/*
  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![-57, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {3, 3, w}>;

primesArray := [
[3, 3, w],
[3, 3, w + 2],
[4, 2, 2],
[5, 5, w + 1],
[5, 5, w + 3],
[11, 11, w + 1],
[11, 11, w + 9],
[17, 17, w + 2],
[17, 17, w + 14],
[19, 19, w],
[19, 19, w + 18],
[37, 37, -w - 4],
[37, 37, w - 5],
[43, 43, w + 16],
[43, 43, w + 26],
[49, 7, -7],
[53, 53, -w - 10],
[53, 53, w - 11],
[61, 61, w + 15],
[61, 61, w + 45],
[71, 71, w + 33],
[71, 71, w + 37],
[83, 83, w + 17],
[83, 83, w + 65],
[97, 97, w + 30],
[97, 97, w + 66],
[103, 103, w + 34],
[103, 103, w + 68],
[149, 149, w + 61],
[149, 149, w + 87],
[151, 151, w + 28],
[151, 151, w + 122],
[167, 167, w + 39],
[167, 167, w + 127],
[169, 13, -13],
[173, 173, 3*w - 20],
[173, 173, -3*w - 17],
[181, 181, w + 24],
[181, 181, w + 156],
[193, 193, 2*w - 7],
[193, 193, -2*w - 5],
[229, 229, 2*w - 1],
[233, 233, w + 27],
[233, 233, w + 205],
[241, 241, -3*w - 26],
[241, 241, 3*w - 29],
[271, 271, w + 29],
[271, 271, w + 241],
[277, 277, w + 50],
[277, 277, w + 226],
[293, 293, w + 112],
[293, 293, w + 180],
[307, 307, w + 132],
[307, 307, w + 174],
[311, 311, w + 88],
[311, 311, w + 222],
[337, 337, w + 71],
[337, 337, w + 265],
[347, 347, 2*w - 25],
[347, 347, -2*w - 23],
[359, 359, 3*w - 14],
[359, 359, -3*w - 11],
[367, 367, w + 43],
[367, 367, w + 323],
[373, 373, w + 64],
[373, 373, w + 308],
[383, 383, -3*w - 10],
[383, 383, 3*w - 13],
[397, 397, w + 82],
[397, 397, w + 314],
[401, 401, w + 35],
[401, 401, w + 365],
[409, 409, w + 67],
[409, 409, w + 341],
[421, 421, w + 46],
[421, 421, w + 374],
[431, 431, w + 119],
[431, 431, w + 311],
[433, 433, w + 175],
[433, 433, w + 257],
[439, 439, 5*w - 34],
[439, 439, -5*w - 29],
[443, 443, -3*w - 7],
[443, 443, 3*w - 10],
[449, 449, -w - 22],
[449, 449, w - 23],
[457, 457, w + 208],
[457, 457, w + 248],
[461, 461, 5*w - 46],
[461, 461, -5*w - 41],
[463, 463, w + 186],
[463, 463, w + 276],
[467, 467, w + 94],
[467, 467, w + 372],
[491, 491, w + 145],
[491, 491, w + 345],
[503, 503, 3*w - 5],
[503, 503, -3*w - 2],
[509, 509, -3*w - 1],
[509, 509, 3*w - 4],
[529, 23, -23],
[541, 541, 3*w - 34],
[541, 541, -3*w - 31],
[557, 557, w + 158],
[557, 557, w + 398],
[569, 569, w + 262],
[569, 569, w + 306],
[587, 587, w + 223],
[587, 587, w + 363],
[593, 593, -w - 25],
[593, 593, w - 26],
[607, 607, -3*w - 32],
[607, 607, 3*w - 35],
[617, 617, 6*w - 41],
[617, 617, -6*w - 35],
[619, 619, -5*w - 26],
[619, 619, 5*w - 31],
[631, 631, w + 144],
[631, 631, w + 486],
[641, 641, w + 44],
[641, 641, w + 596],
[643, 643, 7*w + 43],
[643, 643, 7*w - 50],
[661, 661, w + 77],
[661, 661, w + 583],
[673, 673, w + 58],
[673, 673, w + 614],
[683, 683, w + 101],
[683, 683, w + 581],
[691, 691, -4*w - 13],
[691, 691, 4*w - 17],
[701, 701, w + 196],
[701, 701, w + 504],
[733, 733, w + 47],
[733, 733, w + 685],
[743, 743, w + 136],
[743, 743, w + 606],
[751, 751, w + 318],
[751, 751, w + 432],
[757, 757, w + 353],
[757, 757, w + 403],
[769, 769, w + 209],
[769, 769, w + 559],
[787, 787, w + 84],
[787, 787, w + 702],
[821, 821, w + 354],
[821, 821, w + 466],
[841, 29, -29],
[859, 859, w + 97],
[859, 859, w + 761],
[863, 863, w + 264],
[863, 863, w + 598],
[883, 883, w + 250],
[883, 883, w + 632],
[907, 907, -4*w - 1],
[907, 907, 4*w - 5],
[911, 911, w + 261],
[911, 911, w + 649],
[919, 919, w + 203],
[919, 919, w + 715],
[941, 941, w + 356],
[941, 941, w + 584],
[953, 953, w + 177],
[953, 953, w + 775],
[961, 31, -31],
[967, 967, 3*w - 40],
[967, 967, -3*w - 37],
[971, 971, w + 54],
[971, 971, w + 916],
[977, 977, -7*w - 58],
[977, 977, 7*w - 65],
[991, 991, w + 459],
[991, 991, w + 531],
[997, 997, w + 385],
