/*
  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![-2, 1, 8, -1, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {32, 32, -w^4 + 5*w^2 - 4}>;

primesArray := [
[2, 2, -w],
[13, 13, -w^2 + 3],
[17, 17, -w^5 + w^4 + 5*w^3 - 3*w^2 - 5*w + 1],
[23, 23, w^4 - w^3 - 4*w^2 + 2*w + 1],
[31, 31, w^5 - w^4 - 6*w^3 + 5*w^2 + 7*w - 5],
[32, 2, w^5 - 6*w^3 - w^2 + 8*w + 1],
[43, 43, w^4 - 5*w^2 + 3],
[47, 47, -w^4 + w^3 + 5*w^2 - 2*w - 5],
[49, 7, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w - 1],
[53, 53, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 5*w - 7],
[59, 59, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 3],
[71, 71, w^5 - w^4 - 5*w^3 + 4*w^2 + 4*w - 5],
[79, 79, w^3 - w^2 - 4*w + 1],
[83, 83, w^5 - 6*w^3 - w^2 + 7*w - 1],
[83, 83, 2*w^5 - w^4 - 11*w^3 + 2*w^2 + 12*w + 1],
[89, 89, -w^5 + w^4 + 4*w^3 - 2*w^2 - w - 1],
[89, 89, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w + 3],
[89, 89, w^5 + w^4 - 6*w^3 - 6*w^2 + 6*w + 3],
[89, 89, 2*w^4 - w^3 - 9*w^2 + w + 5],
[101, 101, w^5 - 5*w^3 - w^2 + 5*w - 1],
[107, 107, w^4 - w^3 - 3*w^2 + 2*w - 1],
[107, 107, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w + 1],
[107, 107, -w^5 + 4*w^3 + 2*w^2 + 1],
[113, 113, 2*w^5 - w^4 - 10*w^3 + 2*w^2 + 8*w - 3],
[121, 11, 2*w^5 - w^4 - 10*w^3 + 2*w^2 + 8*w - 1],
[125, 5, -w^3 + 4*w - 1],
[125, 5, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 9*w + 3],
[131, 131, w^5 + 2*w^4 - 7*w^3 - 10*w^2 + 9*w + 5],
[137, 137, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 7],
[137, 137, w^4 - 3*w^2 - 2*w + 1],
[139, 139, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w + 1],
[139, 139, w^5 - w^4 - 5*w^3 + 4*w^2 + 3*w - 3],
[149, 149, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 7],
[151, 151, w^4 - 6*w^2 - w + 5],
[163, 163, -w^5 + w^4 + 4*w^3 - 3*w^2 - w + 3],
[167, 167, -w^5 + w^4 + 5*w^3 - 2*w^2 - 6*w + 1],
[169, 13, -2*w^5 + 12*w^3 + w^2 - 13*w + 1],
[179, 179, -w^5 + 6*w^3 + 2*w^2 - 9*w - 3],
[179, 179, -w^3 + 5*w - 1],
[191, 191, w^4 - 3*w^2 - 1],
[193, 193, -w^5 - 3*w^4 + 6*w^3 + 17*w^2 - 5*w - 15],
[197, 197, 2*w^5 - 11*w^3 - 2*w^2 + 10*w + 1],
[197, 197, w^5 - 5*w^3 - 2*w^2 + 3*w - 1],
[227, 227, w^5 - w^4 - 5*w^3 + 5*w^2 + 4*w - 5],
[229, 229, -2*w^4 + w^3 + 10*w^2 - w - 9],
[241, 241, 3*w^5 - 2*w^4 - 16*w^3 + 6*w^2 + 17*w - 3],
[263, 263, -2*w^5 + 11*w^3 + w^2 - 11*w - 1],
[269, 269, -w^5 + w^4 + 5*w^3 - 4*w^2 - 7*w + 3],
[277, 277, 2*w^5 - w^4 - 10*w^3 + w^2 + 7*w + 1],
