/*
  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, 0, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 4, -w^2 + 1}>;

primesArray := [
[2, 2, w],
[4, 2, -w + 1],
[5, 5, -3/2*w^3 + 1/2*w^2 + 10*w - 4],
[5, 5, 3/2*w^3 + 1/2*w^2 - 10*w - 4],
[25, 5, -1/2*w^3 + 1/2*w^2 + 4*w - 4],
[37, 37, -1/2*w^3 - 1/2*w^2 + 4*w],
[37, 37, 1/2*w^3 - 1/2*w^2 - 4*w],
[43, 43, -3/2*w^3 + 1/2*w^2 + 10*w - 2],
[43, 43, -3/2*w^3 - 1/2*w^2 + 10*w + 2],
[49, 7, -w^2 - w + 3],
[49, 7, -w^2 + w + 3],
[59, 59, 1/2*w^3 + 1/2*w^2 - 5*w - 2],
[59, 59, -1/2*w^3 + 1/2*w^2 + 5*w - 2],
[61, 61, -1/2*w^3 + 1/2*w^2 + 3*w + 2],
[61, 61, 3/2*w^3 + 1/2*w^2 - 8*w - 4],
[73, 73, -1/2*w^3 - 3/2*w^2 + 2*w],
[73, 73, w^2 - w + 1],
[73, 73, w^2 + w + 1],
[73, 73, 1/2*w^3 - 3/2*w^2 - 2*w],
[81, 3, -3],
[83, 83, w^2 - w - 5],
[83, 83, w^2 + w - 5],
[103, 103, -w^3 + 5*w + 1],
[103, 103, -3/2*w^3 + 1/2*w^2 + 9*w - 2],
[103, 103, -3/2*w^3 - 1/2*w^2 + 9*w + 2],
[103, 103, w^3 - 5*w + 1],
[107, 107, 5/2*w^3 + 1/2*w^2 - 17*w - 2],
[107, 107, w^3 - 3*w + 1],
[113, 113, -w^3 + w^2 + 6*w - 7],
[113, 113, -1/2*w^3 - 1/2*w^2 + w + 2],
[113, 113, -1/2*w^3 + 1/2*w^2 + w - 2],
[113, 113, -3/2*w^3 + 3/2*w^2 + 9*w - 8],
[131, 131, 5/2*w^3 - 1/2*w^2 - 16*w + 2],
[131, 131, 5/2*w^3 + 1/2*w^2 - 16*w - 2],
[139, 139, 2*w - 3],
[139, 139, -1/2*w^3 + 1/2*w^2 + 5*w - 4],
[163, 163, 1/2*w^3 + 1/2*w^2 - 2*w - 4],
[163, 163, -1/2*w^3 + 1/2*w^2 + 2*w - 4],
[173, 173, -3/2*w^3 - 1/2*w^2 + 11*w],
[173, 173, 3/2*w^3 - 1/2*w^2 - 11*w],
[197, 197, -1/2*w^3 - 1/2*w^2 + 3*w + 6],
[197, 197, -1/2*w^3 + 1/2*w^2 + 3*w - 6],
[223, 223, -1/2*w^3 - 5/2*w^2 + 2],
[223, 223, 7/2*w^3 + 1/2*w^2 - 24*w - 4],
[223, 223, 7/2*w^3 - 1/2*w^2 - 24*w + 4],
[223, 223, 1/2*w^3 - 5/2*w^2 + 2],
[251, 251, 3/2*w^3 - 3/2*w^2 - 6*w + 4],
[251, 251, 3/2*w^3 + 3/2*w^2 - 6*w - 4],
[269, 269, w^3 - w^2 - 6*w + 1],
[269, 269, -w^3 - w^2 + 6*w + 1],
[277, 277, 1/2*w^3 + 3/2*w^2 - 4*w - 4],
[277, 277, -1/2*w^3 + 3/2*w^2 + 4*w - 4],
[283, 283, -1/2*w^3 + 3/2*w^2 + 2*w - 4],
[283, 283, 1/2*w^3 + 3/2*w^2 - 2*w - 4],
[289, 17, -3/2*w^3 - 1/2*w^2 + 8*w],
[289, 17, 3/2*w^3 - 1/2*w^2 - 8*w],
[307, 307, -1/2*w^3 - 1/2*w^2 + 6*w + 2],
[307, 307, -1/2*w^3 + 1/2*w^2 + 6*w - 2],
[349, 349, w^3 + w^2 - 8*w - 9],
[349, 349, -w^3 + w^2 + 8*w - 9],
[359, 359, -3/2*w^3 + 1/2*w^2 + 8*w - 2],
[359, 359, -1/2*w^3 + 3/2*w^2 + 4*w - 6],
[359, 359, 1/2*w^3 + 3/2*w^2 - 4*w - 6],
[359, 359, -3/2*w^3 - 1/2*w^2 + 8*w + 2],
