/*
  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![29, 16, -15, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9,3,2/23*w^3 - 3/23*w^2 - 43/23*w + 68/23}>;

primesArray := [
[4, 2, -2/23*w^3 + 3/23*w^2 - 3/23*w + 1/23],
[9, 3, -2/23*w^3 + 3/23*w^2 + 43/23*w + 24/23],
[9, 3, -2/23*w^3 + 3/23*w^2 + 43/23*w - 68/23],
[19, 19, -2/23*w^3 + 3/23*w^2 - 3/23*w + 24/23],
[19, 19, -6/23*w^3 + 9/23*w^2 + 83/23*w - 20/23],
[19, 19, 6/23*w^3 - 9/23*w^2 - 83/23*w + 66/23],
[19, 19, -2/23*w^3 + 3/23*w^2 - 3/23*w - 22/23],
[25, 5, -4/23*w^3 + 6/23*w^2 + 40/23*w - 21/23],
[29, 29, w],
[29, 29, -4/23*w^3 + 6/23*w^2 + 63/23*w - 21/23],
[29, 29, 4/23*w^3 - 6/23*w^2 - 63/23*w + 44/23],
[29, 29, -w + 1],
[31, 31, w + 2],
[31, 31, -4/23*w^3 + 6/23*w^2 + 63/23*w + 25/23],
[31, 31, 4/23*w^3 - 6/23*w^2 - 63/23*w + 90/23],
[31, 31, -w + 3],
[49, 7, -2/23*w^3 + 3/23*w^2 + 43/23*w - 22/23],
[59, 59, -20/23*w^3 + 30/23*w^2 + 269/23*w - 128/23],
[59, 59, -8/23*w^3 + 12/23*w^2 + 103/23*w - 88/23],
[59, 59, -8/23*w^3 + 12/23*w^2 + 103/23*w - 19/23],
[59, 59, -4/23*w^3 + 6/23*w^2 + 17/23*w - 44/23],
[109, 109, -9/23*w^3 + 2/23*w^2 + 113/23*w + 62/23],
[109, 109, 4/23*w^3 - 6/23*w^2 - 63/23*w - 71/23],
[109, 109, 7/23*w^3 - 22/23*w^2 - 47/23*w + 100/23],
[109, 109, 9/23*w^3 - 25/23*w^2 - 90/23*w + 168/23],
[121, 11, -2/23*w^3 + 3/23*w^2 + 20/23*w - 91/23],
[121, 11, 2/23*w^3 - 3/23*w^2 - 20/23*w - 70/23],
[131, 131, -5/23*w^3 + 19/23*w^2 + 50/23*w - 101/23],
[131, 131, -3/23*w^3 + 16/23*w^2 + 7/23*w - 125/23],
[131, 131, 3/23*w^3 + 7/23*w^2 - 30/23*w - 105/23],
[131, 131, 5/23*w^3 + 4/23*w^2 - 73/23*w - 37/23],
[139, 139, -2/23*w^3 - 20/23*w^2 + 43/23*w + 323/23],
[139, 139, 17/23*w^3 - 37/23*w^2 - 216/23*w + 302/23],
[139, 139, 17/23*w^3 - 14/23*w^2 - 239/23*w - 66/23],
[139, 139, -2/23*w^3 + 26/23*w^2 - 3/23*w - 344/23],
[149, 149, -1/23*w^3 - 10/23*w^2 + 33/23*w - 34/23],
[149, 149, 1/23*w^3 + 10/23*w^2 - 33/23*w - 196/23],
[149, 149, -1/23*w^3 + 13/23*w^2 + 10/23*w - 218/23],
[149, 149, 1/23*w^3 - 13/23*w^2 - 10/23*w - 12/23],
[169, 13, 6/23*w^3 - 9/23*w^2 - 37/23*w + 20/23],
[169, 13, -10/23*w^3 + 15/23*w^2 + 123/23*w - 64/23],
[199, 199, -1/23*w^3 + 13/23*w^2 - 13/23*w - 195/23],
[199, 199, 3/23*w^3 + 7/23*w^2 - 53/23*w + 33/23],
[199, 199, 3/23*w^3 - 16/23*w^2 - 30/23*w + 10/23],
[199, 199, 1/23*w^3 + 10/23*w^2 - 10/23*w - 196/23],
[251, 251, 6/23*w^3 - 9/23*w^2 - 14/23*w + 43/23],
[251, 251, 14/23*w^3 - 21/23*w^2 - 186/23*w + 62/23],
[251, 251, 4/23*w^3 - 6/23*w^2 - 109/23*w + 228/23],
[251, 251, 6/23*w^3 - 9/23*w^2 - 14/23*w - 26/23],
[271, 271, -8/23*w^3 + 12/23*w^2 + 126/23*w + 27/23],
