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

NN := ideal<ZF | {19, 19, -w^2 - w + 1}>;

primesArray := [
[5, 5, w^3 - 4*w],
[7, 7, w + 1],
[7, 7, -w^3 + 4*w - 1],
[13, 13, -w^2 + 3],
[16, 2, 2],
[17, 17, w^3 + w^2 - 4*w - 2],
[17, 17, -w^3 + 5*w - 2],
[19, 19, -w^2 - w + 4],
[19, 19, -w^2 - w + 1],
[29, 29, 2*w^3 - w^2 - 9*w + 5],
[41, 41, -w^3 + w^2 + 5*w - 3],
[43, 43, w^3 - w^2 - 4*w + 2],
[43, 43, w^3 - 6*w],
[47, 47, -w^3 - w^2 + 5*w],
[49, 7, w^2 + 2*w - 2],
[59, 59, 2*w^3 - 8*w + 3],
[67, 67, w^2 - w - 4],
[79, 79, w^3 - w^2 - 4*w + 1],
[81, 3, -3],
[97, 97, w^3 + w^2 - 5*w + 1],
[107, 107, -2*w^3 + 3*w^2 + 10*w - 12],
[109, 109, 2*w^3 - w^2 - 9*w + 3],
[125, 5, -3*w^3 - w^2 + 12*w + 1],
[127, 127, -2*w^3 + w^2 + 10*w - 8],
[137, 137, -w^3 + w^2 + 3*w - 5],
[139, 139, w^3 - w^2 - 5*w + 1],
[149, 149, 2*w^2 - w - 6],
[149, 149, 2*w^3 + w^2 - 9*w],
[163, 163, w^3 - 5*w - 3],
[163, 163, 2*w^3 - 7*w],
[173, 173, -2*w^3 + 11*w - 3],
[173, 173, -w^3 + w^2 + 4*w - 8],
[179, 179, 2*w^3 - w^2 - 8*w + 5],
[181, 181, 2*w^3 - w^2 - 8*w],
[181, 181, 2*w^3 - 9*w - 3],
[191, 191, 2*w^2 - 5],
[191, 191, -2*w^2 - w + 10],
[193, 193, w^3 + w^2 - 3*w - 4],
[193, 193, -w^3 + 2*w^2 + 3*w - 3],
[197, 197, -2*w^3 + 9*w - 4],
[197, 197, w^2 - 7],
[199, 199, 3*w^3 - 13*w + 1],
[199, 199, w^3 + 2*w^2 - 5*w - 7],
[227, 227, -3*w^3 + w^2 + 13*w - 4],
[227, 227, -w^3 + 2*w^2 + 7*w - 6],
[241, 241, -2*w^3 + 7*w - 1],
[257, 257, -2*w^3 + w^2 + 7*w - 5],
[257, 257, 2*w^3 - w^2 - 7*w + 2],
[263, 263, -w^3 + 7*w - 3],
[269, 269, -2*w^3 + 2*w^2 + 11*w - 8],
[271, 271, w^2 + 3*w - 2],
[271, 271, w^2 + 2*w - 7],
[277, 277, -2*w^3 + 11*w - 2],
[281, 281, 2*w^3 - 11*w + 1],
[283, 283, 2*w^3 - 9*w + 5],
[283, 283, -2*w^3 + w^2 + 11*w - 7],
[289, 17, -w^3 + 2*w^2 + 4*w - 4],
[293, 293, -2*w^3 - w^2 + 7*w + 3],
[307, 307, -3*w^3 + 2*w^2 + 13*w - 9],
[311, 311, -w^3 + w^2 + 3*w - 7],
[313, 313, -w^3 + 2*w^2 + 3*w - 2],
[313, 313, w^2 - w - 8],
[317, 317, -2*w^3 - w^2 + 9*w - 1],
[331, 331, w^3 + 2*w^2 - 5*w - 3],
[337, 337, w^3 + 2*w^2 - 6*w - 4],
[337, 337, w^3 - 2*w^2 - 6*w + 5],
[347, 347, 2*w^3 + 2*w^2 - 9*w - 3],