[373, 373, -1/2*w^3 + 5/2*w^2 + 4*w - 18],
[373, 373, -2*w^2 - 2*w + 3],
[379, 379, 1/2*w^3 - 1/2*w^2 + w + 2],
[379, 379, -1/2*w^3 - 1/2*w^2 - w + 2],
[389, 389, 1/2*w^3 - 1/2*w^2 + 2*w + 2],
[389, 389, -1/2*w^3 - 1/2*w^2 - 2*w + 2],
[401, 401, -4*w + 1],
[401, 401, -3/2*w^3 + 3/2*w^2 + 10*w - 12],
[401, 401, -5/2*w^3 + 5/2*w^2 + 16*w - 14],
[401, 401, -3/2*w^3 + 3/2*w^2 + 5*w - 2],
[419, 419, w^3 + 2*w^2 - 7*w - 5],
[419, 419, w^3 + w^2 - 10*w - 5],
[443, 443, 1/2*w^3 + 3/2*w^2 - 3*w - 6],
[443, 443, 1/2*w^3 - 3/2*w^2 - 3*w + 6],
[449, 449, -5/2*w^3 - 1/2*w^2 + 15*w - 2],
[449, 449, w^3 + w^2 - 6*w + 1],
[449, 449, -1/2*w^3 - 3/2*w^2 + 7*w - 2],
[449, 449, -5/2*w^3 + 1/2*w^2 + 15*w + 2],
[461, 461, 2*w^3 - 12*w - 1],
[461, 461, -2*w^3 + 12*w - 1],
[467, 467, 3/2*w^3 - 1/2*w^2 - 5*w + 2],
[467, 467, 3/2*w^3 + 1/2*w^2 - 5*w - 2],
[487, 487, -5/2*w^3 - 3/2*w^2 + 16*w + 6],
[487, 487, 2*w^3 - w^2 - 11*w + 7],
[487, 487, 2*w^3 + w^2 - 11*w - 7],
[487, 487, -5/2*w^3 + 3/2*w^2 + 16*w - 6],
[491, 491, -w^3 - 2*w^2 + 7*w + 11],
[491, 491, -w^3 + 2*w^2 + 7*w - 11],
[523, 523, 3*w^3 - 21*w - 1],
[523, 523, 3*w^3 - 21*w + 1],
[529, 23, 2*w^2 - 15],
[529, 23, 5/2*w^3 - 5/2*w^2 - 16*w + 16],
[541, 541, -3/2*w^3 - 1/2*w^2 + 10*w - 4],
[541, 541, 3/2*w^3 - 1/2*w^2 - 10*w - 4],
[569, 569, 3/2*w^3 - 1/2*w^2 - 7*w + 2],
[569, 569, 5/2*w^3 - 1/2*w^2 - 15*w],
[569, 569, -5/2*w^3 - 1/2*w^2 + 15*w],
[569, 569, -1/2*w^3 - 1/2*w^2 + 7*w - 2],
[613, 613, 4*w^3 - 26*w + 1],
[613, 613, 4*w^3 - 26*w - 1],
[617, 617, 7/2*w^3 + 1/2*w^2 - 23*w - 6],
[617, 617, 3/2*w^3 - 1/2*w^2 - 6*w + 4],
[617, 617, 3/2*w^3 + 1/2*w^2 - 6*w - 4],
[617, 617, 7/2*w^3 - 1/2*w^2 - 23*w + 6],
[619, 619, 4*w^3 - w^2 - 27*w + 9],
[619, 619, -4*w^3 - w^2 + 27*w + 9],
[631, 631, -3/2*w^3 - 3/2*w^2 + 11*w + 4],
[631, 631, 1/2*w^3 + 3/2*w^2 - w - 6],
[631, 631, -1/2*w^3 + 3/2*w^2 + w - 6],
[631, 631, -3/2*w^3 + 3/2*w^2 + 11*w - 4],
[647, 647, 3/2*w^3 + 1/2*w^2 - 10*w + 2],
[647, 647, -3/2*w^3 + 1/2*w^2 + 11*w + 2],
[647, 647, 3/2*w^3 + 1/2*w^2 - 11*w + 2],
[647, 647, -3/2*w^3 + 1/2*w^2 + 10*w + 2],
[661, 661, 11/2*w^3 - 1/2*w^2 - 36*w + 6],
[661, 661, -11/2*w^3 - 1/2*w^2 + 36*w + 6],
[677, 677, -1/2*w^3 - 1/2*w^2 + 2*w - 4],
[677, 677, 1/2*w^3 - 1/2*w^2 - 2*w - 4],
[701, 701, -1/2*w^3 - 3/2*w^2 + 5*w + 8],
[701, 701, -1/2*w^3 + 3/2*w^2 + 5*w - 8],
[733, 733, -1/2*w^3 + 5/2*w^2 + w - 10],
[733, 733, -1/2*w^3 - 5/2*w^2 + w + 10],
