/*
  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![3, 2, -5, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, w^3 - 2*w^2 - 3*w + 2}>;

primesArray := [
[3, 3, w],
[4, 2, -w^3 + 3*w^2 + w - 2],
[11, 11, -w^3 + 3*w^2 + 2*w - 5],
[13, 13, w^3 - 2*w^2 - 4*w + 2],
[13, 13, -w + 2],
[17, 17, w^3 - 3*w^2 - 2*w + 2],
[19, 19, -w^2 + w + 4],
[19, 19, w^3 - 2*w^2 - 3*w + 2],
[23, 23, w^2 - 2*w - 1],
[27, 3, w^3 - 2*w^2 - 5*w - 1],
[29, 29, -w^3 + 2*w^2 + 5*w - 1],
[37, 37, -w^3 + 2*w^2 + 5*w - 4],
[41, 41, 2*w^3 - 5*w^2 - 6*w + 4],
[53, 53, w^3 - 2*w^2 - 3*w - 2],
[59, 59, w - 4],
[67, 67, 2*w^2 - 3*w - 8],
[73, 73, 2*w^3 - 5*w^2 - 6*w + 5],
[89, 89, -2*w^3 + 6*w^2 + 5*w - 7],
[97, 97, -2*w^3 + 6*w^2 + 3*w - 7],
[97, 97, -w^3 + 3*w^2 + 3*w - 1],
[103, 103, w^2 - 4*w - 4],
[113, 113, 2*w^3 - 5*w^2 - 4*w + 1],
[113, 113, -w^3 + 3*w^2 + w - 5],
[139, 139, 2*w^2 - 3*w - 4],
[139, 139, w^3 - w^2 - 6*w - 1],
[149, 149, 2*w^3 - 5*w^2 - 5*w + 2],
[149, 149, w^3 - 4*w^2 + 10],
[151, 151, -3*w - 1],
[157, 157, w^2 - w - 8],
[157, 157, w^3 - 7*w - 8],
[163, 163, -3*w^3 + 8*w^2 + 8*w - 10],
[163, 163, w^3 - 3*w^2 - 4*w + 7],
[163, 163, -w^3 + 4*w^2 + w - 8],
[163, 163, -2*w^3 + 5*w^2 + 7*w - 4],
[167, 167, -2*w^3 + 5*w^2 + 8*w - 4],
[169, 13, -2*w^3 + 4*w^2 + 7*w - 5],
[173, 173, 2*w^3 - 4*w^2 - 9*w + 4],
[173, 173, -w^3 + 5*w^2 - 2*w - 5],
[181, 181, 3*w^3 - 7*w^2 - 12*w + 7],
[191, 191, w^3 - 3*w^2 - 3*w - 1],
[199, 199, 2*w^3 - 3*w^2 - 10*w - 1],
[199, 199, 2*w^3 - 6*w^2 - 6*w + 13],
[223, 223, -2*w^3 + 6*w^2 + 4*w - 11],
[223, 223, 3*w^3 - 7*w^2 - 12*w + 8],
[223, 223, -2*w^3 + 5*w^2 + 6*w - 1],
[229, 229, -w^3 + 3*w^2 + 2*w + 1],
[229, 229, 3*w^3 - 4*w^2 - 15*w - 5],
[233, 233, -3*w^3 + 7*w^2 + 11*w - 5],
[233, 233, 2*w^3 - 5*w^2 - 4*w + 5],
[241, 241, w^3 - 3*w^2 - 5*w + 7],
[257, 257, w^3 - 3*w^2 - 4*w + 8],
[263, 263, w^2 - 4*w - 5],
[271, 271, -2*w^3 + 6*w^2 + 5*w - 13],
[277, 277, -2*w^3 + 6*w^2 + 3*w - 8],
[277, 277, w^3 - 2*w^2 - 2*w - 2],
[281, 281, 2*w^3 - 5*w^2 - 4*w + 2],
[283, 283, -3*w^2 + 7*w + 10],
[283, 283, -w^3 + 4*w^2 - w - 7],
[293, 293, 2*w^3 - 5*w^2 - 3*w + 4],
[293, 293, w^3 - 8*w - 4],
[307, 307, -w^3 + 2*w^2 + 6*w - 2],
[311, 311, w^2 - 5],
[317, 317, -w^3 + 5*w^2 - 2*w - 11],
[331, 331, -w^3 + 3*w^2 - 5],
[349, 349, -2*w^2 + 5*w + 2],
[353, 353, -2*w^3 + 5*w^2 + 4*w - 4],
[353, 353, 3*w^3 - 7*w^2 - 10*w + 2],
[353, 353, -2*w^3 + 7*w^2 + 2*w - 10],
[353, 353, w^3 + w^2 - 9*w - 13],
[361, 19, -w^3 + w^2 + 5*w - 1],
[373, 373, -w^3 + 3*w^2 + 4*w - 11],
[373, 373, 2*w^3 - 5*w^2 - 5*w - 2],
[383, 383, w^3 - 5*w^2 + w + 13],
