/*
  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, -2, -5, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {43, 43, -w - 3}>;

primesArray := [
[4, 2, -w^3 + 4*w + 1],
[5, 5, w - 1],
[7, 7, -w^2 + w + 2],
[13, 13, -w^3 + w^2 + 4*w],
[13, 13, -w^2 + 3],
[17, 17, -w^2 + 2],
[19, 19, -w^3 + 5*w],
[31, 31, -w^2 + 2*w + 3],
[37, 37, -2*w^3 + w^2 + 8*w - 1],
[43, 43, -w - 3],
[53, 53, -w^3 + 2*w^2 + 3*w - 2],
[53, 53, w^3 - 6*w - 2],
[59, 59, 2*w^3 - w^2 - 10*w - 2],
[61, 61, 2*w^3 - w^2 - 10*w],
[61, 61, 2*w^3 - w^2 - 8*w],
[71, 71, 2*w^3 - 9*w - 2],
[73, 73, -w^3 - w^2 + 6*w + 3],
[81, 3, -3],
[83, 83, -w^3 + 2*w^2 + 3*w - 3],
[101, 101, 2*w^3 - 8*w - 3],
[125, 5, w^3 + w^2 - 4*w - 6],
[127, 127, 2*w^3 - w^2 - 7*w - 2],
[127, 127, w^3 - 7*w - 7],
[131, 131, w^3 - 3*w - 4],
[131, 131, 3*w^3 - 13*w - 7],
[137, 137, w^2 + w - 4],
[137, 137, -w^3 + 2*w^2 + 4*w - 4],
[149, 149, 3*w + 5],
[157, 157, -2*w^3 - w^2 + 7*w + 2],
[157, 157, -2*w^3 + 2*w^2 + 9*w - 1],
[163, 163, -2*w^3 + w^2 + 7*w],
[163, 163, -3*w^3 + w^2 + 15*w + 3],
[163, 163, 2*w^3 - 7*w - 1],
[163, 163, 2*w^3 - 11*w - 5],
[167, 167, -2*w^3 + w^2 + 9*w - 4],
[167, 167, w^3 - w^2 - 7*w + 1],
[167, 167, w^3 - w^2 - 7*w - 1],
[167, 167, w^2 - 2*w - 5],
[169, 13, w^3 + w^2 - 6*w - 2],
[173, 173, -3*w^3 + 13*w + 3],
[181, 181, 2*w^2 + w - 4],
[191, 191, 2*w^3 - 10*w - 1],
[191, 191, -2*w^3 + 2*w^2 + 6*w - 3],
[193, 193, 3*w^3 - 13*w - 6],
[197, 197, -w^3 + w^2 + 2*w - 3],
[197, 197, w^3 + w^2 - 7*w - 7],
[199, 199, -w^2 + 2*w + 6],
[223, 223, -2*w^3 + w^2 + 11*w + 2],
[227, 227, -3*w^3 + 2*w^2 + 15*w - 1],
[229, 229, -w^3 + w^2 + 6*w - 5],
[233, 233, 2*w^3 - 11*w - 4],
[239, 239, -2*w^3 + 11*w + 3],
[251, 251, 2*w^3 - w^2 - 10*w - 4],
[257, 257, -2*w^3 + w^2 + 6*w + 1],
[257, 257, w^3 - 7*w - 2],
[263, 263, 2*w^3 - 9*w],
[269, 269, w^3 - 4*w - 6],
[269, 269, -2*w^3 + 8*w - 3],
[271, 271, 2*w^2 - w - 5],
[277, 277, -3*w^3 + 2*w^2 + 11*w - 2],
[277, 277, 3*w^3 - w^2 - 15*w - 5],
[281, 281, 2*w^2 - 5],
[283, 283, -3*w^3 + 2*w^2 + 12*w + 2],