[739, 739, 7/2*w^3 - 3/2*w^2 - 22*w + 8],
[739, 739, 5/2*w^3 - 1/2*w^2 - 18*w + 4],
[743, 743, w^3 - 9*w + 1],
[743, 743, 3/2*w^3 + 3/2*w^2 - 13*w - 4],
[743, 743, -3/2*w^3 + 3/2*w^2 + 13*w - 4],
[743, 743, -w^3 + 9*w + 1],
[761, 761, 5*w^3 - 2*w^2 - 33*w + 11],
[761, 761, 3*w^2 - w - 17],
[761, 761, -3*w^2 - w + 17],
[761, 761, -5*w^3 - 2*w^2 + 33*w + 11],
[769, 769, -5/2*w^3 + 3/2*w^2 + 17*w - 6],
[769, 769, w^3 - w^2 - 8*w - 1],
[769, 769, -w^3 - w^2 + 8*w - 1],
[769, 769, 5/2*w^3 - 3/2*w^2 - 15*w + 4],
[787, 787, 1/2*w^3 - 1/2*w^2 - 3*w - 4],
[787, 787, -1/2*w^3 - 1/2*w^2 + 3*w - 4],
[797, 797, -5/2*w^3 + 1/2*w^2 + 14*w - 6],
[797, 797, -5/2*w^3 - 1/2*w^2 + 14*w + 6],
[811, 811, -1/2*w^3 + 5/2*w^2 + 2*w - 14],
[811, 811, 1/2*w^3 + 5/2*w^2 - 2*w - 14],
[821, 821, 13/2*w^3 - 1/2*w^2 - 43*w + 4],
[821, 821, -3/2*w^3 + 5/2*w^2 + 6*w - 2],
[829, 829, -w^3 + w^2 + 4*w - 5],
[829, 829, w^3 + w^2 - 4*w - 5],
[853, 853, -5/2*w^3 + 1/2*w^2 + 17*w],
[853, 853, 5/2*w^3 + 1/2*w^2 - 17*w],
[859, 859, -3/2*w^3 - 1/2*w^2 + 12*w + 2],
[859, 859, 3/2*w^3 - 1/2*w^2 - 12*w + 2],
[877, 877, -5*w^3 - w^2 + 34*w + 9],
[877, 877, 5*w^3 - w^2 - 34*w + 9],
[907, 907, 3/2*w^3 - 1/2*w^2 - 6*w],
[907, 907, -3/2*w^3 - 1/2*w^2 + 6*w],
[941, 941, 3/2*w^3 + 3/2*w^2 - 7*w + 4],
[941, 941, 3/2*w^3 - 3/2*w^2 - 7*w - 4],
[947, 947, -2*w^3 + w^2 + 15*w - 1],
[947, 947, -2*w^3 - w^2 + 15*w + 1],
[961, 31, -3/2*w^3 + 3/2*w^2 + 8*w - 8],
[961, 31, 1/2*w^3 + 1/2*w^2 - 6*w - 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - x^5 - 22*x^4 + 12*x^3 + 93*x^2 + 73*x + 13;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-88/377*e^5 + 14/29*e^4 + 1793/377*e^3 - 2877/377*e^2 - 5822/377*e - 112/29, 0, e, -119/377*e^5 + 13/29*e^4 + 2566/377*e^3 - 2541/377*e^2 - 10332/377*e - 307/29, 138/377*e^5 - 18/29*e^4 - 2906/377*e^3 + 3612/377*e^2 + 10518/377*e + 347/29, 398/377*e^5 - 62/29*e^4 - 8015/377*e^3 + 12712/377*e^2 + 25663/377*e + 438/29, -20/377*e^5 - 10/29*e^4 + 596/377*e^3 + 2345/377*e^2 - 4819/377*e - 413/29, -11/29*e^5 + 30/29*e^4 + 206/29*e^3 - 501/29*e^2 - 387/29*e + 108/29, -297/377*e^5 + 40/29*e^4 + 6287/377*e^3 - 7872/377*e^2 - 24079/377*e - 726/29, -540/377*e^5 + 78/29*e^4 + 10814/377*e^3 - 16232/377*e^2 - 32847/377*e - 653/29, 107/377*e^5 - 19/29*e^4 - 1756/377*e^3 + 4325/377*e^2 + 730/377*e - 312/29, 190/377*e^5 - 21/29*e^4 - 3777/377*e^3 + 4301/377*e^2 + 12416/377*e + 226/29, -730/377*e^5 + 99/29*e^4 + 14968/377*e^3 - 20156/377*e^2 - 50918/377*e - 1256/29, 162/377*e^5 - 35/29*e^4 - 3018/377*e^3 + 7584/377*e^2 + 5971/377*e - 155/29, 197/377*e^5 - 32/29*e^4 - 4061/377*e^3 + 6402/377*e^2 + 14310/377*e + 372/29, 437/377*e^5 - 57/29*e^4 - 8574/377*e^3 + 11815/377*e^2 + 24636/377*e + 630/29, -651/377*e^5 + 95/29*e^4 + 13217/377*e^3 - 19334/377*e^2 - 43061/377*e - 1050/29, -53/377*e^5 + 17/29*e^4 + 1127/377*e^3 - 3682/377*e^2 - 3515/377*e + 38/29, -808/377*e^5 + 118/29*e^4 + 16086/377*e^3 - 24394/377*e^2 - 48487/377*e - 799/29, -43/377*e^5 - 7/29*e^4 + 1206/377*e^3 + 1743/377*e^2 - 7703/377*e - 495/29, 414/377*e^5 - 54/29*e^4 - 8718/377*e^3 + 10836/377*e^2 + 32308/377*e + 867/29, 176/377*e^5 - 28/29*e^4 - 3586/377*e^3 + 5754/377*e^2 + 10890/377*e + 253/29, -787/377*e^5 + 114/29*e^4 + 15988/377*e^3 - 23746/377*e^2 - 51099/377*e - 825/29, 161/377*e^5 - 21/29*e^4 - 3139/377*e^3 + 3837/377*e^2 + 8878/377*e + 429/29, -211/377*e^5 + 25/29*e^4 + 4252/377*e^3 - 4572/377*e^2 - 13574/377*e - 722/29, -236/377*e^5 + 27/29*e^4 + 5374/377*e^3 - 4751/377*e^2 - 24216/377*e - 941/29, 34/13*e^5 - 5*e^4 - 683/13*e^3 + 1038/13*e^2 + 2172/13*e + 39, -639/377*e^5 + 72/29*e^4 + 13161/377*e^3 - 14709/377*e^2 - 46277/377*e - 1069/29, -50/377*e^5 + 4/29*e^4 + 1113/377*e^3 - 735/377*e^2 - 4696/377*e + 113/29, -24/13*e^5 + 3*e^4 + 502/13*e^3 - 592/13*e^2 - 1849/13*e - 51, 175/377*e^5 - 14/29*e^4 - 3707/377*e^3 + 2384/377*e^2 + 15682/377*e + 750/29, -50/377*e^5 + 4/29*e^4 + 1113/377*e^3 - 735/377*e^2 - 4696/377*e + 113/29, 199/377*e^5 - 31/29*e^4 - 4196/377*e^3 + 5979/377*e^2 + 14528/377*e + 683/29, 724/377*e^5 - 102/29*e^4 - 14940/377*e^3 + 21048/377*e^2 + 51395/377*e + 1164/29, 85/377*e^5 - 1/29*e^4 - 1779/377*e^3 - 70/377*e^2 + 6626/377*e + 530/29, -394/377*e^5 + 64/29*e^4 + 7745/377*e^3 - 13181/377*e^2 - 22211/377*e - 251/29, -344/377*e^5 + 31/29*e^4 + 7386/377*e^3 - 6037/377*e^2 - 29579/377*e - 857/29, 396/377*e^5 - 63/29*e^4 - 7880/377*e^3 + 13135/377*e^2 + 24691/377*e + 388/29, -504/377*e^5 + 67/29*e^4 + 10269/377*e^3 - 13667/377*e^2 - 32316/377*e - 855/29, -298/377*e^5 + 25/29*e^4 + 6920/377*e^3 - 4456/377*e^2 - 33236/377*e - 1215/29, 757/377*e^5 - 129/29*e^4 - 15094/377*e^3 + 27075/377*e^2 + 45190/377*e + 394/29, 470/377*e^5 - 84/29*e^4 - 9482/377*e^3 + 17088/377*e^2 + 29741/377*e + 527/29, 20/29*e^5 - 15/29*e^4 - 451/29*e^3 + 207/29*e^2 + 1977/29*e + 758/29, 764/377*e^5 - 111/29*e^4 - 15755/377*e^3 + 22767/377*e^2 + 