/*
  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![1, -4, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {29, 29, -w^2 + 2*w + 4}>;

primesArray := [
[3, 3, w + 1],
[3, 3, w - 1],
[7, 7, w^2 - 2],
[8, 2, 2],
[11, 11, -w^2 + w + 1],
[23, 23, -w - 3],
[29, 29, -w^2 + 2*w + 4],
[31, 31, 2*w - 3],
[41, 41, -2*w^2 + 3*w + 6],
[43, 43, w^2 - 3*w + 3],
[47, 47, w^2 + w - 4],
[49, 7, 2*w^2 - 3*w - 3],
[53, 53, w^2 - 3*w - 2],
[59, 59, 2*w^2 - w - 5],
[59, 59, w^2 - w - 7],
[59, 59, -w^2 - w + 7],
[67, 67, 2*w^2 - 3*w - 7],
[73, 73, -w^2 + 4*w - 5],
[79, 79, w^2 - 8],
[79, 79, w^2 - 5*w + 5],
[79, 79, w^2 - 3*w - 3],
[83, 83, -3*w^2 + w + 9],
[89, 89, 2*w^2 - w - 2],
[97, 97, w^2 - 3*w - 6],
[101, 101, 3*w - 4],
[103, 103, 3*w - 5],
[107, 107, -3*w^2 + 10],
[107, 107, w^2 - 3*w - 5],
[109, 109, w^2 + 2*w - 4],
[121, 11, 3*w^2 - 2*w - 9],
[125, 5, -5],
[127, 127, w^2 + w - 10],
[137, 137, 3*w^2 - 3*w - 8],
[139, 139, -5*w^2 + 3*w + 18],
[149, 149, 3*w^2 - 4*w - 6],
[157, 157, w - 6],
[163, 163, -4*w^2 + 2*w + 13],
[163, 163, -w^2 - 3],
[163, 163, 2*w^2 - 4*w - 11],
[181, 181, 3*w^2 - 11],
[193, 193, 3*w^2 - 3*w - 7],
[193, 193, -3*w^2 + 3*w + 16],
[193, 193, w^2 + 2*w - 7],
[197, 197, 2*w^2 + w - 8],
[211, 211, 4*w^2 - 5*w - 10],
[227, 227, -w - 6],
[229, 229, w^2 - 4*w - 7],
[233, 233, 3*w^2 - 5],
[239, 239, 2*w^2 - 3*w - 13],
[251, 251, 3*w^2 - 4*w - 3],
[257, 257, 3*w^2 - w - 6],
[257, 257, 3*w^2 - 5*w - 9],
[257, 257, w^2 + 3*w - 5],
[263, 263, 2*w^2 + w - 11],
[277, 277, 3*w^2 - 2*w - 6],
[281, 281, 3*w^2 - 3*w - 5],
[281, 281, -w^2 - 4],
[281, 281, 4*w^2 - 3*w - 12],
[293, 293, 2*w^2 - 5*w - 5],
[307, 307, 3*w^2 - 2*w - 3],
[317, 317, 3*w^2 - w - 18],
[317, 317, -w^2 + 2*w - 5],
[317, 317, -4*w^2 + 7*w + 4],
[349, 349, -w^2 + 3*w - 6],
[353, 353, 3*w^2 - 6*w - 7],
[353, 353, w^2 - w - 10],
[353, 353, -5*w - 3],
[367, 367, -4*w^2 - w + 7],
[379, 379, 4*w^2 - 3*w - 11],
[383, 383, 4*w^2 - w - 10],
[389, 389, w^2 - 5*w - 4],
[389, 389, -5*w^2 + 7*w + 11],
[389, 389, -7*w^2 + 4*w + 25],
[397, 397, -w^2 + w - 5],
[397, 397, 5*w^2 - 6*w - 22],
[397, 397, -5*w^2 - w + 11],
[401, 401, 2*w^2 - w - 14],
[401, 401, 5*w^2 - 3*w - 16],
[401, 401, w^2 + 3*w - 8],
[409, 409, w^2 + 3*w - 9],
[431, 431, 2*w - 9],
[433, 433, w^2 - 3*w - 11],
[439, 439, w^2 - 11],
[439, 439, 5*w - 6],
[439, 439, 3*w^2 + 2*w - 9],
[443, 443, -6*w^2 + 3*w + 20],
[443, 443, -w^2 - w - 5],
[443, 443, 2*w^2 - 5*w - 11],
[461, 461, 3*w^2 - 4*w - 18],
[463, 463, 3*w^2 - 6*w - 17],
[463, 463, -4*w^2 - 3*w + 9],
[463, 463, 3*w^2 - 6*w - 8],
[467, 467, 2*w^2 - 5*w - 8],
[487, 487, 2*w^2 + 3*w - 7],
[499, 499, 5*w^2 - 8*w - 8],
[503, 503, 3*w^2 + w - 12],
[509, 509, w^2 - 6*w - 11],
[523, 523, 3*w^2 - 9*w - 2],
[529, 23, -w^2 + 4*w - 8],
[541, 541, 2*w^2 + 2*w - 11],
