/*
  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![3, -4, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[3, 3, w],
[3, 3, w^3 - 2*w^2 - 6*w + 1],
[9, 3, w^3 - 2*w^2 - 5*w + 1],
[16, 2, 2],
[17, 17, w^3 - w^2 - 8*w - 5],
[17, 17, -2*w^3 + 4*w^2 + 11*w - 2],
[17, 17, -w^3 + 3*w^2 + 2*w - 2],
[17, 17, w^3 - 2*w^2 - 4*w + 1],
[25, 5, w^3 - 3*w^2 - 3*w + 1],
[25, 5, w^2 - 2*w - 7],
[29, 29, w^3 - 2*w^2 - 6*w - 1],
[29, 29, w - 2],
[53, 53, w^3 - 3*w^2 - 4*w + 4],
[53, 53, -w^3 + 3*w^2 + 4*w - 5],
[61, 61, -w^3 + 2*w^2 + 7*w - 4],
[61, 61, -4*w^3 + 11*w^2 + 14*w - 8],
[101, 101, -w^3 + 2*w^2 + 7*w - 1],
[103, 103, -w^3 + 3*w^2 + 2*w - 5],
[103, 103, -w^3 + w^2 + 8*w + 2],
[113, 113, w^3 - 3*w^2 - 3*w + 7],
[113, 113, w^2 - 2*w - 1],
[127, 127, w^3 - 2*w^2 - 4*w - 2],
[127, 127, -2*w^3 + 4*w^2 + 11*w + 1],
[131, 131, -3*w^3 + 7*w^2 + 14*w - 8],
[131, 131, -w^2 + w + 8],
[131, 131, 2*w^3 - 5*w^2 - 9*w + 1],
[131, 131, -3*w^3 + 6*w^2 + 16*w + 1],
[139, 139, -2*w^3 + 5*w^2 + 9*w - 2],
[139, 139, -w^2 + w + 7],
[157, 157, -2*w^3 + 5*w^2 + 7*w - 5],
[157, 157, -2*w^3 + 3*w^2 + 13*w + 2],
[157, 157, -2*w^3 + 5*w^2 + 6*w - 2],
[157, 157, -3*w^3 + 5*w^2 + 19*w + 4],
[169, 13, -2*w^3 + 4*w^2 + 10*w - 1],
[173, 173, 3*w^3 - 8*w^2 - 10*w + 7],
[173, 173, 2*w^3 - 2*w^2 - 15*w - 8],
[179, 179, -3*w^2 + 6*w + 17],
[179, 179, -2*w^3 + 3*w^2 + 12*w + 1],
[179, 179, -3*w^3 + 7*w^2 + 13*w - 7],
[179, 179, -2*w^3 + 3*w^2 + 14*w + 4],
[191, 191, w^3 - 3*w^2 - w + 4],
[191, 191, -2*w^3 + 3*w^2 + 14*w + 2],
[199, 199, -w^3 + 8*w + 10],
[199, 199, 5*w^3 - 15*w^2 - 14*w + 14],
[211, 211, -w - 4],
[211, 211, -w^3 + 2*w^2 + 6*w - 5],
[211, 211, 2*w^3 - 3*w^2 - 12*w - 4],
[211, 211, -3*w^3 + 7*w^2 + 13*w - 4],
[257, 257, -3*w^3 + 8*w^2 + 10*w - 2],
[257, 257, 2*w^3 - 2*w^2 - 16*w - 7],
[263, 263, 3*w^3 - 6*w^2 - 17*w + 1],
[263, 263, w^3 - 2*w^2 - 3*w - 1],
[269, 269, 5*w^3 - 13*w^2 - 19*w + 7],
[269, 269, -3*w^3 + 6*w^2 + 18*w - 1],
[277, 277, w^3 - 4*w^2 - w + 5],
[277, 277, -w^3 + 4*w^2 + w - 11],
[283, 283, -w^3 + 5*w^2 - w - 14],
[283, 283, 2*w^3 - 7*w^2 - 4*w + 10],
[283, 283, -w^3 + w^2 + 9*w + 1],
[283, 283, w^2 - 4*w - 5],
[311, 311, -w^3 + w^2 + 9*w + 5],
[311, 311, w^2 - 4*w - 1],