[283, 283, -2*w^3 + w^2 + 6*w + 6],
[293, 293, -2*w^3 - w^2 + 12*w + 7],
[293, 293, -2*w^3 + 9*w - 1],
[311, 311, -3*w^3 + w^2 + 12*w + 4],
[313, 313, -w^3 + w^2 + 2*w - 4],
[313, 313, -4*w^3 + 2*w^2 + 16*w - 3],
[331, 331, 3*w^3 - w^2 - 16*w - 5],
[331, 331, 2*w^3 - 2*w^2 - 8*w - 1],
[343, 7, 2*w^3 - 3*w^2 - 9*w + 6],
[359, 359, -2*w^3 + 10*w - 1],
[367, 367, -w^3 + 2*w^2 + w - 4],
[367, 367, 3*w^3 - 11*w - 1],
[373, 373, -w^3 + w^2 + 2*w - 5],
[373, 373, 2*w^3 - 2*w^2 - 11*w - 1],
[389, 389, -2*w^3 + 3*w^2 + 8*w - 6],
[409, 409, 3*w^3 - 2*w^2 - 13*w - 1],
[419, 419, 3*w^3 - w^2 - 12*w - 1],
[419, 419, w^3 - w^2 - 4*w - 4],
[439, 439, w^2 + 2*w - 4],
[439, 439, 3*w^3 + w^2 - 11*w - 5],
[443, 443, 2*w^3 + 2*w^2 - 7*w - 8],
[449, 449, 4*w^3 - w^2 - 16*w - 8],
[457, 457, 2*w^3 + w^2 - 11*w - 4],
[461, 461, -5*w^3 + 2*w^2 + 24*w + 2],
[463, 463, -w^3 + 3*w^2 + 6*w - 5],
[467, 467, w^3 - 3*w^2 - 6*w + 4],
[467, 467, 3*w^3 - w^2 - 12*w - 2],
[479, 479, -w^3 + w^2 + 4*w - 6],
[479, 479, -w^3 + 2*w + 8],
[491, 491, -w^3 + 2*w^2 + 2*w - 6],
[491, 491, w - 5],
[499, 499, 2*w^3 + w^2 - 10*w - 4],
[509, 509, -4*w^3 + w^2 + 20*w + 5],
[521, 521, w^3 - w + 6],
[523, 523, -4*w^3 + 18*w + 5],
[523, 523, -2*w^3 - 2*w^2 + 9*w + 12],
[557, 557, w^2 - 3*w - 6],
[571, 571, 3*w^3 - 11*w - 8],
[577, 577, 2*w^3 + 2*w^2 - 11*w - 7],
[577, 577, -3*w^3 + 4*w^2 + 11*w - 6],
[587, 587, -w^3 - w^2 + 7*w + 11],
[593, 593, w^3 + w^2 - 8*w - 8],
[593, 593, -2*w^3 + 4*w^2 + 6*w - 9],
[599, 599, 4*w^3 - 17*w - 9],
[599, 599, w^3 + w^2 - 6*w - 10],
[601, 601, 3*w^2 - 10],
[607, 607, 4*w^3 - w^2 - 18*w - 8],
[619, 619, w^2 - w - 8],
[641, 641, -w^3 + 3*w^2 + 2*w - 10],
[643, 643, 3*w^3 - w^2 - 16*w + 1],
[653, 653, -2*w^3 + 2*w^2 + 9*w + 3],
[677, 677, w^3 + w^2 - 8*w - 11],
[677, 677, 2*w^3 - w^2 - 12*w - 3],
[683, 683, 3*w^3 - 2*w^2 - 13*w - 2],
[683, 683, 3*w^3 + 2*w^2 - 16*w - 10],
[691, 691, -2*w^3 - 2*w^2 + 12*w + 15],
[691, 691, 3*w^3 - 2*w^2 - 15*w - 2],
[691, 691, w^3 - 2*w^2 - 6*w + 6],