53493/377*e + 946/29, 561/377*e^5 - 82/29*e^4 - 11666/377*e^3 + 16503/377*e^2 + 40414/377*e + 859/29, 57/377*e^5 + 14/29*e^4 - 1774/377*e^3 - 3573/377*e^2 + 12622/377*e + 671/29, 347/377*e^5 - 44/29*e^4 - 7777/377*e^3 + 8230/377*e^2 + 34053/377*e + 932/29, 959/377*e^5 - 144/29*e^4 - 19304/377*e^3 + 29969/377*e^2 + 59291/377*e + 746/29, -114/377*e^5 + 1/29*e^4 + 3171/377*e^3 + 360/377*e^2 - 20343/377*e - 675/29, -738/377*e^5 + 95/29*e^4 + 15131/377*e^3 - 19218/377*e^2 - 49905/377*e - 1166/29, 12/377*e^5 - 23/29*e^4 + 321/377*e^3 + 4625/377*e^2 - 8871/377*e - 309/29, 499/377*e^5 - 84/29*e^4 - 10120/377*e^3 + 17929/377*e^2 + 33656/377*e + 92/29, 944/377*e^5 - 137/29*e^4 - 18857/377*e^3 + 29183/377*e^2 + 57279/377*e + 835/29, -599/377*e^5 + 92/29*e^4 + 11592/377*e^3 - 20153/377*e^2 - 30984/377*e + 366/29, 83/377*e^5 - 2/29*e^4 - 2398/377*e^3 - 401/377*e^2 + 15456/377*e + 1205/29, 1052/377*e^5 - 141/29*e^4 - 21623/377*e^3 + 28961/377*e^2 + 72821/377*e + 1940/29, -1258/377*e^5 + 212/29*e^4 + 24972/377*e^3 - 44581/377*e^2 - 71147/377*e - 449/29, -42/13*e^5 + 6*e^4 + 885/13*e^3 - 1192/13*e^2 - 3291/13*e - 80, 673/377*e^5 - 113/29*e^4 - 13948/377*e^3 + 22598/377*e^2 + 48852/377*e + 788/29, 41/13*e^5 - 7*e^4 - 811/13*e^3 + 1475/13*e^2 + 2324/13*e + 5, -1808/377*e^5 + 256/29*e^4 + 36461/377*e^3 - 53420/377*e^2 - 115263/377*e - 2077/29, -1150/377*e^5 + 208/29*e^4 + 22583/377*e^3 - 44049/377*e^2 - 63145/377*e - 11/29, -695/377*e^5 + 131/29*e^4 + 13925/377*e^3 - 26616/377*e^2 - 42956/377*e - 584/29, 716/377*e^5 - 106/29*e^4 - 13646/377*e^3 + 23117/377*e^2 + 34689/377*e - 196/29, -48/377*e^5 + 5/29*e^4 + 601/377*e^3 - 1535/377*e^2 + 1931/377*e + 163/29, 921/377*e^5 - 134/29*e^4 - 18624/377*e^3 + 27827/377*e^2 + 59296/377*e + 898/29, 110/377*e^5 - 32/29*e^4 - 2147/377*e^3 + 6518/377*e^2 + 3319/377*e + 198/29, 1410/377*e^5 - 194/29*e^4 - 29200/377*e^3 + 39577/377*e^2 + 102795/377*e + 2451/29, -345/377*e^5 + 74/29*e^4 + 6134/377*e^3 - 16193/377*e^2 - 9330/377*e + 771/29, 386/377*e^5 - 39/29*e^4 - 7959/377*e^3 + 8087/377*e^2 + 26617/377*e + 892/29, 389/377*e^5 - 23/29*e^4 - 8350/377*e^3 + 5002/377*e^2 + 33353/377*e + 967/29, 713/377*e^5 - 93/29*e^4 - 14009/377*e^3 + 20170/377*e^2 + 41525/377*e - 10/29, -1869/377*e^5 + 269/29*e^4 + 37751/377*e^3 - 56164/377*e^2 - 120781/377*e - 2268/29, -1128/377*e^5 + 190/29*e^4 + 22229/377*e^3 - 40408/377*e^2 - 64894/377*e - 99/29, -695/377*e^5 + 102/29*e^4 + 13171/377*e^3 - 21338/377*e^2 - 33531/377*e - 584/29, 1949/377*e^5 - 258/29*e^4 - 39758/377*e^3 + 52439/377*e^2 + 131763/377*e + 3108/29, -1212/377*e^5 + 177/29*e^4 + 23752/377*e^3 - 37345/377*e^2 - 68772/377*e - 401/29, 1945/377*e^5 - 260/29*e^4 - 39488/377*e^3 + 53662/377*e^2 + 129065/377*e + 3095/29, -589/377*e^5 + 68/29*e^4 + 12048/377*e^3 - 13597/377*e^2 - 42335/377*e - 863/29, 1047/377*e^5 - 129/29*e^4 - 21474/377*e^3 + 27191/377*e^2 + 71899/377*e + 1148/29, -699/377*e^5 + 129/29*e^4 + 13064/377*e^3 - 28409/377*e^2 - 29443/377*e + 592/29, 1167/377*e^5 - 156/29*e^4 - 23919/377*e^3 + 31594/377*e^2 + 81963/377*e + 2321/29, 428/377*e^5 - 47/29*e^4 - 8909/377*e^3 + 10137/377*e^2 + 32326/377*e + 579/29, -706/377*e^5 + 111/29*e^4 + 14102/377*e^3 - 23724/377*e^2 - 43778/377*e - 163/29, -258/377*e^5 + 16/29*e^4 + 6105/377*e^3 - 3114/377*e^2 - 30384/377*e - 1027/29, -21/29*e^5 + 23/29*e^4 + 446/29*e^3 - 300/29*e^2 - 1593/29*e - 1257/29, 1213/377*e^5 - 162/29*e^4 - 24385/377*e^3 + 33929/377*e^2 + 75667/377*e + 1557/29, 860/377*e^5 - 121/29*e^4 - 18465/377*e^3 + 23198/377*e^2 + 73005/377*e + 2360/29, 1390/377*e^5 - 233/29*e^4 - 27473/377*e^3 + 49462/377*e^2 + 77241/377*e + 443/29, -30/13*e^5 + 5*e^4 + 569/13*e^3 - 1078/13*e^2 - 1372/13*e + 13, -191/377*e^5 + 35/29*e^4 + 4033/377*e^3 - 6917/377*e^2 - 14410/377*e - 715/29, -777/377*e^5 + 119/29*e^4 + 15690/377*e^3 - 24730/377*e^2 - 49632/377*e - 1126/29, -971/377*e^5 + 167/29*e^4 + 18606/377*e^3 - 34971/377*e^2 - 48912/377*e - 350/29, 1110/377*e^5 - 112/29*e^4 - 23276/377*e^3 + 21972/377*e^2 + 85175/377*e + 2781/29, -1280/377*e^5 + 172/29*e^4 + 26834/377*e^3 - 34650/377*e^2 - 95788/377*e - 2391/29, -7/377*e^5 + 11/29*e^4 - 93/377*e^3 - 2478/377*e^2 + 3007/377*e + 231/29, 1202/377*e^5 - 211/29*e^4 - 23831/377*e^3 + 43607/377*e^2 + 68059/377*e + 789/29, 117/29*e^5 - 240/29*e^4 - 2402/29*e^3 + 3805/29*e^2 + 8142/29*e + 1630/29, 41/377*e^5 + 6/29*e^4 - 1448/377*e^3 - 2828/377*e^2 + 12763/377*e + 938/29, 692/377*e^5 - 89/29*e^4 - 13911/377*e^3 + 18014/377*e^2 + 47530/377*e + 1205/29, -1839/377*e^5 + 226/29*e^4 + 38365/377*e^3 - 45167/377*e^2 - 139377/377*e - 3751/29, 568/377*e^5 - 93/29*e^4 - 11196/377*e^3 + 20489/377*e^2 + 30998/377*e - 532/29, -432/377*e^5 + 74/29*e^4 + 9556/377*e^3 - 14569/377*e^2 - 39171/377*e - 795/29, -2144/377*e^5 + 320/29*e^4 + 43307/377*e^3 - 66050/377*e^2 - 136807/377*e - 2328/29, -1007/377*e^5 + 149/29*e^4 + 21036/377*e^3 - 