[313, 313, -4*w^3 + 9*w^2 + 19*w - 10],
[313, 313, -2*w^2 + 5*w + 11],
[337, 337, -3*w^3 + 6*w^2 + 14*w + 2],
[337, 337, -4*w^3 + 8*w^2 + 21*w + 1],
[347, 347, -2*w^3 + 6*w^2 + 7*w - 10],
[347, 347, -w^3 + 4*w^2 + 2*w - 7],
[361, 19, -4*w^3 + 11*w^2 + 13*w - 10],
[361, 19, -2*w^3 + 5*w^2 + 8*w - 10],
[373, 373, 2*w^3 - 5*w^2 - 10*w + 2],
[373, 373, -w^3 + 2*w^2 + 8*w - 2],
[373, 373, -2*w^3 + 4*w^2 + 13*w - 1],
[373, 373, -w^3 + 3*w^2 + 5*w - 8],
[389, 389, -w^3 + 4*w^2 + 2*w - 10],
[389, 389, -2*w^3 + 6*w^2 + 7*w - 7],
[419, 419, -2*w^3 + 5*w^2 + 6*w - 1],
[419, 419, -3*w^3 + 5*w^2 + 19*w + 5],
[433, 433, w^2 - 4*w - 2],
[433, 433, w^3 - w^2 - 9*w - 4],
[439, 439, -w^3 + 4*w^2 + w - 2],
[439, 439, w^3 - 4*w^2 - w + 14],
[443, 443, -2*w^3 + 5*w^2 + 9*w + 1],
[443, 443, -w^2 + w + 10],
[467, 467, w^3 - w^2 - 8*w + 4],
[467, 467, w^3 - 4*w^2 + 2],
[491, 491, -4*w^3 + 12*w^2 + 10*w - 11],
[491, 491, 2*w^3 - w^2 - 18*w - 11],
[491, 491, 3*w^3 - 9*w^2 - 7*w + 11],
[491, 491, 2*w^3 - 20*w - 19],
[503, 503, w^3 - 3*w^2 - 4*w + 10],
[503, 503, -w^3 + w^2 + 10*w - 2],
[523, 523, -4*w^3 + 9*w^2 + 22*w - 7],
[523, 523, w^3 - 3*w^2 - 7*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w - 7],
[547, 547, 2*w^3 - w^2 - 18*w - 14],
[547, 547, 3*w^3 - 8*w^2 - 14*w + 8],
[571, 571, 5*w^3 - 11*w^2 - 25*w + 5],
[571, 571, -4*w^3 + 9*w^2 + 19*w - 11],
[599, 599, -5*w^3 + 12*w^2 + 20*w - 4],
[599, 599, 4*w^3 - 6*w^2 - 25*w - 11],
[601, 601, -4*w^3 + 9*w^2 + 20*w - 5],
[601, 601, w^3 - 9*w - 14],
[601, 601, 3*w^3 - 8*w^2 - 11*w + 2],
[601, 601, -2*w^3 + 3*w^2 + 14*w - 1],
[641, 641, -2*w^3 + 3*w^2 + 15*w - 5],
[641, 641, -4*w^3 + 8*w^2 + 23*w - 2],
[647, 647, -3*w^3 + 8*w^2 + 13*w - 11],
[647, 647, -w^3 + 4*w^2 + 3*w - 7],
[659, 659, 3*w^3 - 7*w^2 - 14*w - 2],
[659, 659, w^3 - w^2 - 6*w - 11],
[673, 673, -3*w^3 + 7*w^2 + 11*w - 1],
[673, 673, -4*w^3 + 7*w^2 + 24*w + 5],
[677, 677, 5*w^3 - 12*w^2 - 21*w + 11],
[677, 677, -2*w^3 + 6*w^2 + 5*w - 1],
[677, 677, -2*w^3 + 6*w^2 + 8*w - 7],
[677, 677, 2*w^3 - 6*w^2 - 8*w + 11],
[719, 719, -2*w^3 + 6*w^2 + 10*w - 7],
[719, 719, -4*w^3 + 10*w^2 + 20*w - 13],
[727, 727, 3*w^3 - 10*w^2 - 8*w + 10],
[727, 727, 3*w^3 - 8*w^2 - 13*w + 10],