[569, 569, 4*w^2 - 3*w - 9],
[587, 587, -w^2 - 4*w + 13],
[599, 599, 3*w - 11],
[601, 601, -3*w^2 + 10*w - 9],
[607, 607, 4*w^2 - 6*w - 15],
[613, 613, w - 9],
[619, 619, 5*w^2 - 2*w - 14],
[631, 631, 4*w^2 - 5*w - 4],
[647, 647, 4*w^2 - 4*w - 7],
[647, 647, w^2 - 12],
[647, 647, 4*w^2 - w - 7],
[653, 653, 4*w^2 - 6*w - 17],
[659, 659, 3*w^2 - 6*w - 10],
[659, 659, 3*w^2 + w - 15],
[659, 659, 4*w^2 - 17],
[661, 661, 2*w^2 - 3*w - 15],
[661, 661, -5*w^2 + 5*w + 26],
[661, 661, 5*w^2 - 12*w - 4],
[673, 673, w^2 + w - 13],
[677, 677, w^2 - 3*w - 12],
[683, 683, 5*w^2 - 4*w - 14],
[709, 709, 4*w^2 - 2*w - 7],
[733, 733, 3*w^2 - 6*w - 11],
[739, 739, -2*w - 9],
[743, 743, 6*w - 7],
[751, 751, w^2 - 6*w - 9],
[757, 757, 4*w^2 - 3*w - 5],
[761, 761, -7*w^2 + 8*w + 31],
[769, 769, w^2 - 6*w - 6],
[773, 773, -w - 9],
[773, 773, w^2 - 4*w - 13],
[773, 773, 5*w^2 - 18],
[787, 787, 3*w^2 - 6*w - 14],
[797, 797, w^2 - 6*w - 8],
[811, 811, 2*w^2 + 3*w - 10],
[811, 811, -2*w^2 - 3*w + 16],
[811, 811, -6*w^2 + 11*w + 6],
[823, 823, w^2 + 5*w - 7],
[829, 829, -5*w^2 - 2*w + 14],
[839, 839, 2*w^2 - 6*w - 9],
[841, 29, 2*w^2 - 7*w - 5],
[853, 853, 5*w^2 - 28],
[857, 857, 7*w^2 - 9*w - 23],
[877, 877, -8*w^2 + 3*w + 31],
[881, 881, 6*w^2 - 9*w - 11],
[907, 907, 4*w^2 - 7*w - 13],
[911, 911, 2*w^2 + 3*w - 15],
[911, 911, -w^2 - 7],
[911, 911, 3*w^2 + 3*w - 10],
[919, 919, w^2 - 7*w - 4],
[941, 941, 6*w^2 - 11*w - 12],
[947, 947, 5*w^2 - w - 11],
[947, 947, 3*w^2 - 8*w - 6],
[947, 947, 2*w^2 + 3*w - 12],
[953, 953, -5*w^2 - 4*w + 11],
[953, 953, 4*w^2 - w - 25],
[953, 953, 6*w^2 - 7*w - 15],
[961, 31, 4*w^2 + 2*w - 13],
[977, 977, -2*w^2 + 3*w - 6],
[991, 991, 5*w^2 - 5*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 2*x^7 - 17*x^6 + 36*x^5 + 65*x^4 - 162*x^3 + 44*x^2 + 38*x - 9;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 124/793*e^7 - 154/793*e^6 - 2148/793*e^5 + 226/61*e^4 + 689/61*e^3 - 13848/793*e^2 + 407/793*e + 2552/793, 62/793*e^7 - 77/793*e^6 - 1074/793*e^5 + 113/61*e^4 + 314/61*e^3 - 6924/793*e^2 + 3772/793*e + 483/793, 37/793*e^7 + 18/793*e^6 - 820/793*e^5 - 28/61*e^4 + 394/61*e^3 + 1598/793*e^2 - 6958/793*e + 608/793, 30/793*e^7 - 114/793*e^6 - 622/793*e^5 + 157/61*e^4 + 270/61*e^3 - 9592/793*e^2 - 2984/793*e + 4608/793, -332/793*e^7 + 310/793*e^6 + 5879/793*e^5 - 428/61*e^4 - 2073/61*e^3 + 25054/793*e^2 + 14361/793*e - 6270/793, 1, -231/793*e^7 + 402/793*e^6 + 4155/793*e^5 - 544/61*e^4 - 1408/61*e^3 + 30402/793*e^2 + 297/793*e - 2810/793, -526/793*e^7 + 730/793*e^6 + 9214/793*e^5 - 1000/61*e^4 - 3026/61*e^3 + 57054/793*e^2 + 2942/793*e - 8472/793, -168/793*e^7 + 4/793*e^6 + 3166/793*e^5 - 13/61*e^4 - 1268/61*e^3 + 1060/793*e^2 + 17662/793*e - 746/793, -124/793*e^7 + 154/793*e^6 + 2148/793*e^5 - 226/61*e^4 - 750/61*e^3 + 13848/793*e^2 + 6730/793*e - 5724/793, 140/793*e^7 - 532/793*e^6 - 2374/793*e^5 + 692/61*e^4 + 650/61*e^3 - 37890/793*e^2 + 10922/793*e + 8816/793, 124/793*e^7 - 154/793*e^6 - 2148/793*e^5 + 226/61*e^4 + 750/61*e^3 - 15434/793*e^2 - 6730/793*e + 12068/793, 124/793*e^7 - 154/793*e^6 - 2148/793*e^5 + 226/61*e^4 + 750/61*e^3 - 14641/793*e^2 - 6730/793*e + 4138/793, -618/793*e^7 + 921/793*e^6 + 10910/793*e^5 - 1270/61*e^4 - 3610/61*e^3 + 72777/793*e^2 - 1018/793*e - 8012/793, -264/793*e^7 + 686/793*e^6 + 4522/793*e^5 - 918/61*e^4 - 1278/61*e^3 + 51738/793*e^2 - 18466/793*e - 9782/793, -446/793*e^7 + 426/793*e^6 + 8084/793*e^5 - 622/61*e^4 - 2916/61*e^3 + 38348/793*e^2 + 19832/793*e - 8872/793, 446/793*e^7 - 426/793*e^6 - 8084/793*e^5 + 622/61*e^4 + 2916/61*e^3 - 38348/793*e^2 - 16660/793*e + 8872/793, 248/793*e^7 - 308/793*e^6 - 4296/793*e^5 + 452/61*e^4 + 1378/61*e^3 - 29282/793*e^2 + 2400/793*e + 14620/793, 204/793*e^7 - 458/793*e^6 - 3278/793*e^5 + 604/61*e^4 + 860/61*e^3 - 34140/793*e^2 + 10160/793*e + 11668/793, 218/793*e^7 - 194/793*e^6 - 3674/793*e^5 + 234/61*e^4 + 1108/61*e^3 - 10967/793*e^2 + 2212/793*e - 4262/793, 402/793*e^7 - 576/793*e^6 - 7066/793*e^5 + 774/61*e^4 + 2398/61*e^3 - 43206/793*e^2 - 7314/793*e + 7506/793, -217/793*e^7 + 666/793*e^6 + 3759/793*e^5 - 914/61*e^4 - 1038/61*e^3 + 52782/793*e^2 - 20339/793*e - 10810/793, -26/61*e^7 + 50/61*e^6 + 474/61*e^5 - 862/61*e^4 - 2188/61*e^3 + 3734/61*e^2 + 626/61*e - 1212/61, 84/793*e^7 - 2/793*e^6 - 1583/793*e^5 - 24/61*e^4 + 573/61*e^3 + 4228/793*e^2 - 901/793*e - 6764/793, 248/793*e^7 - 308/793*e^6 - 4296/793*e^5 + 452/61*e^4 + 1378/61*e^3 - 27696/793*e^2 + 2400/793*e + 8276/793, -388/793*e^7 + 840/793*e^6 + 6670/793*e^5 - 1144/61*e^4 - 2028/61*e^3 + 65586/793*e^2 - 11736/793*e - 15506/793, 291/793*e^7 - 630/793*e^6 - 5399/793*e^5 + 858/61*e^4 + 1948/61*e^3 - 49586/793*e^2 - 9437/793*e + 16784/793, -516/793*e^7 + 692/793*e^6 + 9271/793*e^5 - 968/61*e^4 - 3241/61*e^3 + 58086/793*e^2 + 12785/793*e - 16452/793, 43/61*e^7 - 78/61*e^6 - 737/61*e^5 + 1374/61*e^4 + 3018/61*e^3 - 6030/61*e^2 + 241/61*e + 1310/61, 602/793*e^7 - 543/793*e^6 - 10684/793*e^5 + 743/61*e^4 + 3710/61*e^3 - 43184/793*e^2 - 19806/793*e + 7299/793, -228/793*e^7 + 232/793*e^6 + 4410/793*e^5 - 266/61*e^4 - 1808/61*e^3 + 12314/793*e^2 + 25216/793*e - 6790/793, 12/793*e^7 + 113/793*e^6 - 566/793*e^5 - 169/61*e^4 + 474/61*e^3 + 10120/793*e^2 - 16102/793*e - 4025/793, -1102/793*e^7 + 1650/793*e^6 + 18936/793*e^5 - 2282/61*e^4 - 6014/61*e^3 + 131152/793*e^2 - 4474/793*e - 16694/793, -576/793*e^7 + 920/793*e^6 + 9722/793*e^5 - 1282/61*e^4 - 2927/61*e^3 + 75684/793*e^2 - 14553/793*e - 14566/793, 662/793*e^7 - 771/793*e^6 - 11928/793*e^5 + 1118/61*e^4 + 4128/61*e^3 - 67919/793*e^2 - 13086/793*e + 