/*
  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![11, 6, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3}>;

primesArray := [
[9, 3, 1/3*w^3 - 8/3*w - 2/3],
[9, 3, -w + 1],
[11, 11, w],
[11, 11, -1/3*w^3 + 2/3*w + 5/3],
[11, 11, 1/3*w^3 - 8/3*w + 1/3],
[11, 11, -2/3*w^3 + 13/3*w + 4/3],
[16, 2, 2],
[25, 5, 2/3*w^3 - 10/3*w - 1/3],
[29, 29, -w - 3],
[29, 29, -1/3*w^3 + 8/3*w - 10/3],
[31, 31, w^3 - 6*w + 1],
[31, 31, w^3 - 6*w - 2],
[41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3],
[41, 41, 2/3*w^3 - 13/3*w + 5/3],
[59, 59, 2/3*w^3 - 13/3*w + 8/3],
[59, 59, w^3 + w^2 - 6*w - 4],
[79, 79, 2/3*w^3 + w^2 - 10/3*w - 19/3],
[79, 79, 1/3*w^3 + w^2 - 2/3*w - 17/3],
[89, 89, 4/3*w^3 - 23/3*w - 5/3],
[89, 89, 1/3*w^3 + 2*w^2 - 5/3*w - 32/3],
[101, 101, 5/3*w^3 + w^2 - 31/3*w - 16/3],
[101, 101, 2/3*w^3 - w^2 - 4/3*w + 8/3],
[109, 109, -w^3 - 2*w^2 + 7*w + 7],
[109, 109, -1/3*w^3 + 2*w^2 + 2/3*w - 16/3],
[131, 131, w^3 - w^2 - 7*w + 5],
[131, 131, -2/3*w^3 - w^2 + 4/3*w + 13/3],
[139, 139, -2/3*w^3 - w^2 + 13/3*w + 4/3],
[139, 139, w^2 - w - 7],
[151, 151, -5/3*w^3 + 28/3*w - 5/3],
[151, 151, -1/3*w^3 + w^2 + 2/3*w - 16/3],
[151, 151, w^3 + w^2 - 6*w - 3],
[151, 151, -w^3 + w^2 + 7*w - 6],
[179, 179, 5/3*w^3 + w^2 - 28/3*w - 19/3],
[179, 179, -1/3*w^3 + w^2 + 11/3*w - 4/3],
[181, 181, -4/3*w^3 - 2*w^2 + 20/3*w + 29/3],
[181, 181, 1/3*w^3 + w^2 - 2/3*w - 20/3],
[181, 181, w^3 + w^2 - 5*w - 7],
[181, 181, 5/3*w^3 + w^2 - 28/3*w - 16/3],
[191, 191, 2*w - 1],
[191, 191, 2/3*w^3 - 16/3*w - 1/3],
[199, 199, 1/3*w^3 - 11/3*w + 4/3],
[199, 199, 1/3*w^3 - 11/3*w - 2/3],
[211, 211, -2/3*w^3 - w^2 + 19/3*w + 16/3],
[211, 211, 2/3*w^3 + w^2 - 19/3*w - 7/3],
[229, 229, 1/3*w^3 - 11/3*w + 1/3],
[239, 239, -5/3*w^3 - w^2 + 25/3*w + 19/3],
[239, 239, -4/3*w^3 + w^2 + 20/3*w - 7/3],
[251, 251, -2/3*w^3 + w^2 + 16/3*w - 14/3],
[251, 251, -1/3*w^3 - w^2 - 1/3*w + 14/3],
[269, 269, 2/3*w^3 + w^2 - 7/3*w - 19/3],
[269, 269, 1/3*w^3 + 2*w^2 - 2/3*w - 32/3],
[281, 281, w^3 - w^2 - 5*w + 1],
[281, 281, 4/3*w^3 + w^2 - 20/3*w - 23/3],
[289, 17, 2/3*w^3 + w^2 - 16/3*w - 4/3],
[289, 17, 1/3*w^3 + w^2 - 11/3*w - 20/3],
[331, 331, 1/3*w^3 + 2*w^2 - 8/3*w - 23/3],
[331, 331, 2/3*w^3 + 2*w^2 - 13/3*w - 28/3],
[349, 349, 4/3*w^3 - 17/3*w + 1/3],
[349, 349, -5/3*w^3 + 28/3*w - 2/3],
[361, 19, -1/3*w^3 + 5/3*w + 14/3],
[361, 19, 4/3*w^3 - 20/3*w - 5/3],
[379, 379, 4/3*w^3 + 2*w^2 - 23/3*w - 41/3],
[379, 379, -5/3*w^3 + 31/3*w - 5/3],
[389, 389, 5/3*w^3 + w^2 - 25/3*w - 16/3],
[389, 389, -4/3*w^3 + w^2 + 20/3*w - 10/3],
[401, 401, -1/3*w^3 - 3*w^2 + 5/3*w + 26/3],
[401, 401, 1/3*w^3 + w^2 - 11/3*w - 23/3],
[401, 401, -2/3*w^3 - 3*w^2 + 10/3*w + 52/3],
[401, 401, 2/3*w^3 + w^2 - 16/3*w - 1/3],
[409, 409, -2*w^3 - w^2 + 11*w + 2],