[727, 727, 2*w^3 - 8*w^2 - 3*w + 23],
[727, 727, -w^3 + 4*w^2 + 3*w - 8],
[751, 751, -7*w^3 + 18*w^2 + 25*w - 7],
[751, 751, w^3 - 7*w^2 + 5*w + 26],
[757, 757, -4*w^3 + 9*w^2 + 18*w - 7],
[757, 757, -3*w^3 + 5*w^2 + 17*w + 1],
[797, 797, 2*w^2 - 4*w - 5],
[797, 797, 2*w^3 - 6*w^2 - 6*w + 11],
[823, 823, -2*w^3 + 2*w^2 + 15*w + 14],
[823, 823, -3*w^3 + 8*w^2 + 10*w - 1],
[829, 829, -6*w^3 + 16*w^2 + 22*w - 7],
[829, 829, 5*w^3 - 13*w^2 - 20*w + 8],
[829, 829, -w^3 + 7*w + 4],
[829, 829, -5*w^3 + 12*w^2 + 23*w - 14],
[841, 29, -3*w^3 + 6*w^2 + 15*w - 2],
[859, 859, -4*w^3 + 7*w^2 + 24*w - 1],
[859, 859, -5*w^3 + 15*w^2 + 14*w - 13],
[881, 881, 3*w^3 - 7*w^2 - 16*w + 4],
[881, 881, -w^3 + 3*w^2 + 6*w - 7],
[883, 883, -2*w^3 + 3*w^2 + 12*w + 10],
[883, 883, -3*w^3 + 7*w^2 + 13*w + 2],
[887, 887, 4*w^3 - 11*w^2 - 11*w + 11],
[887, 887, 2*w^3 - 7*w^2 - 3*w + 7],
[887, 887, 4*w^3 - 5*w^2 - 29*w - 10],
[887, 887, 3*w^2 - 7*w - 16],
[907, 907, -w^3 + 3*w^2 + 8*w - 5],
[907, 907, -5*w^3 + 11*w^2 + 28*w - 8],
[911, 911, -4*w^3 + 10*w^2 + 18*w - 17],
[911, 911, 2*w^2 - 2*w - 1],
[919, 919, 4*w^3 - 7*w^2 - 25*w - 4],
[919, 919, 2*w^3 - 5*w^2 - 5*w + 1],
[937, 937, -w^3 + 3*w^2 + w - 8],
[937, 937, -2*w^3 + 3*w^2 + 14*w - 2],
[953, 953, -2*w^3 + w^2 + 17*w + 19],
[953, 953, 4*w^3 - 11*w^2 - 13*w + 4],
[961, 31, 2*w^3 - 3*w^2 - 11*w - 8],
[961, 31, w^3 - 4*w^2 + 2*w + 11],
[971, 971, 2*w^2 - 5*w - 4],
[971, 971, -w^3 + 4*w^2 - 11],
[991, 991, -2*w^3 + 6*w^2 + 9*w - 8],
[991, 991, -2*w^3 + 3*w^2 + 13*w - 1],
[991, 991, 3*w^3 - 8*w^2 - 14*w + 11],
[991, 991, -2*w^3 + 5*w^2 + 7*w - 8],
[997, 997, -2*w^3 + 5*w^2 + 6*w + 1],
[997, 997, -3*w^3 + 5*w^2 + 19*w + 7],
[997, 997, -6*w^3 + 12*w^2 + 33*w - 4],
[997, 997, 4*w^3 - 11*w^2 - 15*w + 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 13*x^4 - 2*x^3 + 40*x^2 + 21*x - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 4/19*e^5 + 3/19*e^4 - 45/19*e^3 - 18/19*e^2 + 99/19*e + 11/19, 2/19*e^5 - 8/19*e^4 - 32/19*e^3 + 67/19*e^2 + 97/19*e - 23/19, -5/19*e^5 + 1/19*e^4 + 61/19*e^3 - 25/19*e^2 - 157/19*e + 29/19, -6/19*e^5 + 5/19*e^4 + 58/19*e^3 - 68/19*e^2 - 82/19*e + 69/19, 2/19*e^5 - 8/19*e^4 - 32/19*e^3 + 67/19*e^2 + 97/19*e - 61/19, 1, -2/19*e^5 + 8/19*e^4 + 32/19*e^3 - 67/19*e^2 - 97/19*e - 15/19, -12/19*e^5 - 9/19*e^4 + 116/19*e^3 + 54/19*e^2 - 183/19*e + 5/19, 2/19*e^5 + 11/19*e^4 - 13/19*e^3 - 85/19*e^2 - 17/19*e + 34/19, -6/19*e^5 + 5/19*e^4 + 77/19*e^3 - 30/19*e^2 - 196/19*e - 45/19, 5/19*e^5 - 1/19*e^4 - 23/19*e^3 + 25/19*e^2 - 90/19*e - 86/19, 10/19*e^5 - 2/19*e^4 - 122/19*e^3 - 7/19*e^2 + 333/19*e + 189/19, -5/19*e^5 + 1/19*e^4 + 42/19*e^3 - 6/19*e^2 - 5/19*e - 104/19, 6/19*e^5 + 14/19*e^4 - 58/19*e^3 - 103/19*e^2 + 101/19*e - 12/19, -10/19*e^5 + 40/19*e^4 + 141/19*e^3 - 335/19*e^2 - 447/19*e + 210/19, 4/19*e^5 + 3/19*e^4 - 7/19*e^3 + 58/19*e^2 - 186/19*e - 274/19, 2/19*e^5 + 11/19*e^4 + 6/19*e^3 - 123/19*e^2 - 131/19*e + 262/19, -2/19*e^5 + 27/19*e^4 + 51/19*e^3 - 200/19*e^2 - 211/19*e + 4/19, 13/19*e^5 - 14/19*e^4 - 132/19*e^3 + 84/19*e^2 + 184/19*e - 7/19, 16/19*e^5 - 26/19*e^4 - 199/19*e^3 + 213/19*e^2 + 510/19*e - 203/19, 2/19*e^5 - 27/19*e^4 - 89/19*e^3 + 162/19*e^2 + 553/19*e + 15/19, -21/19*e^5 - 11/19*e^4 + 241/19*e^3 + 66/19*e^2 - 534/19*e - 148/19, 46/19*e^5 - 13/19*e^4 - 470/19*e^3 + 173/19*e^2 + 882/19*e - 54/19, 21/19*e^5 + 11/19*e^4 - 222/19*e^3 - 66/19*e^2 + 401/19*e + 300/19, 15/19*e^5 + 16/19*e^4 - 126/19*e^3 - 77/19*e^2 + 167/19*e - 201/19, 11/19*e^5 - 6/19*e^4 - 81/19*e^3 + 93/19*e^2 - 84/19*e - 212/19, 2/19*e^5 - 8/19*e^4 + 6/19*e^3 + 86/19*e^2 - 150/19*e - 327/19, -27/19*e^5 + 32/19*e^4 + 318/19*e^3 - 287/19*e^2 - 844/19*e + 73/19, -15/19*e^5 - 16/19*e^4 + 145/19*e^3 + 96/19*e^2 - 243/19*e - 122/19, -25/19*e^5 - 14/19*e^4 + 210/19*e^3 + 103/19*e^2 - 158/19*e - 178/19, -3/19*e^5 - 7/19*e^4 + 10/19*e^3 - 34/19*e^2 + 92/19*e + 386/19, -3/19*e^5 - 7/19*e^4 - 9/19*e^3 + 61/19*e^2 + 206/19*e - 70/19, -12/19*e^5 - 9/19*e^4 + 116/19*e^3 + 16/19*e^2 - 202/19*e + 138/19, -10/19*e^5 + 21/19*e^4 + 103/19*e^3 - 240/19*e^2 - 200/19*e + 191/19, -e^3 - e^2 + 8*e + 10, -2/19*e^5 - 11/19*e^4 - 44/19*e^3 + 47/19*e^2 + 359/19*e + 80/19, -31/19*e^5 + 10/19*e^4 + 325/19*e^3 - 136/19*e^2 - 658/19*e - 52/19, 15/19*e^5 - 41/19*e^4 - 164/19*e^3 + 322/19*e^2 + 319/19*e - 106/19, 7/19*e^5 - 28/19*e^4 - 55/19*e^3 + 263/19*e^2 - 12/19*e - 242/19, 22/19*e^5 - 50/19*e^4 - 295/19*e^3 + 433/19*e^2 + 915/19*e - 158/19, 30/19*e^5 - 25/19*e^4 - 347/19*e^3 + 207/19*e^2 + 809/19*e + 225/19, -7/19*e^5 - 29/19*e^4 - 2/19*e^3 + 250/19*e^2 + 373/19*e - 157/19, 7/19*e^5 - 47/19*e^4 - 112/19*e^3 + 415/19*e^2 + 254/19*e - 318/19, -6/19*e^5 + 62/19*e^4 + 96/19*e^3 - 543/19*e^2 - 329/19*e + 430/19, 8/19*e^5 - 32/19*e^4 - 71/19*e^3 + 287/19*e^2 + 46/19*e - 415/19, 30/19*e^5 - 6/19*e^4 - 290/19*e^3 + 93/19*e^2 + 448/19*e - 136/19, -12/19*e^5 - 9/19*e^4 + 97/19*e^3 + 73/19*e^2 - 31/19*e - 71/19, -9/19*e^5 + 36/19*e^4 + 163/19*e^3 - 292/19*e^2 - 598/19*e - 248/19, 12/19*e^5 + 28/19*e^4 - 116/19*e^3 - 301/19*e^2 + 278/19*e + 489/19, -2/19*e^5 - 30/19*e^4 + 51/19*e^3 + 237/19*e^2 - 230/19*e - 129/19, -29/19*e^5 + 78/19*e^4 + 407/19*e^3 - 677/19*e^2 - 1207/19*e + 324/19, 36/19*e^5 - 30/19*e^4 - 424/19*e^3 + 275/19*e^2 + 986/19*e + 42/19, 16/19*e^5 - 45/19*e^4 - 199/19*e^3 + 384/19*e^2 + 567/19*e - 374/19, -20/19*e^5 - 15/19*e^4 + 206/19*e^3 + 166/19*e^2 - 533/19*e - 416/19, 22/19*e^5 + 7/19*e^4 - 200/19*e^3 + 53/19*e^2 + 174/19*e - 234/19, -11/19*e^5 - 13/19*e^4 + 81/19*e^3 + 2/19*e^2 - 11/19*e + 516/19, -23/19*e^5 - 22/19*e^4 + 254/19*e^3 + 189/19*e^2 - 745/19*e - 334/19, 66/19*e^5 + 2/19*e^4 - 657/19*e^3 + 121/19*e^2 + 1054/19*e - 227/19, 48/19*e^5 - 40/19*e^4 - 559/19*e^3 + 373/19*e^2 + 1207/19*e - 210/19, 39/19*e^5 - 23/19*e^4 - 434/19*e^3 + 252/19*e^2 + 913/19*e + 188/19, -20/19*e^5 - 34/19*e^4 + 168/19*e^3 + 204/19*e^2 - 134/19*e - 74/19, -4/19*e^5 - 60/19*e^4 - 69/19*e^3 + 455/19*e^2 + 642/19*e - 220/19, -31/19*e^5 + 29/19*e^4 + 382/19*e^3 - 174/19*e^2 - 1038/19*e - 280/19, 11/19*e^5 - 44/19*e^4 - 176/19*e^3 + 397/19*e^2 + 524/19*e - 649/19, e^4 + 5*e^3 - 8*e^2 - 34*e - 4, -8/19*e^5 + 32/19*e^4 + 90/19*e^3 - 249/19*e^2 - 103/19*e - 60/19, -12/19*e^5 - 9/19*e^4 + 59/19*e^3 + 35/19*e^2 + 292/19*e + 100/19, 20/19*e^5 - 4/19*e^4 - 244/19*e^3 + 81/19*e^2 + 761/19*e + 36/19, 30/19*e^5 + 13/19*e^4 - 347/19*e^3 - 78/19*e^2 + 866/19*e + 320/19, 21/19*e^5 - 65/19*e^4 - 298/19*e^3 + 580/19*e^2 + 914/19*e - 270/19, -2*e^5 + 2*e^4 + 22*e^3 - 14*e^2 - 47*e - 10, -21/19*e^5 + 65/19*e^4 + 317/19*e^3 - 542/19*e^2 - 1104/19*e + 80/19, -28/19*e^5 - 21/19*e^4 + 220/19*e^3 + 164/19*e^2 - 142/19*e - 267/19, 16/19*e^5 - 45/19*e^4 - 199/19*e^3 + 403/19*e^2 + 567/19*e - 336/19, 17/19*e^5 - 30/19*e^4 - 253/19*e^3 + 256/19*e^2 + 777/19*e - 110/19, -33/19*e^5 - 20/19*e^4 + 376/19*e^3 + 158/19*e^2 - 1078/19*e - 257/19, 14/19*e^5 - 37/19*e^4 - 186/19*e^3 + 355/19*e^2 + 527/19*e - 294/19, -48/19*e^5 + 59/19*e^4 + 521/19*e^3 - 582/19*e^2 - 1131/19*e + 495/19, -21/19*e^5 + 8/19*e^4 + 222/19*e^3 - 105/19*e^2 - 496/19*e + 137/19, 39/19*e^5 - 4/19*e^4 - 358/19*e^3 + 119/19*e^2 + 514/19*e - 249/19, 37/19*e^5 + 4/19*e^4 - 345/19*e^3 + 71/19*e^2 + 569/19*e - 454/19, 17/19*e^5 + 8/19*e^4 - 177/19*e^3 + 28/19*e^2 + 188/19*e - 585/19, 32/19*e^5 - 71/19*e^4 - 417/19*e^3 + 692/19*e^2 + 1153/19*e - 444/19, 2*e^5 - 4*e^4 - 24*e^3 + 34*e^2 + 58*e - 14, -16/19*e^5 + 26/19*e^4 + 294/19*e^3 - 156/19*e^2 - 1327/19*e - 348/19, 50/19*e^5 + 9/19*e^4 - 553/19*e^3 - 54/19*e^2 + 1247/19*e + 71/19, -5/19*e^5 + 77/19*e^4 + 80/19*e^3 - 614/19*e^2 - 214/19*e + 352/19, 22/19*e^5 - 69/19*e^4 - 276/19*e^3 + 661/19*e^2 + 630/19*e - 671/19, 34/19*e^5 - 3/19*e^4 - 411/19*e^3 + 75/19*e^2 + 1079/19*e + 122/19, -12/19*e^5 + 67/19*e^4 + 173/19*e^3 - 573/19*e^2 - 544/19*e + 632/19, 22/19*e^5 - 69/19*e^4 - 352/19*e^3 + 490/19*e^2 + 1200/19*e + 203/19, -50/19*e^5 + 29/19*e^4 + 553/19*e^3 - 212/19*e^2 - 1323/19*e - 356/19, 34/19*e^5 - 22/19*e^4 - 411/19*e^3 + 189/19*e^2 + 1060/19*e + 559/19, -27/19*e^5 - 25/19*e^4 + 185/19*e^3 + 93/19*e^2 + 220/19*e + 206/19, -24/19*e^5 - 18/19*e^4 + 175/19*e^3 + 51/19*e^2 + 90/19*e - 199/19, -68/19*e^5 + 44/19*e^4 + 765/19*e^3 - 454/19*e^2 - 1664/19*e - 130/19, -59/19*e^5 + 103/19*e^4 + 773/19*e^3 - 827/19*e^2 - 2111/19*e + 42/19, -37/19*e^5 + 53/19*e^4 + 459/19*e^3 - 451/19*e^2 - 1139/19*e + 74/19, 10/19*e^5 + 17/19*e^4 - 46/19*e^3 - 64/19*e^2 - 28/19*e - 210/19, -15/19*e^5 + 22/19*e^4 + 259/19*e^3 - 208/19*e^2 - 870/19*e + 296/19, 37/19*e^5 - 72/19*e^4 - 459/19*e^3 + 660/19*e^2 + 1082/19*e - 340/19, -20/19*e^5 + 42/19*e^4 + 263/19*e^3 - 290/19*e^2 - 856/19*e - 74/19, -64/19*e^5 + 28/19*e^4 + 777/19*e^3 - 301/19*e^2 - 1869/19*e - 62/19, 34/19*e^5 - 60/19*e^4 - 373/19*e^3 + 645/19*e^2 + 661/19*e - 771/19, 20/19*e^5 - 23/19*e^4 - 206/19*e^3 + 138/19*e^2 + 172/19*e + 112/19, e^5 - 2*e^4 - 15*e^3 + 19*e^2 + 53*e - 26, 17/19*e^5 - 49/19*e^4 - 196/19*e^3 + 522/19*e^2 + 492/19*e - 699/19, -4/19*e^5 - 60/19*e^4 - 12/19*e^3 + 379/19*e^2 + 262/19*e + 293/19, 77/19*e^5 - 4/19*e^4 - 795/19*e^3 + 119/19*e^2 + 1597/19*e - 40/19, 4/19*e^5 + 3/19*e^4 - 7/19*e^3 + 115/19*e^2 - 224/19*e - 692/19, -80/19*e^5 + 54/19*e^4 + 862/19*e^3 - 476/19*e^2 - 1790/19*e - 106/19, -81/19*e^5 + 20/19*e^4 + 935/19*e^3 - 215/19*e^2 - 2152/19*e - 275/19, 8/19*e^5 + 25/19*e^4 + 43/19*e^3 - 245/19*e^2 - 714/19*e + 174/19, 8/19*e^5 + 6/19*e^4 + 5/19*e^3 + 2/19*e^2 - 448/19*e - 54/19, -40/19*e^5 - 11/19*e^4 + 412/19*e^3 + 85/19*e^2 - 933/19*e - 490/19, 30/19*e^5 + 13/19*e^4 - 195/19*e^3 + 74/19*e^2 - 274/19*e - 535/19, -12/19*e^5 + 10/19*e^4 + 97/19*e^3 - 117/19*e^2 + 26/19*e + 62/19, -55/19*e^5 + 30/19*e^4 + 614/19*e^3 - 389/19*e^2 - 1195/19*e + 224/19, 17/19*e^5 + 46/19*e^4 - 6/19*e^3 - 276/19*e^2 - 743/19*e - 262/19, 3/19*e^5 - 31/19*e^4 - 48/19*e^3 + 281/19*e^2 + 364/19*e - 405/19, 8/19*e^5 - 13/19*e^4 - 33/19*e^3 + 40/19*e^2 - 220/19*e + 231/19, 8/19*e^5 + 25/19*e^4 - 71/19*e^3 - 188/19*e^2 + 274/19*e - 54/19, -12/19*e^5 - 28/19*e^4 + 78/19*e^3 + 320/19*e^2 + 235/19*e - 736/19, 5/19*e^5 - 96/19*e^4 - 61/19*e^3 + 880/19*e^2 + 119/19*e - 846/19, 14/19*e^5 - 56/19*e^4 - 262/19*e^3 + 526/19*e^2 + 1078/19*e - 617/19, -25/19*e^5 + 43/19*e^4 + 248/19*e^3 - 486/19*e^2 - 405/19*e + 734/19, -26/19*e^5 + 28/19*e^4 + 321/19*e^3 - 130/19*e^2 - 767/19*e - 822/19, 22/19*e^5 + 45/19*e^4 - 238/19*e^3 - 327/19*e^2 + 763/19*e + 70/19, 7/19*e^5 + 48/19*e^4 - 74/19*e^3 - 478/19*e^2 + 482/19*e + 594/19, -42/19*e^5 + 16/19*e^4 + 444/19*e^3 - 305/19*e^2 - 783/19*e + 768/19, 30/19*e^5 - 6/19*e^4 - 385/19*e^3 - 2/19*e^2 + 961/19*e + 510/19, -32/19*e^5 + 71/19*e^4 + 493/19*e^3 - 483/19*e^2 - 1647/19*e - 297/19, -16/19*e^5 + 45/19*e^4 + 142/19*e^3 - 517/19*e^2 - 92/19*e + 716/19, 24/19*e^5 - 39/19*e^4 - 251/19*e^3 + 329/19*e^2 + 404/19*e - 447/19, -2/19*e^5 - 30/19*e^4 - 25/19*e^3 + 256/19*e^2 + 74/19*e - 224/19, 66/19*e^5 + 21/19*e^4 - 714/19*e^3 - 69/19*e^2 + 1548/19*e + 115/19, -18/19*e^5 - 4/19*e^4 + 155/19*e^3 - 90/19*e^2 - 113/19*e + 340/19, -30/19*e^5 + 82/19*e^4 + 423/19*e^3 - 758/19*e^2 - 1189/19*e + 668/19, 43/19*e^5 - 58/19*e^4 - 688/19*e^3 + 348/19*e^2 + 2456/19*e + 408/19, 12/19*e^5 - 29/19*e^4 - 249/19*e^3 + 117/19*e^2 + 1038/19*e + 470/19, 12/19*e^5 - 67/19*e^4 - 230/19*e^3 + 573/19*e^2 + 1000/19*e - 442/19, -22/19*e^5 - 45/19*e^4 + 200/19*e^3 + 327/19*e^2 - 136/19*e + 25/19, -15/19*e^5 + 41/19*e^4 + 202/19*e^3 - 512/19*e^2 - 528/19*e + 923/19, 54/19*e^5 + 12/19*e^4 - 541/19*e^3 + 42/19*e^2 + 814/19*e - 32/19, 6/19*e^5 - 5/19*e^4 - 39/19*e^3 + 106/19*e^2 - 184/19*e - 449/19, 62/19*e^5 + 37/19*e^4 - 479/19*e^3 - 51/19*e^2 + 195/19*e - 1036/19, -110/19*e^5 - 35/19*e^4 + 1133/19*e^3 + 77/19*e^2 - 2067/19*e - 46/19, 8/19*e^5 - 32/19*e^4 - 166/19*e^3 + 135/19*e^2 + 901/19*e + 592/19, 9/19*e^5 + 78/19*e^4 - 68/19*e^3 - 620/19*e^2 + 66/19*e + 20/19, -22/19*e^5 - 7/19*e^4 + 86/19*e^3 - 110/19*e^2 + 415/19*e + 918/19, -59/19*e^5 + 141/19*e^4 + 792/19*e^3 - 1074/19*e^2 - 2206/19*e - 129/19, -62/19*e^5 - 18/19*e^4 + 498/19*e^3 - 44/19*e^2 - 233/19*e + 656/19, -4*e^5 + 3*e^4 + 48*e^3 - 25*e^2 - 115*e - 16, 2*e^5 - e^4 - 20*e^3 + 16*e^2 + 20*e - 34, 46/19*e^5 + 25/19*e^4 - 565/19*e^3 - 188/19*e^2 + 1623/19*e + 364/19, 79/19*e^5 - 31/19*e^4 - 846/19*e^3 + 224/19*e^2 + 1675/19*e + 564/19, 60/19*e^5 - 69/19*e^4 - 713/19*e^3 + 623/19*e^2 + 1618/19*e - 196/19, 32/19*e^5 - 109/19*e^4 - 436/19*e^3 + 882/19*e^2 + 1134/19*e - 349/19, 62/19*e^5 - 20/19*e^4 - 650/19*e^3 + 158/19*e^2 + 1240/19*e - 238/19, 6/19*e^5 + 14/19*e^4 - 96/19*e^3 - 217/19*e^2 + 766/19*e + 862/19, 54/19*e^5 - 121/19*e^4 - 731/19*e^3 + 897/19*e^2 + 2011/19*e + 234/19, 24/19*e^5 + 113/19*e^4 - 61/19*e^3 - 925/19*e^2 - 679/19*e + 807/19, 1/19*e^5 - 42/19*e^4 - 54/19*e^3 + 290/19*e^2 + 400/19*e + 169/19, 13/19*e^5 - 33/19*e^4 - 18/19*e^3 + 407/19*e^2 - 500/19*e - 767/19, 5/19*e^5 + 37/19*e^4 - 99/19*e^3 - 222/19*e^2 + 328/19*e - 276/19, 12/19*e^5 + 85/19*e^4 - 78/19*e^3 - 567/19*e^2 - 26/19*e - 62/19, -31/19*e^5 + 48/19*e^4 + 553/19*e^3 - 307/19*e^2 - 1969/19*e - 584/19, -25/19*e^5 + 81/19*e^4 + 400/19*e^3 - 638/19*e^2 - 1260/19*e + 183/19];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17,17,w^3 - 3*w^2 - 2*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;