9378/793, -678/793*e^7 + 1149/793*e^6 + 12154/793*e^5 - 1584/61*e^4 - 4150/61*e^3 + 91961/793*e^2 + 4950/793*e - 15642/793, 446/793*e^7 - 426/793*e^6 - 8084/793*e^5 + 622/61*e^4 + 2916/61*e^3 - 38348/793*e^2 - 23004/793*e + 5700/793, -50/793*e^7 + 190/793*e^6 + 508/793*e^5 - 282/61*e^4 + 160/61*e^3 + 18630/793*e^2 - 18288/793*e - 9266/793, -34/793*e^7 - 188/793*e^6 + 282/793*e^5 + 306/61*e^4 + 60/61*e^3 - 22858/793*e^2 - 4601/793*e + 19202/793, -16/793*e^7 + 378/793*e^6 + 226/793*e^5 - 466/61*e^4 - 22/61*e^3 + 22456/793*e^2 - 3378/793*e - 1506/793, 18/61*e^7 - 44/61*e^6 - 300/61*e^5 + 822/61*e^4 + 1069/61*e^3 - 3852/61*e^2 + 1223/61*e + 886/61, -848/793*e^7 + 1002/793*e^6 + 15150/793*e^5 - 1396/61*e^4 - 5192/61*e^3 + 82347/793*e^2 + 16044/793*e - 19550/793, 18/61*e^7 - 44/61*e^6 - 300/61*e^5 + 822/61*e^4 + 1069/61*e^3 - 3852/61*e^2 + 1223/61*e + 1130/61, 18/61*e^7 - 44/61*e^6 - 300/61*e^5 + 822/61*e^4 + 1130/61*e^3 - 3852/61*e^2 + 430/61*e + 1252/61, 120/793*e^7 - 456/793*e^6 - 1695/793*e^5 + 628/61*e^4 + 165/61*e^3 - 35196/793*e^2 + 25335/793*e + 10502/793, -184/793*e^7 + 382/793*e^6 + 3392/793*e^5 - 540/61*e^4 - 1168/61*e^3 + 33032/793*e^2 + 1596/793*e - 8596/793, 716/793*e^7 - 1452/793*e^6 - 12096/793*e^5 + 1974/61*e^4 + 3638/61*e^3 - 110402/793*e^2 + 18338/793*e + 15452/793, -528/793*e^7 + 579/793*e^6 + 9044/793*e^5 - 799/61*e^4 - 2922/61*e^3 + 46380/793*e^2 + 1132/793*e - 10841/793, 268/793*e^7 - 384/793*e^6 - 4975/793*e^5 + 516/61*e^4 + 1741/61*e^3 - 28804/793*e^2 - 2497/793*e + 3418/793, 506/793*e^7 - 654/793*e^6 - 9328/793*e^5 + 936/61*e^4 + 3334/61*e^3 - 57532/793*e^2 - 13112/793*e + 8572/793, 944/793*e^7 - 1684/793*e^6 - 16506/793*e^5 + 2362/61*e^4 + 5324/61*e^3 - 138576/793*e^2 + 10568/793*e + 22242/793, 42/793*e^7 - 1/793*e^6 - 1188/793*e^5 - 12/61*e^4 + 744/61*e^3 + 2907/793*e^2 - 19086/793*e - 1796/793, 212/793*e^7 + 146/793*e^6 - 4184/793*e^5 - 200/61*e^4 + 1908/61*e^3 + 11728/793*e^2 - 44454/793*e - 1060/793, -14/61*e^7 - 20/61*e^6 + 274/61*e^5 + 296/61*e^4 - 1638/61*e^3 - 1030/61*e^2 + 3190/61*e + 192/61, -462/793*e^7 + 804/793*e^6 + 8310/793*e^5 - 1088/61*e^4 - 2938/61*e^3 + 63183/793*e^2 + 13282/793*e - 19894/793, 74/793*e^7 + 36/793*e^6 - 1640/793*e^5 - 56/61*e^4 + 788/61*e^3 + 3196/793*e^2 - 13916/793*e + 10732/793, 644/793*e^7 - 544/793*e^6 - 11872/793*e^5 + 792/61*e^4 + 4454/61*e^3 - 49000/793*e^2 - 38892/793*e + 11054/793, 401/793*e^7 - 1048/793*e^6 - 7151/793*e^5 + 1454/61*e^4 + 2328/61*e^3 - 84228/793*e^2 + 7641/793*e + 24164/793, -820/793*e^7 + 1530/793*e^6 + 14358/793*e^5 - 2136/61*e^4 - 4696/61*e^3 + 126314/793*e^2 + 941/793*e - 26034/793, -866/793*e^7 + 1229/793*e^6 + 15206/793*e^5 - 1661/61*e^4 - 4988/61*e^3 + 94922/793*e^2 + 1340/793*e - 12323/793, 58/61*e^7 - 74/61*e^6 - 1048/61*e^5 + 1327/61*e^4 + 