[409, 409, -5/3*w^3 - 2*w^2 + 28/3*w + 37/3],
[409, 409, 1/3*w^3 - 11/3*w - 17/3],
[409, 409, 5/3*w^3 + 3*w^2 - 25/3*w - 37/3],
[419, 419, -1/3*w^3 + 3*w^2 - 1/3*w - 37/3],
[419, 419, 2*w^3 + 3*w^2 - 12*w - 13],
[421, 421, -1/3*w^3 + 3*w^2 + 2/3*w - 31/3],
[421, 421, -5/3*w^3 - w^2 + 25/3*w + 13/3],
[421, 421, -w^3 + 7*w - 4],
[421, 421, -4/3*w^3 + w^2 + 20/3*w - 13/3],
[439, 439, -4/3*w^3 + w^2 + 14/3*w - 7/3],
[439, 439, 4/3*w^3 - 17/3*w - 5/3],
[439, 439, -7/3*w^3 - w^2 + 41/3*w + 17/3],
[439, 439, 5/3*w^3 - 28/3*w - 4/3],
[449, 449, w^3 + 3*w^2 - 6*w - 17],
[449, 449, 1/3*w^3 + 3*w^2 - 8/3*w - 26/3],
[449, 449, -4/3*w^3 + 26/3*w + 17/3],
[449, 449, 7/3*w^3 + w^2 - 38/3*w - 8/3],
[461, 461, -5/3*w^3 + w^2 + 28/3*w - 14/3],
[461, 461, -2/3*w^3 - 2*w^2 + 10/3*w + 43/3],
[461, 461, -2*w^3 - w^2 + 12*w + 4],
[461, 461, -w^3 + 8*w - 5],
[479, 479, w^3 + w^2 - 7*w - 2],
[479, 479, -4/3*w^3 - w^2 + 23/3*w + 5/3],
[499, 499, -4/3*w^3 - 2*w^2 + 23/3*w + 17/3],
[499, 499, -4/3*w^3 - w^2 + 29/3*w + 8/3],
[521, 521, 2*w^3 - 11*w + 2],
[521, 521, 4/3*w^3 - w^2 - 14/3*w + 1/3],
[569, 569, w^3 + 3*w^2 - 6*w - 9],
[569, 569, 4/3*w^3 + w^2 - 26/3*w - 32/3],
[569, 569, -w^3 + 6*w - 6],
[569, 569, -1/3*w^3 - 3*w^2 + 8/3*w + 50/3],
[571, 571, 2/3*w^3 + 3*w^2 - 10/3*w - 46/3],
[571, 571, 2*w^2 - 15],
[599, 599, -w^3 + 2*w^2 + 4*w - 10],
[599, 599, -w^2 - 3*w + 8],
[601, 601, 5/3*w^3 + 3*w^2 - 28/3*w - 34/3],
[601, 601, 1/3*w^3 - 3*w^2 - 2/3*w + 43/3],
[619, 619, 2/3*w^3 - w^2 - 13/3*w + 2/3],
[619, 619, 2/3*w^3 + w^2 - 7/3*w - 25/3],
[619, 619, 2*w^3 + w^2 - 11*w - 6],
[619, 619, -2/3*w^3 + 13/3*w - 17/3],
[631, 631, 2/3*w^3 + 2*w^2 - 19/3*w - 31/3],
[631, 631, w^3 + 2*w^2 - 8*w - 6],
[631, 631, 2/3*w^3 + 2*w^2 - 4/3*w - 31/3],
[631, 631, -2/3*w^3 + 2*w^2 + 16/3*w - 23/3],
[659, 659, -2/3*w^3 + 1/3*w + 16/3],
[659, 659, 5/3*w^3 - 34/3*w - 13/3],
[661, 661, w^3 - 8*w + 2],
[661, 661, 4/3*w^3 + w^2 - 23/3*w + 1/3],
[661, 661, 5/3*w^3 + w^2 - 31/3*w - 10/3],
[661, 661, -3*w - 1],
[691, 691, -4/3*w^3 + 29/3*w + 2/3],
[691, 691, 1/3*w^3 + 4/3*w - 5/3],
[691, 691, -5/3*w^3 + w^2 + 25/3*w - 5/3],
[691, 691, 2*w^3 + w^2 - 10*w - 7],
[701, 701, -2/3*w^3 - 3*w^2 + 19/3*w + 46/3],
[701, 701, w^3 - 2*w^2 - 6*w + 8],
[701, 701, 4/3*w^3 + 2*w^2 - 17/3*w - 29/3],
[701, 701, 5/3*w^3 + 3*w^2 - 28/3*w - 52/3],
[709, 709, 1/3*w^3 - 14/3*w + 10/3],
[709, 709, -2/3*w^3 + 19/3*w + 7/3],
[719, 719, 3*w^2 - w - 10],
[719, 719, -4/3*w^3 - 3*w^2 + 23/3*w + 47/3],
[739, 739, 5/3*w^3 + w^2 - 34/3*w - 10/3],
[739, 739, 4/3*w^3 + 2*w^2 - 23/3*w - 14/3],
[761, 761, 7/3*w^3 + 2*w^2 - 41/3*w - 26/3],
[761, 761, 2*w^3 + 3*w^2 - 12*w - 15],
[769, 769, -2/3*w^3 + 19/3*w - 8/3],
[769, 769, -4/3*w^3 + 2*w^2 + 17/3*w - 31/3],
[769, 769, 1/3*w^3 - 14/3*w - 5/3],
[769, 769, -1/3*w^3 + 3*w^2 + 2/3*w - 34/3],