[389, 389, 3*w^3 - 10*w^2 - 4*w + 16],
[397, 397, -w^3 + 2*w^2 + 3*w + 5],
[397, 397, -3*w^3 + 7*w^2 + 10*w - 7],
[419, 419, -w^3 + 2*w^2 + 4*w + 4],
[419, 419, 3*w^3 - 7*w^2 - 12*w + 5],
[421, 421, -w^2 + 5*w - 2],
[421, 421, 3*w^3 - 7*w^2 - 10*w + 5],
[431, 431, w^2 - 10],
[439, 439, w^3 - w^2 - 7*w - 7],
[461, 461, -2*w^3 + 7*w^2 + 5*w - 10],
[467, 467, -3*w^3 + 6*w^2 + 14*w - 4],
[479, 479, 3*w^2 - 6*w - 5],
[479, 479, -4*w^3 + 11*w^2 + 10*w - 10],
[491, 491, -w^3 + w^2 + 6*w - 1],
[499, 499, -4*w^3 + 9*w^2 + 15*w - 10],
[499, 499, 2*w^3 - 5*w^2 - 7*w + 2],
[503, 503, -2*w^3 + 7*w^2 - 8],
[521, 521, 2*w^2 - w - 7],
[521, 521, -3*w^3 + 7*w^2 + 11*w - 11],
[569, 569, 2*w^2 - 5*w - 1],
[569, 569, -4*w - 1],
[571, 571, -w^3 + 3*w^2 - 7],
[577, 577, -w^3 + w^2 + 8*w - 4],
[587, 587, -3*w^3 + 8*w^2 + 7*w - 8],
[593, 593, 3*w^2 - 4*w - 11],
[593, 593, -3*w^3 + 9*w^2 + 6*w - 14],
[601, 601, -3*w^3 + 9*w^2 + 4*w - 10],
[601, 601, 5*w^3 - 9*w^2 - 24*w - 2],
[607, 607, 2*w^2 - 7*w - 5],
[613, 613, w^3 - 6*w - 2],
[613, 613, -w^2 + 3*w + 8],
[617, 617, -w^2 + 4*w - 5],
[619, 619, 3*w^3 - 6*w^2 - 13*w + 5],
[625, 5, -5],
[631, 631, -2*w^3 + 4*w^2 + 8*w - 7],
[647, 647, -2*w^3 + 4*w^2 + 5*w - 1],
[647, 647, w^3 - 3*w^2 - 2*w - 2],
[653, 653, w^3 - 4*w^2 - w + 13],
[653, 653, w^3 - 2*w^2 - 7*w + 5],
[659, 659, -4*w^3 + 10*w^2 + 15*w - 14],
[659, 659, 5*w^3 - 12*w^2 - 17*w + 14],
[661, 661, w^2 - 5*w - 2],
[661, 661, -w^3 + 5*w^2 - 2*w - 7],
[673, 673, -w^3 + w^2 + 6*w - 2],
[683, 683, 2*w^3 - 8*w^2 + w + 8],
[683, 683, 5*w^3 - 12*w^2 - 19*w + 14],
[701, 701, -2*w^3 + 6*w^2 + 5*w - 4],
[709, 709, -w^2 + 3*w - 4],
[709, 709, -2*w^3 + 5*w^2 + 9*w - 8],
[719, 719, w^2 - 7],
[719, 719, -2*w^3 + 3*w^2 + 8*w + 5],
[733, 733, -2*w^3 + 7*w^2 + 3*w - 10],
[739, 739, 4*w^3 - 11*w^2 - 13*w + 14],
[739, 739, -5*w^3 + 10*w^2 + 21*w - 4],
[739, 739, 3*w^3 - 8*w^2 - 6*w + 8],
[739, 739, -w^3 + w^2 + 8*w - 5],
[743, 743, -w - 5],
[743, 743, w^3 - w^2 - 8*w - 8],
[751, 751, -2*w^3 + 5*w^2 + 8*w - 2],
[751, 751, -2*w^3 + 6*w^2 - 1],
[757, 757, -2*w^3 + 6*w^2 + 3*w - 10],
[757, 757, 5*w^2 - 9*w - 22],
[761, 761, -2*w^3 + 4*w^2 + 9*w - 7],
[761, 761, 2*w^3 - 3*w^2 - 12*w - 1],
[769, 769, 5*w^3 - 13*w^2 - 14*w + 10],
[773, 773, -3*w^3 + 5*w^2 + 16*w + 1],
[773, 773, w^3 - 11*w + 2],
[787, 787, -w^3 + 4*w^2 - w - 11],
[797, 797, -5*w^3 + 12*w^2 + 17*w - 11],
[809, 809, -2*w^3 + 7*w^2 + 4*w - 11],
[809, 809, 4*w^3 - 9*w^2 - 14*w + 4],
[821, 821, -w^3 + 5*w^2 - 2*w - 17],
[827, 827, -3*w^3 + 8*w^2 + 9*w - 7],
[829, 829, 2*w^3 - 6*w^2 - 5*w + 2],