4834/61*e^3 - 6068/61*e^2 - 1678/61*e + 2028/61, 20/793*e^7 - 76/793*e^6 + 114/793*e^5 + 64/61*e^4 - 308/61*e^3 + 478/793*e^2 + 10170/793*e - 9616/793, 124/793*e^7 - 154/793*e^6 - 2148/793*e^5 + 104/61*e^4 + 750/61*e^3 + 426/793*e^2 - 6730/793*e - 620/793, -422/793*e^7 + 652/793*e^6 + 6952/793*e^5 - 838/61*e^4 - 1968/61*e^3 + 45900/793*e^2 - 13958/793*e - 5820/793, 670/793*e^7 - 960/793*e^6 - 11248/793*e^5 + 1290/61*e^4 + 3468/61*e^3 - 72010/793*e^2 + 5256/793*e + 10924/793, -433/793*e^7 + 218/793*e^6 + 7603/793*e^5 - 312/61*e^4 - 2494/61*e^3 + 16534/793*e^2 + 3049/793*e + 7716/793, -35/61*e^7 + 72/61*e^6 + 563/61*e^5 - 1334/61*e^4 - 1838/61*e^3 + 6148/61*e^2 - 2761/61*e - 618/61, 12/793*e^7 - 680/793*e^6 + 227/793*e^5 + 868/61*e^4 - 441/61*e^3 - 46976/793*e^2 + 27513/793*e + 23730/793, -38/61*e^7 + 59/61*e^6 + 674/61*e^5 - 1044/61*e^4 - 2982/61*e^3 + 4655/61*e^2 + 990/61*e - 1030/61, 408/793*e^7 - 916/793*e^6 - 6556/793*e^5 + 1330/61*e^4 + 1476/61*e^3 - 82554/793*e^2 + 48868/793*e + 16992/793, 452/793*e^7 - 766/793*e^6 - 7574/793*e^5 + 1056/61*e^4 + 2238/61*e^3 - 63422/793*e^2 + 10974/793*e + 15186/793, -174/793*e^7 + 344/793*e^6 + 2656/793*e^5 - 508/61*e^4 - 468/61*e^3 + 32478/793*e^2 - 29797/793*e - 2302/793, 848/793*e^7 - 1002/793*e^6 - 14357/793*e^5 + 1396/61*e^4 + 4399/61*e^3 - 79968/793*e^2 + 8539/793*e + 518/793, 248/793*e^7 - 308/793*e^6 - 4296/793*e^5 + 513/61*e^4 + 1256/61*e^3 - 34040/793*e^2 + 15088/793*e - 9170/793, 160/793*e^7 - 608/793*e^6 - 3053/793*e^5 + 756/61*e^4 + 1135/61*e^3 - 38998/793*e^2 - 6663/793*e + 11888/793, 644/793*e^7 - 544/793*e^6 - 11872/793*e^5 + 731/61*e^4 + 4332/61*e^3 - 39484/793*e^2 - 23032/793*e - 4806/793, -446/793*e^7 + 426/793*e^6 + 8084/793*e^5 - 622/61*e^4 - 2916/61*e^3 + 35176/793*e^2 + 20625/793*e + 3816/793, -154/793*e^7 + 268/793*e^6 + 2770/793*e^5 - 322/61*e^4 - 1020/61*e^3 + 15510/793*e^2 + 12093/793*e - 5574/793, -18/13*e^7 + 32/13*e^6 + 316/13*e^5 - 44*e^4 - 100*e^3 + 2526/13*e^2 - 378/13*e - 352/13, -227/793*e^7 + 704/793*e^6 + 4495/793*e^5 - 946/61*e^4 - 1738/61*e^3 + 53336/793*e^2 + 8675/793*e - 9174/793, 550/793*e^7 - 504/793*e^6 - 10346/793*e^5 + 784/61*e^4 + 3852/61*e^3 - 51088/793*e^2 - 19286/793*e + 19454/793, -152/793*e^7 - 374/793*e^6 + 2940/793*e^5 + 514/61*e^4 - 1368/61*e^3 - 29326/793*e^2 + 40865/793*e + 15034/793, 58/61*e^7 - 74/61*e^6 - 1048/61*e^5 + 1266/61*e^4 + 4712/61*e^3 - 5214/61*e^2 - 702/61*e - 46/61, -1072/793*e^7 + 1536/793*e^6 + 18314/793*e^5 - 2186/61*e^4 - 5744/61*e^3 + 131076/793*e^2 - 9837/793*e - 23188/793, -894/793*e^7 + 1494/793*e^6 + 15998/793*e^5 - 2080/61*e^4 - 5362/61*e^3 + 123118/793*e^2 - 3382/793*e - 33594/793, 902/793*e^7 - 890/793*e^6 - 16904/793*e^5 + 1276/61*e^4 + 6288/61*e^3 - 75664/793*e^2 - 38544/793*e + 5006/793, 1790/793*e^7 - 2044/793*e^6 - 31826/793*e^5 + 2922/61*e^4 + 10864/61*e^3 - 175294/793*e^2 - 32662/793*e + 33872/793, 158/793*e^7 + 34/793*e^6 - 2430/793*e^5 + 42/61*e^4 + 446/61*e^3 - 8436/793*e^2 + 27212/793*e - 5548/793, -30/793*e^7 + 114/793*e^6 + 622/793*e^5 - 218/61*e^4 - 392/61*e^3 + 19108/793*e^2 + 15672/793*e - 25226/793, -324/793*e^7 + 121/793*e^6 + 5766/793*e^5 - 256/61*e^4 - 2062/61*e^3 + 22549/793*e^2 + 14464/793*e - 9482/793, -259/793*e^7 - 126/793*e^6 + 4947/793*e^5 + 74/61*e^4 - 1904/61*e^3 + 1502/793*e^2 + 16193/793*e - 1084/793, -592/793*e^7 + 1298/793*e^6 + 9948/793*e^5 - 1870/61*e^4 - 2766/61*e^3 + 114000/793*e^2 - 40928/793*e - 25588/793, -478/793*e^7 + 1182/793*e^6 + 8536/793*e^5 - 1554/61*e^4 - 2716/61*e^3 + 84846/793*e^2 - 13886/793*e - 7126/793, 284/793*e^7 - 762/793*e^6 - 4408/793*e^5 + 982/61*e^4 + 726/61*e^3 - 51260/793*e^2 + 50840/793*e + 6510/793, 1116/793*e^7 - 1386/793*e^6 - 19332/793*e^5 + 1912/61*e^4 + 6384/61*e^3 - 111944/793*e^2 - 17748/793*e + 30898/793, -740/793*e^7 + 1226/793*e^6 + 13228/793*e^5 - 1636/61*e^4 - 4586/61*e^3 + 90162/793*e^2 + 18624/793*e - 16918/793, 232/793*e^7 - 723/793*e^6 - 4070/793*e^5 + 962/61*e^4 + 1234/61*e^3 - 52027/793*e^2 + 10124/793*e - 7504/793, -860/793*e^7 + 1682/793*e^6 + 14130/793*e^5 - 2264/61*e^4 - 3836/61*e^3 + 126944/793*e^2 - 51912/793*e - 24248/793, 774/793*e^7 - 1038/793*e^6 - 13510/793*e^5 + 1452/61*e^4 + 4404/61*e^3 - 86336/793*e^2 - 1335/793*e + 26264/793, 56/61*e^7 - 42/61*e^6 - 974/61*e^5 + 768/61*e^4 + 4295/61*e^3 - 3444/61*e^2 - 2207/61*e + 330/61, -8/793*e^7 - 604/793*e^6 + 906/793*e^5 + 682/61*e^4 - 926/61*e^3 - 27629/793*e^2 + 41926/793*e - 11062/793, 1270/793*e^7 - 1654/793*e^6 - 22102/793*e^5 + 2356/61*e^4 + 7404/61*e^3 - 140142/793*e^2 - 27462/793*e + 19026/793, -1026/793*e^7 + 1044/793*e^6 + 17466/793*e^5 - 1502/61*e^4 - 5452/61*e^3 + 91098/793*e^2 - 12615/793*e - 7558/793, -20/793*e^7 + 76/793*e^6 - 114/793*e^5 - 64/61*e^4 + 552/61*e^3 + 2694/793*e^2 - 35546/793*e - 7830/793, 396/793*e^7 - 236/793*e^6 - 7576/793*e^5 + 340/61*e^4 + 3076/61*e^3 - 18132/793*e^2 - 39706/793*e - 8324/793, 492/793*e^7 - 918/793*e^6 - 8932/793*e^5 + 1184/61*e^4 + 3330/61*e^3 - 62466/793*e^2 - 27368/793*e + 11814/793, -920/793*e^7 + 324/793*e^6 + 16960/793*e^5 - 626/61*e^4 - 6450/61*e^3 + 49382/793*e^2 + 63490/793*e - 20776/793, 498/793*e^7 - 1258/793*e^6 - 8422/793*e^5 + 1618/61*e^4 + 2286/61*e^3 - 85954/793*e^2 + 39916/793*e + 10198/793, 896/793*e^7 - 2136/793*e^6 - 15828/793*e^5 + 2916/61*e^4 + 5136/61*e^3 - 166368/793*e^2 + 16294/793*e + 27240/793, -600/793*e^7 + 694/793*e^6 + 10854/793*e^5 - 822/61*e^4 - 3936/61*e^3 + 36412/793*e^2 + 27960/793*e + 15688/793, -2289/793*e^7 + 2830/793*e^6 + 40163/793*e^5 - 3982/61*e^4 - 13342/61*e^3 + 232914/793*e^2 + 14045/793*e - 