[277, 277, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 5],
[283, 283, -w^4 + w^3 + 4*w^2 - 3],
[289, 17, -w^5 - w^4 + 7*w^3 + 4*w^2 - 10*w - 1],
[293, 293, -3*w^5 + 16*w^3 + 3*w^2 - 15*w - 1],
[293, 293, -2*w^5 - w^4 + 10*w^3 + 7*w^2 - 6*w - 5],
[307, 307, -2*w^5 + w^4 + 9*w^3 - w^2 - 4*w - 1],
[311, 311, w^4 - w^3 - 5*w^2 + 4*w + 1],
[311, 311, -3*w^5 + 17*w^3 + 4*w^2 - 18*w - 7],
[311, 311, -w^5 - w^4 + 5*w^3 + 6*w^2 - 2*w - 5],
[311, 311, 2*w^4 - w^3 - 10*w^2 + w + 5],
[313, 313, w^5 - 5*w^3 - 3*w^2 + 3*w + 3],
[313, 313, -w^5 - w^4 + 5*w^3 + 7*w^2 - 3*w - 3],
[317, 317, -w^5 + 2*w^4 + 3*w^3 - 8*w^2 + w + 5],
[337, 337, 3*w^5 + w^4 - 16*w^3 - 10*w^2 + 14*w + 11],
[337, 337, -w^5 - 2*w^4 + 6*w^3 + 10*w^2 - 6*w - 7],
[337, 337, 2*w^5 + w^4 - 11*w^3 - 7*w^2 + 10*w + 3],
[337, 337, w^4 - w^3 - 6*w^2 + 4*w + 7],
[347, 347, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 7*w + 3],
[347, 347, -w^5 - w^4 + 5*w^3 + 7*w^2 - 2*w - 7],
[347, 347, -w^5 - w^4 + 6*w^3 + 7*w^2 - 5*w - 7],
[349, 349, 3*w^5 - w^4 - 17*w^3 + 3*w^2 + 19*w - 3],
[349, 349, w^4 + w^3 - 6*w^2 - 4*w + 7],
[359, 359, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 13*w + 1],
[373, 373, -2*w^5 + 10*w^3 + 2*w^2 - 9*w - 1],
[379, 379, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 9],
[389, 389, -2*w^5 + w^4 + 11*w^3 - 2*w^2 - 13*w - 1],
[397, 397, w^5 - 3*w^4 - 5*w^3 + 14*w^2 + 7*w - 11],
[419, 419, w^5 - 6*w^3 + w^2 + 7*w - 1],
[419, 419, -w^5 + w^4 + 3*w^3 - 2*w^2 + w - 1],
[433, 433, -w^5 + w^4 + 4*w^3 - 4*w^2 - w + 5],
[433, 433, w^5 + w^4 - 7*w^3 - 4*w^2 + 9*w - 1],
[439, 439, w^4 - 2*w^3 - 4*w^2 + 7*w + 3],
[443, 443, -w^3 + w^2 + 3*w - 5],
[443, 443, w^5 - w^4 - 5*w^3 + 2*w^2 + 5*w + 3],
[449, 449, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 12*w - 5],
[457, 457, -2*w^5 - 2*w^4 + 11*w^3 + 13*w^2 - 10*w - 11],
[461, 461, w^5 + 2*w^4 - 6*w^3 - 10*w^2 + 5*w + 7],
[479, 479, -2*w^5 + 10*w^3 + w^2 - 6*w + 1],
[479, 479, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 1],
[491, 491, w^5 - w^4 - 4*w^3 + w^2 + w + 3],
[499, 499, -w^3 + 5*w - 3],
[499, 499, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 1],
[509, 509, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 9*w + 11],
[521, 521, -w^5 + 4*w^3 + 2*w^2 - w - 3],
[521, 521, 2*w^5 + 2*w^4 - 11*w^3 - 13*w^2 + 9*w + 13],
[521, 521, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3],
[523, 523, -w^5 + w^4 + 6*w^3 - 5*w^2 - 8*w + 3],
[523, 523, w^5 + w^4 - 5*w^3 - 8*w^2 + 3*w + 7],
[541, 541, w^5 + 3*w^4 - 7*w^3 - 16*w^2 + 8*w + 13],