[829, 829, -w^3 + 4*w^2 - w - 10],
[839, 839, w^2 - 8],
[853, 853, -5*w^3 + 12*w^2 + 18*w - 16],
[853, 853, w^3 - 6*w - 8],
[863, 863, 5*w^3 - 11*w^2 - 19*w + 7],
[881, 881, -3*w^3 + 9*w^2 + 9*w - 11],
[907, 907, -w^3 + 2*w^2 + 7*w - 4],
[907, 907, -2*w^3 + 3*w^2 + 7*w + 4],
[929, 929, -w^3 + 5*w^2 - 3*w - 11],
[937, 937, 2*w^3 - 5*w^2 - 6*w + 11],
[937, 937, -2*w^3 + 6*w^2 + 3*w - 11],
[941, 941, -3*w^3 + 8*w^2 + 11*w - 8],
[953, 953, -4*w^3 + 10*w^2 + 10*w - 13],
[953, 953, w^3 - 5*w^2 - w + 13],
[967, 967, 3*w^3 - 7*w^2 - 9*w + 7],
[967, 967, -2*w^3 + 5*w^2 + 8*w + 1],
[967, 967, 4*w^3 - 13*w^2 - 6*w + 20],
[967, 967, -2*w^2 + 3*w + 13],
[971, 971, -6*w^3 + 13*w^2 + 25*w - 8],
[977, 977, w^3 - w^2 - 4*w - 10],
[977, 977, 2*w^3 - 6*w^2 - 7*w + 13],
[983, 983, -4*w^3 + 10*w^2 + 11*w - 13],
[983, 983, -w^3 + 5*w^2 - w - 7],
[983, 983, 3*w^3 - 9*w^2 - 7*w + 19],
[983, 983, -2*w^3 + 3*w^2 + 10*w - 1],
[997, 997, 5*w^3 - 12*w^2 - 17*w + 13],
[997, 997, -2*w^3 + 3*w^2 + 13*w - 4],
[997, 997, w^3 - 7*w - 2],
[997, 997, w^3 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^15 + 9*x^14 + 11*x^13 - 119*x^12 - 357*x^11 + 364*x^10 + 2321*x^9 + 560*x^8 - 6394*x^7 - 4638*x^6 + 8032*x^5 + 7969*x^4 - 3706*x^3 - 4532*x^2 - 122*x + 121;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1149312/38730541*e^14 - 7184077/38730541*e^13 + 14436223/38730541*e^12 + 165423074/38730541*e^11 + 69266219/38730541*e^10 - 1338913939/38730541*e^9 - 1645455201/38730541*e^8 + 4610167070/38730541*e^7 + 7655474852/38730541*e^6 - 6888753029/38730541*e^5 - 14012738698/38730541*e^4 + 3343795565/38730541*e^3 + 9183536556/38730541*e^2 + 609599068/38730541*e - 336254426/38730541, 6478798/38730541*e^14 + 47609852/38730541*e^13 - 4213167/38730541*e^12 - 736514880/38730541*e^11 - 1057543996/38730541*e^10 + 3820615808/38730541*e^9 + 7812559839/38730541*e^8 - 8972256441/38730541*e^7 - 22729602193/38730541*e^6 + 9828866711/38730541*e^5 + 30196561786/38730541*e^4 - 3828563311/38730541*e^3 - 15632457265/38730541*e^2 - 527508772/38730541*e + 578461068/38730541, -4209305/38730541*e^14 - 31351880/38730541*e^13 - 2365132/38730541*e^12 + 464903034/38730541*e^11 + 737556412/38730541*e^10 - 2210350397/38730541*e^9 - 5028290935/38730541*e^8 + 4431276384/38730541*e^7 + 13305414971/38730541*e^6 - 3757335805/38730541*e^5 - 15465636833/38730541*e^4 + 1347881528/38730541*e^3 + 6636828440/38730541*e^2 - 630409979/38730541*e - 250540177/38730541, 4021775/38730541*e^14 + 31226025/38730541*e^13 + 8127810/38730541*e^12 - 466161468/38730541*e^11 - 827781584/38730541*e^10 + 2228079589/38730541*e^9 + 