[347, 347, -2*w^3 + w^2 + 7*w - 4],
[349, 349, w^2 - 2*w - 4],
[353, 353, -w^3 + 2*w^2 + 2*w - 6],
[359, 359, -w^3 + w^2 + 7*w - 6],
[359, 359, -w^2 - w - 2],
[361, 19, 2*w^3 - w^2 - 11*w],
[367, 367, -2*w^3 + 2*w^2 + 8*w - 5],
[373, 373, w^3 + w^2 - 2*w - 4],
[379, 379, 2*w^3 - w^2 - 11*w + 4],
[389, 389, 3*w^2 + w - 11],
[397, 397, 3*w^3 - 13*w + 5],
[401, 401, w - 5],
[409, 409, 3*w^3 - 12*w + 4],
[409, 409, 3*w^3 + 2*w^2 - 14*w - 4],
[421, 421, 2*w^3 - w^2 - 10*w + 1],
[421, 421, w^3 + 2*w^2 - 5*w - 4],
[439, 439, -2*w^3 + 10*w - 5],
[443, 443, -2*w^3 + 12*w - 5],
[443, 443, -4*w^3 + w^2 + 18*w - 9],
[443, 443, -2*w^3 + w^2 + 7*w - 9],
[443, 443, 3*w^3 + 2*w^2 - 14*w - 5],
[449, 449, w^3 + 2*w^2 - 2*w - 6],
[449, 449, -2*w^3 + w^2 + 11*w - 5],
[461, 461, 2*w^3 + 2*w^2 - 9*w - 4],
[467, 467, -w^3 + w^2 + 7*w - 5],
[487, 487, 2*w^2 + 4*w - 7],
[491, 491, -w^3 + w^2 + 7*w - 4],
[503, 503, 2*w^3 + w^2 - 9*w + 2],
[521, 521, w^2 - 3*w - 3],
[523, 523, -w^3 + 2*w^2 + 5*w - 4],
[523, 523, -w^3 + 8*w - 5],
[563, 563, w^3 + 3*w^2 - w - 8],
[563, 563, w^3 - 2*w^2 - 8*w + 4],
[563, 563, -w^3 + w^2 + 2*w - 5],
[563, 563, -3*w^3 - w^2 + 10*w - 2],
[577, 577, 3*w^3 - 16*w],
[587, 587, w^3 + 2*w^2 - 6],
[587, 587, -w^3 + 4*w - 6],
[601, 601, -w - 5],
[607, 607, w^3 - 2*w^2 - 4*w + 1],
[613, 613, 4*w^3 - 17*w],
[617, 617, -3*w^3 + w^2 + 14*w - 2],
[617, 617, -4*w^3 + 17*w - 4],
[619, 619, -3*w^3 + 12*w - 2],
[643, 643, 3*w^3 - w^2 - 12*w + 3],
[643, 643, -2*w^3 - w^2 + 11*w - 2],
[647, 647, 2*w^3 + w^2 - 9*w + 3],
[653, 653, w^3 + w^2 - 7*w - 4],
[661, 661, -w^3 + 4*w^2 + 6*w - 15],
[661, 661, 3*w^3 - 2*w^2 - 12*w + 10],
[677, 677, -3*w^2 - 2*w + 12],
[691, 691, -2*w^3 + 3*w^2 + 11*w - 10],
[691, 691, -3*w^3 + 16*w - 3],
[701, 701, 3*w^3 - 13*w + 6],
[719, 719, -3*w^2 + w + 10],
[727, 727, w^3 - 3*w^2 - 4*w + 6],
[743, 743, -w^3 + 2*w^2 + 5*w - 3],
[761, 761, -w^3 + 3*w^2 + 3*w - 12],
[761, 761, 3*w^3 - 2*w^2 - 14*w + 7],
[769, 769, -3*w^3 + 3*w^2 + 16*w - 13],
[769, 769, w^3 + 2*w^2 - 7*w - 6],
[773, 773, w^3 - 8*w + 4],
[773, 773, 4*w^2 + w - 16],
[787, 787, w^2 + 4*w - 2],
[797, 797, -2*w^3 + 3*w^2 + 10*w - 10],