[691, 691, -4*w^3 + 3*w^2 + 15*w - 2],
[719, 719, 2*w^3 - 7*w - 8],
[719, 719, w^2 + 2*w - 5],
[727, 727, -w^3 + 3*w^2 + 4*w - 8],
[727, 727, 2*w^3 - 9*w - 9],
[733, 733, w^3 - 8*w - 2],
[739, 739, -w^3 - 2*w^2 + 4*w + 10],
[743, 743, w^3 - w^2 - 2*w + 8],
[743, 743, 4*w^3 - w^2 - 17*w - 4],
[751, 751, -2*w^3 + 2*w^2 + 6*w - 5],
[751, 751, w^3 - 5*w - 7],
[769, 769, 3*w^3 - 2*w^2 - 10*w],
[787, 787, -2*w^3 + 3*w^2 + 8*w],
[787, 787, 3*w^3 - w^2 - 10*w - 4],
[809, 809, 3*w^3 + w^2 - 16*w - 6],
[811, 811, 3*w^3 - 14*w - 2],
[811, 811, -w^3 + 3*w^2 + 5*w - 5],
[811, 811, w^3 - 2*w - 6],
[811, 811, 3*w^3 - w^2 - 11*w - 3],
[823, 823, w^3 + 2*w^2 - 3*w - 8],
[827, 827, -w^3 + 3*w^2 + 3*w - 7],
[829, 829, w^3 + w^2 - 3*w - 7],
[853, 853, 3*w^3 + w^2 - 14*w - 6],
[857, 857, 3*w^3 + w^2 - 13*w - 5],
[857, 857, 3*w^3 - 3*w^2 - 11*w - 1],
[859, 859, 4*w^3 - 2*w^2 - 18*w - 3],
[859, 859, 3*w^3 - 11*w - 3],
[863, 863, 3*w^3 + w^2 - 17*w - 9],
[877, 877, -w^3 - 2*w^2 + 9*w + 1],
[877, 877, 5*w^3 + w^2 - 23*w - 11],
[911, 911, w^2 - 4*w - 6],
[911, 911, -2*w^3 + 2*w^2 + 10*w + 3],
[919, 919, -4*w^3 + 3*w^2 + 18*w - 1],
[929, 929, 2*w^3 - 13*w - 12],
[929, 929, -w^3 + 2*w^2 + 7*w - 5],
[953, 953, -3*w^3 + 3*w^2 + 13*w - 1],
[967, 967, -3*w^3 + 4*w^2 + 13*w - 7],
[971, 971, 2*w^2 - 3*w - 12],
[971, 971, 3*w^3 - w^2 - 16*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 - 36*x^8 + 466*x^6 - 2588*x^4 + 5293*x^2 - 384;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/64*e^9 - 31/64*e^7 + 319/64*e^5 - 1209/64*e^3 + 137/8*e, 1/64*e^9 - 31/64*e^7 + 311/64*e^5 - 1049/64*e^3 + 27/4*e, -1/32*e^8 + 31/32*e^6 - 311/32*e^4 + 1033/32*e^2 - 8, 1/8*e^5 - 5/2*e^3 + 91/8*e, 1/32*e^9 - 31/32*e^7 + 315/32*e^5 - 1145/32*e^3 + 227/8*e, -1/2*e^3 + 11/2*e, -7/64*e^8 + 201/64*e^6 - 1809/64*e^4 + 5119/64*e^2 - 2, -1/64*e^9 + 31/64*e^7 - 319/64*e^5 + 1209/64*e^3 - 137/8*e, -1, 1/64*e^9 - 31/64*e^7 + 335/64*e^5 - 1561/64*e^3 + 363/8*e, 5/32*e^8 - 151/32*e^6 + 1443/32*e^4 - 4353/32*e^2 + 6, -3/32*e^8 + 89/32*e^6 - 837/32*e^4 + 2559/32*e^2 - 12, 1/16*e^8 - 27/16*e^6 + 223/16*e^4 - 549/16*e^2 - 8, 1/32*e^9 - 31/32*e^7 + 307/32*e^5 - 985/32*e^3 + 37/8*e, 1/4*e^5 - 5*e^3 + 91/4*e, 1/8*e^5 - 5/2*e^3 + 99/8*e, 5/64*e^8 - 139/64*e^6 + 1187/64*e^4 - 3117/64*e^2 - 2, 1/32*e^9 - 31/32*e^7 + 311/32*e^5 - 1033/32*e^3 + 8*e, 3/8*e^5 - 8*e^3 + 293/8*e, 1/32*e^9 - 27/32*e^7 + 211/32*e^5 - 349/32*e^3 - 219/8*e, 1/32*e^9 - 27/32*e^7 + 207/32*e^5 - 221/32*e^3 - 213/4*e, -5/32*e^8 + 147/32*e^6 - 1379/32*e^4 + 4261/32*e^2 - 28, -7/64*e^9 + 209/64*e^7 - 1985/64*e^5 + 6103/64*e^3 - 59/4*e, -1/4*e^5 + 4*e^3 - 39/4*e, -1/16*e^9 + 29/16*e^7 - 257/16*e^5 + 619/16*e^3 + 309/8*e, 3/32*e^9 - 89/32*e^7 + 833/32*e^5 - 2447/32*e^3 - 35/8*e, -1/8*e^6 + 5/2*e^4 - 91/8*e^2, -1/16*e^9 + 29/16*e^7 - 265/16*e^5 + 771/16*e^3 - 27/8*e, -1/16*e^8 + 29/16*e^6 - 263/16*e^4 + 779/16*e^2 - 16, -1/8*e^9 + 29/8*e^7 - 261/8*e^5 + 691/8*e^3 + 147/4*e, -1/16*e^8 + 33/16*e^6 - 351/16*e^4 + 1215/16*e^2 - 16, 1/32*e^9 - 31/32*e^7 + 303/32*e^5 - 905/32*e^3 - 31/4*e, -1/16*e^8 + 27/16*e^6 - 223/16*e^4 + 613/16*e^2 - 20, 1/16*e^8 - 29/16*e^6 + 255/16*e^4 - 627/16*e^2 - 12, 1/32*e^8 - 35/32*e^6 + 391/32*e^4 - 1461/32*e^2 + 24, 17/64*e^8 - 511/64*e^6 + 4855/64*e^4 - 14617/64*e^2 + 18, -1/4*e^8 + 15/2*e^6 - 287/4*e^4 + 445/2*e^2 - 36, -1/8*e^9 + 15/4*e^7 - 73/2*e^5 + 497/4*e^3 - 635/8*e, 1/64*e^9 - 31/64*e^7 + 319/64*e^5 - 1177/64*e^3 + 77/8*e, -1/32*e^9 + 27/32*e^7 - 211/32*e^5 + 333/32*e^3 + 231/8*e, -1/32*e^9 + 27/32*e^7 - 215/32*e^5 + 445/32*e^3 + 21/2*e, 3/32*e^8 - 85/32*e^6 + 757/32*e^4 - 2259/32*e^2 + 24, -7/64*e^9 + 209/64*e^7 - 1977/64*e^5 + 6007/64*e^3 - 115/8*e, -9/64*e^8 + 255/64*e^6 - 2271/64*e^4 + 6393/64*e^2 + 6, -11/64*e^8 + 333/64*e^6 - 3213/64*e^4 + 9915/64*e^2 - 24, 1/4*e^8 - 29/4*e^6 + 263/4*e^4 - 755/4*e^2 + 20, 3/16*e^8 - 89/16*e^6 + 837/16*e^4 - 2495/16*e^2 + 4, -17/64*e^8 + 503/64*e^6 - 4727/64*e^4 + 14305/64*e^2 - 18, 1/8*e^7 - 27/8*e^5 + 231/8*e^3 - 637/8*e, 7/32*e^8 - 201/32*e^6 + 1809/32*e^4 - 5055/32*e^2 - 12, -13/64*e^8 + 387/64*e^6 - 3643/64*e^4 + 10773/64*e^2 + 18, 7/16*e^8 - 207/16*e^6 + 1937/16*e^4 - 5801/16*e^2 + 36, 5/32*e^9 - 147/32*e^7 + 1359/32*e^5 - 3893/32*e^3 - 119/8*e, 2*e^2 - 12, 1/32*e^9 - 31/32*e^7 + 311/32*e^5 - 1097/32*e^3 + 28*e, 11/32*e^8 - 325/32*e^6 + 3037/32*e^4 - 9075/32*e^2 + 18, -1/4*e^8 + 29/4*e^6 - 265/4*e^4 + 777/4*e^2 - 24, -7/64*e^9 + 201/64*e^7 - 1745/64*e^5 + 3871/64*e^3 + 175/2*e, -3/32*e^9 + 89/32*e^7 - 809/32*e^5 + 1951/32*e^3 + 561/8*e, 1/64*e^8 - 39/64*e^6 + 519/64*e^4 - 2353/64*e^2 + 28, -1/16*e^9 + 33/16*e^7 - 365/16*e^5 + 1495/16*e^3 - 765/8*e, -5/32*e^8 + 147/32*e^6 - 1363/32*e^4 + 4021/32*e^2 - 8, 1/64*e^9 - 23/64*e^7 + 103/64*e^5 + 671/64*e^3 - 72*e, 3/16*e^8 - 91/16*e^6 + 877/16*e^4 - 2709/16*e^2 + 18, -5/32*e^9 + 151/32*e^7 - 1455/32*e^5 + 4561/32*e^3 - 225/8*e, 3/32*e^9 - 93/32*e^7 + 925/32*e^5 - 3051/32*e^3 + 135/4*e, -5/32*e^8 + 147/32*e^6 - 1347/32*e^4 + 3781/32*e^2 - 4, -3/32*e^9 + 89/32*e^7 - 849/32*e^5 + 2799/32*e^3 - 385/8*e, -3/64*e^8 + 85/64*e^6 - 789/64*e^4 + 2483/64*e^2 + 14, 3/32*e^8 - 93/32*e^6 + 933/32*e^4 - 3035/32*e^2 - 4, -1/16*e^9 + 29/16*e^7 - 255/16*e^5 + 563/16*e^3 + 54*e, -1/8*e^9 + 29/8*e^7 - 261/8*e^5 + 691/8*e^3 + 163/4*e, -13/32*e^8 + 391/32*e^6 - 3707/32*e^4 + 11025/32*e^2 - 8, -5/32*e^9 + 147/32*e^7 - 1355/32*e^5 + 3829/32*e^3 + 87/4*e, 5/32*e^8 - 151/32*e^6 + 1427/32*e^4 - 4081/32*e^2 - 14, -7/64*e^8 + 209/64*e^6 - 1969/64*e^4 + 6039/64*e^2 - 14, 1/32*e^9 - 23/32*e^7 + 99/32*e^5 + 623/32*e^3 - 811/8*e, 7/64*e^8 - 201/64*e^6 + 1841/64*e^4 - 5535/64*e^2 + 8, 3/32*e^9 - 89/32*e^7 + 861/32*e^5 - 2991/32*e^3 + 275/4*e, 3/8*e^8 - 89/8*e^6 + 841/8*e^4 - 2571/8*e^2 + 48, e^5 - 21*e^3 + 98*e, 1/16*e^9 - 31/16*e^7 + 323/16*e^5 - 1273/16*e^3 + 329/4*e, 11/64*e^9 - 317/64*e^7 + 2797/64*e^5 - 6731/64*e^3 - 207/2*e, -5/32*e^9 + 151/32*e^7 - 1455/32*e^5 + 4577/32*e^3 - 269/8*e, 1/8*e^9 - 7/2*e^7 + 59/2*e^5 - 125/2*e^3 - 765/8*e, 1/4*e^8 - 59/8*e^6 + 273/4*e^4 - 1569/8*e^2 - 12, 1/8*e^7 - 13/4*e^5 + 195/8*e^3 - 205/4*e, 1/32*e^9 - 35/32*e^7 + 