5752887182/38730541*e^8 - 4368368014/38730541*e^7 - 16067144819/38730541*e^6 + 3117216021/38730541*e^5 + 20317813783/38730541*e^4 + 92512539/38730541*e^3 - 9817435739/38730541*e^2 - 532448099/38730541*e + 300617288/38730541, -2813770/38730541*e^14 - 20434755/38730541*e^13 + 2760889/38730541*e^12 + 311854874/38730541*e^11 + 428702275/38730541*e^10 - 1565637859/38730541*e^9 - 3031404482/38730541*e^8 + 3427006833/38730541*e^7 + 7985337449/38730541*e^6 - 3313988033/38730541*e^5 - 8915746809/38730541*e^4 + 1010872182/38730541*e^3 + 3568654330/38730541*e^2 + 234629224/38730541*e - 224188721/38730541, -3881433/38730541*e^14 - 28122699/38730541*e^13 + 1032750/38730541*e^12 + 413579383/38730541*e^11 + 613201479/38730541*e^10 - 1927406960/38730541*e^9 - 4097200599/38730541*e^8 + 3690728832/38730541*e^7 + 10108392637/38730541*e^6 - 2850206563/38730541*e^5 - 9992166316/38730541*e^4 + 1028753663/38730541*e^3 + 2742856697/38730541*e^2 - 692518215/38730541*e + 59268219/38730541, 1, 4157170/38730541*e^14 + 30599200/38730541*e^13 - 3552544/38730541*e^12 - 484931053/38730541*e^11 - 705232774/38730541*e^10 + 2581099380/38730541*e^9 + 5489815036/38730541*e^8 - 5999252862/38730541*e^7 - 16647353138/38730541*e^6 + 5981267252/38730541*e^5 + 22676585465/38730541*e^4 - 1800385783/38730541*e^3 - 11559720313/38730541*e^2 - 197547048/38730541*e + 152397933/38730541, -11093623/38730541*e^14 - 85184009/38730541*e^13 - 23012600/38730541*e^12 + 1237953834/38730541*e^11 + 2214196187/38730541*e^10 - 5594979691/38730541*e^9 - 14753667597/38730541*e^8 + 9783760495/38730541*e^7 + 38944055503/38730541*e^6 - 4901990310/38730541*e^5 - 45712255198/38730541*e^4 - 1676551142/38730541*e^3 + 20144907143/38730541*e^2 + 357017736/38730541*e - 732134231/38730541, 422489/38730541*e^14 + 6976140/38730541*e^13 + 27381786/38730541*e^12 - 46324328/38730541*e^11 - 454382987/38730541*e^10 - 394577216/38730541*e^9 + 2012584340/38730541*e^8 + 3477315889/38730541*e^7 - 2656352415/38730541*e^6 - 8124636232/38730541*e^5 - 1335538241/38730541*e^4 + 5515895686/38730541*e^3 + 3340334873/38730541*e^2 + 856361955/38730541*e + 89879071/38730541, 299665/38730541*e^14 - 431903/38730541*e^13 - 17178502/38730541*e^12 - 27716981/38730541*e^11 + 170512690/38730541*e^10 + 396197785/38730541*e^9 - 485502803/38730541*e^8 - 1344340550/38730541*e^7 + 674995080/38730541*e^6 + 1384580022/38730541*e^5 - 1254161855/38730541*e^4 - 246620783/38730541*e^3 + 1377661938/38730541*e^2 + 221228000/38730541*e - 117674591/38730541, 1772671/38730541*e^14 + 12781161/38730541*e^13 - 3126020/38730541*e^12 - 206434076/38730541*e^11 - 