/*
  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![5, -1, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, -w^2 + w + 1}>;

primesArray := [
[5, 5, w],
[5, 5, -w^2 + w + 2],
[7, 7, -w^2 + 2],
[13, 13, -w^2 + 3],
[13, 13, w^2 - w - 4],
[16, 2, 2],
[19, 19, -w^2 + w + 1],
[23, 23, -w^3 + 4*w + 2],
[25, 5, -w^3 + w^2 + 3*w - 1],
[29, 29, w^3 - w^2 - 4*w + 1],
[29, 29, -w + 3],
[31, 31, -w^3 + w^2 + 5*w - 2],
[37, 37, w^3 - 4*w - 1],
[37, 37, w^3 - 3*w + 1],
[53, 53, -w^3 + 2*w^2 + 4*w - 6],
[59, 59, w^2 + w - 4],
[61, 61, -2*w^2 + w + 8],
[73, 73, -w^3 + 5*w - 1],
[79, 79, 2*w^2 + w - 6],
[79, 79, 2*w^3 - w^2 - 9*w + 3],
[81, 3, -3],
[83, 83, -w^3 + w^2 + 3*w - 4],
[97, 97, -w^3 + 6*w + 2],
[97, 97, -2*w^3 + w^2 + 10*w - 1],
[97, 97, 3*w^3 - 4*w^2 - 15*w + 13],
[97, 97, w^3 - w^2 - 6*w + 2],
[103, 103, -2*w^3 + 3*w^2 + 11*w - 9],
[109, 109, w^2 + 3*w - 3],
[137, 137, -w^2 + 7],
[149, 149, w^3 - 2*w^2 - 5*w + 4],
[149, 149, w^3 + 2*w^2 - 6*w - 7],
[151, 151, w^3 + w^2 - 5*w - 3],
[163, 163, w^3 - 3*w - 4],
[163, 163, -w^3 + 6*w - 3],
[167, 167, w^3 - 2*w^2 - 6*w + 4],
[169, 13, w^2 + 2*w - 4],
[191, 191, w^3 - w^2 - 5*w - 2],
[191, 191, 2*w^3 - 9*w - 2],
[197, 197, w^3 - 4*w + 4],
[211, 211, -w^3 + 2*w^2 + 6*w - 8],
[223, 223, -w^3 + 3*w^2 + 4*w - 6],
[229, 229, w^3 - w^2 - 5*w - 1],
[251, 251, 2*w^3 - 2*w^2 - 7*w + 3],
[257, 257, 3*w^2 - 14],
[263, 263, w^2 - 3*w - 1],
[263, 263, w^2 - 2*w - 7],
[269, 269, -w^3 + w^2 + 2*w - 4],
[269, 269, 2*w^3 - 11*w - 1],
[271, 271, -w^3 + w^2 + 4*w - 7],
[277, 277, w^3 - w^2 - 3*w + 8],
[311, 311, w^3 + 2*w^2 - 7*w - 11],
[311, 311, -3*w^2 - 2*w + 8],
[317, 317, -2*w^3 + 2*w^2 + 7*w - 4],
[331, 331, w^2 - w - 8],
[331, 331, 3*w^3 - 2*w^2 - 12*w + 9],
[337, 337, -2*w^3 + 2*w^2 + 11*w - 8],
[343, 7, w^3 - 3*w^2 - 4*w + 11],
[347, 347, -2*w^3 + 2*w^2 + 9*w - 4],
[349, 349, -w^3 + 2*w^2 + 2*w - 6],
[353, 353, 2*w^2 - w - 12],
[353, 353, 2*w^3 - w^2 - 9*w - 1],
[367, 367, -2*w^3 + w^2 + 10*w - 6],
[367, 367, 2*w^3 - 11*w - 3],
[379, 379, -3*w - 1],
[379, 379, w^3 - 2*w - 3],
[383, 383, -w^3 + 2*w^2 + 4*w - 11],
[383, 383, -w^3 - w^2 + 7*w + 4],
[389, 389, 2*w^3 - w^2 - 11*w + 4],
[401, 401, w^3 + 2*w^2 - 7*w - 6],
[401, 401, 2*w^3 - 11*w - 6],
[419, 419, -2*w^3 + w^2 + 8*w - 1],
[421, 421, -w^3 + 3*w^2 + 3*w - 12],
[421, 421, -w^3 - 2*w^2 + 2*w + 6],
[433, 433, w^3 - 2*w^2 - 6*w + 11],
[433, 433, w^2 - 8],
[439, 439, -w^3 + 3*w^2 + 3*w - 7],
[443, 443, w^3 - w^2 - 2*w - 4],
[443, 443, 2*w^3 + w^2 - 7*w - 2],
[443, 443, 2*w^3 - w^2 - 8*w + 2],
[443, 443, -w^2 - 2],
[467, 467, 3*w^3 - 5*w^2 - 15*w + 19],
[467, 467, 2*w^3 - 3*w^2 - 8*w + 13],
[467, 467, -w^3 + 2*w^2 + 3*w - 9],
[467, 467, -w^3 + 3*w^2 + 3*w - 8],
[479, 479, -w^3 - 3*w^2 + 5*w + 13],
[479, 479, -w^3 + w^2 + w - 4],
[487, 487, 3*w^3 - w^2 - 13*w - 2],
[491, 491, w^2 + w - 9],
[499, 499, -3*w^3 + 3*w^2 + 13*w - 7],
[503, 503, 2*w^2 + w - 9],
[509, 509, -2*w^2 - 3*w + 6],
[509, 509, -w^3 + w^2 - 4],
[521, 521, -3*w^2 + w + 12],
[523, 523, 2*w^3 - 8*w - 3],
[529, 23, 2*w^3 - w^2 - 6*w - 1],
[541, 541, -w^3 + w^2 + 7*w - 4],
[547, 547, 2*w^3 - 3*w^2 - 9*w + 7],
[557, 557, 3*w^2 - 2*w - 9],
[557, 557, w^3 - 6*w - 7],
[563, 563, -2*w^3 + w^2 + 6*w - 6],
[577, 577, -w^3 + 4*w^2 + 2*w - 13],
[587, 587, w^2 - 2*w + 3],
[587, 587, w^2 + 2*w - 6],