30373/377*e^2 - 75833/377*e - 1337/29, 4*e^5 - 7*e^4 - 82*e^3 + 108*e^2 + 282*e + 113, -678/377*e^5 + 67/29*e^4 + 14097/377*e^3 - 12681/377*e^2 - 52790/377*e - 1435/29, -891/377*e^5 + 149/29*e^4 + 17730/377*e^3 - 30779/377*e^2 - 51125/377*e - 583/29, 788/377*e^5 - 99/29*e^4 - 15867/377*e^3 + 20330/377*e^2 + 51208/377*e + 1372/29, -1197/377*e^5 + 170/29*e^4 + 24059/377*e^3 - 35051/377*e^2 - 76185/377*e - 1302/29, 828/377*e^5 - 137/29*e^4 - 16682/377*e^3 + 27704/377*e^2 + 55568/377*e + 1299/29, 823/377*e^5 - 125/29*e^4 - 17287/377*e^3 + 25934/377*e^2 + 61809/377*e + 913/29, 1375/377*e^5 - 197/29*e^4 - 28157/377*e^3 + 39628/377*e^2 + 93325/377*e + 2301/29, -8/13*e^5 - e^4 + 215/13*e^3 + 275/13*e^2 - 1418/13*e - 79, -628/377*e^5 + 63/29*e^4 + 12984/377*e^3 - 12700/377*e^2 - 45832/377*e - 1490/29, -357/377*e^5 + 39/29*e^4 + 8452/377*e^3 - 7246/377*e^2 - 41929/377*e - 1356/29, -1306/377*e^5 + 188/29*e^4 + 26327/377*e^3 - 38576/377*e^2 - 81280/377*e - 1794/29, 1058/377*e^5 - 167/29*e^4 - 20897/377*e^3 + 34855/377*e^2 + 61788/377*e + 640/29, -359/377*e^5 + 67/29*e^4 + 7456/377*e^3 - 14363/377*e^2 - 25559/377*e + 450/29, -911/377*e^5 + 139/29*e^4 + 18326/377*e^3 - 28057/377*e^2 - 57075/377*e - 938/29, 1119/377*e^5 - 209/29*e^4 - 21810/377*e^3 + 44008/377*e^2 + 60143/377*e + 193/29, 414/377*e^5 - 112/29*e^4 - 7587/377*e^3 + 23654/377*e^2 + 14212/377*e - 525/29, -977/377*e^5 + 106/29*e^4 + 20519/377*e^3 - 21261/377*e^2 - 75579/377*e - 1428/29, 648/377*e^5 - 111/29*e^4 - 12449/377*e^3 + 23550/377*e^2 + 33686/377*e + 772/29, 867/377*e^5 - 103/29*e^4 - 17995/377*e^3 + 20775/377*e^2 + 64720/377*e + 1259/29, -61/29*e^5 + 111/29*e^4 + 1232/29*e^3 - 1758/29*e^2 - 4068/29*e - 1468/29, 840/377*e^5 - 131/29*e^4 - 17492/377*e^3 + 27051/377*e^2 + 59515/377*e + 1570/29, 1772/377*e^5 - 245/29*e^4 - 36670/377*e^3 + 49347/377*e^2 + 127173/377*e + 3729/29, 12/29*e^5 - 38/29*e^4 - 259/29*e^3 + 710/29*e^2 + 844/29*e - 827/29, -2005/377*e^5 + 259/29*e^4 + 40899/377*e^3 - 52659/377*e^2 - 135982/377*e - 3058/29, 1141/377*e^5 - 169/29*e^4 - 22164/377*e^3 + 35208/377*e^2 + 62164/377*e + 1265/29, 1269/377*e^5 - 163/29*e^4 - 26280/377*e^3 + 31510/377*e^2 + 92327/377*e + 3073/29, -2671/377*e^5 + 390/29*e^4 + 53809/377*e^3 - 81073/377*e^2 - 170876/377*e - 3004/29, -1003/377*e^5 + 93/29*e^4 + 21520/377*e^3 - 17270/377*e^2 - 88592/377*e - 2861/29, 1633/377*e^5 - 213/29*e^4 - 33885/377*e^3 + 42365/377*e^2 + 123332/377*e + 3676/29, 