[271, 271, 8/23*w^3 - 35/23*w^2 - 80/23*w + 226/23],
[271, 271, 13/23*w^3 - 31/23*w^2 - 153/23*w + 189/23],
[271, 271, 8/23*w^3 - 12/23*w^2 - 126/23*w + 157/23],
[281, 281, -w^2 + 2*w + 9],
[281, 281, 13/23*w^3 - 8/23*w^2 - 153/23*w + 28/23],
[281, 281, -13/23*w^3 + 31/23*w^2 + 130/23*w - 120/23],
[281, 281, 11/23*w^3 - 28/23*w^2 - 87/23*w + 190/23],
[289, 17, -11/23*w^3 + 51/23*w^2 + 110/23*w - 581/23],
[289, 17, -8/23*w^3 + 35/23*w^2 + 80/23*w - 410/23],
[311, 311, 5/23*w^3 + 4/23*w^2 - 50/23*w - 106/23],
[311, 311, -7/23*w^3 + 22/23*w^2 + 70/23*w - 100/23],
[311, 311, 7/23*w^3 + 1/23*w^2 - 93/23*w - 15/23],
[311, 311, -5/23*w^3 + 19/23*w^2 + 27/23*w - 147/23],
[389, 389, -8/23*w^3 + 35/23*w^2 + 80/23*w - 295/23],
[389, 389, -4/23*w^3 + 29/23*w^2 - 6/23*w - 136/23],
[389, 389, -4/23*w^3 - 17/23*w^2 + 40/23*w + 117/23],
[389, 389, 2/23*w^3 + 20/23*w^2 - 20/23*w - 185/23],
[401, 401, 11/23*w^3 - 5/23*w^2 - 179/23*w - 17/23],
[401, 401, 2/23*w^3 + 20/23*w^2 - 43/23*w - 93/23],
[401, 401, 1/23*w^3 + 10/23*w^2 + 36/23*w - 104/23],
[401, 401, 11/23*w^3 - 28/23*w^2 - 156/23*w + 190/23],
[419, 419, -6/23*w^3 - 14/23*w^2 + 83/23*w + 72/23],
[419, 419, 3/23*w^3 + 7/23*w^2 - 76/23*w + 79/23],
[419, 419, -6/23*w^3 + 32/23*w^2 + 37/23*w - 296/23],
[419, 419, -1/23*w^3 + 13/23*w^2 + 56/23*w - 172/23],
[421, 421, -8/23*w^3 + 12/23*w^2 + 57/23*w - 65/23],
[421, 421, -12/23*w^3 + 18/23*w^2 + 143/23*w - 109/23],
[421, 421, 12/23*w^3 - 18/23*w^2 - 143/23*w + 40/23],
[421, 421, 8/23*w^3 - 12/23*w^2 - 57/23*w - 4/23],
[439, 439, 7/23*w^3 - 22/23*w^2 - 93/23*w + 123/23],
[439, 439, 1/23*w^3 + 10/23*w^2 + 13/23*w - 127/23],
[439, 439, 5/23*w^3 + 4/23*w^2 - 73/23*w + 9/23],
[439, 439, 7/23*w^3 + 1/23*w^2 - 116/23*w - 15/23],
[449, 449, -11/23*w^3 + 5/23*w^2 + 133/23*w + 63/23],
[449, 449, -9/23*w^3 + 2/23*w^2 + 90/23*w + 16/23],
[449, 449, 9/23*w^3 - 25/23*w^2 - 67/23*w + 99/23],
[449, 449, -11/23*w^3 + 28/23*w^2 + 110/23*w - 190/23],
[479, 479, 10/23*w^3 - 15/23*w^2 - 123/23*w - 74/23],
[479, 479, 6/23*w^3 - 9/23*w^2 - 37/23*w + 158/23],
[479, 479, -6/23*w^3 + 9/23*w^2 + 37/23*w + 118/23],
[479, 479, 10/23*w^3 - 15/23*w^2 - 123/23*w + 202/23],
[529, 23, -4/23*w^3 + 29/23*w^2 + 40/23*w - 274/23],
[529, 23, w^2 - 6],
[541, 541, 6/23*w^3 + 14/23*w^2 - 83/23*w - 141/23],
[541, 541, -6/23*w^3 + 32/23*w^2 + 37/23*w - 227/23],
[541, 541, 6/23*w^3 + 14/23*w^2 - 83/23*w - 164/23],
[541, 541, -6/23*w^3 + 32/23*w^2 + 37/23*w - 204/23],
[569, 569, -5/23*w^3 - 4/23*w^2 + 50/23*w - 78/23],
[569, 569, -7/23*w^3 + 22/23*w^2 + 70/23*w - 284/23],
[569, 569, -5/23*w^3 + 19/23*w^2 + 50/23*w - 262/23],
