/*
  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, 8, -12, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, 2/5*w^3 - 14/5*w - 3/5}>;

primesArray := [
[5, 5, -w + 2],
[5, 5, -3/5*w^3 - w^2 + 26/5*w + 42/5],
[9, 3, 2/5*w^3 - 19/5*w - 3/5],
[9, 3, -1/5*w^3 + 2/5*w + 4/5],
[11, 11, -2/5*w^3 - w^2 + 14/5*w + 28/5],
[11, 11, 1/5*w^3 - w^2 - 7/5*w + 36/5],
[16, 2, 2],
[29, 29, -w],
[29, 29, 1/5*w^3 - 12/5*w + 1/5],
[41, 41, -2/5*w^3 - w^2 + 14/5*w + 38/5],
[41, 41, -w^2 + 10],
[41, 41, 11/5*w^3 + 3*w^2 - 102/5*w - 149/5],
[41, 41, -1/5*w^3 + w^2 + 7/5*w - 26/5],
[49, 7, 1/5*w^3 + w^2 - 12/5*w - 19/5],
[49, 7, 1/5*w^3 + w^2 - 12/5*w - 44/5],
[59, 59, -4/5*w^3 - w^2 + 33/5*w + 36/5],
[59, 59, -2*w^3 - 3*w^2 + 17*w + 26],
[61, 61, -1/5*w^3 + w^2 + 12/5*w - 51/5],
[61, 61, 4/5*w^3 + w^2 - 28/5*w - 41/5],
[71, 71, 1/5*w^3 + w^2 - 7/5*w - 49/5],
[71, 71, -2*w^3 - 3*w^2 + 18*w + 28],
[89, 89, -3/5*w^3 + 16/5*w + 7/5],
[89, 89, 3/5*w^3 + w^2 - 26/5*w - 57/5],
[121, 11, -3/5*w^3 + 21/5*w + 2/5],
[131, 131, w^2 - w - 8],
[131, 131, 2/5*w^3 + w^2 - 19/5*w - 23/5],
[149, 149, 2/5*w^3 - w^2 - 14/5*w + 22/5],
[149, 149, 3/5*w^3 + w^2 - 21/5*w - 42/5],
[179, 179, -6/5*w^3 - w^2 + 57/5*w + 69/5],
[179, 179, -2/5*w^3 - 2*w^2 + 14/5*w + 73/5],
[179, 179, 2*w^2 - 11],
[179, 179, -3/5*w^3 + 2*w^2 + 16/5*w - 48/5],
[181, 181, -7/5*w^3 - 2*w^2 + 59/5*w + 78/5],
[181, 181, -2/5*w^3 + 24/5*w - 27/5],
[191, 191, -3/5*w^3 + 2*w^2 + 21/5*w - 78/5],
[191, 191, 2/5*w^3 - 2*w^2 - 9/5*w + 57/5],
[199, 199, -1/5*w^3 + 17/5*w - 16/5],
[199, 199, 1/5*w^3 - 17/5*w - 14/5],
[211, 211, 2/5*w^3 - w^2 + 1/5*w + 7/5],
[211, 211, -1/5*w^3 + w^2 - 3/5*w - 26/5],
[211, 211, 4/5*w^3 + w^2 - 38/5*w - 36/5],
[211, 211, -1/5*w^3 - 2*w^2 + 7/5*w + 44/5],
[239, 239, 3/5*w^3 + w^2 - 16/5*w - 37/5],
[239, 239, 2/5*w^3 + w^2 - 14/5*w - 53/5],
[251, 251, 2/5*w^3 - 24/5*w - 13/5],
[251, 251, 2*w - 3],
[251, 251, w^2 + w - 10],
[251, 251, -4/5*w^3 - w^2 + 28/5*w + 46/5],
[269, 269, 1/5*w^3 - 12/5*w - 24/5],
[269, 269, w - 5],
[271, 271, -3/5*w^3 + 31/5*w - 33/5],
[271, 271, -8/5*w^3 - 2*w^2 + 66/5*w + 77/5],
[281, 281, 4/5*w^3 - w^2 - 38/5*w + 39/5],
[281, 281, -14/5*w^3 - 3*w^2 + 128/5*w + 161/5],
[311, 311, 1/5*w^3 - 2*w^2 - 2/5*w + 51/5],
[311, 311, 1/5*w^3 - w^2 - 17/5*w + 21/5],