[547, 547, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 7*w - 3],
[557, 557, 3*w^5 - w^4 - 15*w^3 + w^2 + 11*w - 1],
[557, 557, 2*w^5 - w^4 - 11*w^3 + 2*w^2 + 11*w + 1],
[563, 563, w^5 - 3*w^4 - 4*w^3 + 13*w^2 + 3*w - 9],
[569, 569, -w^5 - w^4 + 5*w^3 + 6*w^2 - 4*w - 1],
[571, 571, -3*w^5 + 2*w^4 + 15*w^3 - 7*w^2 - 10*w + 3],
[577, 577, 2*w^5 - 11*w^3 - 2*w^2 + 9*w + 3],
[587, 587, w^4 - 6*w^2 - 2*w + 7],
[599, 599, -2*w^5 + 2*w^4 + 10*w^3 - 7*w^2 - 10*w + 3],
[599, 599, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + 3*w - 7],
[601, 601, 2*w^2 - 5],
[607, 607, -w^5 - w^4 + 4*w^3 + 7*w^2 - 5],
[613, 613, -w^5 + 5*w^3 + w^2 - 3*w + 3],
[617, 617, -2*w^4 + 11*w^2 + 2*w - 9],
[619, 619, -2*w^5 + 10*w^3 + w^2 - 7*w + 1],
[619, 619, -2*w^5 + 4*w^4 + 8*w^3 - 15*w^2 - 5*w + 7],
[643, 643, -2*w^5 - w^4 + 12*w^3 + 6*w^2 - 12*w - 3],
[647, 647, 3*w^5 - w^4 - 18*w^3 + w^2 + 23*w + 3],
[647, 647, 2*w^5 + w^4 - 10*w^3 - 9*w^2 + 9*w + 9],
[653, 653, -w^4 + 7*w^2 + w - 7],
[673, 673, -2*w^4 + w^3 + 8*w^2 - w - 5],
[673, 673, -2*w^5 + 11*w^3 + w^2 - 9*w + 1],
[677, 677, 4*w^5 - w^4 - 20*w^3 + w^2 + 15*w - 3],
[683, 683, 3*w^5 - 2*w^4 - 14*w^3 + 6*w^2 + 11*w - 5],
[683, 683, 2*w^5 - w^4 - 11*w^3 + 3*w^2 + 10*w - 3],
[683, 683, -w^5 + 5*w^3 + w^2 - 4*w + 3],
[691, 691, w^5 + w^4 - 6*w^3 - 6*w^2 + 8*w + 3],
[709, 709, -2*w^5 + 2*w^4 + 11*w^3 - 7*w^2 - 13*w + 3],
[719, 719, 2*w^5 - 2*w^4 - 10*w^3 + 8*w^2 + 9*w - 9],
[719, 719, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 12*w + 3],
[727, 727, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 1],
[729, 3, -3],
[739, 739, 2*w^4 - w^3 - 9*w^2 + 5],
[739, 739, -w^5 + w^4 + 4*w^3 - 3*w^2 - 3*w - 1],
[751, 751, -2*w^4 + w^3 + 9*w^2 - 3*w - 3],
[761, 761, -w^5 - 2*w^4 + 7*w^3 + 12*w^2 - 9*w - 13],
[773, 773, -w^5 + 5*w^3 + 2*w^2 - 2*w - 3],
[787, 787, 2*w^5 - 12*w^3 - 2*w^2 + 13*w + 3],
[787, 787, -2*w^5 + w^4 + 10*w^3 - 3*w^2 - 10*w + 1],
[797, 797, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 7*w + 3],
[811, 811, -2*w^5 + 12*w^3 + w^2 - 15*w - 1],
[821, 821, 4*w^5 + w^4 - 22*w^3 - 8*w^2 + 21*w + 3],
[821, 821, w^5 - 2*w^4 - 3*w^3 + 8*w^2 - 3*w - 3],
[821, 821, -w^5 + 7*w^3 - 9*w - 1],
[827, 827, w^5 - 7*w^3 - w^2 + 10*w + 1],
[829, 829, -2*w^4 + 10*w^2 + w - 7],
[839, 839, -w^5 + 2*w^4 + 6*w^3 - 10*w^2 - 8*w + 9],
[853, 853, -w^5 + w^4 + 3*w^3 - 3*w^2 + 2*w + 1],
[859, 859, -w^5 + 6*w^3 - 9*w + 1],
[859, 859, -2*w^5 + 3*w^4 + 8*w^3 - 10*w^2 - 6*w + 5],