[569, 569, 5/23*w^3 - 19/23*w^2 - 27/23*w - 37/23],
[619, 619, -2/23*w^3 - 43/23*w^2 + 66/23*w + 622/23],
[619, 619, 17/23*w^3 - 37/23*w^2 - 216/23*w + 233/23],
[619, 619, 17/23*w^3 - 14/23*w^2 - 239/23*w + 3/23],
[619, 619, -2/23*w^3 + 49/23*w^2 - 26/23*w - 643/23],
[641, 641, -1/23*w^3 + 13/23*w^2 - 59/23*w + 127/23],
[641, 641, -14/23*w^3 + 44/23*w^2 + 117/23*w - 246/23],
[641, 641, -11/23*w^3 + 5/23*w^2 + 179/23*w + 201/23],
[641, 641, -1/23*w^3 - 10/23*w^2 - 36/23*w - 80/23],
[691, 691, 5/23*w^3 - 65/23*w^2 - 4/23*w + 837/23],
[691, 691, -2/23*w^3 + 26/23*w^2 + 20/23*w - 321/23],
[691, 691, -2/23*w^3 - 20/23*w^2 + 66/23*w + 277/23],
[691, 691, -5/23*w^3 - 50/23*w^2 + 119/23*w + 773/23],
[701, 701, 2/23*w^3 + 20/23*w^2 - 66/23*w - 185/23],
[701, 701, w^2 - 7],
[701, 701, -2/23*w^3 + 26/23*w^2 + 20/23*w - 183/23],
[701, 701, -2/23*w^3 + 26/23*w^2 + 20/23*w - 229/23],
[709, 709, -18/23*w^3 + 27/23*w^2 + 226/23*w - 152/23],
[709, 709, -33/23*w^3 + 38/23*w^2 + 445/23*w - 41/23],
[709, 709, -33/23*w^3 + 61/23*w^2 + 422/23*w - 409/23],
[709, 709, 18/23*w^3 - 27/23*w^2 - 226/23*w + 83/23],
[719, 719, -4/23*w^3 + 29/23*w^2 + 40/23*w - 136/23],
[719, 719, 12/23*w^3 - 18/23*w^2 - 166/23*w + 155/23],
[719, 719, w^2 - 12],
[719, 719, 4/23*w^3 + 17/23*w^2 - 86/23*w - 71/23],
[809, 809, -10/23*w^3 + 15/23*w^2 + 123/23*w - 248/23],
[809, 809, 7/23*w^3 - 22/23*w^2 - 93/23*w + 307/23],
[809, 809, 7/23*w^3 + 1/23*w^2 - 116/23*w - 199/23],
[809, 809, 10/23*w^3 - 15/23*w^2 - 123/23*w - 120/23],
[811, 811, -7/23*w^3 - 1/23*w^2 + 93/23*w - 31/23],
[811, 811, -7/23*w^3 + 22/23*w^2 + 93/23*w - 100/23],
[811, 811, 7/23*w^3 + 1/23*w^2 - 116/23*w + 8/23],
[811, 811, 1/23*w^3 + 10/23*w^2 + 13/23*w - 150/23],
[821, 821, -4/23*w^3 + 29/23*w^2 + 17/23*w - 182/23],
[821, 821, 4/23*w^3 + 17/23*w^2 - 63/23*w - 186/23],
[821, 821, -4/23*w^3 + 29/23*w^2 + 17/23*w - 228/23],
[821, 821, 4/23*w^3 + 17/23*w^2 - 63/23*w - 140/23],
[839, 839, -16/23*w^3 + 70/23*w^2 + 160/23*w - 751/23],
[839, 839, 22/23*w^3 - 33/23*w^2 - 289/23*w + 196/23],
[839, 839, -22/23*w^3 + 33/23*w^2 + 289/23*w - 104/23],
[839, 839, -16/23*w^3 - 22/23*w^2 + 252/23*w + 537/23],
[859, 859, -7/23*w^3 + 22/23*w^2 + 47/23*w - 192/23],
[859, 859, -9/23*w^3 + 2/23*w^2 + 113/23*w - 30/23],
[859, 859, -9/23*w^3 + 25/23*w^2 + 90/23*w - 76/23],
[859, 859, 7/23*w^3 + 1/23*w^2 - 70/23*w - 130/23],
[971, 971, -10/23*w^3 + 15/23*w^2 + 146/23*w + 97/23],
[971, 971, -14/23*w^3 - 2/23*w^2 + 186/23*w + 99/23],
[971, 971, 14/23*w^3 - 44/23*w^2 - 140/23*w + 269/23],