[331, 331, 1/5*w^3 + 3/5*w - 4/5],
[331, 331, 3/5*w^3 - 31/5*w - 2/5],
[359, 359, -w^3 + 6*w - 1],
[359, 359, -6/5*w^3 + 47/5*w - 6/5],
[359, 359, 3/5*w^3 + w^2 - 31/5*w - 27/5],
[359, 359, 3/5*w^3 + w^2 - 26/5*w - 22/5],
[361, 19, 4/5*w^3 - 28/5*w - 11/5],
[361, 19, 1/5*w^3 - 7/5*w - 24/5],
[379, 379, 9/5*w^3 + 3*w^2 - 78/5*w - 121/5],
[379, 379, -4/5*w^3 + 3*w^2 + 28/5*w - 99/5],
[389, 389, -3/5*w^3 - 2*w^2 + 26/5*w + 42/5],
[389, 389, -2*w^2 + w + 17],
[401, 401, 4/5*w^3 - w^2 - 23/5*w + 24/5],
[401, 401, 3/5*w^3 + w^2 - 26/5*w - 7/5],
[401, 401, -4/5*w^3 - w^2 + 38/5*w + 66/5],
[401, 401, 6/5*w^3 + w^2 - 52/5*w - 39/5],
[409, 409, -6/5*w^3 + 52/5*w + 19/5],
[409, 409, 1/5*w^3 + 2*w^2 - 7/5*w - 39/5],
[421, 421, -1/5*w^3 - w^2 + 17/5*w + 44/5],
[421, 421, 2/5*w^3 + w^2 - 24/5*w - 18/5],
[431, 431, -3/5*w^3 - w^2 + 36/5*w + 47/5],
[431, 431, -13/5*w^3 - 4*w^2 + 116/5*w + 182/5],
[431, 431, -2/5*w^3 + w^2 + 24/5*w - 47/5],
[431, 431, 2/5*w^3 - w^2 + 6/5*w - 8/5],
[439, 439, 6/5*w^3 + w^2 - 47/5*w - 49/5],
[439, 439, -2/5*w^3 + 2*w^2 + 9/5*w - 42/5],
[439, 439, 4/5*w^3 - w^2 - 23/5*w + 14/5],
[439, 439, -w^3 - 2*w^2 + 8*w + 17],
[461, 461, w^3 + w^2 - 10*w - 9],
[461, 461, -2*w^3 - 3*w^2 + 17*w + 27],
[461, 461, -1/5*w^3 + w^2 - 8/5*w - 16/5],
[461, 461, -3*w^3 - 4*w^2 + 26*w + 35],
[479, 479, 2*w - 1],
[479, 479, 2/5*w^3 - 24/5*w - 3/5],
[491, 491, w^3 - 2*w^2 - 6*w + 11],
[491, 491, w^3 + w^2 - 10*w - 15],
[499, 499, -1/5*w^3 + 2*w^2 + 2/5*w - 36/5],
[499, 499, 4/5*w^3 + 2*w^2 - 33/5*w - 91/5],
[509, 509, 8/5*w^3 + w^2 - 66/5*w - 42/5],
[509, 509, 1/5*w^3 - 2*w^2 + 8/5*w + 16/5],
[521, 521, w^3 - 9*w - 4],
[521, 521, 3/5*w^3 + w^2 - 16/5*w - 42/5],
[521, 521, 3/5*w^3 - w^2 - 26/5*w + 23/5],
[521, 521, -3/5*w^3 + 11/5*w + 22/5],
[529, 23, 3/5*w^3 + w^2 - 36/5*w - 62/5],
[529, 23, -4/5*w^3 + 2*w^2 + 23/5*w - 64/5],
[541, 541, -w^3 - w^2 + 9*w + 7],
[541, 541, 1/5*w^3 + 2*w^2 - 7/5*w - 64/5],
[541, 541, 3/5*w^3 + w^2 - 26/5*w - 12/5],
[571, 571, -w^3 - w^2 + 6*w + 9],
[571, 571, -w^3 + w^2 + 8*w - 4],
[619, 619, -w^3 - 2*w^2 + 8*w + 11],
[619, 619, -2/5*w^3 + 2*w^2 + 9/5*w - 72/5],
[619, 619, 2/5*w^3 - 2*w^2 - 4/5*w + 57/5],
[619, 619, -6/5*w^3 - 2*w^2 + 52/5*w + 69/5],
[631, 631, 1/5*w^3 + w^2 + 3/5*w - 24/5],
[631, 631, 2/5*w^3 - w^2 - 24/5*w + 42/5],