28998/793, -922/793*e^7 + 966/793*e^6 + 16790/793*e^5 - 1340/61*e^4 - 6224/61*e^3 + 76772/793*e^2 + 60094/793*e - 16008/793, 1372/793*e^7 - 1090/793*e^6 - 24534/793*e^5 + 1560/61*e^4 + 8688/61*e^3 - 92186/793*e^2 - 47758/793*e + 9000/793, -90/61*e^7 + 98/61*e^6 + 1622/61*e^5 - 1670/61*e^4 - 7419/61*e^3 + 7060/61*e^2 + 2791/61*e - 892/61, 262/793*e^7 - 837/793*e^6 - 4692/793*e^5 + 1180/61*e^4 + 1260/61*e^3 - 69549/793*e^2 + 37274/793*e + 3448/793, -1282/793*e^7 + 2334/793*e^6 + 22668/793*e^5 - 3102/61*e^4 - 7512/61*e^3 + 169672/793*e^2 - 2430/793*e - 12622/793, 2088/793*e^7 - 2542/793*e^6 - 36630/793*e^5 + 3656/61*e^4 + 12082/61*e^3 - 220034/793*e^2 - 10388/793*e + 40312/793, -788/793*e^7 + 774/793*e^6 + 13906/793*e^5 - 1204/61*e^4 - 4835/61*e^3 + 79816/793*e^2 + 31487/793*e - 42054/793, -1020/793*e^7 + 1497/793*e^6 + 17976/793*e^5 - 2166/61*e^4 - 6008/61*e^3 + 133429/793*e^2 + 6296/793*e - 31378/793, 1866/793*e^7 - 2650/793*e^6 - 33296/793*e^5 + 3580/61*e^4 + 11426/61*e^3 - 204246/793*e^2 - 28908/793*e + 47766/793, 464/793*e^7 - 1446/793*e^6 - 7347/793*e^5 + 1924/61*e^4 + 1797/61*e^3 - 105640/793*e^2 + 36901/793*e + 21470/793, -174/793*e^7 + 344/793*e^6 + 2656/793*e^5 - 508/61*e^4 - 346/61*e^3 + 27720/793*e^2 - 46450/793*e + 11972/793, -406/793*e^7 + 274/793*e^6 + 6726/793*e^5 - 372/61*e^4 - 2068/61*e^3 + 17100/793*e^2 - 1064/793*e + 21062/793, 740/793*e^7 - 1226/793*e^6 - 13228/793*e^5 + 1636/61*e^4 + 4586/61*e^3 - 88576/793*e^2 - 23382/793*e + 5816/793, -113/793*e^7 + 588/793*e^6 + 1497/793*e^5 - 874/61*e^4 - 102/61*e^3 + 57488/793*e^2 - 24551/793*e - 27190/793, 24/793*e^7 + 226/793*e^6 - 339/793*e^5 - 216/61*e^4 + 33/61*e^3 + 7552/793*e^2 + 1895/793*e - 15980/793, -1356/793*e^7 + 1505/793*e^6 + 24308/793*e^5 - 2070/61*e^4 - 8544/61*e^3 + 122861/793*e^2 + 41620/793*e - 39214/793, -22/61*e^7 - 14/61*e^6 + 448/61*e^5 + 256/61*e^4 - 2818/61*e^3 - 1209/61*e^2 + 5466/61*e - 134/61, -6/793*e^7 + 340/793*e^6 + 1076/793*e^5 - 312/61*e^4 - 1152/61*e^3 + 9214/793*e^2 + 52459/793*e - 15830/793, 666/793*e^7 - 1262/793*e^6 - 11588/793*e^5 + 1692/61*e^4 + 3676/61*e^3 - 96530/793*e^2 + 15910/793*e + 18874/793, 1944/793*e^7 - 2312/793*e^6 - 34596/793*e^5 + 3244/61*e^4 + 11884/61*e^3 - 193183/793*e^2 - 42376/793*e + 48962/793, 154/793*e^7 - 268/793*e^6 - 2770/793*e^5 + 322/61*e^4 + 1264/61*e^3 - 12338/793*e^2 - 35090/793*e - 11872/793, -372/793*e^7 + 462/793*e^6 + 6444/793*e^5 - 556/61*e^4 - 1884/61*e^3 + 25684/793*e^2 - 17874/793*e - 2898/793, -466/793*e^7 + 502/793*e^6 + 7970/793*e^5 - 808/61*e^4 - 2608/61*e^3 + 55316/793*e^2 + 8076/793*e - 23046/793, 2472/793*e^7 - 3684/793*e^6 - 43640/793*e^5 + 4958/61*e^4 + 14684/61*e^3 - 281592/793*e^2 - 32406/793*e + 63768/793, 918/793*e^7 - 1268/793*e^6 - 17130/793*e^5 + 1742/61*e^4 + 6432/61*e^3 - 