[859, 859, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 2*w - 7],
[859, 859, 2*w^5 - 2*w^4 - 10*w^3 + 6*w^2 + 10*w - 3],
[863, 863, 3*w^5 - 4*w^4 - 14*w^3 + 14*w^2 + 13*w - 7],
[877, 877, w^3 + w^2 - 4*w - 1],
[881, 881, -w^5 + w^4 + 4*w^3 - 4*w^2 - 2*w + 5],
[881, 881, -2*w^5 + 10*w^3 + 4*w^2 - 7*w - 5],
[907, 907, -w^4 + w^3 + 4*w^2 - w + 1],
[937, 937, w^5 - 7*w^3 + w^2 + 11*w - 3],
[941, 941, -2*w^5 + w^4 + 11*w^3 - w^2 - 12*w - 1],
[947, 947, -w^5 + 5*w^3 + 3*w^2 - 5*w - 7],
[953, 953, 2*w^5 - w^4 - 11*w^3 + 4*w^2 + 10*w - 7],
[971, 971, w^5 - w^4 - 5*w^3 + 3*w^2 + 7*w + 1],
[983, 983, -2*w^5 + 9*w^3 + 3*w^2 - 5*w - 3],
[997, 997, 3*w^5 - 18*w^3 - 2*w^2 + 22*w + 1]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 12*x^5 + 22*x^4 - 204*x^3 - 891*x^2 - 1064*x - 240;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, -2/31*e^5 - 29/62*e^4 + 21/31*e^3 + 601/62*e^2 + 467/31*e - 36/31, 3/124*e^5 + 45/124*e^4 + 77/124*e^3 - 939/124*e^2 - 1381/62*e - 281/31, -3/124*e^5 - 45/124*e^4 - 77/124*e^3 + 939/124*e^2 + 1381/62*e + 157/31, -3/124*e^5 - 45/124*e^4 - 77/124*e^3 + 939/124*e^2 + 1443/62*e + 281/31, -9/124*e^5 - 73/124*e^4 + 79/124*e^3 + 1515/124*e^2 + 537/31*e - 56/31, 9/124*e^5 + 73/124*e^4 - 79/124*e^3 - 1515/124*e^2 - 599/31*e - 192/31, -21/124*e^5 - 191/124*e^4 + 19/124*e^3 + 3845/124*e^2 + 2214/31*e + 882/31, 13/124*e^5 + 133/124*e^4 + 3/124*e^3 - 2891/124*e^2 - 2781/62*e - 19/31, -1/124*e^5 - 15/124*e^4 - 5/124*e^3 + 313/124*e^2 + 70/31*e - 51/31, 21/124*e^5 + 191/124*e^4 - 19/124*e^3 - 3845/124*e^2 - 2245/31*e - 882/31, -19/124*e^5 - 161/124*e^4 + 91/124*e^3 + 3343/124*e^2 + 3373/62*e + 395/31, -3/31*e^5 - 59/62*e^4 + 1/62*e^3 + 1227/62*e^2 + 2455/62*e + 225/31, 1/31*e^5 + 15/31*e^4 + 36/31*e^3 - 282/31*e^2 - 1117/31*e - 571/31, -23/124*e^5 - 221/124*e^4 + 9/124*e^3 + 4595/124*e^2 + 2385/31*e + 656/31, 21/124*e^5 + 191/124*e^4 - 19/124*e^3 - 3845/124*e^2 - 2245/31*e - 913/31, 9/124*e^5 + 73/124*e^4 - 17/124*e^3 - 1391/124*e^2 - 1973/62*e - 657/31, -2/31*e^5 - 30/31*e^4 - 41/31*e^3 + 657/31*e^2 + 1645/31*e + 336/31, 15/124*e^5 + 163/124*e^4 + 75/124*e^3 - 3393/124*e^2 - 2042/31*e - 785/31, -9/124*e^5 - 73/124*e^4 + 79/124*e^3 + 1515/124*e^2 + 537/31*e + 68/31, 6/31*e^5 + 59/31*e^4 - 1/31*e^3 - 1258/31*e^2 - 2579/31*e - 419/31, 3/124*e^5 + 45/124*e^4 + 139/124*e^3 - 815/124*e^2 - 1047/31*e - 622/31, 1/124*e^5 + 15/124*e^4 - 57/124*e^3 - 561/124*e^2 + 387/62*e + 671/31, -9/124*e^5 - 73/124*e^4 + 79/124*e^3 + 1515/124*e^2 + 568/31*e - 56/31, 1/62*e^5 + 15/62*e^4 + 5/62*e^3 - 313/62*e^2 - 140/31*e - 53/31, -9/124*e^5 - 73/124*e^4 + 79/124*e^3 + 1515/124*e^2 + 537/31*e + 99/31, -3/62*e^5 - 45/62*e^4 - 77/62*e^3 + 939/62*e^2 + 1443/31*e + 562/31, 13/124*e^5 + 133/124*e^4 + 65/124*e^3 - 2643/124*e^2 - 1654/31*e - 918/31, 3/124*e^5 + 45/124*e^4 + 15/124*e^3 - 1187/124*e^2 - 334/31*e + 618/31, 7/62*e^5 + 43/62*e^4 - 89/62*e^3 - 765/62*e^2 - 670/31*e - 774/31, -1/124*e^5 - 15/124*e^4 - 5/124*e^3 + 437/124*e^2 + 70/31*e - 640/31, -7/124*e^5 - 43/124*e^4 + 89/124*e^3 + 889/124*e^2 + 428/31*e - 78/31, -23/124*e^5 - 221/124*e^4 + 9/124*e^3 + 4595/124*e^2 + 2509/31*e + 811/31, -5/124*e^5 - 75/124*e^4 - 149/124*e^3 + 1565/124*e^2 + 1311/31*e + 489/31, 13/124*e^5 + 71/124*e^4 - 307/124*e^3 - 1589/124*e^2 + 113/31*e + 694/31, 25/124*e^5 + 189/124*e^4 - 247/124*e^3 - 3919/124*e^2 - 1502/31*e - 306/31, 43/124*e^5 + 397/124*e^4 - 95/124*e^3 - 8251/124*e^2 - 8407/62*e - 1093/31, 45/124*e^5 + 427/124*e^4 - 23/124*e^3 - 8753/124*e^2 - 4700/31*e - 1456/31, 2/31*e^5 + 29/62*e^4 - 11/62*e^3 - 477/62*e^2 - 1647/62*e - 863/31, 7/31*e^5 + 117/62*e^4 - 85/62*e^3 - 2429/62*e^2 - 4633/62*e - 587/31, -13/62*e^5 - 133/62*e^4 - 65/62*e^3 + 2643/62*e^2 + 3432/31*e + 1650/31, -7/31*e^5 - 74/31*e^4 - 35/31*e^3 + 1509/31*e^2 + 3696/31*e + 1610/31, 25/62*e^5 + 110/31*e^4 - 46/31*e^3 - 2285/31*e^2 - 9201/62*e - 1263/31, 6/31*e^5 + 59/31*e^4 - 1/31*e^3 - 1258/31*e^2 - 2579/31*e - 140/31, -7/31*e^5 - 117/62*e^4 + 27/31*e^3 + 2305/62*e^2 + 2518/31*e + 1114/31, 45/124*e^5 + 427/124*e^4 - 23/124*e^3 - 8877/124*e^2 - 4824/31*e - 1487/31, 8/31*e^5 + 89/31*e^4 + 111/62*e^3 - 1822/31*e^2 - 8975/62*e - 1871/31, 5/62*e^5 + 22/31*e^4 - 37/62*e^3 - 488/31*e^2 - 700/31*e - 110/31, -15/62*e^5 - 66/31*e^4 + 40/31*e^3 + 1371/31*e^2 + 4727/62*e + 361/31, -13/124*e^5 - 195/124*e^4 - 313/124*e^3 + 4069/124*e^2 + 2832/31*e + 1011/31, -1/31*e^5 - 15/31*e^4 - 36/31*e^3 + 313/31*e^2 + 1117/31*e + 44/31, -6/31*e^5 - 59/31*e^4 + 1/31*e^3 + 1227/31*e^2 + 2579/31*e + 512/31, -1/31*e^5 + 1/62*e^4 + 83/62*e^3 - 25/62*e^2 - 711/62*e - 607/31, -9/62*e^5 - 73/62*e^4 + 17/62*e^3 + 1391/62*e^2 + 1911/31*e + 1190/31, -15/124*e^5 - 101/124*e^4 + 111/124*e^3 + 1843/124*e^2 + 2441/62*e + 909/31, 35/124*e^5 + 277/124*e^4 - 321/124*e^3 - 5871/124*e^2 - 2326/31*e - 168/31, 4/31*e^5 + 29/31*e^4 - 42/31*e^3 - 601/31*e^2 - 872/31*e - 114/31, 1/124*e^5 + 15/124*e^4 + 5/124*e^3 - 561/124*e^2 - 194/31*e + 795/31, -15/124*e^5 - 163/124*e^4 + 49/124*e^3 + 3765/124*e^2 + 1205/31*e - 796/31, -1/62*e^5 - 15/62*e^4 - 5/62*e^3 + 313/62*e^2 + 140/31*e + 270/31, -3/62*e^5 - 45/62*e^4 - 77/62*e^3 + 939/62*e^2 + 1288/31*e + 66/31, 17/62*e^5 + 81/31*e^4 - 39/62*e^3 - 1746/31*e^2 - 3186/31*e - 2/31, -15/124*e^5 - 163/124*e^4 - 75/124*e^3 + 3393/124*e^2 + 1949/31*e + 10/31, -2/31*e^5 - 29/62*e^4 + 73/62*e^3 + 601/62*e^2 + 97/62*e - 67/31, -45/124*e^5 - 427/124*e^4 + 23/124*e^3 + 8753/124*e^2 + 4917/31*e + 2076/31, 25/124*e^5 + 189/124*e^4 - 247/124*e^3 - 3919/124*e^2 - 1409/31*e + 314/31, 21/124*e^5 + 191/124*e^4 - 19/124*e^3 - 3969/124*e^2 - 2245/31*e - 293/31, 17/124*e^5 + 131/124*e^4 - 163/124*e^3 - 2841/124*e^2 - 1128/31*e + 402/31, 3/31*e^5 + 45/31*e^4 + 77/31*e^3 - 939/31*e^2 - 2731/31*e - 876/31, 2/31*e^5 + 29/62*e^4 - 21/31*e^3 - 539/62*e^2 - 436/31*e - 274/31, 13/31*e^5 + 133/31*e^4 + 34/31*e^3 - 2767/31*e^2 - 6337/31*e - 2184/31, -6/31*e^5 - 59/31*e^4 + 1/31*e^3 + 1196/31*e^2 + 2579/31*e + 1349/31, 1/62*e^5 + 15/62*e^4 + 5/62*e^3 - 251/62*e^2 - 140/31*e - 549/31, 1/62*e^5 + 15/62*e^4 + 5/62*e^3 - 313/62*e^2 - 140/31*e - 270/31, 4/31*e^5 + 27/62*e^4 - 135/31*e^3 - 613/62*e^2 + 1081/31*e + 1281/31, 59/124*e^5 + 513/124*e^4 - 263/124*e^3 - 10655/124*e^2 - 10399/62*e - 1331/31, 1/31*e^5 + 61/62*e^4 + 227/62*e^3 - 1153/62*e^2 - 5303/62*e - 1873/31, -11/31*e^5 - 103/31*e^4 + 7/31*e^3 + 2079/31*e^2 + 4754/31*e + 2282/31, 10/31*e^5 + 145/62*e^4 - 105/31*e^3 - 2943/62*e^2 - 2366/31*e - 998/31, 9/124*e^5 + 73/124*e^4 + 45/124*e^3 - 1267/124*e^2 - 1312/31*e - 874/31, -33/124*e^5 - 247/124*e^4 + 331/124*e^3 + 4997/124*e^2 + 2062/31*e + 1014/31, -25/124*e^5 - 189/124*e^4 + 247/124*e^3 + 3919/124*e^2 + 1533/31*e + 58/31, -17/124*e^5 - 193/124*e^4 - 209/124*e^3 + 3895/124*e^2 + 3019/31*e + 1210/31, -3/31*e^5 - 59/62*e^4 - 15/31*e^3 + 1165/62*e^2 + 1739/31*e + 814/31, 12/31*e^5 + 87/31*e^4 - 126/31*e^3 - 1803/31*e^2 - 2802/31*e + 402/31, -51/124*e^5 - 393/124*e^4 + 489/124*e^3 + 8275/124*e^2 + 3260/31*e + 96/31, 9/124*e^5 + 135/124*e^4 + 169/124*e^3 - 2941/124*e^2 - 1591/31*e - 68/31, -5/62*e^5 - 13/62*e^4 + 161/62*e^3 + 263/62*e^2 - 571/31*e - 634/31, -21/124*e^5 - 191/124*e^4 + 81/124*e^3 + 4217/124*e^2 + 4025/62*e - 141/31, -21/124*e^5 - 191/124*e^4 + 19/124*e^3 + 3969/124*e^2 + 2369/31*e + 355/31, -3/31*e^5 - 45/31*e^4 - 77/31*e^3 + 877/31*e^2 + 2638/31*e + 2054/31, 23/62*e^5 + 95/31*e^4 - 133/62*e^3 - 1941/31*e^2 - 3871/31*e - 1560/31, -25/62*e^5 - 110/31*e^4 + 77/31*e^3 + 2347/31*e^2 + 7651/62*e - 163/31, 17/124*e^5 + 193/124*e^4 + 209/124*e^3 - 3771/124*e^2 - 3019/31*e - 