[659, 659, -13/5*w^3 - 4*w^2 + 111/5*w + 177/5],
[659, 659, -11/5*w^3 - 3*w^2 + 92/5*w + 119/5],
[659, 659, -8/5*w^3 - w^2 + 61/5*w + 27/5],
[659, 659, -8/5*w^3 + 71/5*w + 2/5],
[661, 661, -w^3 - w^2 + 7*w + 11],
[661, 661, 4/5*w^3 - w^2 - 28/5*w + 9/5],
[691, 691, 1/5*w^3 - w^2 - 17/5*w + 41/5],
[691, 691, 3/5*w^3 + 2*w^2 - 26/5*w - 67/5],
[701, 701, 12/5*w^3 + 4*w^2 - 99/5*w - 158/5],
[701, 701, -17/5*w^3 - 4*w^2 + 149/5*w + 193/5],
[709, 709, -1/5*w^3 + w^2 + 12/5*w - 6/5],
[709, 709, 1/5*w^3 + w^2 - 2/5*w - 59/5],
[719, 719, 2/5*w^3 + w^2 - 29/5*w - 8/5],
[719, 719, -w^3 + w^2 + 7*w - 4],
[719, 719, 6/5*w^3 + w^2 - 42/5*w - 44/5],
[719, 719, -2/5*w^3 - w^2 + 29/5*w + 53/5],
[739, 739, 14/5*w^3 + 4*w^2 - 123/5*w - 171/5],
[739, 739, -1/5*w^3 + 2*w^2 + 12/5*w - 56/5],
[739, 739, -6/5*w^3 + 3*w^2 + 37/5*w - 96/5],
[739, 739, 2/5*w^3 + 2*w^2 - 9/5*w - 73/5],
[751, 751, -9/5*w^3 - 2*w^2 + 73/5*w + 86/5],
[751, 751, -w^3 + 2*w^2 + 5*w - 8],
[761, 761, 1/5*w^3 + w^2 - 22/5*w + 16/5],
[761, 761, -16/5*w^3 - 4*w^2 + 147/5*w + 214/5],
[769, 769, 13/5*w^3 + 3*w^2 - 121/5*w - 172/5],
[769, 769, 3/5*w^3 - w^2 - 41/5*w + 78/5],
[769, 769, -12/5*w^3 - 4*w^2 + 99/5*w + 153/5],
[769, 769, -6/5*w^3 + 3*w^2 + 27/5*w - 61/5],
[809, 809, 1/5*w^3 + 3*w^2 - 22/5*w - 104/5],
[809, 809, w^3 + 3*w^2 - 10*w - 17],
[811, 811, -5*w + 11],
[811, 811, -16/5*w^3 - 5*w^2 + 137/5*w + 214/5],
[821, 821, -1/5*w^3 + w^2 + 17/5*w - 36/5],
[821, 821, w^2 + 2*w - 6],
[829, 829, 3*w^2 + w - 17],
[829, 829, 2/5*w^3 + 3*w^2 - 9/5*w - 108/5],
[841, 29, w^3 - 7*w - 3],
[859, 859, 6/5*w^3 + w^2 - 37/5*w - 34/5],
[859, 859, 2/5*w^3 + w^2 - 14/5*w - 68/5],
[881, 881, 1/5*w^3 + 2*w^2 - 17/5*w - 79/5],
[881, 881, 3/5*w^3 + 2*w^2 - 31/5*w - 47/5],
[911, 911, 4/5*w^3 - 3*w^2 - 28/5*w + 89/5],
[911, 911, 1/5*w^3 - w^2 - 7/5*w + 61/5],
[911, 911, w^3 - 6*w - 3],
[911, 911, -11/5*w^3 - 3*w^2 + 92/5*w + 139/5],
[919, 919, 7/5*w^3 + 2*w^2 - 54/5*w - 88/5],
[919, 919, 4/5*w^3 - 2*w^2 - 23/5*w + 39/5],
[929, 929, -7/5*w^3 + w^2 + 54/5*w - 17/5],
[929, 929, -1/5*w^3 + w^2 + 22/5*w - 71/5],
[941, 941, 9/5*w^3 + w^2 - 73/5*w - 31/5],
[941, 941, 1/5*w^3 - w^2 - 2/5*w + 66/5],
[941, 941, 2/5*w^3 + 2*w^2 - 19/5*w - 103/5],
[941, 941, -w^3 + w^2 + 5*w - 3],
[961, 31, w^3 - 7*w - 2],
[961, 31, 2/5*w^3 - 14/5*w - 33/5],