101292/793*e^2 - 44682/793*e + 39818/793, -1048/793*e^7 + 1762/793*e^6 + 18768/793*e^5 - 2402/61*e^4 - 6382/61*e^3 + 135456/793*e^2 + 12676/793*e - 21722/793, -1446/793*e^7 + 1054/793*e^6 + 26174/793*e^5 - 1504/61*e^4 - 9476/61*e^3 + 87404/793*e^2 + 68018/793*e - 13388/793, 1578/793*e^7 - 2190/793*e^6 - 27642/793*e^5 + 3122/61*e^4 + 8895/61*e^3 - 188608/793*e^2 + 12585/793*e + 31760/793, -634/793*e^7 + 506/793*e^6 + 11136/793*e^5 - 638/61*e^4 - 3876/61*e^3 + 35758/793*e^2 + 25738/793*e - 14276/793, -756/793*e^7 + 1604/793*e^6 + 11868/793*e^5 - 2285/61*e^4 - 2778/61*e^3 + 136408/793*e^2 - 70398/793*e - 26354/793, -1424/793*e^7 + 1922/793*e^6 + 24872/793*e^5 - 2556/61*e^4 - 8424/61*e^3 + 139792/793*e^2 + 32418/793*e - 10326/793, -484/793*e^7 + 1522/793*e^6 + 8026/793*e^5 - 1988/61*e^4 - 2038/61*e^3 + 108334/793*e^2 - 43106/793*e - 21370/793, 1424/793*e^7 - 1922/793*e^6 - 24872/793*e^5 + 2556/61*e^4 + 7936/61*e^3 - 139792/793*e^2 + 16748/793*e + 810/793, -966/793*e^7 + 816/793*e^6 + 17808/793*e^5 - 1310/61*e^4 - 6620/61*e^3 + 90153/793*e^2 + 50408/793*e - 39578/793, -314/793*e^7 + 876/793*e^6 + 5030/793*e^5 - 1200/61*e^4 - 1118/61*e^3 + 75126/793*e^2 - 41512/793*e - 30150/793, 22/13*e^7 - 16/13*e^6 - 392/13*e^5 + 22*e^4 + 140*e^3 - 1263/13*e^2 - 1072/13*e + 150/13, 516/793*e^7 - 692/793*e^6 - 8478/793*e^5 + 1090/61*e^4 + 2204/61*e^3 - 75532/793*e^2 + 41932/793*e + 32312/793, 1766/793*e^7 - 2270/793*e^6 - 30694/793*e^5 + 3077/61*e^4 + 9916/61*e^3 - 174916/793*e^2 - 458/793*e + 18132/793, -440/793*e^7 + 86/793*e^6 + 8594/793*e^5 - 310/61*e^4 - 3594/61*e^3 + 29134/793*e^2 + 47466/793*e - 7316/793, -684/793*e^7 + 696/793*e^6 + 13230/793*e^5 - 1042/61*e^4 - 5180/61*e^3 + 67076/793*e^2 + 46307/793*e - 28300/793, 544/793*e^7 - 164/793*e^6 - 9270/793*e^5 + 289/61*e^4 + 3188/61*e^3 - 18084/793*e^2 - 32646/793*e - 16994/793, -242/793*e^7 - 825/793*e^6 + 4806/793*e^5 + 958/61*e^4 - 2178/61*e^3 - 40993/793*e^2 + 47438/793*e - 6720/793, 2868/793*e^7 - 3920/793*e^6 - 51216/793*e^5 + 5420/61*e^4 + 17394/61*e^3 - 313998/793*e^2 - 35634/793*e + 61788/793, 2432/793*e^7 - 3532/793*e^6 - 42282/793*e^5 + 4830/61*e^4 + 13775/61*e^3 - 276204/793*e^2 - 5959/793*e + 56038/793, 924/793*e^7 - 1608/793*e^6 - 16620/793*e^5 + 2176/61*e^4 + 5510/61*e^3 - 120022/793*e^2 + 11500/793*e - 4620/793, 1696/793*e^7 - 2004/793*e^6 - 30300/793*e^5 + 2792/61*e^4 + 10628/61*e^3 - 165487/793*e^2 - 63808/793*e + 47030/793, -1004/793*e^7 + 1912/793*e^6 + 17750/793*e^5 - 2554/61*e^4 - 5864/61*e^3 + 145072/793*e^2 + 158/793*e - 53662/793, 766/793*e^7 - 1642/793*e^6 - 12604/793*e^5 + 2378/61*e^4 + 3356/61*e^3 - 146478/793*e^2 + 62002/793*e + 29476/793];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, -w^2 + 2*w + 4}>] := -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;