[797, 797, -w^3 - w^2 + 4*w - 4],
[797, 797, w^3 + 2*w^2 - 3*w - 9],
[797, 797, -3*w^3 + 3*w^2 + 11*w - 10],
[821, 821, w^2 + w - 9],
[827, 827, w^3 - 8*w + 1],
[827, 827, -2*w^3 + w^2 + 9*w - 11],
[827, 827, 3*w^3 - 14*w + 5],
[827, 827, 2*w^3 + w^2 - 7*w - 5],
[829, 829, -2*w^3 - 2*w^2 + 9*w - 1],
[839, 839, -3*w^3 + 15*w - 5],
[853, 853, -4*w^3 + 15*w - 8],
[853, 853, -w^3 + 2*w^2 + 6*w - 3],
[859, 859, -w^3 + w^2 + w - 4],
[859, 859, w^3 + w^2 - w - 5],
[881, 881, -w^3 + w^2 + 2*w - 6],
[881, 881, w^3 + w^2 - 2*w - 6],
[883, 883, w^3 + 3*w^2 - 3*w - 6],
[887, 887, w^3 + 3*w^2 - 3*w - 8],
[919, 919, 2*w^3 - w^2 - 13*w],
[929, 929, -w^3 - 3*w^2 + 2*w + 10],
[929, 929, w^3 + w^2 - 5*w - 8],
[941, 941, w^3 + 2*w^2 - 3*w - 10],
[953, 953, -w^3 + 5*w - 7],
[953, 953, -2*w^3 + 3*w^2 + 9*w - 9],
[953, 953, -2*w^3 - w^2 + 10*w - 3],
[953, 953, -3*w^3 + 11*w - 1],
[961, 31, -w^3 + 3*w^2 + 2*w - 10],
[961, 31, -3*w^2 + 2*w + 7],
[967, 967, w^3 + 3*w^2 - 3*w - 7],
[967, 967, 4*w^3 + w^2 - 20*w + 1],
[983, 983, 2*w^3 - w^2 - 12*w + 8],
[983, 983, 2*w^3 + w^2 - 6*w - 4],
[991, 991, 2*w^3 - 12*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 - 7*x^4 + 5*x^3 + 52*x^2 - 101*x + 25;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -2/5*e^4 + 7/5*e^3 + 17/5*e^2 - 52/5*e - 1, -e^4 + 4*e^3 + 7*e^2 - 32*e + 10, 6/5*e^4 - 26/5*e^3 - 36/5*e^2 + 201/5*e - 15, 2/5*e^4 - 7/5*e^3 - 12/5*e^2 + 52/5*e - 10, 1/5*e^4 - 6/5*e^3 - 6/5*e^2 + 51/5*e, 3/5*e^4 - 13/5*e^3 - 18/5*e^2 + 103/5*e - 7, 1/5*e^4 - 6/5*e^3 - 1/5*e^2 + 51/5*e - 9, -1, 1/5*e^4 - 1/5*e^3 - 11/5*e^2 + 6/5*e + 6, 11/5*e^4 - 41/5*e^3 - 81/5*e^2 + 311/5*e - 8, e^3 - 2*e^2 - 8*e + 8, 7/5*e^4 - 27/5*e^3 - 52/5*e^2 + 222/5*e - 11, 6/5*e^4 - 26/5*e^3 - 46/5*e^2 + 206/5*e, 6/5*e^4 - 26/5*e^3 - 36/5*e^2 + 191/5*e - 17, 14/5*e^4 - 59/5*e^3 - 89/5*e^2 + 454/5*e - 29, -2*e^4 + 8*e^3 + 13*e^2 - 65*e + 29, -2*e^4 + 8*e^3 + 15*e^2 - 64*e + 8, -7/5*e^4 + 32/5*e^3 + 42/5*e^2 - 252/5*e + 14, -12/5*e^4 + 47/5*e^3 + 82/5*e^2 - 372/5*e + 19, -2*e^4 + 7*e^3 + 13*e^2 - 53*e + 33, 14/5*e^4 - 49/5*e^3 - 104/5*e^2 + 374/5*e - 17, 17/5*e^4 - 62/5*e^3 - 127/5*e^2 + 487/5*e - 22, -19/5*e^4 + 79/5*e^3 + 119/5*e^2 - 624/5*e + 52, 6/5*e^4 - 21/5*e^3 - 56/5*e^2 + 166/5*e + 14, 2*e^3 - 2*e^2 - 20*e + 15, 8/5*e^4 - 38/5*e^3 - 48/5*e^2 + 298/5*e - 14, -11/5*e^4 + 46/5*e^3 + 71/5*e^2 - 381/5*e + 35, -3*e^4 + 12*e^3 + 20*e^2 - 95*e + 30, 4*e^4 - 17*e^3 - 28*e^2 + 136*e - 31, -6/5*e^4 + 31/5*e^3 + 16/5*e^2 - 236/5*e + 38, -6/5*e^4 + 31/5*e^3 + 26/5*e^2 - 236/5*e + 16, -11/5*e^4 + 51/5*e^3 + 66/5*e^2 - 411/5*e + 36, -3*e^2 + 6*e + 19, 3/5*e^4 - 8/5*e^3 - 38/5*e^2 + 78/5*e + 10, -12/5*e^4 + 47/5*e^3 + 92/5*e^2 - 377/5*e + 2, 2*e^4 - 8*e^3 - 14*e^2 + 57*e - 5, -4/5*e^4 + 19/5*e^3 + 29/5*e^2 - 159/5*e + 2, 4/5*e^4 - 14/5*e^3 - 24/5*e^2 + 109/5*e - 6, -1/5*e^4 - 4/5*e^3 + 21/5*e^2 + 44/5*e - 14, -e^4 + 2*e^3 + 12*e^2 - 16*e - 19, 3/5*e^4 - 13/5*e^3 - 28/5*e^2 + 103/5*e + 12, -2/5*e^4 + 2/5*e^3 + 37/5*e^2 - 17/5*e - 14, -3*e^4 + 12*e^3 + 17*e^2 - 91*e + 48, -3*e^4 + 13*e^3 + 16*e^2 - 98*e + 55, -11/5*e^4 + 41/5*e^3 + 71/5*e^2 - 316/5*e + 25, 2/5*e^4 - 17/5*e^3 + 8/5*e^2 + 132/5*e - 37, -6/5*e^4 + 21/5*e^3 + 51/5*e^2 - 141/5*e - 17, 18/5*e^4 - 73/5*e^3 - 113/5*e^2 + 563/5*e - 50, 4/5*e^4 - 4/5*e^3 - 44/5*e^2 - 21/5*e + 18, 18/5*e^4 - 78/5*e^3 - 118/5*e^2 + 648/5*e - 47, 7/5*e^4 - 17/5*e^3 - 67/5*e^2 + 127/5*e + 17, -6*e^4 + 26*e^3 + 37*e^2 - 207*e + 83, -e^4 + 2*e^3 + 13*e^2 - 16*e - 21, -e^4 + 3*e^3 + 8*e^2 - 23*e + 2, 2*e^4 - 8*e^3 - 12*e^2 + 63*e - 21, -3/5*e^4 + 8/5*e^3 + 23/5*e^2 - 48/5*e + 14, -17/5*e^4 + 57/5*e^3 + 132/5*e^2 - 407/5*e + 1, 28/5*e^4 - 108/5*e^3 - 203/5*e^2 + 838/5*e - 33, 2*e^4 - 7*e^3 - 17*e^2 + 57*e + 2, 12/5*e^4 - 52/5*e^3 - 82/5*e^2 + 442/5*e - 25, -7/5*e^4 + 32/5*e^3 + 57/5*e^2 - 262/5*e - 15, 12/5*e^4 - 62/5*e^3 - 62/5*e^2 + 487/5*e - 42, -13/5*e^4 + 48/5*e^3 + 113/5*e^2 - 388/5*e - 5, 13/5*e^4 - 58/5*e^3 - 68/5*e^2 + 443/5*e - 43, -3/5*e^4 + 18/5*e^3 + 13/5*e^2 - 178/5*e + 18, -e^4 + 6*e^3 + 5*e^2 - 49*e + 7, -8/5*e^4 + 43/5*e^3 + 48/5*e^2 - 368/5*e + 20, -5*e^4 + 20*e^3 + 36*e^2 - 160*e + 34, -18/5*e^4 + 63/5*e^3 + 143/5*e^2 - 503/5*e + 17, 3*e^4 - 9*e^3 - 25*e^2 + 66*e, -32/5*e^4 + 132/5*e^3 + 212/5*e^2 - 1027/5*e + 