439/32*e^5 - 2309/32*e^3 + 130*e, -1/16*e^9 + 31/16*e^7 - 319/16*e^5 + 1241/16*e^3 - 181/2*e, 1/8*e^8 - 31/8*e^6 + 303/8*e^4 - 897/8*e^2 - 24, -1/8*e^9 + 15/4*e^7 - 285/8*e^5 + 429/4*e^3 - 29/4*e, 7/64*e^8 - 193/64*e^6 + 1649/64*e^4 - 4455/64*e^2 + 30, 5/64*e^8 - 179/64*e^6 + 2019/64*e^4 - 6981/64*e^2 - 6, -3/64*e^9 + 101/64*e^7 - 1125/64*e^5 + 4355/64*e^3 - 181/4*e, -3/16*e^8 + 89/16*e^6 - 845/16*e^4 + 2647/16*e^2 - 42, -3/32*e^8 + 101/32*e^6 - 1109/32*e^4 + 4067/32*e^2 - 54, 5/32*e^9 - 151/32*e^7 + 1483/32*e^5 - 5105/32*e^3 + 385/4*e, 5/32*e^9 - 143/32*e^7 + 1243/32*e^5 - 2825/32*e^3 - 447/4*e, -3/32*e^8 + 89/32*e^6 - 853/32*e^4 + 2703/32*e^2 - 18, -3/64*e^9 + 101/64*e^7 - 1141/64*e^5 + 4739/64*e^3 - 73*e, -1/16*e^9 + 25/16*e^7 - 153/16*e^5 - 209/16*e^3 + 1313/8*e, -3/32*e^9 + 89/32*e^7 - 849/32*e^5 + 2703/32*e^3 - 137/8*e, -5/64*e^8 + 163/64*e^6 - 1667/64*e^4 + 5173/64*e^2 + 6, -11/32*e^8 + 329/32*e^6 - 3101/32*e^4 + 9231/32*e^2 - 12, -5/32*e^9 + 159/32*e^7 - 1671/32*e^5 + 6361/32*e^3 - 1367/8*e, 1/8*e^9 - 31/8*e^7 + 313/8*e^5 - 1097/8*e^3 + 303/4*e, -3/32*e^8 + 85/32*e^6 - 757/32*e^4 + 2131/32*e^2 - 12, 3/8*e^5 - 6*e^3 + 165/8*e, -1/8*e^8 + 29/8*e^6 - 263/8*e^4 + 747/8*e^2 + 8, 1/4*e^8 - 15/2*e^6 + 289/4*e^4 - 226*e^2 + 16, 7/32*e^8 - 201/32*e^6 + 1825/32*e^4 - 5295/32*e^2 + 18, 7/64*e^8 - 209/64*e^6 + 2001/64*e^4 - 6263/64*e^2 + 26, 5/16*e^8 - 147/16*e^6 + 1371/16*e^4 - 4109/16*e^2 + 18, 17/32*e^8 - 499/32*e^6 + 4647/32*e^4 - 14005/32*e^2 + 72, -1/16*e^8 + 33/16*e^6 - 343/16*e^4 + 1015/16*e^2 + 36, 5/64*e^8 - 147/64*e^6 + 1347/64*e^4 - 3461/64*e^2 - 30, -1/32*e^9 + 35/32*e^7 - 439/32*e^5 + 2325/32*e^3 - 273/2*e, -1/8*e^6 + 2*e^4 - 55/8*e^2 + 16, -1/8*e^6 + 3*e^4 - 143/8*e^2 + 40, 3/16*e^9 - 93/16*e^7 + 941/16*e^5 - 3323/16*e^3 + 255/2*e, -1/4*e^9 + 61/8*e^7 - 151/2*e^5 + 2067/8*e^3 - 589/4*e, 17/64*e^9 - 511/64*e^7 + 4887/64*e^5 - 15193/64*e^3 + 99/2*e, 5/32*e^9 - 139/32*e^7 + 1131/32*e^5 - 1869/32*e^3 - 765/4*e, -3/32*e^9 + 85/32*e^7 - 693/32*e^5 + 915/32*e^3 + 162*e, -21/32*e^8 + 