[971, 971, -2/5*w^3 - 2*w^2 + 24/5*w + 93/5],
[971, 971, 2/5*w^3 + 2*w^2 - 24/5*w - 33/5],
[991, 991, -w^3 + 2*w^2 + 7*w - 10],
[991, 991, -7/5*w^3 - 2*w^2 + 49/5*w + 78/5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - x^7 - 38*x^6 + 24*x^5 + 367*x^4 - 275*x^3 - 738*x^2 + 796*x - 200;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 1, e, e, 113/1440*e^7 - 61/1440*e^6 - 121/40*e^5 + 67/120*e^4 + 42683/1440*e^3 - 1443/160*e^2 - 257/4*e + 2521/72, 113/1440*e^7 - 61/1440*e^6 - 121/40*e^5 + 67/120*e^4 + 42683/1440*e^3 - 1443/160*e^2 - 257/4*e + 2521/72, -37/480*e^7 + 17/480*e^6 + 119/40*e^5 - 11/40*e^4 - 14023/480*e^3 + 957/160*e^2 + 1277/20*e - 653/24, -23/720*e^7 - 1/360*e^6 + 49/40*e^5 + 67/120*e^4 - 8447/720*e^3 - 19/5*e^2 + 517/20*e - 25/9, -23/720*e^7 - 1/360*e^6 + 49/40*e^5 + 67/120*e^4 - 8447/720*e^3 - 19/5*e^2 + 517/20*e - 25/9, 1/16*e^7 - 19/8*e^5 - 7/8*e^4 + 353/16*e^3 + 39/8*e^2 - 165/4*e + 21/2, -1/288*e^7 + 43/1440*e^6 + 1/10*e^5 - 59/60*e^4 - 1211/1440*e^3 + 1217/160*e^2 + 11/5*e - 451/72, -1/288*e^7 + 43/1440*e^6 + 1/10*e^5 - 59/60*e^4 - 1211/1440*e^3 + 1217/160*e^2 + 11/5*e - 451/72, 1/16*e^7 - 19/8*e^5 - 7/8*e^4 + 353/16*e^3 + 39/8*e^2 - 165/4*e + 21/2, -59/720*e^7 + 43/720*e^6 + 63/20*e^5 - 61/60*e^4 - 22649/720*e^3 + 949/80*e^2 + 149/2*e - 1495/36, -59/720*e^7 + 43/720*e^6 + 63/20*e^5 - 61/60*e^4 - 22649/720*e^3 + 949/80*e^2 + 149/2*e - 1495/36, 83/720*e^7 - 11/144*e^6 - 22/5*e^5 + 19/15*e^4 + 30701/720*e^3 - 1349/80*e^2 - 917/10*e + 1975/36, 83/720*e^7 - 11/144*e^6 - 22/5*e^5 + 19/15*e^4 + 30701/720*e^3 - 1349/80*e^2 - 917/10*e + 1975/36, 31/480*e^7 - 11/480*e^6 - 97/40*e^5 - 7/40*e^4 + 10789/480*e^3 - 111/160*e^2 - 861/20*e + 527/24, 31/480*e^7 - 11/480*e^6 - 97/40*e^5 - 7/40*e^4 + 10789/480*e^3 - 111/160*e^2 - 861/20*e + 527/24, -13/45*e^7 + 13/90*e^6 + 11*e^5 - 4/3*e^4 - 4729/45*e^3 + 49/2*e^2 + 1077/5*e - 968/9, -13/45*e^7 + 13/90*e^6 + 11*e^5 - 4/3*e^4 - 4729/45*e^3 + 49/2*e^2 + 1077/5*e - 968/9, -17/90*e^7 + 43/360*e^6 + 36/5*e^5 - 49/30*e^4 - 6263/90*e^3 + 801/40*e^2 + 1467/10*e - 1315/18, -17/90*e^7 + 43/360*e^6 + 36/5*e^5 - 49/30*e^4 - 6263/90*e^3 + 801/40*e^2 + 1467/10*e - 1315/18, -253/720*e^7 + 13/90*e^6 + 107/8*e^5 - 11/24*e^4 - 91549/720*e^3 + 165/8*e^2 + 5113/20*e - 2089/18, -125/288*e^7 + 281/1440*e^6 + 