58, -11/5*e^4 + 36/5*e^3 + 91/5*e^2 - 246/5*e - 2, -13/5*e^4 + 58/5*e^3 + 93/5*e^2 - 483/5*e + 17, -2*e^4 + 10*e^3 + 14*e^2 - 81*e + 8, -6/5*e^4 + 11/5*e^3 + 61/5*e^2 - 66/5*e - 15, -19/5*e^4 + 84/5*e^3 + 104/5*e^2 - 654/5*e + 62, 2/5*e^4 - 7/5*e^3 - 7/5*e^2 + 42/5*e - 3, -4/5*e^4 + 19/5*e^3 + 9/5*e^2 - 104/5*e + 33, 21/5*e^4 - 81/5*e^3 - 126/5*e^2 + 586/5*e - 65, 12/5*e^4 - 62/5*e^3 - 57/5*e^2 + 472/5*e - 48, -11/5*e^4 + 36/5*e^3 + 91/5*e^2 - 276/5*e + 24, -2*e^4 + 11*e^3 + 8*e^2 - 95*e + 55, 16/5*e^4 - 66/5*e^3 - 96/5*e^2 + 506/5*e - 42, -4*e^4 + 16*e^3 + 26*e^2 - 127*e + 33, 3/5*e^4 + 2/5*e^3 - 33/5*e^2 - 47/5*e + 9, 3/5*e^4 - 8/5*e^3 - 38/5*e^2 + 38/5*e + 25, 19/5*e^4 - 84/5*e^3 - 114/5*e^2 + 674/5*e - 55, 3*e^4 - 10*e^3 - 23*e^2 + 72*e - 12, 12/5*e^4 - 22/5*e^3 - 122/5*e^2 + 127/5*e + 26, -2*e^4 + 10*e^3 + 10*e^2 - 75*e + 14, 3*e^2 - 7*e - 18, -26/5*e^4 + 106/5*e^3 + 161/5*e^2 - 806/5*e + 71, -7/5*e^4 + 22/5*e^3 + 47/5*e^2 - 187/5*e + 37, 31/5*e^4 - 136/5*e^3 - 191/5*e^2 + 1111/5*e - 94, -1/5*e^4 - 19/5*e^3 + 46/5*e^2 + 189/5*e - 41, 29/5*e^4 - 129/5*e^3 - 189/5*e^2 + 1014/5*e - 56, e^4 - 4*e^3 - 10*e^2 + 39*e + 13, -3/5*e^4 + 28/5*e^3 - 17/5*e^2 - 228/5*e + 33, -16/5*e^4 + 56/5*e^3 + 116/5*e^2 - 411/5*e, -13/5*e^4 + 43/5*e^3 + 83/5*e^2 - 293/5*e + 40, 28/5*e^4 - 118/5*e^3 - 188/5*e^2 + 928/5*e - 46, -24/5*e^4 + 89/5*e^3 + 179/5*e^2 - 704/5*e + 10, 5*e^4 - 23*e^3 - 29*e^2 + 189*e - 84, 21/5*e^4 - 81/5*e^3 - 171/5*e^2 + 626/5*e + 4, 1/5*e^4 + 4/5*e^3 - 16/5*e^2 - 74/5*e + 4, -13/5*e^4 + 38/5*e^3 + 113/5*e^2 - 278/5*e - 9, -3*e^4 + 10*e^3 + 23*e^2 - 69*e - 2, -29/5*e^4 + 124/5*e^3 + 154/5*e^2 - 954/5*e + 99, 4/5*e^4 - 39/5*e^3 + 21/5*e^2 + 299/5*e - 39, 4/5*e^4 + 1/5*e^3 - 39/5*e^2 - 81/5*e + 18, -6*e^4 + 28*e^3 + 35*e^2 - 216*e + 78, 18/5*e^4 - 68/5*e^3 - 128/5*e^2 + 493/5*e + 1, 4*e^4 - 18*e^3 - 20*e^2 + 143*e - 81, -24/5*e^4 + 114/5*e^3 + 134/5*e^2 - 879/5*e + 80, 2*e^3 - 7*e^2 - 6*e + 38, 4/5*e^4 - 9/5*e^3 - 24/5*e^2 + 59/5*e - 24, -4/5*e^4 + 19/5*e^3 + 9/5*e^2 - 174/5*e + 18, -28/5*e^4 + 118/5*e^3 + 148/5*e^2 - 903/5*e + 95, -7/5*e^4 + 22/5*e^3 + 57/5*e^2 - 142/5*e - 14, 13/5*e^4 - 48/5*e^3 - 123/5*e^2 + 408/5*e + 12, -44/5*e^4 + 169/5*e^3 + 314/5*e^2 - 1334/5*e + 64, 21/5*e^4 - 86/5*e^3 - 111/5*e^2 + 656/5*e - 73, -18/5*e^4 + 63/5*e^3 + 168/5*e^2 - 523/5*e - 19, -27/5*e^4 + 107/5*e^3 + 162/5*e^2 - 812/5*e + 86, 13/5*e^4 - 48/5*e^3 - 83/5*e^2 + 338/5*e - 52, -37/5*e^4 + 167/5*e^3 + 227/5*e^2 - 1312/5*e + 106, 41/5*e^4 - 166/5*e^3 - 271/5*e^2 + 1291/5*e - 54, -8*e^4 + 32*e^3 + 58*e^2 - 254*e + 51, 8*e^4 - 33*e^3 - 54*e^2 + 253*e - 65, 19/5*e^4 - 74/5*e^3 - 114/5*e^2 + 574/5*e - 53, -56/5*e^4 + 241/5*e^3 + 341/5*e^2 - 1881/5*e + 149, 4*e^4 - 20*e^3 - 21*e^2 + 153*e - 53, -36/5*e^4 + 141/5*e^3 + 271/5*e^2 - 1166/5*e + 36, -2*e^4 + 10*e^3 + 6*e^2 - 68*e + 52, e^4 - 5*e^3 - 4*e^2 + 46*e - 31, -51/5*e^4 + 211/5*e^3 + 331/5*e^2 - 1661/5*e + 117, 34/5*e^4 - 124/5*e^3 - 239/5*e^2 + 914/5*e - 24, -3*e^4 + 11*e^3 + 23*e^2 - 98*e + 15, 1/5*e^4 - 16/5*e^3 + 19/5*e^2 + 151/5*e - 53, -4*e^3 + 11*e^2 + 30*e - 42, 44/5*e^4 - 179/5*e^3 - 319/5*e^2 + 1404/5*e - 50, 4*e^4 - 12*e^3 - 34*e^2 + 79*e + 27, 2*e^4 - 7*e^3 - 16*e^2 + 54*e + 26, -6/5*e^4 + 36/5*e^3 + 21/5*e^2 - 261/5*e + 32, -2*e^4 + 9*e^3 + 9*e^2 - 62*e + 46, -7/5*e^4 + 37/5*e^3 + 17/5*e^2 - 262/5*e + 45, 29/5*e^4 - 129/5*e^3 - 169/5*e^2 + 959/5*e - 76, -26/5*e^4 + 111/5*e^3 + 166/5*e^2 - 841/5*e + 37, 23/5*e^4 - 88/5*e^3 - 153/5*e^2 + 653/5*e - 44, -24/5*e^4 + 109/5*e^3 + 159/5*e^2 - 874/5*e + 41, -33/5*e^4 + 143/5*e^3 + 208/5*e^2 - 1193/5*e + 90, 9/5*e^4 - 44/5*e^3 - 69/5*e^2 + 399/5*e + 11, 12/5*e^4 - 42/5*e^3 - 52/5*e^2 + 292/5*e - 70, 32/5*e^4 - 142/5*e^3 - 197/5*e^2 + 1097/5*e - 79, -2*e^4 + 5*e^3 + 18*e^2 - 37*e + 30, 4*e^4 - 15*e^3 - 30*e^2 + 121*e - 6, 2*e^4 - 8*e^3 - 17*e^2 + 66*e - 14, 11/5*e^4 - 41/5*e^3 - 66/5*e^2 + 376/5*e - 49, 39/5*e^4 - 144/5*e^3 - 279/5*e^2 + 1079/5*e - 47, 13/5*e^4 - 58/5*e^3 - 98/5*e^2 + 463/5*e + 4, 59/5*e^4 - 234/5*e^3 - 409/5*e^2 + 1814/5*e - 68, -4*e^4 + 17*e^3 + 28*e^2 - 123*e + 11, 2*e^4 - 5*e^3 - 23*e^2 + 44*e + 36, 21/5*e^4 - 111/5*e^3 - 91/5*e^2 + 836/5*e - 89];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, -w^2 - w + 1}>] := 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;