[809, 809, -4/3*w^3 + w^2 + 26/3*w - 7/3],
[809, 809, w^3 + w^2 - 3*w - 7],
[811, 811, 4/3*w^3 + w^2 - 17/3*w - 23/3],
[811, 811, -1/3*w^3 - 2*w^2 + 2/3*w + 38/3],
[821, 821, 8/3*w^3 + 3*w^2 - 43/3*w - 34/3],
[821, 821, 2/3*w^3 + 2*w^2 - 7/3*w - 34/3],
[821, 821, -w^3 - w^2 + 4*w + 8],
[821, 821, 2/3*w^3 + 4*w^2 - 7/3*w - 58/3],
[829, 829, 4/3*w^3 + 3*w^2 - 26/3*w - 50/3],
[829, 829, 7/3*w^3 - 41/3*w - 5/3],
[839, 839, 2*w^3 + w^2 - 10*w - 3],
[839, 839, 5/3*w^3 - w^2 - 25/3*w + 17/3],
[841, 29, 5/3*w^3 - 25/3*w - 7/3],
[859, 859, -7/3*w^3 - w^2 + 32/3*w + 29/3],
[859, 859, 2*w^3 + w^2 - 14*w - 6],
[859, 859, 1/3*w^3 + 2*w^2 - 2/3*w - 47/3],
[859, 859, -5/3*w^3 - 2*w^2 + 31/3*w + 49/3],
[881, 881, w^3 + 4*w^2 - 4*w - 20],
[881, 881, 8/3*w^3 + 2*w^2 - 43/3*w - 34/3],
[919, 919, -w^3 + 3*w - 7],
[919, 919, 1/3*w^3 + 3*w^2 - 5/3*w - 35/3],
[919, 919, -5/3*w^3 + 31/3*w - 23/3],
[919, 919, 2/3*w^3 + 3*w^2 - 10/3*w - 43/3],
[929, 929, w^2 - 2*w - 9],
[929, 929, w^3 + w^2 - 7*w + 1],
[961, 31, 2/3*w^3 - 10/3*w - 19/3],
[971, 971, -w^3 - 3*w^2 + 7*w + 16],
[971, 971, -5/3*w^3 + w^2 + 25/3*w - 8/3],
[971, 971, 2*w^3 + w^2 - 10*w - 6],
[971, 971, 1/3*w^3 + 2*w^2 - 8/3*w - 47/3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 8*x^5 - 11*x^4 - 200*x^3 - 204*x^2 + 1175*x + 2060;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -20/1541*e^5 - 178/1541*e^4 + 16/67*e^3 + 4023/1541*e^2 - 2624/1541*e - 20314/1541, -341/1541*e^5 - 1648/1541*e^4 + 380/67*e^3 + 39082/1541*e^2 - 52136/1541*e - 217372/1541, 136/1541*e^5 + 594/1541*e^4 - 149/67*e^3 - 12871/1541*e^2 + 19076/1541*e + 62318/1541, 30/1541*e^5 + 267/1541*e^4 - 24/67*e^3 - 6805/1541*e^2 + 2395/1541*e + 38176/1541, 176/1541*e^5 + 950/1541*e^4 - 181/67*e^3 - 22458/1541*e^2 + 22783/1541*e + 121438/1541, -146/1541*e^5 - 683/1541*e^4 + 157/67*e^3 + 15653/1541*e^2 - 20388/1541*e - 87885/1541, -205/1541*e^5 - 1054/1541*e^4 + 231/67*e^3 + 26211/1541*e^2 - 31519/1541*e - 155054/1541, 10/1541*e^5 + 89/1541*e^4 - 8/67*e^3 - 2782/1541*e^2 - 1770/1541*e + 17862/1541, 350/1541*e^5 + 1574/1541*e^4 - 414/67*e^3 - 37271/1541*e^2 + 61330/1541*e + 206018/1541, 155/1541*e^5 + 609/1541*e^4 - 191/67*e^3 - 13842/1541*e^2 + 28041/1541*e + 71908/1541, 370/1541*e^5 + 1752/1541*e^4 - 430/67*e^3 - 42835/1541*e^2 + 63954/1541*e + 244824/1541, 1, -291/1541*e^5 - 1203/1541*e^4 + 340/67*e^3 + 26713/1541*e^2 - 50199/1541*e - 140390/1541, -409/1541*e^5 - 1945/1541*e^4 + 421/67*e^3 + 43206/1541*e^2 - 50887/1541*e - 225416/1541, 281/1541*e^5 + 1114/1541*e^4 - 332/67*e^3 - 25472/1541*e^2 + 47346/1541*e + 131774/1541, 525/1541*e^5 + 2361/1541*e^4 - 621/67*e^3 - 56677/1541*e^2 + 90454/1541*e + 313650/1541, -28/1541*e^5 + 59/1541*e^4 + 76/67*e^3 + 701/1541*e^2 - 19700/1541*e - 16728/1541, -642/1541*e^5 - 2940/1541*e^4 + 728/67*e^3 + 70118/1541*e^2 - 102106/1541*e - 400280/1541, -8/67*e^5 - 31/67*e^4 + 241/67*e^3 + 765/67*e^2 - 1599/67*e - 4320/67, 459/1541*e^5 + 2390/1541*e^4 - 528/67*e^3 - 61739/1541*e^2 + 77480/1541*e + 364038/1541, -264/1541*e^5 - 1425/1541*e^4 + 305/67*e^3 + 36769/1541*e^2 - 45732/1541*e - 233010/1541, 573/1541*e^5 + 2480/1541*e^4 - 646/67*e^3 - 55237/1541*e^2 + 88122/1541*e + 276724/1541, -9/67*e^5 - 60/67*e^4 + 246/67*e^3 + 1673/67*e^2 - 1757/67*e - 10622/67, -193/1541*e^5 - 639/1541*e^4 + 208/67*e^3 + 9620/1541*e^2 - 27479/1541*e - 17120/1541, -456/1541*e^5 - 1901/1541*e^4 + 539/67*e^3 + 44878/1541*e^2 - 76470/1541*e - 248652/1541, 231/1541*e^5 + 669/1541*e^4 - 292/67*e^3 - 14644/1541*e^2 + 40786/1541*e + 67120/1541, -1257/1541*e^5 - 6102/1541*e^4 + 1421/67*e^3 + 147210/1541*e^2 - 201286/1541*e - 831540/1541, -983/1541*e^5 - 4588/1541*e^4 + 1108/67*e^3 + 109200/1541*e^2 - 154242/1541*e - 608406/1541, -246/1541*e^5 - 1573/1541*e^4 + 237/67*e^3 + 37309/1541*e^2 - 25803/1541*e - 190996/1541, 216/1541*e^5 + 1306/1541*e^4 - 213/67*e^3 - 32045/1541*e^2 + 23408/1541*e + 168230/1541, 1499/1541*e^5 + 7023/1541*e^4 - 1695/67*e^3 - 166147/1541*e^2 + 238584/1541*e + 921082/1541, -1062/1541*e^5 - 5137/1541*e^4 + 1198/67*e^3 + 122240/1541*e^2 - 164915/1541*e - 669692/1541, 323/1541*e^5 + 1796/1541*e^4 - 312/67*e^3 - 39622/1541*e^2 + 39912/1541*e + 193850/1541, -223/1541*e^5 - 906/1541*e^4 + 232/67*e^3 + 16425/1541*e^2 - 28333/1541*e - 61460/1541, 654/1541*e^5 + 3355/1541*e^4 - 751/67*e^3 - 86709/1541*e^2 + 106146/1541*e + 513558/1541, 313/1541*e^5 + 1707/1541*e^4 - 371/67*e^3 - 46086/1541*e^2 + 58633/1541*e + 290022/1541, -565/1541*e^5 - 2717/1541*e^4 + 586/67*e^3 + 60100/1541*e^2 - 74128/1541*e - 295720/1541, 126/1541*e^5 + 505/1541*e^4 - 141/67*e^3 - 10089/1541*e^2 + 22387/1541*e + 38292/1541, 177/1541*e^5 + 1113/1541*e^4 - 222/67*e^3 - 30133/1541*e^2 + 39557/1541*e + 193802/1541, 586/1541*e^5 + 3058/1541*e^4 - 643/67*e^3 - 76421/1541*e^2 + 84280/1541*e + 422300/1541, -523/1541*e^5 - 2035/1541*e^4 + 606/67*e^3 + 44409/1541*e^2 - 89267/1541*e - 239808/1541, 333/1541*e^5 + 1885/1541*e^4 - 320/67*e^3 - 39322/1541*e^2 + 39683/1541*e + 162400/1541, -89/1541*e^5 - 638/1541*e^4 + 98/67*e^3 + 18904/1541*e^2 - 10444/1541*e - 119214/1541, 1142/1541*e^5 + 5849/1541*e^4 - 1262/67*e^3 - 142955/1541*e^2 + 173870/1541*e + 803342/1541, 479/1541*e^5 + 2568/1541*e^4 - 477/67*e^3 - 59598/1541*e^2 + 58530/1541*e + 313466/1541, 984/1541*e^5 + 4751/1541*e^4 - 1149/67*e^3 - 116875/1541*e^2 + 172557/1541*e + 671524/1541, -119/1541*e^5 - 905/1541*e^4 + 122/67*e^3 + 25709/1541*e^2 - 20544/1541*e - 163554/1541, -1267/1541*e^5 - 6191/1541*e^4 + 1429/67*e^3 + 149992/1541*e^2 - 205680/1541*e - 858648/1541, 680/1541*e^5 + 2970/1541*e^4 - 812/67*e^3 - 73601/1541*e^2 + 