663/40*e^5 - 161/120*e^4 - 229027/1440*e^3 + 5279/160*e^2 + 6583/20*e - 11789/72, -125/288*e^7 + 281/1440*e^6 + 663/40*e^5 - 161/120*e^4 - 229027/1440*e^3 + 5279/160*e^2 + 6583/20*e - 11789/72, -121/360*e^7 + 139/720*e^6 + 513/40*e^5 - 281/120*e^4 - 22313/180*e^3 + 2727/80*e^2 + 1057/4*e - 5065/36, -121/360*e^7 + 139/720*e^6 + 513/40*e^5 - 281/120*e^4 - 22313/180*e^3 + 2727/80*e^2 + 1057/4*e - 5065/36, -439/720*e^7 + 179/720*e^6 + 233/10*e^5 - 31/30*e^4 - 160801/720*e^3 + 3233/80*e^2 + 4643/10*e - 8195/36, 89/720*e^7 - 1/180*e^6 - 187/40*e^5 - 181/120*e^4 + 31481/720*e^3 + 241/40*e^2 - 1841/20*e + 575/18, 89/720*e^7 - 1/180*e^6 - 187/40*e^5 - 181/120*e^4 + 31481/720*e^3 + 241/40*e^2 - 1841/20*e + 575/18, -439/720*e^7 + 179/720*e^6 + 233/10*e^5 - 31/30*e^4 - 160801/720*e^3 + 3233/80*e^2 + 4643/10*e - 8195/36, 589/1440*e^7 - 67/288*e^6 - 78/5*e^5 + 169/60*e^4 + 217363/1440*e^3 - 6637/160*e^2 - 1639/5*e + 11951/72, 589/1440*e^7 - 67/288*e^6 - 78/5*e^5 + 169/60*e^4 + 217363/1440*e^3 - 6637/160*e^2 - 1639/5*e + 11951/72, 83/360*e^7 - 23/180*e^6 - 177/20*e^5 + 89/60*e^4 + 31043/360*e^3 - 106/5*e^2 - 375/2*e + 812/9, 83/360*e^7 - 23/180*e^6 - 177/20*e^5 + 89/60*e^4 + 31043/360*e^3 - 106/5*e^2 - 375/2*e + 812/9, 19/120*e^7 - 19/240*e^6 - 49/8*e^5 + 7/8*e^4 + 3641/60*e^3 - 261/16*e^2 - 2759/20*e + 925/12, 19/120*e^7 - 19/240*e^6 - 49/8*e^5 + 7/8*e^4 + 3641/60*e^3 - 261/16*e^2 - 2759/20*e + 925/12, -563/1440*e^7 + 41/288*e^6 + 299/20*e^5 - 2/15*e^4 - 206501/1440*e^3 + 3679/160*e^2 + 1498/5*e - 10621/72, 193/480*e^7 - 107/480*e^6 - 153/10*e^5 + 49/20*e^4 + 14099/96*e^3 - 5891/160*e^2 - 3067/10*e + 3683/24, 193/480*e^7 - 107/480*e^6 - 153/10*e^5 + 49/20*e^4 + 14099/96*e^3 - 5891/160*e^2 - 3067/10*e + 3683/24, -563/1440*e^7 + 41/288*e^6 + 299/20*e^5 - 2/15*e^4 - 206501/1440*e^3 + 3679/160*e^2 + 1498/5*e - 10621/72, -16/45*e^7 + 83/720*e^6 + 109/8*e^5 + 7/24*e^4 - 47099/360*e^3 + 323/16*e^2 + 5451/20*e - 5345/36, -16/45*e^7 + 83/720*e^6 + 109/8*e^5 + 7/24*e^4 - 47099/360*e^3 + 323/16*e^2 + 5451/20*e - 5345/36, 679/1440*e^7 - 299/1440*e^6 - 721/40*e^5 + 167/120*e^4 + 249781/1440*e^3 - 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6/5*e^4 - 8951/80*e^3 + 1583/80*e^2 + 2321/10*e - 405/4, -5/16*e^7 + 13/80*e^6 + 59/5*e^5 - 6/5*e^4 - 8951/80*e^3 + 1583/80*e^2 + 2321/10*e - 405/4, 1/8*e^7 - 1/8*e^6 - 19/4*e^5 + 11/4*e^4 + 377/8*e^3 - 221/8*e^2 - 120*e + 135/2, -37/60*e^7 + 1/3*e^6 + 237/10*e^5 - 43/10*e^4 - 13849/60*e^3 + 689/10*e^2 + 2448/5*e - 800/3, -79/120*e^7 + 19/60*e^6 + 127/5*e^5 - 31/10*e^4 - 29929/120*e^3 + 1191/20*e^2 + 1087/2*e - 805/3, -37/60*e^7 + 1/3*e^6 + 237/10*e^5 - 43/10*e^4 - 13849/60*e^3 + 689/10*e^2 + 2448/5*e - 800/3, -79/120*e^7 + 19/60*e^6 + 127/5*e^5 - 31/10*e^4 - 29929/120*e^3 + 1191/20*e^2 + 1087/2*e - 805/3, 119/240*e^7 - 4/15*e^6 - 761/40*e^5 + 119/40*e^4 + 44411/240*e^3 - 1827/40*e^2 - 7833/20*e + 1097/6, 119/240*e^7 - 4/15*e^6 - 761/40*e^5 + 119/40*e^4 + 44411/240*e^3 - 1827/40*e^2 - 7833/20*e + 1097/6, -751/1440*e^7 + 407/1440*e^6 + 797/40*e^5 - 359/120*e^4 - 276061/1440*e^3 + 6961/160*e^2 + 1571/4*e - 13211/72, -751/1440*e^7 + 407/1440*e^6 + 797/40*e^5 - 359/120*e^4 - 276061/1440*e^3 + 6961/160*e^2 + 1571/4*e - 13211/72, 73/360*e^7 - 19/72*e^6 - 39/5*e^5 + 103/15*e^4 + 28351/360*e^3 - 2629/40*e^2 - 927/5*e + 2495/18, -17/45*e^7 + 17/90*e^6 + 29/2*e^5 - 13/6*e^4 - 12697/90*e^3 + 40*e^2 + 1558/5*e - 1450/9, -17/45*e^7 + 17/90*e^6 + 29/2*e^5 - 13/6*e^4 - 12697/90*e^3 + 40*e^2 + 1558/5*e - 1450/9, 73/360*e^7 - 19/72*e^6 - 39/5*e^5 + 103/15*e^4 + 28351/360*e^3 - 2629/40*e^2 - 927/5*e + 2495/18, -11/120*e^7 - 1/15*e^6 + 18/5*e^5 + 31/10*e^4 - 4061/120*e^3 - 69/5*e^2 + 129/2*e - 80/3, -11/120*e^7 - 1/15*e^6 + 18/5*e^5 + 31/10*e^4 - 4061/120*e^3 - 69/5*e^2 + 129/2*e - 80/3, 23/45*e^7 - 7/45*e^6 - 197/10*e^5 - 31/30*e^4 + 3449/18*e^3 - 169/10*e^2 - 2044/5*e + 1498/9, 23/45*e^7 - 7/45*e^6 - 197/10*e^5 - 31/30*e^4 + 3449/18*e^3 - 169/10*e^2 - 2044/5*e + 1498/9, 517/1440*e^7 - 191/1440*e^6 - 55/4*e^5 + 1/6*e^4 + 191011/1440*e^3 - 705/32*e^2 - 1434/5*e + 10871/72, 517/1440*e^7 - 191/1440*e^6 - 55/4*e^5 + 1/6*e^4 + 191011/1440*e^3 - 705/32*e^2 - 1434/5*e + 10871/72, 51/80*e^7 - 21/80*e^6 - 491/20*e^5 + 29/20*e^4 + 19169/80*e^3 - 3723/80*e^2 - 2619/5*e + 985/4, 51/80*e^7 - 21/80*e^6 - 491/20*e^5 + 29/20*e^4 + 19169/80*e^3 - 3723/80*e^2 - 2619/5*e + 985/4, 1093/1440*e^7 - 407/1440*e^6 - 291/10*e^5 + 19/60*e^4 + 80783/288*e^3 - 6517/160*e^2 - 2942/5*e + 22391/72, 281/360*e^7 - 263/720*e^6 - 1197/40*e^5 + 349/120*e^4 + 10451/36*e^3 - 4883/80*e^2 - 12589/20*e + 10925/36, 281/360*e^7 - 263/720*e^6 - 