619/32*e^6 - 5779/32*e^4 + 17293/32*e^2 - 56, -1/8*e^8 + 13/4*e^6 - 199/8*e^4 + 227/4*e^2 - 16, 1/4*e^9 - 57/8*e^7 + 247/4*e^5 - 1155/8*e^3 - 287/2*e, -9/32*e^8 + 259/32*e^6 - 2367/32*e^4 + 7125/32*e^2 - 48, 1/64*e^9 - 31/64*e^7 + 327/64*e^5 - 1433/64*e^3 + 71/2*e, 1/4*e^8 - 15/2*e^6 + 287/4*e^4 - 445/2*e^2 + 32, -1/8*e^8 + 29/8*e^6 - 267/8*e^4 + 791/8*e^2 + 8, 1/8*e^9 - 15/4*e^7 + 139/4*e^5 - 365/4*e^3 - 463/8*e, -11/16*e^8 + 321/16*e^6 - 2949/16*e^4 + 8495/16*e^2 + 16, -5/32*e^9 + 143/32*e^7 - 1211/32*e^5 + 2201/32*e^3 + 753/4*e, -1/8*e^9 + 15/4*e^7 - 73/2*e^5 + 501/4*e^3 - 691/8*e, 1/32*e^9 - 27/32*e^7 + 215/32*e^5 - 413/32*e^3 - 61/2*e, -3/64*e^9 + 85/64*e^7 - 805/64*e^5 + 3219/64*e^3 - 329/4*e, 3/16*e^9 - 91/16*e^7 + 885/16*e^5 - 2901/16*e^3 + 179/2*e, 1/64*e^9 - 23/64*e^7 + 71/64*e^5 + 1375/64*e^3 - 245/2*e, 1/32*e^9 - 31/32*e^7 + 343/32*e^5 - 1689/32*e^3 + 197/2*e, 1/32*e^9 - 35/32*e^7 + 463/32*e^5 - 2821/32*e^3 + 797/4*e, -3/64*e^9 + 93/64*e^7 - 893/64*e^5 + 2379/64*e^3 + 321/8*e, 1/16*e^9 - 29/16*e^7 + 249/16*e^5 - 435/16*e^3 - 757/8*e, -3/64*e^9 + 85/64*e^7 - 733/64*e^5 + 1587/64*e^3 + 369/8*e, -19/32*e^8 + 557/32*e^6 - 5157/32*e^4 + 15163/32*e^2 - 30, -3/16*e^8 + 91/16*e^6 - 869/16*e^4 + 2637/16*e^2 - 32, -3/32*e^9 + 97/32*e^7 - 1021/32*e^5 + 3655/32*e^3 - 227/4*e, -17/64*e^8 + 487/64*e^6 - 4311/64*e^4 + 11665/64*e^2 + 30, 1/32*e^9 - 35/32*e^7 + 411/32*e^5 - 1717/32*e^3 + 347/8*e, 1/16*e^9 - 29/16*e^7 + 249/16*e^5 - 451/16*e^3 - 765/8*e, 19/32*e^8 - 561/32*e^6 + 5205/32*e^4 - 15079/32*e^2, 3/8*e^8 - 43/4*e^6 + 777/8*e^4 - 1115/4*e^2 + 12, 7/32*e^8 - 217/32*e^6 + 2161/32*e^4 - 7119/32*e^2 + 56, -e^4 + 13*e^2 - 24, -1/8*e^7 + 21/8*e^5 - 111/8*e^3 + 75/8*e, -1/8*e^6 + 3*e^4 - 143/8*e^2 - 6, -1/32*e^9 + 31/32*e^7 - 303/32*e^5 + 793/32*e^3 + 169/4*e, 1/8*e^8 - 7/2*e^6 + 251/8*e^4 - 95*e^2 + 36, 7/32*e^8 - 197/32*e^6 + 1729/32*e^4 - 4883/32*e^2 + 36];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {43, 43, -w - 3}>] := 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;