118495/1541*e + 441034/1541, -1575/1541*e^5 - 7083/1541*e^4 + 1863/67*e^3 + 174654/1541*e^2 - 271362/1541*e - 1011836/1541, 625/1541*e^5 + 3251/1541*e^4 - 701/67*e^3 - 81415/1541*e^2 + 105115/1541*e + 476860/1541, 118/1541*e^5 + 742/1541*e^4 - 81/67*e^3 - 18034/1541*e^2 + 3770/1541*e + 122010/1541, 9/1541*e^5 - 74/1541*e^4 - 34/67*e^3 + 3352/1541*e^2 + 6112/1541*e - 60666/1541, -592/1541*e^5 - 2495/1541*e^4 + 755/67*e^3 + 63913/1541*e^2 - 117120/1541*e - 388020/1541, 1006/1541*e^5 + 5255/1541*e^4 - 1113/67*e^3 - 128543/1541*e^2 + 153253/1541*e + 728696/1541, -1412/1541*e^5 - 6711/1541*e^4 + 1679/67*e^3 + 170298/1541*e^2 - 250901/1541*e - 1014400/1541, -191/1541*e^5 - 313/1541*e^4 + 260/67*e^3 + 5057/1541*e^2 - 41702/1541*e - 23410/1541, -59/1541*e^5 - 371/1541*e^4 + 74/67*e^3 + 10558/1541*e^2 - 11131/1541*e - 50218/1541, -330/1541*e^5 - 1396/1541*e^4 + 465/67*e^3 + 37871/1541*e^2 - 86444/1541*e - 238098/1541, -957/1541*e^5 - 4973/1541*e^4 + 1114/67*e^3 + 126931/1541*e^2 - 166549/1541*e - 742570/1541, -1489/1541*e^5 - 6934/1541*e^4 + 1687/67*e^3 + 163365/1541*e^2 - 238813/1541*e - 893974/1541, -1113/1541*e^5 - 5745/1541*e^4 + 1212/67*e^3 + 137661/1541*e^2 - 169757/1541*e - 772808/1541, 997/1541*e^5 + 3788/1541*e^4 - 1213/67*e^3 - 85665/1541*e^2 + 190289/1541*e + 468834/1541, -1811/1541*e^5 - 8567/1541*e^4 + 2092/67*e^3 + 206099/1541*e^2 - 309722/1541*e - 1178806/1541, 40/1541*e^5 + 356/1541*e^4 - 32/67*e^3 - 4964/1541*e^2 + 6789/1541*e - 11766/1541, 143/1541*e^5 + 194/1541*e^4 - 235/67*e^3 - 3415/1541*e^2 + 47116/1541*e + 1778/1541, -879/1541*e^5 - 4587/1541*e^4 + 998/67*e^3 + 112320/1541*e^2 - 149535/1541*e - 655024/1541, -2308/1541*e^5 - 10987/1541*e^4 + 2637/67*e^3 + 266698/1541*e^2 - 374312/1541*e - 1525040/1541, -1045/1541*e^5 - 5448/1541*e^4 + 1171/67*e^3 + 133537/1541*e^2 - 164842/1541*e - 767846/1541, 535/1541*e^5 + 2450/1541*e^4 - 696/67*e^3 - 64082/1541*e^2 + 113340/1541*e + 368496/1541, 1078/1541*e^5 + 6204/1541*e^4 - 1117/67*e^3 - 149498/1541*e^2 + 149755/1541*e + 825866/1541, 1456/1541*e^5 + 7719/1541*e^4 - 1607/67*e^3 - 189011/1541*e^2 + 227703/1541*e + 1085596/1541, -370/1541*e^5 - 1752/1541*e^4 + 430/67*e^3 + 38212/1541*e^2 - 67036/1541*e - 173938/1541, 962/1541*e^5 + 4247/1541*e^4 - 1118/67*e^3 - 99043/1541*e^2 + 162582/1541*e + 534220/1541, 75/67*e^5 + 366/67*e^4 - 1916/67*e^3 - 8604/67*e^2 + 11649/67*e + 47066/67, 1073/1541*e^5 + 5389/1541*e^4 - 1247/67*e^3 - 134238/1541*e^2 + 181460/1541*e + 784574/1541, -58/67*e^5 - 275/67*e^4 + 1563/67*e^3 + 6635/67*e^2 - 9767/67*e - 36010/67, -8/67*e^5 - 31/67*e^4 + 241/67*e^3 + 832/67*e^2 - 1331/67*e - 5258/67, -912/1541*e^5 - 3802/1541*e^4 + 1078/67*e^3 + 91297/1541*e^2 - 154481/1541*e - 528124/1541, 1228/1541*e^5 + 5998/1541*e^4 - 1371/67*e^3 - 143457/1541*e^2 + 189468/1541*e + 804088/1541, 88/1541*e^5 + 475/1541*e^4 - 124/67*e^3 - 11229/1541*e^2 + 30654/1541*e + 77670/1541, 2012/1541*e^5 + 8969/1541*e^4 - 2427/67*e^3 - 223184/1541*e^2 + 366605/1541*e + 1309456/1541, -887/1541*e^5 - 4350/1541*e^4 + 1058/67*e^3 + 110539/1541*e^2 - 160447/1541*e - 679176/1541, -784/1541*e^5 - 4512/1541*e^4 + 788/67*e^3 + 107465/1541*e^2 - 97005/1541*e - 594746/1541, -1121/1541*e^5 - 5508/1541*e^4 + 1272/67*e^3 + 132798/1541*e^2 - 180669/1541*e - 732238/1541, 560/1541*e^5 + 3443/1541*e^4 - 582/67*e^3 - 87988/1541*e^2 + 71931/1541*e + 491742/1541, -24/67*e^5 - 160/67*e^4 + 522/67*e^3 + 3903/67*e^2 - 2452/67*e - 21402/67, -1468/1541*e^5 - 8134/1541*e^4 + 1563/67*e^3 + 196356/1541*e^2 - 214792/1541*e - 1100250/1541, -1968/1541*e^5 - 9502/1541*e^4 + 2298/67*e^3 + 235291/1541*e^2 - 340491/1541*e - 1373868/1541, -1199/1541*e^5 - 5894/1541*e^4 + 1388/67*e^3 + 148950/1541*e^2 - 196142/1541*e - 878342/1541, 533/1541*e^5 + 3665/1541*e^4 - 480/67*e^3 - 88798/1541*e^2 + 50513/1541*e + 476492/1541, 488/1541*e^5 + 2494/1541*e^4 - 511/67*e^3 - 57787/1541*e^2 + 72347/1541*e + 308276/1541, 341/1541*e^5 + 1648/1541*e^4 - 380/67*e^3 - 42164/1541*e^2 + 45972/1541*e + 226618/1541, -809/1541*e^5 - 3964/1541*e^4 + 1009/67*e^3 + 105174/1541*e^2 - 160384/1541*e - 640942/1541, 1026/1541*e^5 + 5433/1541*e^4 - 1062/67*e^3 - 126402/1541*e^2 + 135844/1541*e + 684288/1541, -1643/1541*e^5 - 7380/1541*e^4 + 1971/67*e^3 + 180319/1541*e^2 - 296310/1541*e - 1022962/1541, 68/1541*e^5 + 297/1541*e^4 - 108/67*e^3 - 13370/1541*e^2 + 20325/1541*e + 109750/1541, 408/1541*e^5 + 1782/1541*e^4 - 447/67*e^3 - 41695/1541*e^2 + 57228/1541*e + 230102/1541, -847/1541*e^5 - 3994/1541*e^4 + 959/67*e^3 + 97870/1541*e^2 - 127461/1541*e - 552252/1541, -418/1541*e^5 - 1871/1541*e^4 + 455/67*e^3 + 39854/1541*e^2 - 52376/1541*e - 164750/1541, -2007/1541*e^5 - 9695/1541*e^4 + 2289/67*e^3 + 234121/1541*e^2 - 319719/1541*e - 1323640/1541, -1191/1541*e^5 - 6131/1541*e^4 + 1261/67*e^3 + 144567/1541*e^2 - 163656/1541*e - 801796/1541, 49/67*e^5 + 215/67*e^4 - 1317/67*e^3 - 5029/67*e^2 + 8211/67*e + 27532/67, 2270/1541*e^5 + 10957/1541*e^4 - 2486/67*e^3 - 253969/1541*e^2 + 337890/1541*e + 1364088/1541, 2257/1541*e^5 + 10379/1541*e^4 - 2623/67*e^3 - 252818/1541*e^2 + 371011/1541*e + 1437334/1541, 999/1541*e^5 + 4114/1541*e^4 - 1228/67*e^3 - 101015/1541*e^2 + 191476/1541*e + 613562/1541, -294/1541*e^5 - 1692/1541*e^4 + 329/67*e^3 + 45115/1541*e^2 - 48127/1541*e - 249612/1541, -291/1541*e^5 - 1203/1541*e^4 + 340/67*e^3 + 25172/1541*e^2 - 44035/1541*e - 118816/1541, 1325/1541*e^5 + 6399/1541*e^4 - 1529/67*e^3 - 154416/1541*e^2 + 223152/1541*e + 895060/1541, -1599/1541*e^5 - 7913/1541*e^4 + 1775/67*e^3 + 190885/1541*e^2 - 242458/1541*e - 1087374/1541, 1095/1541*e^5 + 5893/1541*e^4 - 1144/67*e^3 - 142824/1541*e^2 + 146746/1541*e + 792434/1541, 418/1541*e^5 + 1871/1541*e^4 - 455/67*e^3 - 39854/1541*e^2 + 61622/1541*e + 167832/1541, 1210/1541*e^5 + 4605/1541*e^4 - 1571/67*e^3 - 111636/1541*e^2 + 251212/1541*e + 635712/1541, 717/1541*e^5 + 2837/1541*e^4 - 855/67*e^3 - 67868/1541*e^2 + 122733/1541*e + 397096/1541, 486/1541*e^5 + 2168/1541*e^4 - 563/67*e^3 - 51683/1541*e^2 + 86570/1541*e + 299156/1541, -1268/1541*e^5 - 6354/1541*e^4 + 1537/67*e^3 + 163831/1541*e^2 - 233241/1541*e - 974160/1541, -7/67*e^5 - 2/67*e^4 + 169/67*e^3 - 411/67*e^2 - 704/67*e + 5198/67, 1394/1541*e^5 + 6859/1541*e^4 - 1544/67*e^3 - 164674/1541*e^2 + 206316/1541*e + 910746/1541, 841/1541*e^5 + 4557/1541*e^4 - 914/67*e^3 - 115001/1541*e^2 + 119277/1541*e + 654336/1541, 993/1541*e^5 + 4677/1541*e^4 - 1116/67*e^3 - 105818/1541*e^2 + 158636/1541*e + 555382/1541, -1252/1541*e^5 - 5287/1541*e^4 + 1417/67*e^3 + 113458/1541*e^2 - 194466/1541*e - 537524/1541, -154/1541*e^5 - 446/1541*e^4 + 284/67*e^3 + 21577/1541*e^2 - 54415/1541*e - 193710/1541, -123/1541*e^5 - 1557/1541*e^4 + 85/67*e^3 + 42540/1541*e^2 - 13672/1541*e - 283500/1541, 1309/1541*e^5 + 6873/1541*e^4 - 1409/67*e^3 - 164142/1541*e^2 + 189000/1541*e + 933052/1541, 1209/1541*e^5 + 5983/1541*e^4 - 1262/67*e^3 - 136322/1541*e^2 + 162011/1541*e + 720530/1541, 199/1541*e^5 + 1617/1541*e^4 - 119/67*e^3 - 38719/1541*e^2 + 6384/1541*e + 223236/1541, 1597/1541*e^5 + 7587/1541*e^4 - 1760/67*e^3 - 173994/1541*e^2 + 232025/1541*e + 911826/1541, 1418/1541*e^5 + 6148/1541*e^4 - 1657/67*e^3 - 145462/1541*e^2 + 235970/1541*e + 782872/1541, -665/1541*e^5 - 3607/1541*e^4 + 666/67*e^3 + 86379/1541*e^2 - 78002/1541*e - 443520/1541, -438/1541*e^5 - 2049/1541*e^4 + 471/67*e^3 + 43877/1541*e^2 - 62705/1541*e - 197392/1541, 1370/1541*e^5 + 6029/1541*e^4 - 1632/67*e^3 - 142279/1541*e^2 + 238302/1541*e + 767404/1541, -1253/1541*e^5 - 6991/1541*e^4 + 1391/67*e^3 + 176609/1541*e^2 - 191207/1541*e - 992056/1541, -1958/1541*e^5 - 9413/1541*e^4 + 2290/67*e^3 + 238673/1541*e^2 - 333015/1541*e - 1402236/1541, 1180/1541*e^5 + 5879/1541*e^4 - 1413/67*e^3 - 149520/1541*e^2 + 214915/1541*e + 887244/1541, 17/67*e^5 + 91/67*e^4 - 487/67*e^3 - 2438/67*e^2 + 3289/67*e + 14406/67, 746/1541*e^5 + 2941/1541*e^4 - 905/67*e^3 - 63916/1541*e^2 + 137633/1541*e + 335170/1541, 1018/1541*e^5 + 4129/1541*e^4 - 1270/67*e^3 - 100445/1541*e^2 + 192736/1541*e + 595414/1541, -471/1541*e^5 - 2805/1541*e^4 + 484/67*e^3 + 67543/1541*e^2 - 75356/1541*e - 375610/1541, -1509/1541*e^5 - 7112/1541*e^4 + 1636/67*e^3 + 165847/1541*e^2 - 210617/1541*e - 898878/1541, 1318/1541*e^5 + 6799/1541*e^4 - 1443/67*e^3 - 165413/1541*e^2 + 196653/1541*e + 940190/1541, 2289/1541*e^5 + 10972/1541*e^4 - 2662/67*e^3 - 274973/1541*e^2 + 380757/1541*e + 1607910/1541, -22/1541*e^5 - 504/1541*e^4 - 36/67*e^3 + 13209/1541*e^2 + 10058/1541*e - 72582/1541, 57/1541*e^5 + 45/1541*e^4 - 126/67*e^3 - 5995/1541*e^2 + 26895/1541*e + 81164/1541, -1372/1541*e^5 - 6355/1541*e^4 + 1580/67*e^3 + 157629/1541*e^2 - 220997/1541*e - 924460/1541, -1400/1541*e^5 - 6296/1541*e^4 + 1656/67*e^3 + 150625/1541*e^2 - 248402/1541*e - 833318/1541, -2730/1541*e^5 - 13510/1541*e^4 + 3122/67*e^3 + 329547/1541*e^2 - 455259/1541*e - 1892950/1541, 1693/1541*e^5 + 7825/1541*e^4 - 1944/67*e^3 - 186524/1541*e^2 + 272050/1541*e + 1010566/1541, -1525/1541*e^5 - 6638/1541*e^4 + 1823/67*e^3 + 159203/1541*e^2 - 280212/1541*e - 922526/1541, 1259/1541*e^5 + 6428/1541*e^4 - 1436/67*e^3 - 157937/1541*e^2 + 200932/1541*e + 880726/1541, -1071/1541*e^5 - 3522/1541*e^4 + 1433/67*e^3 + 86527/1541*e^2 - 234208/1541*e - 501156/1541, 123/1541*e^5 + 1557/1541*e^4 - 85/67*e^3 - 47163/1541*e^2 + 1344/1541*e + 289664/1541, -1533/1541*e^5 - 7942/1541*e^4 + 1682/67*e^3 + 191324/1541*e^2 - 235648/1541*e - 1057630/1541, -1216/1541*e^5 - 5583/1541*e^4 + 1415/67*e^3 + 129948/1541*e^2 - 203920/1541*e - 703138/1541, 1515/1541*e^5 + 8090/1541*e^4 - 1614/67*e^3 - 196487/1541*e^2 + 206473/1541*e + 1086502/1541, 2639/1541*e^5 + 12546/1541*e^4 - 3009/67*e^3 - 306080/1541*e^2 + 428218/1541*e + 1780026/1541, -2212/1541*e^5 - 10749/1541*e^4 + 2520/67*e^3 + 261873/1541*e^2 - 348156/1541*e - 1460202/1541, -381/1541*e^5 - 2004/1541*e^4 + 546/67*e^3 + 59456/1541*e^2 - 106696/1541*e - 387444/1541, -1791/1541*e^5 - 8389/1541*e^4 + 2076/67*e^3 + 202076/1541*e^2 - 304016/1541*e - 1155410/1541, 55/1541*e^5 - 281/1541*e^4 - 111/67*e^3 + 7814/1541*e^2 + 16462/1541*e - 78974/1541, 2375/1541*e^5 + 11121/1541*e^4 - 2771/67*e^3 - 269311/1541*e^2 + 414847/1541*e + 1540852/1541, 675/1541*e^5 + 3696/1541*e^4 - 674/67*e^3 - 89161/1541*e^2 + 79314/1541*e + 523022/1541, -1621/1541*e^5 - 6876/1541*e^4 + 1940/67*e^3 + 159405/1541*e^2 - 294040/1541*e - 864084/1541, 44/1541*e^5 - 533/1541*e^4 - 129/67*e^3 + 16730/1541*e^2 + 30737/1541*e - 132216/1541, 2423/1541*e^5 + 11240/1541*e^4 - 2863/67*e^3 - 275576/1541*e^2 + 427925/1541*e + 1584058/1541, 365/1541*e^5 + 2478/1541*e^4 - 292/67*e^3 - 56854/1541*e^2 + 32478/1541*e + 295992/1541, 1411/1541*e^5 + 6548/1541*e^4 - 1571/67*e^3 - 147213/1541*e^2 + 218717/1541*e + 775608/1541, -2378/1541*e^5 - 11610/1541*e^4 + 2626/67*e^3 + 275385/1541*e^2 - 365004/1541*e - 1548368/1541, -266/1541*e^5 - 1751/1541*e^4 + 253/67*e^3 + 39791/1541*e^2 - 23804/1541*e - 171244/1541, -3292/1541*e^5 - 15738/1541*e^4 + 3786/67*e^3 + 385114/1541*e^2 - 548410/1541*e - 2233548/1541, 691/1541*e^5 + 3222/1541*e^4 - 794/67*e^3 - 76353/1541*e^2 + 121171/1541*e + 401816/1541];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3}>] := -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;