1197/40*e^5 + 349/120*e^4 + 10451/36*e^3 - 4883/80*e^2 - 12589/20*e + 10925/36, -79/160*e^7 + 9/32*e^6 + 189/10*e^5 - 67/20*e^4 - 29473/160*e^3 + 7663/160*e^2 + 2021/5*e - 1581/8, -79/160*e^7 + 9/32*e^6 + 189/10*e^5 - 67/20*e^4 - 29473/160*e^3 + 7663/160*e^2 + 2021/5*e - 1581/8, 329/1440*e^7 + 47/1440*e^6 - 353/40*e^5 - 509/120*e^4 + 122099/1440*e^3 + 4081/160*e^2 - 717/4*e + 3349/72, 937/1440*e^7 - 473/1440*e^6 - 991/40*e^5 + 317/120*e^4 + 68447/288*e^3 - 7183/160*e^2 - 9897/20*e + 16229/72, 937/1440*e^7 - 473/1440*e^6 - 991/40*e^5 + 317/120*e^4 + 68447/288*e^3 - 7183/160*e^2 - 9897/20*e + 16229/72, 329/1440*e^7 + 47/1440*e^6 - 353/40*e^5 - 509/120*e^4 + 122099/1440*e^3 + 4081/160*e^2 - 717/4*e + 3349/72, -41/240*e^7 + 13/240*e^6 + 13/2*e^5 + 1/2*e^4 - 14543/240*e^3 + 9/16*e^2 + 1059/10*e - 445/12, -41/240*e^7 + 13/240*e^6 + 13/2*e^5 + 1/2*e^4 - 14543/240*e^3 + 9/16*e^2 + 1059/10*e - 445/12, 37/180*e^7 - 8/45*e^6 - 157/20*e^5 + 239/60*e^4 + 6899/90*e^3 - 919/20*e^2 - 858/5*e + 965/9, 37/180*e^7 - 8/45*e^6 - 157/20*e^5 + 239/60*e^4 + 6899/90*e^3 - 919/20*e^2 - 858/5*e + 965/9, 209/480*e^7 - 97/480*e^6 - 133/8*e^5 + 13/8*e^4 + 76547/480*e^3 - 1217/32*e^2 - 6631/20*e + 4573/24, 239/480*e^7 - 127/480*e^6 - 153/8*e^5 + 25/8*e^4 + 89717/480*e^3 - 1639/32*e^2 - 8141/20*e + 5083/24, 209/480*e^7 - 97/480*e^6 - 133/8*e^5 + 13/8*e^4 + 76547/480*e^3 - 1217/32*e^2 - 6631/20*e + 4573/24, 239/480*e^7 - 127/480*e^6 - 153/8*e^5 + 25/8*e^4 + 89717/480*e^3 - 1639/32*e^2 - 8141/20*e + 5083/24, -829/720*e^7 + 41/72*e^6 + 1763/40*e^5 - 631/120*e^4 - 308053/720*e^3 + 507/5*e^2 + 18707/20*e - 4037/9, 169/360*e^7 - 31/180*e^6 - 18*e^5 + 1/6*e^4 + 62287/360*e^3 - 101/4*e^2 - 3539/10*e + 1843/9, -631/1440*e^7 + 203/1440*e^6 + 135/8*e^5 + 11/24*e^4 - 237133/1440*e^3 + 705/32*e^2 + 7343/20*e - 12719/72, -631/1440*e^7 + 203/1440*e^6 + 135/8*e^5 + 11/24*e^4 - 237133/1440*e^3 + 705/32*e^2 + 7343/20*e - 12719/72, 121/120*e^7 - 17/30*e^6 - 77/2*e^5 + 7*e^4 + 44563/120*e^3 - 104*e^2 - 7773/10*e + 1204/3, 121/120*e^7 - 17/30*e^6 - 77/2*e^5 + 7*e^4 + 44563/120*e^3 - 104*e^2 - 7773/10*e + 1204/3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, -w + 2}>] := -1;
ALEigenvalues[ideal<ZF | {5, 5, -3/5*w^3 - w^2 + 26/5*w + 42/5}>] := -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;