[971, 971, -10/23*w^3 + 15/23*w^2 + 146/23*w - 248/23]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 149*x^6 + 7288*x^4 - 119797*x^2 + 105644;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [2597/1674161*e^6 - 275874/1674161*e^4 + 7110480/1674161*e^2 - 2404723/1674161, 4840/1674161*e^6 - 512854/1674161*e^4 + 13285889/1674161*e^2 - 9962486/1674161, -1, -39706/82033889*e^7 + 4371959/82033889*e^5 - 123883356/82033889*e^3 + 403045207/82033889*e, e, -39706/82033889*e^7 + 4371959/82033889*e^5 - 123883356/82033889*e^3 + 403045207/82033889*e, e, -116/1674161*e^6 + 31662/1674161*e^4 - 1479266/1674161*e^2 + 4951326/1674161, 1120/1674161*e^6 - 132513/1674161*e^4 + 3544846/1674161*e^2 + 7960974/1674161, 1120/1674161*e^6 - 132513/1674161*e^4 + 3544846/1674161*e^2 + 7960974/1674161, 2481/1674161*e^6 - 244212/1674161*e^4 + 5631214/1674161*e^2 + 872442/1674161, 2481/1674161*e^6 - 244212/1674161*e^4 + 5631214/1674161*e^2 + 872442/1674161, -47841/82033889*e^7 + 4773908/82033889*e^5 - 100646808/82033889*e^3 - 606225098/82033889*e, -47841/82033889*e^7 + 4773908/82033889*e^5 - 100646808/82033889*e^3 - 606225098/82033889*e, 5557/82033889*e^7 - 506504/82033889*e^5 + 10739021/82033889*e^3 - 38286480/82033889*e, 5557/82033889*e^7 - 506504/82033889*e^5 + 10739021/82033889*e^3 - 38286480/82033889*e, 1535/1674161*e^6 - 159192/1674161*e^4 + 4305267/1674161*e^2 - 2796394/1674161, 73855/82033889*e^7 - 8237414/82033889*e^5 + 237027691/82033889*e^3 - 849837823/82033889*e, 73855/82033889*e^7 - 8237414/82033889*e^5 + 237027691/82033889*e^3 - 849837823/82033889*e, 36184/82033889*e^7 - 3814743/82033889*e^5 + 100676143/82033889*e^3 - 186958589/82033889*e, 36184/82033889*e^7 - 3814743/82033889*e^5 + 100676143/82033889*e^3 - 186958589/82033889*e, 10977/1674161*e^6 - 1177668/1674161*e^4 + 31421240/1674161*e^2 - 49567342/1674161, 10977/1674161*e^6 - 1177668/1674161*e^4 + 31421240/1674161*e^2 - 49567342/1674161, -5484/1674161*e^6 + 630903/1674161*e^4 - 17919125/1674161*e^2 + 25558566/1674161, -5484/1674161*e^6 + 630903/1674161*e^4 - 17919125/1674161*e^2 + 25558566/1674161, -946/1674161*e^6 + 85020/1674161*e^4 - 1325947/1674161*e^2 - 27107090/1674161, -946/1674161*e^6 + 85020/1674161*e^4 - 1325947/1674161*e^2 - 27107090/1674161, -58544/82033889*e^7 + 6280896/82033889*e^5 - 178741020/82033889*e^3 + 924536645/82033889*e, -58544/82033889*e^7 + 6280896/82033889*e^5 - 178741020/82033889*e^3 + 924536645/82033889*e, -5146/82033889*e^7 + 1000484/82033889*e^5 - 67355191/82033889*e^3 + 1492475263/82033889*e, -5146/82033889*e^7 + 1000484/82033889*e^5 - 67355191/82033889*e^3 + 1492475263/82033889*e, -5731/11719127*e^7 + 553997/11719127*e^5 - 9609598/11719127*e^3 - 161882495/11719127*e, -5731/11719127*e^7 + 553997/11719127*e^5 - 9609598/11719127*e^3 - 161882495/11719127*e, -12738/82033889*e^7 + 1456276/82033889*e^5 - 65626020/82033889*e^3 + 1352961605/82033889*e, -12738/82033889*e^7 + 1456276/82033889*e^5 - 65626020/82033889*e^3 + 1352961605/82033889*e, 5316/1674161*e^6 - 527318/1674161*e^4 + 12197499/1674161*e^2 - 3147042/1674161, -14694/1674161*e^6 + 1586055/1674161*e^4 - 42076566/1674161*e^2 + 62426862/1674161, 5316/1674161*e^6 - 527318/1674161*e^4 + 12197499/1674161*e^2 - 3147042/1674161, -14694/1674161*e^6 + 1586055/1674161*e^4 - 42076566/1674161*e^2 + 62426862/1674161, -2539/1674161*e^6 + 260043/1674161*e^4 - 6370847/1674161*e^2 + 9974026/1674161, -2539/1674161*e^6 + 260043/1674161*e^4 - 6370847/1674161*e^2 + 9974026/1674161, 44319/82033889*e^7 - 4216692/82033889*e^5 + 77439595/82033889*e^3 + 822311716/82033889*e, -186741/82033889*e^7 + 20460493/82033889*e^5 - 584337322/82033889*e^3 + 2223977586/82033889*e, 44319/82033889*e^7 - 4216692/82033889*e^5 + 77439595/82033889*e^3 + 822311716/82033889*e, -186741/82033889*e^7 + 20460493/82033889*e^5 - 584337322/82033889*e^3 + 2223977586/82033889*e, 32114/82033889*e^7 - 3916167/82033889*e^5 + 125612527/82033889*e^3 - 706626643/82033889*e, 32114/82033889*e^7 - 3916167/82033889*e^5 + 125612527/82033889*e^3 - 706626643/82033889*e, -189730/82033889*e^7 + 19861958/82033889*e^5 - 493745583/82033889*e^3 - 195734093/82033889*e, -189730/82033889*e^7 + 19861958/82033889*e^5 - 493745583/82033889*e^3 - 195734093/82033889*e, 130364/82033889*e^7 - 14569022/82033889*e^5 + 428236903/82033889*e^3 - 1952174606/82033889*e, 80366/82033889*e^7 - 8196095/82033889*e^5 + 169643165/82033889*e^3 + 1517855016/82033889*e, 130364/82033889*e^7 - 14569022/82033889*e^5 + 428236903/82033889*e^3 - 1952174606/82033889*e, 80366/82033889*e^7 - 8196095/82033889*e^5 + 169643165/82033889*e^3 + 1517855016/82033889*e, 7260/1674161*e^6 - 769281/1674161*e^4 + 19091753/1674161*e^2 + 10168686/1674161, 7260/1674161*e^6 - 769281/1674161*e^4 + 19091753/1674161*e^2 + 10168686/1674161, 2420/1674161*e^6 - 256427/1674161*e^4 + 5805864/1674161*e^2 + 3389562/1674161, 2420/1674161*e^6 - 256427/1674161*e^4 + 5805864/1674161*e^2 + 3389562/1674161, -3189/1674161*e^6 + 322000/1674161*e^4 - 7501356/1674161*e^2 + 22304698/1674161, -3189/1674161*e^6 + 322000/1674161*e^4 - 7501356/1674161*e^2 + 22304698/1674161, -5731/11719127*e^7 + 553997/11719127*e^5 - 9609598/11719127*e^3 - 150163368/11719127*e, -131186/82033889*e^7 + 13581062/82033889*e^5 - 315004563/82033889*e^3 - 956202960/82033889*e, -5731/11719127*e^7 + 553997/11719127*e^5 - 