281505568/38730541*e^10 + 1133084400/38730541*e^9 + 2296335203/38730541*e^8 - 2732396831/38730541*e^7 - 7164666175/38730541*e^6 + 2741844951/38730541*e^5 + 9783473640/38730541*e^4 - 380598901/38730541*e^3 - 4690413853/38730541*e^2 - 789499607/38730541*e + 47281923/38730541, -87270/38730541*e^14 - 10913777/38730541*e^13 - 81297412/38730541*e^12 - 42321182/38730541*e^11 + 1057071187/38730541*e^10 + 2190289953/38730541*e^9 - 3580976680/38730541*e^8 - 11956634342/38730541*e^7 + 2361119996/38730541*e^6 + 23966710602/38730541*e^5 + 5801369294/38730541*e^4 - 17073482565/38730541*e^3 - 6950141563/38730541*e^2 + 1021834840/38730541*e + 239596668/38730541, 5364734/38730541*e^14 + 42279285/38730541*e^13 + 18114455/38730541*e^12 - 597586842/38730541*e^11 - 1144156922/38730541*e^10 + 2566418544/38730541*e^9 + 7201119706/38730541*e^8 - 4205213812/38730541*e^7 - 17975601276/38730541*e^6 + 2227344096/38730541*e^5 + 19732829207/38730541*e^4 - 520332576/38730541*e^3 - 7889126799/38730541*e^2 + 1017412964/38730541*e + 72976851/38730541, -7371788/38730541*e^14 - 58329117/38730541*e^13 - 25060976/38730541*e^12 + 841466104/38730541*e^11 + 1655558377/38730541*e^10 - 3670513475/38730541*e^9 - 11002165406/38730541*e^8 + 5373361548/38730541*e^7 + 29101710115/38730541*e^6 + 735845112/38730541*e^5 - 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2132611500/38730541, -390440/38730541*e^14 - 14659227/38730541*e^13 - 55037482/38730541*e^12 + 222984860/38730541*e^11 + 1117901842/38730541*e^10 - 970759066/38730541*e^9 - 7646105124/38730541*e^8 + 764268855/38730541*e^7 + 23006386559/38730541*e^6 + 1853412247/38730541*e^5 - 30507893667/38730541*e^4 + 875101321/38730541*e^3 + 13466710100/38730541*e^2 - 4666455977/38730541*e + 760829913/38730541, -1005049/38730541*e^14 - 18833644/38730541*e^13 - 65647942/38730541*e^12 + 242325575/38730541*e^11 + 1453072639/38730541*e^10 - 218006575/38730541*e^9 - 9636717500/38730541*e^8 - 6830282946/38730541*e^7 + 24568170517/38730541*e^6 + 24508270132/38730541*e^5 - 22143685034/38730541*e^4 - 22919499224/38730541*e^3 + 1439205842/38730541*e^2 + 1583788798/38730541*e + 1738919503/38730541, 36201528/38730541*e^14 + 263519058/38730541*e^13 - 47550339/38730541*e^12 - 4133422299/38730541*e^11 - 5555595911/38730541*e^10 + 21985256101/38730541*e^9 + 41928722769/38730541*e^8 - 53465012174/38730541*e^7 - 122858384127/38730541*e^6 + 60437219550/38730541*e^5 + 162581559266/38730541*e^4 - 22136519892/38730541*e^3 - 81802291684/38730541*e^2 - 5057175686/38730541*e + 1227294132/38730541, 12709699/38730541*e^14 + 98022202/38730541*e^13 + 