1985/31, 1/62*e^5 + 15/62*e^4 + 5/62*e^3 - 437/62*e^2 - 202/31*e + 660/31, -11/124*e^5 - 165/124*e^4 - 303/124*e^3 + 3319/124*e^2 + 2816/31*e + 1888/31, 17/62*e^5 + 81/31*e^4 - 39/62*e^3 - 1684/31*e^2 - 3186/31*e - 622/31, -33/124*e^5 - 247/124*e^4 + 331/124*e^3 + 5245/124*e^2 + 2186/31*e - 288/31, 1/31*e^5 - 1/62*e^4 - 26/31*e^3 + 87/62*e^2 - 156/31*e - 230/31, 5/124*e^5 - 49/124*e^4 - 471/124*e^3 + 915/124*e^2 + 1634/31*e + 999/31, -3/124*e^5 + 17/124*e^4 + 109/124*e^3 - 611/124*e^2 - 7/31*e + 188/31, -37/124*e^5 - 307/124*e^4 + 187/124*e^3 + 6125/124*e^2 + 3179/31*e + 1554/31, -39/124*e^5 - 337/124*e^4 + 239/124*e^3 + 7247/124*e^2 + 6421/62*e - 5/31, -15/31*e^5 - 132/31*e^4 + 49/31*e^3 + 2680/31*e^2 + 5564/31*e + 1900/31, 15/124*e^5 + 101/124*e^4 - 173/124*e^3 - 2091/124*e^2 - 864/31*e - 537/31, 45/124*e^5 + 427/124*e^4 - 85/124*e^3 - 9125/124*e^2 - 8811/62*e - 309/31, 29/62*e^5 + 249/62*e^4 - 103/62*e^3 - 5047/62*e^2 - 5424/31*e - 2064/31, 11/62*e^5 + 103/62*e^4 - 7/62*e^3 - 2017/62*e^2 - 2191/31*e - 1358/31, -41/124*e^5 - 429/124*e^4 - 81/124*e^3 + 9175/124*e^2 + 4823/31*e + 544/31, 13/31*e^5 + 133/31*e^4 + 65/31*e^3 - 2674/31*e^2 - 6864/31*e - 3021/31, -3/62*e^5 - 45/62*e^4 - 77/62*e^3 + 1001/62*e^2 + 1505/31*e + 35/31, -7/62*e^5 - 43/62*e^4 + 89/62*e^3 + 827/62*e^2 + 794/31*e + 743/31, 37/124*e^5 + 307/124*e^4 - 187/124*e^3 - 6125/124*e^2 - 3179/31*e - 1523/31, -35/124*e^5 - 339/124*e^4 - 51/124*e^3 + 7049/124*e^2 + 4093/31*e + 1036/31, -9/31*e^5 - 73/31*e^4 + 79/31*e^3 + 1515/31*e^2 + 2148/31*e + 272/31, -89/124*e^5 - 777/124*e^4 + 423/124*e^3 + 16387/124*e^2 + 7687/31*e + 1134/31, 7/31*e^5 + 74/31*e^4 + 4/31*e^3 - 1602/31*e^2 - 3107/31*e - 494/31, -15/124*e^5 - 101/124*e^4 + 173/124*e^3 + 2215/124*e^2 + 895/31*e - 858/31, -29/62*e^5 - 249/62*e^4 + 67/31*e^3 + 5233/62*e^2 + 10073/62*e + 359/31, -16/31*e^5 - 147/31*e^4 + 13/31*e^3 + 2993/31*e^2 + 6929/31*e + 2595/31, -35/124*e^5 - 401/124*e^4 - 299/124*e^3 + 8227/124*e^2 + 5023/31*e + 1656/31, 7/31*e^5 + 117/62*e^4 - 85/62*e^3 - 2305/62*e^2 - 4323/62*e - 1517/31, 3/62*e^5 - 17/62*e^4 - 171/62*e^3 + 487/62*e^2 + 851/31*e - 810/31, -49/124*e^5 - 487/124*e^4 + 3/124*e^3 + 10377/124*e^2 + 5104/31*e + 229/31, 75/124*e^5 + 629/124*e^4 - 493/124*e^3 - 13183/124*e^2 - 5715/31*e - 887/31, -6/31*e^5 - 59/31*e^4 + 32/31*e^3 + 1320/31*e^2 + 1742/31*e - 1193/31, -3/124*e^5 + 17/124*e^4 + 109/124*e^3 - 611/124*e^2 - 193/31*e + 312/31, 17/62*e^5 + 131/62*e^4 - 163/62*e^3 - 2593/62*e^2 - 2008/31*e - 932/31, 75/124*e^5 + 629/124*e^4 - 493/124*e^3 - 13183/124*e^2 - 5715/31*e - 577/31, 23/62*e^5 + 221/62*e^4 - 9/62*e^3 - 4595/62*e^2 - 4770/31*e - 444/31, -9/124*e^5 - 135/124*e^4 - 169/124*e^3 + 3189/124*e^2 + 1715/31*e - 490/31, 25/62*e^5 + 189/62*e^4 - 247/62*e^3 - 3919/62*e^2 - 2942/31*e - 178/31, 14/31*e^5 + 117/31*e^4 - 85/31*e^3 - 2429/31*e^2 - 4478/31*e - 1174/31, -39/124*e^5 - 337/124*e^4 + 177/124*e^3 + 6999/124*e^2 + 3567/31*e + 1018/31, -13/62*e^5 - 71/62*e^4 + 245/62*e^3 + 1403/62*e^2 + 363/31*e - 303/31, -17/124*e^5 - 69/124*e^4 + 411/124*e^3 + 1415/124*e^2 - 174/31*e - 898/31, 9/31*e^5 + 104/31*e^4 + 76/31*e^3 - 2166/31*e^2 - 5341/31*e - 2318/31, 17/62*e^5 + 81/31*e^4 + 27/31*e^3 - 1622/31*e^2 - 8883/62*e - 1831/31, -9/62*e^5 - 73/62*e^4 + 79/62*e^3 + 1577/62*e^2 + 1322/31*e + 570/31, 47/62*e^5 + 213/31*e^4 - 137/62*e^3 - 4457/31*e^2 - 8998/31*e - 2181/31, 1/62*e^5 - 8/31*e^4 - 119/62*e^3 + 169/31*e^2 + 1038/31*e + 970/31, -43/124*e^5 - 521/124*e^4 - 463/124*e^3 + 10731/124*e^2 + 6792/31*e + 3108/31, -23/124*e^5 - 159/124*e^4 + 257/124*e^3 + 3045/124*e^2 + 1238/31*e + 1214/31, 19/62*e^5 + 223/62*e^4 + 157/62*e^3 - 4645/62*e^2 - 5419/31*e - 1596/31, -15/124*e^5 - 101/124*e^4 + 173/124*e^3 + 2091/124*e^2 + 740/31*e + 196/31, -33/62*e^5 - 139/31*e^4 + 88/31*e^3 + 2886/31*e^2 + 11317/62*e + 1625/31, -37/62*e^5 - 369/62*e^4 - 61/62*e^3 + 7613/62*e^2 + 8714/31*e + 2953/31, 5/124*e^5 + 75/124*e^4 + 25/124*e^3 - 1937/124*e^2 - 598/31*e + 1154/31, -49/124*e^5 - 487/124*e^4 - 245/124*e^3 + 9633/124*e^2 + 6778/31*e + 3515/31, 7/62*e^5 + 43/62*e^4 - 89/62*e^3 - 765/62*e^2 - 732/31*e - 278/31, 13/62*e^5 + 133/62*e^4 + 3/62*e^3 - 2829/62*e^2 - 2719/31*e - 410/31, 33/62*e^5 + 139/31*e^4 - 88/31*e^3 - 2886/31*e^2 - 11193/62*e - 1377/31, -9/31*e^5 - 73/31*e^4 + 17/31*e^3 + 1422/31*e^2 + 4070/31*e + 2070/31, 1/31*e^5 - 16/31*e^4 - 88/31*e^3 + 400/31*e^2 + 991/31*e - 292/31, -11/62*e^5 - 36/31*e^4 + 131/62*e^3 + 714/31*e^2 + 1106/31*e + 118/31, -9/31*e^5 - 104/31*e^4 - 76/31*e^3 + 2166/31*e^2 + 5093/31*e + 1264/31, 10/31*e^5 + 88/31*e^4 - 43/31*e^3 - 1859/31*e^2 - 3513/31*e - 533/31, -9/62*e^5 - 73/62*e^4 + 79/62*e^3 + 1577/62*e^2 + 1322/31*e + 229/31, 15/62*e^5 + 163/62*e^4 + 199/62*e^3 - 3021/62*e^2 - 5386/31*e - 3058/31, 55/124*e^5 + 577/124*e^4 + 337/124*e^3 - 11759/124*e^2 - 15357/62*e - 3643/31, -3/62*e^5 - 38/31*e^4 - 85/31*e^3 + 857/31*e^2 + 4033/62*e - 213/31];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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