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

NN := ideal<ZF | {17, 17, -w^2 + w + 4}>;

primesArray := [
[3, 3, w],
[3, 3, w + 1],
[3, 3, w + 2],
[8, 2, 2],
[11, 11, -w^2 + 5],
[13, 13, w^2 - w - 7],
[17, 17, -w^2 + w + 4],
[19, 19, -w^2 - w + 4],
[29, 29, 2*w^2 - 2*w - 11],
[31, 31, w^2 - 2*w - 4],
[37, 37, -2*w^2 + 3*w + 7],
[41, 41, w^2 - 2],
[59, 59, 2*w^2 - 13],
[61, 61, w^2 + w - 10],
[67, 67, 2*w^2 - w - 11],
[73, 73, -w^2 - 1],
[83, 83, w^2 - w - 10],
[89, 89, -w^2 + 4*w - 2],
[97, 97, w^2 - 2*w - 7],
[97, 97, -w^2 - 4*w - 5],
[97, 97, 2*w^2 + w - 8],
[101, 101, -4*w^2 + 2*w + 25],
[103, 103, w^2 + w - 7],
[107, 107, -3*w - 4],
[109, 109, 5*w^2 - 3*w - 31],
[121, 11, 2*w^2 - 3*w - 4],
[125, 5, -5],
[137, 137, -5*w^2 + 3*w + 34],
[139, 139, w^2 + 2*w - 4],
[151, 151, 3*w^2 - 23],
[151, 151, w^2 - 3*w - 5],
[151, 151, 3*w^2 - 3*w - 17],
[157, 157, 4*w^2 - 9*w - 8],
[157, 157, w^2 + 3*w - 2],
[157, 157, 2*w^2 - w - 2],
[167, 167, 2*w^2 + w - 11],
[167, 167, -w^2 + 11],
[167, 167, -3*w^2 + 19],
[169, 13, w^2 - 4*w - 4],
[173, 173, 2*w^2 - 3*w - 10],
[179, 179, -3*w + 7],
[191, 191, -w^2 + 5*w - 5],
[193, 193, -w^2 + w - 2],
[199, 199, 3*w - 2],
[211, 211, w^2 - 3*w - 11],
[223, 223, 3*w^2 - 20],
[223, 223, w^2 - 5*w - 4],
[223, 223, 2*w^2 - w - 8],
[229, 229, 4*w^2 - 3*w - 26],
[229, 229, 2*w^2 + w - 20],
[229, 229, 2*w^2 - 2*w - 5],
[239, 239, -6*w^2 + 3*w + 38],
[241, 241, -w^2 + 4*w - 5],
[257, 257, -2*w^2 + 5*w - 1],
[263, 263, -2*w^2 - w + 17],
[269, 269, -w^2 - w + 13],
[269, 269, 3*w - 4],
[269, 269, -4*w^2 + w + 31],
[271, 271, 2*w^2 - 7],
[271, 271, 3*w - 5],
[271, 271, w^2 - 3*w - 8],
[289, 17, 2*w^2 + w - 5],
[307, 307, 2*w^2 - w - 5],
[311, 311, 3*w^2 - 3*w - 19],
[311, 311, -7*w^2 + 13*w + 19],
[311, 311, w^2 - 2*w - 13],
[313, 313, w^2 + 2*w - 7],
[317, 317, -5*w^2 + w + 35],
[331, 331, -w^2 + 3*w - 4],
[337, 337, 3*w^2 - 3*w - 13],
[343, 7, -7],
[347, 347, 2*w^2 - 3*w - 13],
[349, 349, 3*w^2 - 3*w - 20],
[359, 359, -w^2 + 5*w - 2],
[361, 19, -2*w^2 + 7*w - 1],
[389, 389, -2*w^2 + w - 1],
[397, 397, w^2 - 6*w + 10],
[401, 401, 6*w^2 - 3*w - 44],
[419, 419, -3*w^2 + 6*w + 10],
[419, 419, 4*w^2 - 2*w - 31],
[419, 419, -4*w^2 + 4*w + 19],
[421, 421, 6*w^2 - 12*w - 13],
[431, 431, 4*w^2 - 7*w - 10],
[433, 433, w^2 - 4*w - 7],
[443, 443, 2*w^2 + 4*w - 5],
[443, 443, 5*w^2 - 11*w - 11],
[443, 443, 3*w^2 - 14],
[449, 449, -2*w^2 + 2*w - 1],
[457, 457, w^2 - 6*w - 5],
[457, 457, w^2 + 3*w - 5],
[457, 457, 6*w^2 - 3*w - 37],
[463, 463, w^2 + 5*w - 1],
[467, 467, w^2 - w - 13],
[479, 479, 5*w^2 - 9*w - 16],
[487, 487, -6*w - 5],
[499, 499, 3*w^2 - 3*w - 4],
[499, 499, 3*w^2 + 3*w - 1],
[499, 499, w^2 - 5*w + 8],
[503, 503, 4*w^2 - 5*w - 16],
[509, 509, 2*w^2 - 5*w - 8],
[521, 521, 4*w^2 - w - 22],
[523, 523, w^2 - 5*w - 16],
[547, 547, 5*w^2 - 2*w - 38],
[547, 547, 8*w^2 - 3*w - 58],
[547, 547, 4*w^2 - 29],
[557, 557, 7*w^2 - 4*w - 43],
[557, 557, 3*w - 11],
[557, 557, 2*w^2 - 4*w - 11],