[593, 593, -2*w^3 + 2*w^2 + 11*w - 3],
[599, 599, w^3 + 3*w^2 - 6*w - 12],
[601, 601, -2*w^3 + w^2 + 9*w + 2],
[607, 607, -w^3 + 3*w^2 + 4*w - 9],
[617, 617, -2*w^3 + w^2 + 7*w - 3],
[619, 619, -2*w^3 + 4*w^2 + 9*w - 11],
[631, 631, 3*w^2 - w - 7],
[631, 631, -w^3 - w^2 + 6*w + 1],
[641, 641, -w^3 + 2*w^2 + 5*w - 1],
[641, 641, -w^3 + 2*w^2 + 2*w - 7],
[643, 643, -3*w^3 + w^2 + 14*w - 4],
[647, 647, -3*w^3 + 5*w^2 + 14*w - 18],
[647, 647, -w^3 + 7*w + 2],
[647, 647, -2*w^3 + 7*w - 1],
[647, 647, 2*w^3 - w^2 - 10*w - 1],
[653, 653, -2*w^3 - w^2 + 11*w + 4],
[659, 659, -w^3 - w^2 + 5*w - 1],
[673, 673, -w^3 + w^2 + 7*w - 2],
[673, 673, -w^3 + w^2 + 2*w - 8],
[683, 683, 2*w^3 - 11*w - 8],
[683, 683, 2*w^3 - 7*w - 3],
[701, 701, 2*w^3 - 4*w^2 - 10*w + 17],
[727, 727, -w^3 + w^2 + 4*w - 8],
[727, 727, 3*w^2 - 11],
[739, 739, 2*w^3 + 3*w^2 - 8*w - 13],
[743, 743, 2*w^2 + w - 12],
[743, 743, 3*w^2 - 2*w - 14],
[751, 751, w^3 + w^2 - 3*w - 7],
[757, 757, -2*w^3 + 10*w - 1],
[761, 761, -2*w^3 + w^2 + 7*w - 1],
[761, 761, 3*w^2 + w - 7],
[773, 773, -2*w^3 + 2*w^2 + 8*w - 1],
[797, 797, -2*w^3 + w^2 + 9*w + 3],
[797, 797, 3*w^3 - 14*w - 8],
[809, 809, 5*w^3 - 5*w^2 - 24*w + 16],
[821, 821, -4*w^3 + 3*w^2 + 18*w - 9],
[821, 821, w^3 - w - 3],
[823, 823, 3*w^2 + w - 8],
[829, 829, -2*w^3 + 3*w^2 + 10*w - 7],
[829, 829, 3*w^2 - 2*w - 16],
[841, 29, 2*w^2 + w - 11],
[853, 853, 3*w^3 - 2*w^2 - 14*w + 3],
[853, 853, 3*w^3 - 5*w^2 - 14*w + 17],
[857, 857, 3*w^2 - w - 8],
[857, 857, 3*w^3 - w^2 - 13*w + 3],
[859, 859, w^3 - 3*w^2 - 6*w + 12],
[877, 877, w^3 + 2*w^2 - 5*w - 4],
[877, 877, w^2 - w - 9],
[881, 881, w^3 + 2*w^2 - 4*w - 12],
[881, 881, w^2 - 4*w - 4],
[883, 883, -3*w^3 + 2*w^2 + 11*w - 8],
[887, 887, -2*w^3 + w^2 + 6*w - 1],
[911, 911, 3*w^2 - w - 17],
[911, 911, 2*w^3 - 2*w^2 - 12*w + 3],
[919, 919, 5*w^3 - 6*w^2 - 22*w + 24],
[919, 919, -2*w^3 + w^2 + 6*w - 7],
[929, 929, 2*w^3 - 7*w - 1],
[937, 937, -w^3 - w^2 + 8*w - 2],
[947, 947, 3*w^3 + w^2 - 15*w - 6],
[947, 947, -2*w^3 + 9*w + 9],
[953, 953, w^3 - 2*w^2 - 6*w + 1],
[977, 977, w^3 - w^2 - 3*w + 9],
[977, 977, w^3 + w^2 - 3*w - 8],
[983, 983, w^3 - 5*w - 7],
[991, 991, -w^3 + 2*w^2 + 2*w - 8],
[991, 991, w^2 + 4*w - 1],
[997, 997, 2*w^3 + w^2 - 12*w - 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^15 + 3*x^14 - 41*x^13 - 126*x^12 + 577*x^11 + 1842*x^10 - 3282*x^9 - 11653*x^8 + 5986*x^7 + 32458*x^6 + 5471*x^5 - 32416*x^4 - 16081*x^3 + 4894*x^2 + 2029*x - 307;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [3162430915711524925/176032594393538610431*e^14 + 5750830796066504970/176032594393538610431*e^13 - 136889882048059095141/176032594393538610431*e^12 - 237892221062320161354/176032594393538610431*e^11 + 2123547747933403536917/176032594393538610431*e^10 + 305618292678556320762/16002963126685328221*e^9 - 14601130426756441839548/176032594393538610431*e^8 - 20196285876703578289412/176032594393538610431*e^7 + 4025678478814968289191/16002963126685328221*e^6 + 53482827211778684311211/176032594393538610431*e^5 - 49658264682731537084253/176032594393538610431*e^4 - 50924847260840685145636/176032594393538610431*e^3 + 12781014641035012119256/176032594393538610431*e^2 + 591614286664879062461/16002963126685328221*e - 1730514429536245571254/176032594393538610431, e, -902889607633207885/176032594393538610431*e^14 - 1780459797448078957/176032594393538610431*e^13 + 39038107276128793001/176032594393538610431*e^12 + 74243526574289736508/176032594393538610431*e^11 - 605084516177433216897/176032594393538610431*e^10 - 96894163532743072563/16002963126685328221*e^9 + 4169185533141650567560/176032594393538610431*e^8 + 6578348831896240251895/176032594393538610431*e^7 - 1164255849621792830758/16002963126685328221*e^6 - 18141601862724730274803/176032594393538610431*e^5 + 15048094916221795340139/176032594393538610431*e^4 + 18824309477577438385875/176032594393538610431*e^3 - 4762398656507202416837/176032594393538610431*e^2 - 367915082304750279859/16002963126685328221*e + 458763606796914292081/176032594393538610431, -6967803644563202358/176032594393538610431*e^14 - 13170172993958463992/176032594393538610431*e^13 + 300216348085720217658/176032594393538610431*e^12 + 544978297017205955964/176032594393538610431*e^11 - 4620281158882123264083/176032594393538610431*e^10 - 701128725071401466872/16002963126685328221*e^9 + 31318329006238786491989/176032594393538610431*e^8 + 46434311713939758461066/176032594393538610431*e^7 - 8385049696720318373616/16002963126685328221*e^6 - 123073621686124207104398/176032594393538610431*e^5 + 94897572220344919015682/176032594393538610431*e^4 + 116631880731872608484265/176032594393538610431*e^3 - 13947834973168722462009/176032594393538610431*e^2 - 1246150712701764784865/16002963126685328221*e + 1306904759341667564878/176032594393538610431, 1907810865514467839/176032594393538610431*e^14 + 3648249136095234934/176032594393538610431*e^13 - 82659661897866597925/176032594393538610431*e^12 - 151903272953279544292/176032594393538610431*e^11 + 1282982425654087434909/176032594393538610431*e^10 + 197501666269121810298/16002963126685328221*e^9 - 8815331050719676692316/176032594393538610431*e^8 - 13282386586170699554592/176032594393538610431*e^7 + 2417703611057400622400/16002963126685328221*e^6 + 35681927888647175866582/176032594393538610431*e^5 - 29135858802085078277238/176032594393538610431*e^4 - 33602522199102383838053/176032594393538610431*e^3 + 6824864833965665322584/176032594393538610431*e^2 + 265494696027366123961/16002963126685328221*e - 1266410002768660346257/176032594393538610431, 7419881652059144/16002963126685328221*e^14 - 82200434296324311/16002963126685328221*e^13 - 433610994850770212/16002963126685328221*e^12 + 3738360988067794873/16002963126685328221*e^11 + 9831483949143457550/16002963126685328221*e^10 - 61894768177873133072/16002963126685328221*e^9 - 106607339531822518834/16002963126685328221*e^8 + 453842379590582198174/16002963126685328221*e^7 + 575749538867760180378/16002963126685328221*e^6 - 1402351107810512075791/16002963126685328221*e^5 - 1508048763553984459064/16002963126685328221*e^4 + 1370015168568322531746/16002963126685328221*e^3 + 1448364663161411815255/16002963126685328221*e^2 - 15986642365458464040/16002963126685328221*e - 119174667030063938959/16002963126685328221, 1, -6698997389936674574/176032594393538610431*e^14 - 12448590695314194716/176032594393538610431*e^13 + 288304956664906360039/176032594393538610431*e^12 + 514356303793911811258/176032594393538610431*e^11 - 4425399515643133308447/176032594393538610431*e^10 - 659869231949177812205/16002963126685328221*e^9 + 29799306499174572959406/176032594393538610431*e^8 + 43516408709317531809408/176032594393538610431*e^7 - 7833284125423149973943/16002963126685328221*e^6 - 114772834896402936925737/176032594393538610431*e^5 + 83267034739074212423845/176032594393538610431*e^4 + 107265917969316249607454/176032594393538610431*e^3 - 5896787070408382155174/176032594393538610431*e^2 - 946191772341520916084/16002963126685328221*e + 984036013863709507527/176032594393538610431, 5040599570655079297/176032594393538610431*e^14 + 9194111058407361692/176032594393538610431*e^13 - 218328714449501551823/176032594393538610431*e^12 - 381061644428925348326/176032594393538610431*e^11 + 3390155490263946110261/176032594393538610431*e^10 + 490980872261877575576/16002963126685328221*e^9 - 23342358969352775940338/176032594393538610431*e^8 - 32572383304038873961048/176032594393538610431*e^7 + 6449498335169770091959/16002963126685328221*e^6 + 86491226687693828587516/176032594393538610431*e^5 - 80055922032280125188495/176032594393538610431*e^4 - 82186504624767147498712/176032594393538610431*e^3 + 21533126342232186402412/176032594393538610431*e^2 + 959065543687005074952/16002963126685328221*e - 2434812184924848186732/176032594393538610431, -4511321606862537778/176032594393538610431*e^14 - 9468715246923101514/176032594393538610431*e^13 + 194286434345106521368/176032594393538610431*e^12 + 397972923043380416636/176032594393538610431*e^11 - 2981062176251610727285/176032594393538610431*e^10 - 525979606868250753656/16002963126685328221*e^9 + 20002234874681849419642/176032594393538610431*e^8 + 36217629276423698364770/176032594393538610431*e^7 - 5169923391949591992217/16002963126685328221*e^6 - 99526841233543841316861/176032594393538610431*e^5 + 49674575382087682732448/176032594393538610431*e^4 + 95053570176545897232028/176032594393538610431*e^3 + 5140829827988372085693/176032594393538610431*e^2 - 809702845023910381103/16002963126685328221*e - 291051232906969659953/176032594393538610431, 4947456538620553316/176032594393538610431*e^14 + 9311576845289540175/176032594393538610431*e^13 - 213663154926512222060/176032594393538610431*e^12 - 385989974122319266597/176032594393538610431*e^11 + 3299858945656293650585/176032594393538610431*e^10 + 498277121760410384354/16002963126685328221*e^9 - 22489878615030213943441/176032594393538610431*e^8 - 33221108434854350767346/176032594393538610431*e^7 + 6074696769620862519275/16002963126685328221*e^6 + 89196542868718082917615/176032594393538610431*e^5 - 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4672936278795320752/16002963126685328221*e^13 + 113226582723100473912/16002963126685328221*e^12 + 193951362881876146532/16002963126685328221*e^11 - 1762111051134065814375/16002963126685328221*e^10 - 2754642060807346323939/16002963126685328221*e^9 + 12163091267689720870943/16002963126685328221*e^8 + 16682700308542775327809/16002963126685328221*e^7 - 37022887595929537018972/16002963126685328221*e^6 - 44690374337456908459364/16002963126685328221*e^5 + 41710083184434879839342/16002963126685328221*e^4 + 43239419812566171427642/16002963126685328221*e^3 - 11127189764470685787192/16002963126685328221*e^2 - 5881103560338508498251/16002963126685328221*e + 1816379016568349953466/16002963126685328221];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, -w^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;