2 + 14/19*e - 12/19, -230/19*e^5 - 761/19*e^4 + 1429/19*e^3 + 4390/19*e^2 + 1046/19*e - 864/19, -e^5 - e^4 + 13*e^3 + 8*e^2 - 18*e - 28, 25/19*e^5 + 98/19*e^4 - 107/19*e^3 - 592/19*e^2 - 288/19*e + 179/19, 5*e^5 + 20*e^4 - 23*e^3 - 128*e^2 - 63*e + 57, -162/19*e^5 - 594/19*e^4 + 837/19*e^3 + 3482/19*e^2 + 1411/19*e - 672/19, -116/19*e^5 - 438/19*e^4 + 536/19*e^3 + 2528/19*e^2 + 1331/19*e - 484/19, -122/19*e^5 - 460/19*e^4 + 643/19*e^3 + 2873/19*e^2 + 1201/19*e - 1081/19, 35/19*e^5 + 8/19*e^4 - 545/19*e^3 + 49/19*e^2 + 1512/19*e + 110/19, -29/19*e^5 - 157/19*e^4 + 1/19*e^3 + 993/19*e^2 + 1107/19*e - 292/19, 9/19*e^5 - 43/19*e^4 - 208/19*e^3 + 461/19*e^2 + 784/19*e + 12/19, -155/19*e^5 - 524/19*e^4 + 880/19*e^3 + 2975/19*e^2 + 1208/19*e - 1068/19, 157/19*e^5 + 639/19*e^4 - 694/19*e^3 - 3907/19*e^2 - 1855/19*e + 906/19, 25/19*e^5 - 16/19*e^4 - 335/19*e^3 + 339/19*e^2 + 453/19*e - 315/19, -135/19*e^5 - 457/19*e^4 + 840/19*e^3 + 2756/19*e^2 + 723/19*e - 921/19, -136/19*e^5 - 600/19*e^4 + 424/19*e^3 + 3678/19*e^2 + 2519/19*e - 726/19, -2/19*e^5 + 75/19*e^4 + 156/19*e^3 - 759/19*e^2 - 664/19*e + 314/19, 82/19*e^5 + 307/19*e^4 - 354/19*e^3 - 1694/19*e^2 - 1333/19*e + 122/19, 43/19*e^5 + 107/19*e^4 - 371/19*e^3 - 563/19*e^2 + 292/19*e - 25/19, -151/19*e^5 - 503/19*e^4 + 910/19*e^3 + 2745/19*e^2 + 769/19*e + 356/19, -28/19*e^5 + 81/19*e^4 + 569/19*e^3 - 955/19*e^2 - 1848/19*e + 387/19, -10/19*e^5 - 100/19*e^4 - 75/19*e^3 + 765/19*e^2 + 765/19*e - 615/19, -166/19*e^5 - 520/19*e^4 + 1111/19*e^3 + 2914/19*e^2 + 273/19*e - 196/19, -134/19*e^5 - 523/19*e^4 + 629/19*e^3 + 3183/19*e^2 + 1606/19*e - 679/19, 11/19*e^5 + 72/19*e^4 + 54/19*e^3 - 357/19*e^2 - 775/19*e - 340/19, 85/19*e^5 + 318/19*e^4 - 436/19*e^3 - 1857/19*e^2 - 584/19*e + 12/19, -153/19*e^5 - 428/19*e^4 + 1142/19*e^3 + 2271/19*e^2 - 294/19*e - 375/19, 6*e^5 + 25*e^4 - 28*e^3 - 167*e^2 - 71*e + 72, 33/19*e^5 + 178/19*e^4 + 10/19*e^3 - 1071/19*e^2 - 1299/19*e - 51/19, 176/19*e^5 + 696/19*e^4 - 846/19*e^3 - 4306/19*e^2 - 1779/19*e + 1191/19, 158/19*e^5 + 421/19*e^4 - 1228/19*e^3 - 2169/19*e^2 + 529/19*e + 597/19, -75/19*e^5 - 237/19*e^4 + 492/19*e^3 + 1301/19*e^2 + 313/19*e - 214/19, 104/19*e^5 + 356/19*e^4 - 683/19*e^3 - 2180/19*e^2 + 157/19*e + 677/19, -105/19*e^5 - 385/19*e^4 + 609/19*e^3 + 2513/19*e^2 + 670/19*e - 1299/19, 1/19*e^5 - 28/19*e^4 - 40/19*e^3 + 256/19*e^2 - 219/19*e - 81/19, 15/19*e^5 + 36/19*e^4 - 30/19*e^3 - 17/19*e^2 - 663/19*e - 379/19, -188/19*e^5 - 607/19*e^4 + 1212/19*e^3 + 3476/19*e^2 + 531/19*e - 561/19, -24/19*e^5 - 107/19*e^4 + 48/19*e^3 + 544/19*e^2 + 696/19*e + 690/19, 5*e^5 + 18*e^4 - 25*e^3 - 97*e^2 - 52*e - 21, 121/19*e^5 + 469/19*e^4 - 546/19*e^3 - 2711/19*e^2 - 1476/19*e + 288/19, 208/19*e^5 + 750/19*e^4 - 1157/19*e^3 - 4398/19*e^2 - 1396/19*e + 518/19, 7/19*e^5 - 6/19*e^4 - 204/19*e^3 - 32/19*e^2 + 937/19*e + 98/19];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19,19,-2/5*w^3 + 2/5*w^2 + 17/5*w}>] := -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;