[569, 569, 5*w^2 - 4*w - 32],
[569, 569, 3*w^2 - 3*w - 11],
[569, 569, w^2 + 2*w - 16],
[571, 571, w^2 - 4*w - 10],
[593, 593, 2*w^2 - 6*w - 7],
[601, 601, -w^2 - 3*w + 14],
[607, 607, 3*w^2 + 3*w - 7],
[619, 619, 3*w^2 - 6*w - 11],
[631, 631, -w^2 - w - 5],
[641, 641, 3*w^2 - 3*w - 5],
[647, 647, -w^2 + 6*w - 1],
[653, 653, -5*w^2 + 4*w + 35],
[661, 661, 3*w^2 - 3*w - 10],
[673, 673, w^2 + 4*w - 4],
[677, 677, 5*w^2 - 6*w - 25],
[677, 677, w^2 - 14],
[677, 677, 4*w^2 - 3*w - 20],
[691, 691, 5*w^2 - 2*w - 29],
[701, 701, 5*w^2 - 10*w - 14],
[719, 719, -3*w - 11],
[727, 727, -2*w^2 + 9*w - 8],
[733, 733, -w^2 - 4*w - 8],
[739, 739, w^2 + 3*w - 8],
[743, 743, 2*w^2 - 4*w - 17],
[757, 757, -w^2 + w - 5],
[769, 769, 3*w^2 - 3*w - 7],
[769, 769, -w^2 - 3*w - 7],
[769, 769, 5*w^2 - 3*w - 28],
[773, 773, 7*w^2 - 6*w - 38],
[787, 787, 2*w^2 - w - 20],
[797, 797, -5*w^2 + 14*w - 1],
[821, 821, -w^2 + 6*w - 4],
[823, 823, 7*w^2 - 6*w - 41],
[839, 839, 4*w^2 - 6*w - 11],
[841, 29, 5*w^2 - 4*w - 26],
[857, 857, 3*w^2 - 11],
[863, 863, 2*w^2 + 2*w - 17],
[881, 881, 5*w^2 - 5*w - 29],
[883, 883, -4*w^2 + 8*w + 13],
[907, 907, 4*w^2 - 9*w - 11],
[911, 911, 2*w^2 + 3*w - 10],
[947, 947, 2*w^2 + 5*w - 5],
[953, 953, 2*w^2 - 5*w - 11],
[961, 31, w^2 - 5*w - 13],
[967, 967, 2*w^2 - 3*w - 22],
[967, 967, 5*w^2 - 9*w - 10],
[967, 967, 3*w^2 - 10],
[977, 977, 5*w^2 - 37],
[983, 983, 7*w^2 - 5*w - 40],
[997, 997, -7*w^2 + 5*w + 49]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^14 - 30*x^12 + 8*x^11 + 329*x^10 - 175*x^9 - 1602*x^8 + 1255*x^7 + 3262*x^6 - 3415*x^5 - 1605*x^4 + 2650*x^3 - 744*x^2 - 8*x + 16;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -9028111661/115289117408*e^13 - 1907258453/57644558704*e^12 + 134605531889/57644558704*e^11 + 10479091471/28822279352*e^10 - 2950583297437/115289117408*e^9 + 321907883097/115289117408*e^8 + 3640427020883/28822279352*e^7 - 5055938038387/115289117408*e^6 - 7831853709287/28822279352*e^5 + 17004131629123/115289117408*e^4 + 21031540883095/115289117408*e^3 - 3474031368921/28822279352*e^2 + 70471169307/7205569838*e + 7071029893/14411139676, 12904312733/57644558704*e^13 + 2405167993/28822279352*e^12 - 192468332349/28822279352*e^11 - 10001552507/14411139676*e^10 + 4219870264173/57644558704*e^9 - 689132794625/57644558704*e^8 - 5207249635083/14411139676*e^7 + 8443037500059/57644558704*e^6 + 11212637902365/14411139676*e^5 - 27347634665211/57644558704*e^4 - 30305133927951/57644558704*e^3 + 5739221784653/14411139676*e^2 - 169875131613/7205569838*e - 92269231367/7205569838, 17397364377/57644558704*e^13 + 3154357589/28822279352*e^12 - 259432951445/28822279352*e^11 - 12035621523/14411139676*e^10 + 5684375261273/57644558704*e^9 - 1001804405421/57644558704*e^8 - 7001657384413/14411139676*e^7 + 11811615053415/57644558704*e^6 + 14995241540321/14411139676*e^5 - 37986553568823/57644558704*e^4 - 39566454233443/57644558704*e^3 + 7926916614071/14411139676*e^2 - 177263608645/3602784919*e - 83327836587/7205569838, -18409573371/115289117408*e^13 - 4847857823/57644558704*e^12 + 273389557599/57644558704*e^11 + 34771295893/28822279352*e^10 - 5975306851179/115289117408*e^9 + 115264376631/115289117408*e^8 + 7377151683171/28822279352*e^7 - 7927852220325/115289117408*e^6 - 16056629573275/28822279352*e^5 + 30474203069077/115289117408*e^4 + 45890990108041/115289117408*e^3 - 6721345128093/28822279352*e^2 - 33042336911/7205569838*e + 95628138503/14411139676, 40578477861/115289117408*e^13 + 6584280729/57644558704*e^12 - 607006390921/57644558704*e^11 - 18225560763/28822279352*e^10 + 13350477494005/115289117408*e^9 - 2682809286073/115289117408*e^8 - 16524552402261/28822279352*e^7 + 28785261391115/115289117408*e^6 + 35659829233657/28822279352*e^5 - 90195798286427/115289117408*e^4 - 96089778873767/115289117408*e^3 + 18616360423891/28822279352*e^2 - 168540750441/3602784919*e - 235252825109/14411139676, -1, -24152480653/57644558704*e^13 - 4939450585/28822279352*e^12 + 359558708873/28822279352*e^11 + 24674149887/14411139676*e^10 - 7866586571709/57644558704*e^9 + 1059376102897/57644558704*e^8 + 9682970441469/14411139676*e^7 - 14837717331347/57644558704*e^6 - 20772911501081/14411139676*e^5 + 49429076826211/57644558704*e^4 + 55530633906815/57644558704*e^3 - 10403763657255/14411139676*e^2 + 199681764405/3602784919*e + 117971091457/7205569838, 4610458619/14411139676*e^13 + 298224315/3602784919*e^12 - 68862763039/7205569838*e^11 + 444250839/3602784919*e^10 + 1509920910579/14411139676*e^9 - 427680470177/14411139676*e^8 - 3713429155881/7205569838*e^7 + 3943082601937/14411139676*e^6 + 7883896203649/7205569838*e^5 - 11832768533085/14411139676*e^4 - 9925303196907/14411139676*e^3 + 4814522256143/7205569838*e^2 - 313019567846/3602784919*e - 62377177502/3602784919, -56475814873/115289117408*e^13 - 5723103557/57644558704*e^12 + 846629247157/57644558704*e^11 - 26542910097/28822279352*e^10 - 18641509267337/115289117408*e^9 + 6040843572237/115289117408*e^8 + 23035541524353/28822279352*e^7 - 51626715150695/115289117408*e^6 - 49243045707641/28822279352*e^5 + 150941796220151/115289117408*e^4 + 126510281019859/115289117408*e^3 - 30495096493575/28822279352*e^2 + 818739642325/7205569838*e + 350707378421/14411139676, 66835582663/115289117408*e^13 + 9222426367/57644558704*e^12 - 1001429190179/57644558704*e^11 - 5209991341/28822279352*e^10 + 22061413322711/115289117408*e^9 - 5553037161035/115289117408*e^8 - 27339851372657/28822279352*e^7 + 53386143768377/115289117408*e^6 + 58973861246965/28822279352*e^5 - 162719430451337/115289117408*e^4 - 157552188268517/115289117408*e^3 + 33565853370259/28822279352*e^2 - 619931528957/7205569838*e - 569999805527/14411139676, -28233094735/57644558704*e^13 - 4486100345/28822279352*e^12 + 422203704991/28822279352*e^11 + 11075663479/14411139676*e^10 - 9280340412575/57644558704*e^9 + 1948063604335/57644558704*e^8 + 5736759289323/7205569838*e^7 - 20512360462001/57644558704*e^6 - 6176329287112/3602784919*e^5 + 64144629144833/57644558704*e^4 + 66265422392681/57644558704*e^3 - 3343701437043/3602784919*e^2 + 225187027020/3602784919*e + 199306033021/7205569838, 1161886343/14411139676*e^13 + 329634322/3602784919*e^12 - 17096443913/7205569838*e^11 - 7389220994/3602784919*e^10 + 373409179819/14411139676*e^9 + 216445215831/14411139676*e^8 - 941391003327/7205569838*e^7 - 583042948487/14411139676*e^6 + 2208663723199/7205569838*e^5 + 396887998419/14411139676*e^4 - 4125354077395/14411139676*e^3 + 2084445311/7205569838*e^2 + 246813716478/3602784919*e + 2706545220/3602784919, -26155103601/57644558704*e^13 - 2505716709/28822279352*e^12 + 391663149141/28822279352*e^11 - 15170091233/14411139676*e^10 - 8608476734577/57644558704*e^9 + 2964595726069/57644558704*e^8 + 10603647131609/14411139676*e^7 - 24999949600511/57644558704*e^6 - 22511464033885/14411139676*e^5 + 73061151381215/57644558704*e^4 + 56294375401755/57644558704*e^3 - 14915610904987/14411139676*e^2 + 484494459204/3602784919*e + 196556192545/7205569838, -37118059333/115289117408*e^13 - 5725718593/57644558704*e^12 + 557137667433/57644558704*e^11 + 12234002779/28822279352*e^10 - 12315029981845/115289117408*e^9 + 2660733088425/115289117408*e^8 + 15359459797817/28822279352*e^7 - 27474815204427/115289117408*e^6 - 33574725762005/28822279352*e^5 + 85226255126331/115289117408*e^4 + 93897142118935/115289117408*e^3 - 17522906183815/28822279352*e^2 + 32470218514/3602784919*e + 331000571885/14411139676, -3331071197/57644558704*e^13 - 1019192713/28822279352*e^12 + 50304686449/28822279352*e^11 + 9781266787/14411139676*e^10 - 1125694512477/57644558704*e^9 - 204171911535/57644558704*e^8 + 1441235612301/14411139676*e^7 + 87183210685/57644558704*e^6 - 3355305701301/14411139676*e^5 + 1282119161459/57644558704*e^4 + 11523422576527/57644558704*e^3 - 305316903539/14411139676*e^2 - 95631594520/3602784919*e - 9788415839/7205569838, -20386895045/28822279352*e^13 - 2412510395/14411139676*e^12 + 305679727717/14411139676*e^11 - 4549712793/7205569838*e^10 - 6737606581365/28822279352*e^9 + 1972191947581/28822279352*e^8 + 4174919267942/3602784919*e^7 - 17711176620355/28822279352*e^6 - 8993790580343/3602784919*e^5 + 52817411242467/28822279352*e^4 + 47732159616203/28822279352*e^3 - 5406170874432/3602784919*e^2 + 395781935215/3602784919*e + 184700836851/3602784919, 28031520077/57644558704*e^13 + 4177762807/28822279352*e^12 - 419874628573/28822279352*e^11 - 6917509245/14411139676*e^10 + 9245671589341/57644558704*e^9 - 2119620257029/57644558704*e^8 - 2861953895867/3602784919*e^7 + 21380133338531/57644558704*e^6 + 12315886955087/7205569838*e^5 - 66247625679859/57644558704*e^4 - 65024402977395/57644558704*e^3 + 6908704747615/7205569838*e^2 - 323260818475/3602784919*e - 215405186263/7205569838, -105431360533/115289117408*e^13 - 17418864073/57644558704*e^12 + 1574700517161/57644558704*e^11 + 47546244179/28822279352*e^10 - 34565622480773/115289117408*e^9 + 7198950516329/115289117408*e^8 + 