1011/377*e^5 - 118/29*e^4 - 20175/377*e^3 + 25003/377*e^2 + 63828/377*e + 973/29, 1/29*e^5 - 8/29*e^4 - 24/29*e^3 + 93/29*e^2 + 109/29*e + 673/29, -1104/377*e^5 + 173/29*e^4 + 21363/377*e^3 - 36436/377*e^2 - 57754/377*e - 21/29, 1409/377*e^5 - 180/29*e^4 - 28567/377*e^3 + 36961/377*e^2 + 92507/377*e + 2687/29, 503/377*e^5 - 82/29*e^4 - 10013/377*e^3 + 17460/377*e^2 + 29945/377*e + 627/29, 1889/377*e^5 - 259/29*e^4 - 37593/377*e^3 + 54573/377*e^2 + 113536/377*e + 1434/29, -2167/377*e^5 + 323/29*e^4 + 42786/377*e^3 - 68160/377*e^2 - 124988/377*e - 1743/29, -1047/377*e^5 + 187/29*e^4 + 20343/377*e^3 - 40386/377*e^2 - 54180/377*e + 157/29, -18/377*e^5 + 20/29*e^4 + 461/377*e^3 - 2979/377*e^2 - 3093/377*e - 682/29, -607/377*e^5 + 117/29*e^4 + 11755/377*e^3 - 26001/377*e^2 - 28463/377*e + 746/29, -64/29*e^5 + 106/29*e^4 + 1362/29*e^3 - 1515/29*e^2 - 5294/29*e - 2646/29, 703/377*e^5 - 127/29*e^4 - 14465/377*e^3 + 25678/377*e^2 + 47975/377*e + 755/29, 1875/377*e^5 - 295/29*e^4 - 37779/377*e^3 + 61304/377*e^2 + 118419/377*e + 1577/29, -486/377*e^5 + 76/29*e^4 + 9431/377*e^3 - 16343/377*e^2 - 25076/377*e - 86/29, 448/377*e^5 - 66/29*e^4 - 8751/377*e^3 + 14201/377*e^2 + 23950/377*e + 122/29, 816/377*e^5 - 114/29*e^4 - 16249/377*e^3 + 22702/377*e^2 + 49736/377*e + 1231/29, -633/377*e^5 + 75/29*e^4 + 12756/377*e^3 - 14470/377*e^2 - 41099/377*e - 1789/29, 2365/377*e^5 - 369/29*e^4 - 46726/377*e^3 + 77555/377*e^2 + 132621/377*e + 1067/29, 85/377*e^5 - 1/29*e^4 - 3664/377*e^3 - 1955/377*e^2 + 36409/377*e + 1371/29, 159/377*e^5 - 22/29*e^4 - 3004/377*e^3 + 4637/377*e^2 + 6775/377*e + 118/29, -285/377*e^5 + 46/29*e^4 + 5477/377*e^3 - 9656/377*e^2 - 13723/377*e - 136/29, -713/377*e^5 + 122/29*e^4 + 13255/377*e^3 - 26956/377*e^2 - 28707/377*e + 648/29, 1477/377*e^5 - 204/29*e^4 - 29764/377*e^3 + 42937/377*e^2 + 93133/377*e + 1458/29, 38/13*e^5 - 6*e^4 - 771/13*e^3 + 1206/13*e^2 + 2491/13*e + 48, 898/377*e^5 - 160/29*e^4 - 18014/377*e^3 + 34011/377*e^2 + 54527/377*e - 199/29, 142/377*e^5 - 45/29*e^4 - 3176/377*e^3 + 10306/377*e^2 + 13593/377*e - 858/29, 1211/377*e^5 - 192/29*e^4 - 23873/377*e^3 + 38876/377*e^2 + 68286/377*e + 1391/29, -53/29*e^5 + 163/29*e^4 + 982/29*e^3 - 2667/29*e^2 - 2065/29*e + 1161/29, 875/377*e^5 - 157/29*e^4 - 17404/377*e^3 + 32655/377*e^2 + 51643/377*e - 339/29];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, -w + 1}>] := 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;