9609598/11719127*e^3 - 150163368/11719127*e, -131186/82033889*e^7 + 13581062/82033889*e^5 - 315004563/82033889*e^3 - 956202960/82033889*e, -7263/1674161*e^6 + 741235/1674161*e^4 - 18177470/1674161*e^2 + 28378474/1674161, -7263/1674161*e^6 + 741235/1674161*e^4 - 18177470/1674161*e^2 + 28378474/1674161, 17053/1674161*e^6 - 1854697/1674161*e^4 + 48057080/1674161*e^2 + 401294/1674161, 17053/1674161*e^6 - 1854697/1674161*e^4 + 48057080/1674161*e^2 + 401294/1674161, 25732/1674161*e^6 - 2751508/1674161*e^4 + 71648995/1674161*e^2 - 64286494/1674161, 4431/1674161*e^6 - 430083/1674161*e^4 + 9022741/1674161*e^2 + 4060290/1674161, 25732/1674161*e^6 - 2751508/1674161*e^4 + 71648995/1674161*e^2 - 64286494/1674161, 4431/1674161*e^6 - 430083/1674161*e^4 + 9022741/1674161*e^2 + 4060290/1674161, 100559/82033889*e^7 - 10272823/82033889*e^5 + 225067441/82033889*e^3 + 1104875069/82033889*e, 100559/82033889*e^7 - 10272823/82033889*e^5 + 225067441/82033889*e^3 + 1104875069/82033889*e, -210319/82033889*e^7 + 23259097/82033889*e^5 - 654496238/82033889*e^3 + 2052576151/82033889*e, -210319/82033889*e^7 + 23259097/82033889*e^5 - 654496238/82033889*e^3 + 2052576151/82033889*e, -14224/1674161*e^6 + 1515499/1674161*e^4 - 37988068/1674161*e^2 - 7686186/1674161, -26913/1674161*e^6 + 2871806/1674161*e^4 - 76693352/1674161*e^2 + 135854046/1674161, -14224/1674161*e^6 + 1515499/1674161*e^4 - 37988068/1674161*e^2 - 7686186/1674161, -26913/1674161*e^6 + 2871806/1674161*e^4 - 76693352/1674161*e^2 + 135854046/1674161, 30094/82033889*e^7 - 2152488/82033889*e^5 - 23861830/82033889*e^3 + 2815261744/82033889*e, 60295/82033889*e^7 - 7769098/82033889*e^5 + 284634011/82033889*e^3 - 2569321562/82033889*e, 30094/82033889*e^7 - 2152488/82033889*e^5 - 23861830/82033889*e^3 + 2815261744/82033889*e, 60295/82033889*e^7 - 7769098/82033889*e^5 + 284634011/82033889*e^3 - 2569321562/82033889*e, 22723/1674161*e^6 - 2420909/1674161*e^4 + 62863811/1674161*e^2 - 47817538/1674161, 22723/1674161*e^6 - 2420909/1674161*e^4 + 62863811/1674161*e^2 - 47817538/1674161, -8328/1674161*e^6 + 829871/1674161*e^4 - 20068400/1674161*e^2 + 11286650/1674161, -8328/1674161*e^6 + 829871/1674161*e^4 - 20068400/1674161*e^2 + 11286650/1674161, -32251/82033889*e^7 + 3751507/82033889*e^5 - 134085100/82033889*e^3 + 1425060754/82033889*e, -32251/82033889*e^7 + 3751507/82033889*e^5 - 134085100/82033889*e^3 + 1425060754/82033889*e, 148522/82033889*e^7 - 16696462/82033889*e^5 + 496129371/82033889*e^3 - 2460920802/82033889*e, 148522/82033889*e^7 - 16696462/82033889*e^5 + 496129371/82033889*e^3 - 2460920802/82033889*e, -18301/1674161*e^6 + 1906688/1674161*e^4 - 47749899/1674161*e^2 + 23541462/1674161, -18301/1674161*e^6 + 1906688/1674161*e^4 - 47749899/1674161*e^2 + 23541462/1674161, 15820/1674161*e^6 - 1662476/1674161*e^4 + 43792846/1674161*e^2 - 74638734/1674161, -22546/1674161*e^6 + 2401462/1674161*e^4 - 63233356/1674161*e^2 + 103967210/1674161, 15820/1674161*e^6 - 1662476/1674161*e^4 + 43792846/1674161*e^2 - 74638734/1674161, -22546/1674161*e^6 + 2401462/1674161*e^4 - 63233356/1674161*e^2 + 103967210/1674161, -8972/1674161*e^6 + 947920/1674161*e^4 - 24701636/1674161*e^2 + 20186086/1674161, 5301/1674161*e^6 - 667548/1674161*e^4 + 21791397/1674161*e^2 - 71580358/1674161, -8972/1674161*e^6 + 947920/1674161*e^4 - 24701636/1674161*e^2 + 20186086/1674161, 5301/1674161*e^6 - 667548/1674161*e^4 + 21791397/1674161*e^2 - 71580358/1674161, -109369/82033889*e^7 + 11061066/82033889*e^5 - 215432309/82033889*e^3 - 2399469885/82033889*e, 177525/82033889*e^7 - 19561433/82033889*e^5 + 541918515/82033889*e^3 - 1087102599/82033889*e, -109369/82033889*e^7 + 11061066/82033889*e^5 - 215432309/82033889*e^3 - 2399469885/82033889*e, 177525/82033889*e^7 - 19561433/82033889*e^5 + 541918515/82033889*e^3 - 1087102599/82033889*e, -37658/1674161*e^6 + 3986150/1674161*e^4 - 103481899/1674161*e^2 + 98507330/1674161, -37658/1674161*e^6 + 3986150/1674161*e^4 - 103481899/1674161*e^2 + 98507330/1674161, 14694/1674161*e^6 - 1586055/1674161*e^4 + 42076566/1674161*e^2 - 28943642/1674161, 14694/1674161*e^6 - 1586055/1674161*e^4 + 42076566/1674161*e^2 - 28943642/1674161, -192440/82033889*e^7 + 20197540/82033889*e^5 - 494878807/82033889*e^3 - 740892631/82033889*e, -192440/82033889*e^7 + 20197540/82033889*e^5 - 494878807/82033889*e^3 - 740892631/82033889*e, 206396/82033889*e^7 - 21986267/82033889*e^5 + 552598866/82033889*e^3 + 132842739/82033889*e, 206396/82033889*e^7 - 21986267/82033889*e^5 + 552598866/82033889*e^3 + 132842739/82033889*e, -1883/1674161*e^6 + 254178/1674161*e^4 - 7068904/1674161*e^2 - 45967746/1674161, 17523/1674161*e^6 - 1925253/1674161*e^4 + 52145578/1674161*e^2 - 76408398/1674161, -1883/1674161*e^6 + 254178/1674161*e^4 - 7068904/1674161*e^2 - 45967746/1674161, 17523/1674161*e^6 - 1925253/1674161*e^4 + 52145578/1674161*e^2 - 76408398/1674161, -23132/1674161*e^6 + 2503680/1674161*e^4 - 67126959/1674161*e^2 + 142200042/1674161, -21191/1674161*e^6 + 2233671/1674161*e^4 - 57644261/1674161*e^2 + 46736762/1674161, -23132/1674161*e^6 + 2503680/1674161*e^4 - 67126959/1674161*e^2 + 142200042/1674161, -21191/1674161*e^6 + 2233671/1674161*e^4 - 57644261/1674161*e^2 + 46736762/1674161, 8636/11719127*e^7 - 740750/11719127*e^5 + 3213418/11719127*e^3 + 543626872/11719127*e, 8636/11719127*e^7 - 740750/11719127*e^5 + 3213418/11719127*e^3 + 543626872/11719127*e, -186736/82033889*e^7 + 21065290/82033889*e^5 - 610973542/82033889*e^3 + 2172009500/82033889*e, -186736/82033889*e^7 + 21065290/82033889*e^5 - 610973542/82033889*e^3 + 2172009500/82033889*e, -33871/1674161*e^6 + 3674116/1674161*e^4 - 99092394/1674161*e^2 + 148298598/1674161, -33871/1674161*e^6 + 3674116/1674161*e^4 - 99092394/1674161*e^2 + 148298598/1674161, -26556/1674161*e^6 + 2860958/1674161*e^4 - 74998403/1674161*e^2 + 28796842/1674161, -26556/1674161*e^6 + 2860958/1674161*e^4 - 74998403/1674161*e^2 + 28796842/1674161, 90115/82033889*e^7 - 10250906/82033889*e^5 + 325860924/82033889*e^3 - 2172784379/82033889*e, 90115/82033889*e^7 - 10250906/82033889*e^5 + 325860924/82033889*e^3 - 2172784379/82033889*e, 34682/82033889*e^7 - 5021206/82033889*e^5 + 226943287/82033889*e^3 - 3082590913/82033889*e, 34682/82033889*e^7 - 5021206/82033889*e^5 + 226943287/82033889*e^3 - 3082590913/82033889*e, 3018/1674161*e^6 - 246461/1674161*e^4 + 4368174/1674161*e^2 - 41705742/1674161, 12921/1674161*e^6 - 1419631/1674161*e^4 + 36641333/1674161*e^2 + 30714826/1674161, 3018/1674161*e^6 - 246461/1674161*e^4 + 4368174/1674161*e^2 - 41705742/1674161, 12921/1674161*e^6 - 1419631/1674161*e^4 + 36641333/1674161*e^2 + 30714826/1674161, 40528/82033889*e^7 - 3383999/82033889*e^5 + 10651016/82033889*e^3 + 2423298470/82033889*e, 221971/82033889*e^7 - 24823059/82033889*e^5 + 763137012/82033889*e^3 - 4816915494/82033889*e, 40528/82033889*e^7 - 3383999/82033889*e^5 + 10651016/82033889*e^3 + 2423298470/82033889*e, 221971/82033889*e^7 - 24823059/82033889*e^5 + 763137012/82033889*e^3 - 4816915494/82033889*e, -27501/82033889*e^7 + 4071434/82033889*e^5 - 172056288/82033889*e^3 + 1849949677/82033889*e, -144584/82033889*e^7 + 17238023/82033889*e^5 - 556174548/82033889*e^3 + 3154851547/82033889*e, -27501/82033889*e^7 + 4071434/82033889*e^5 - 172056288/82033889*e^3 + 1849949677/82033889*e, -144584/82033889*e^7 + 17238023/82033889*e^5 - 556174548/82033889*e^3 + 3154851547/82033889*e, 12748/82033889*e^7 - 246682/82033889*e^5 - 69680309/82033889*e^3 + 2644796673/82033889*e, 12748/82033889*e^7 - 246682/82033889*e^5 - 69680309/82033889*e^3 + 2644796673/82033889*e, 121285/82033889*e^7 - 13505302/82033889*e^5 + 394290669/82033889*e^3 - 1123564285/82033889*e, 121285/82033889*e^7 - 13505302/82033889*e^5 + 394290669/82033889*e^3 - 1123564285/82033889*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9,3,2/23*w^3 - 3/23*w^2 - 43/23*w + 68/23}>] := 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;