42654494001317/28822279352*e^7 - 77279345624251/115289117408*e^6 - 91433686993025/28822279352*e^5 + 242822540801963/115289117408*e^4 + 239676503893431/115289117408*e^3 - 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1615219390479/28822279352*e^11 - 74681246053/14411139676*e^10 + 35473785329379/57644558704*e^9 - 6131695649935/57644558704*e^8 - 43820369355539/14411139676*e^7 + 72446966685821/57644558704*e^6 + 94154918949147/14411139676*e^5 - 232777307387245/57644558704*e^4 - 249597453460753/57644558704*e^3 + 48375197580765/14411139676*e^2 - 1069504547089/3602784919*e - 571026570695/7205569838, -320232859803/115289117408*e^13 - 45769367675/57644558704*e^12 + 4787089970983/57644558704*e^11 + 38712249241/28822279352*e^10 - 105142588527051/115289117408*e^9 + 26454400550175/115289117408*e^8 + 129744476070097/28822279352*e^7 - 257103412487973/115289117408*e^6 - 277749712520701/28822279352*e^5 + 785552042776437/115289117408*e^4 + 723747332387537/115289117408*e^3 - 161699897019843/28822279352*e^2 + 3924322005149/7205569838*e + 2045001629019/14411139676, 119968072885/115289117408*e^13 + 19268719137/57644558704*e^12 - 1786939314841/57644558704*e^11 - 44329486763/28822279352*e^10 + 39042368252613/115289117408*e^9 - 8695159100633/115289117408*e^8 - 47782146784369/28822279352*e^7 + 90882026226939/115289117408*e^6 + 100785770613069/28822279352*e^5 - 284220638133675/115289117408*e^4 - 251672582729991/115289117408*e^3 + 58959323311863/28822279352*e^2 - 941485904370/3602784919*e - 411611938221/14411139676, -29388792591/57644558704*e^13 - 5580165983/28822279352*e^12 + 440604576635/28822279352*e^11 + 28064329937/14411139676*e^10 - 9717876851199/57644558704*e^9 + 1146282930067/57644558704*e^8 + 12068659993977/14411139676*e^7 - 16058897024385/57644558704*e^6 - 26147451310437/14411139676*e^5 + 52660347071185/57644558704*e^4 + 71324129428797/57644558704*e^3 - 10549311260663/14411139676*e^2 + 135093415083/3602784919*e + 111368721451/7205569838, 30484993891/57644558704*e^13 - 846170773/28822279352*e^12 - 461551836687/28822279352*e^11 + 71781911815/14411139676*e^10 + 10266108693923/57644558704*e^9 - 5723121024743/57644558704*e^8 - 12800941370845/14411139676*e^7 + 39717079630029/57644558704*e^6 + 27479191667573/14411139676*e^5 - 106093548242381/57644558704*e^4 - 68809214294169/57644558704*e^3 + 20376375625443/14411139676*e^2 - 637285005536/3602784919*e - 253827601287/7205569838, -44570096459/115289117408*e^13 - 7276411775/57644558704*e^12 + 668863923487/57644558704*e^11 + 24612897173/28822279352*e^10 - 14785802101371/115289117408*e^9 + 2511755304007/115289117408*e^8 + 18473721441603/28822279352*e^7 - 27913782272885/115289117408*e^6 - 40720283742315/28822279352*e^5 + 86687766188069/115289117408*e^4 + 117492046085561/115289117408*e^3 - 17155144570333/28822279352*e^2 + 42496209727/7205569838*e - 65358710521/14411139676];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17, 17, -w^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;