24781257/38730541*e^12 - 1426894438/38730541*e^11 - 2413262821/38730541*e^10 + 6815598154/38730541*e^9 + 15770632326/38730541*e^8 - 15756228321/38730541*e^7 - 43017804978/38730541*e^6 + 20876962802/38730541*e^5 + 56093310815/38730541*e^4 - 16449437749/38730541*e^3 - 29436626137/38730541*e^2 + 6873160682/38730541*e + 910801425/38730541, 45865850/38730541*e^14 + 302382654/38730541*e^13 - 330398881/38730541*e^12 - 5595538850/38730541*e^11 - 4054535551/38730541*e^10 + 37484989986/38730541*e^9 + 49038499826/38730541*e^8 - 115989287542/38730541*e^7 - 186765121964/38730541*e^6 + 165324420450/38730541*e^5 + 304863569019/38730541*e^4 - 79059351462/38730541*e^3 - 183777774601/38730541*e^2 - 14322579216/38730541*e + 4180010490/38730541, 9794594/38730541*e^14 + 48474115/38730541*e^13 - 196545306/38730541*e^12 - 1276143699/38730541*e^11 + 709938020/38730541*e^10 + 11453021446/38730541*e^9 + 5767066747/38730541*e^8 - 42628773635/38730541*e^7 - 40534197238/38730541*e^6 + 66800106729/38730541*e^5 + 82727396234/38730541*e^4 - 30070866603/38730541*e^3 - 55705488601/38730541*e^2 - 11057291134/38730541*e + 1286744489/38730541, 17506699/38730541*e^14 + 113317361/38730541*e^13 - 160385297/38730541*e^12 - 2273387085/38730541*e^11 - 1280046830/38730541*e^10 + 16876905396/38730541*e^9 + 20437990318/38730541*e^8 - 58469322035/38730541*e^7 - 90306071611/38730541*e^6 + 94573024928/38730541*e^5 + 167051955110/38730541*e^4 - 56898693212/38730541*e^3 - 112265489788/38730541*e^2 - 607339138/38730541*e + 3880261062/38730541, 6430159/38730541*e^14 + 73974821/38730541*e^13 + 192840041/38730541*e^12 - 688970550/38730541*e^11 - 3778814124/38730541*e^10 - 1137267907/38730541*e^9 + 18180530735/38730541*e^8 + 21612085153/38730541*e^7 - 29365997256/38730541*e^6 - 55072663978/38730541*e^5 + 3238494828/38730541*e^4 + 36792985149/38730541*e^3 + 18520409074/38730541*e^2 + 7297033100/38730541*e - 142995297/38730541, 23220920/38730541*e^14 + 164531263/38730541*e^13 - 72497589/38730541*e^12 - 2726603880/38730541*e^11 - 3084649541/38730541*e^10 + 16136280097/38730541*e^9 + 26691982357/38730541*e^8 - 46501034684/38730541*e^7 - 90637239321/38730541*e^6 + 65505535364/38730541*e^5 + 142955524522/38730541*e^4 - 32404017883/38730541*e^3 - 86387068576/38730541*e^2 - 5425149488/38730541*e + 1601846865/38730541, 35828993/38730541*e^14 + 265116452/38730541*e^13 + 362334/38730541*e^12 - 3975429949/38730541*e^11 - 5952652244/38730541*e^10 + 19459435527/38730541*e^9 + 41139322966/38730541*e^8 - 41711844484/38730541*e^7 - 110646833494/38730541*e^6 + 39137419227/38730541*e^5 + 131669520731/38730541*e^4 - 10456410198/38730541*e^3 - 57825637535/38730541*e^2 - 2582074429/38730541*e + 1752383557/38730541, 1502917/38730541*e^14 + 24242211/38730541*e^13 + 127089893/38730541*e^12 + 77530234/38730541*e^11 - 1497167161/38730541*e^10 - 4315620160/38730541*e^9 + 1818661824/38730541*e^8 + 21502444565/38730541*e^7 + 14919956792/38730541*e^6 - 37421490329/38730541*e^5 - 43093150344/38730541*e^4 + 20650064169/38730541*e^3 + 32714334147/38730541*e^2 + 1587659298/38730541*e - 1695138547/38730541, -5016594/38730541*e^14 - 34055002/38730541*e^13 + 29777874/38730541*e^12 + 634948457/38730541*e^11 + 674223775/38730541*e^10 - 3938207024/38730541*e^9 - 7263481476/38730541*e^8 + 9117947599/38730541*e^7 + 23828946048/38730541*e^6 - 6466856329/38730541*e^5 - 30090195793/38730541*e^4 - 58406121/38730541*e^3 + 12016087369/38730541*e^2 - 1785384664/38730541*e + 653326282/38730541, -42882205/38730541*e^14 - 311317060/38730541*e^13 + 99433768/38730541*e^12 + 5213205582/38730541*e^11 + 6774897895/38730541*e^10 - 30202306990/38730541*e^9 - 58485697995/38730541*e^8 + 77779049396/38730541*e^7 + 191725510727/38730541*e^6 - 86322751436/38730541*e^5 - 280989903629/38730541*e^4 + 22400686785/38730541*e^3 + 156054646322/38730541*e^2 + 15668050118/38730541*e - 4189784326/38730541, 22155776/38730541*e^14 + 186896134/38730541*e^13 + 180990879/38730541*e^12 - 2377120836/38730541*e^11 - 6128005181/38730541*e^10 + 7456091034/38730541*e^9 + 34910808727/38730541*e^8 - 504096871/38730541*e^7 - 80128319091/38730541*e^6 - 24238859867/38730541*e^5 + 80642835605/38730541*e^4 + 20541794771/38730541*e^3 - 30031248128/38730541*e^2 + 3528575837/38730541*e + 988908953/38730541, 27968935/38730541*e^14 + 205922670/38730541*e^13 - 41791217/38730541*e^12 - 3387133263/38730541*e^11 - 4744013068/38730541*e^10 + 18983488287/38730541*e^9 + 39357332004/38730541*e^8 - 46149704452/38730541*e^7 - 125098263238/38730541*e^6 + 46199182048/38730541*e^5 + 177483968018/38730541*e^4 - 8620649904/38730541*e^3 - 96067039042/38730541*e^2 - 7619456331/38730541*e + 4003956790/38730541, -43864154/38730541*e^14 - 314001090/38730541*e^13 + 85208240/38730541*e^12 + 4990876308/38730541*e^11 + 6464158786/38730541*e^10 - 26965547283/38730541*e^9 - 50707635503/38730541*e^8 + 65944994446/38730541*e^7 + 151982059784/38730541*e^6 - 74662090178/38730541*e^5 - 205073620375/38730541*e^4 + 29354191976/38730541*e^3 + 105774326265/38730541*e^2 + 4290061593/38730541*e - 1424797193/38730541];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, w^3 - 2*w^2 - 3*w + 2}>] := -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;