[997, 997, w + 611]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^14 + 3*x^13 + 20*x^12 + 51*x^11 + 254*x^10 + 580*x^9 + 1571*x^8 + 2141*x^7 + 3899*x^6 + 4095*x^5 + 6462*x^4 + 4354*x^3 + 2604*x^2 + 392*x + 49;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [57463151712704/18580847037014549*e^13 + 329940423221169/37161694074029098*e^12 + 2267213063705377/37161694074029098*e^11 + 5604077071498749/37161694074029098*e^10 + 2049925008524500/2654406719573507*e^9 + 63543686338458539/37161694074029098*e^8 + 87621667763474479/18580847037014549*e^7 + 115025971620877732/18580847037014549*e^6 + 436286977162939755/37161694074029098*e^5 + 62406175242339603/5308813439147014*e^4 + 362105392051378503/18580847037014549*e^3 + 432318868008653613/37161694074029098*e^2 + 21451631341948629/2654406719573507*e + 6467542780550369/5308813439147014, e, 5377788494839091/1542210304072207567*e^13 + 13044027075055360/1542210304072207567*e^12 + 98057369182260157/1542210304072207567*e^11 + 208058648774374906/1542210304072207567*e^10 + 1204018141720641594/1542210304072207567*e^9 + 2275335700787006548/1542210304072207567*e^8 + 6580047697117077252/1542210304072207567*e^7 + 6019888102377854619/1542210304072207567*e^6 + 23949066879417573/2654406719573507*e^5 + 1176988594746421944/220315757724601081*e^4 + 22223501562655283291/1542210304072207567*e^3 - 3243346928120068/220315757724601081*e^2 - 4028773411896345/220315757724601081*e - 781310660590377947/220315757724601081, 2303790926312614/1542210304072207567*e^13 + 7155879937931383/1542210304072207567*e^12 + 52294688312791965/1542210304072207567*e^11 + 128778512285957413/1542210304072207567*e^10 + 97270478017672171/220315757724601081*e^9 + 1508268139197733874/1542210304072207567*e^8 + 4729133345434432671/1542210304072207567*e^7 + 6421222000898592501/1542210304072207567*e^6 + 13527638477382858427/1542210304072207567*e^5 + 1683739501556569403/220315757724601081*e^4 + 24310981817944368705/1542210304072207567*e^3 + 14648248272232895554/1542210304072207567*e^2 + 2562244479320316771/220315757724601081*e + 39377962568054652/220315757724601081, -17802210105619914/1542210304072207567*e^13 - 114420960495687277/3084420608144415134*e^12 - 723232217695261613/3084420608144415134*e^11 - 1938422096497856635/3084420608144415134*e^10 - 4619744271100015579/1542210304072207567*e^9 - 22098259720722542745/3084420608144415134*e^8 - 28991732793199669436/1542210304072207567*e^7 - 41474952203691131480/1542210304072207567*e^6 - 20487200106456053105/440631515449202162*e^5 - 22847658253792145865/440631515449202162*e^4 - 118284486407146564360/1542210304072207567*e^3 - 24496825478127502259/440631515449202162*e^2 - 6534859279947327842/220315757724601081*e - 1963416827483278349/440631515449202162, -21236940226845453/1542210304072207567*e^13 - 62050552424124574/1542210304072207567*e^12 - 417281503050081745/1542210304072207567*e^11 - 1045727861391166025/1542210304072207567*e^10 - 5286729024651695843/1542210304072207567*e^9 - 11832197220260007679/1542210304072207567*e^8 - 4592391331344780974/220315757724601081*e^7 - 42299869596251044335/1542210304072207567*e^6 - 79469316507773826549/1542210304072207567*e^5 - 11461523937668817921/220315757724601081*e^4 - 132421402925894265531/1542210304072207567*e^3 - 77533417076727631489/1542210304072207567*e^2 - 7826210628103999885/220315757724601081*e - 1179645724726869225/220315757724601081, -17397663185925209/1542210304072207567*e^13 - 101151966811114407/3084420608144415134*e^12 - 696953181787476343/3084420608144415134*e^11 - 1719295462740163541/3084420608144415134*e^10 - 630971559439695676/220315757724601081*e^9 - 19559293954826445629/3084420608144415134*e^8 - 27311752338177784483/1542210304072207567*e^7 - 35695056418034561164/1542210304072207567*e^6 - 136201704759854247069/3084420608144415134*e^5 - 19118499832769681933/440631515449202162*e^4 - 113601837155819489937/1542210304072207567*e^3 - 135711323443043284395/3084420608144415134*e^2 - 5983291504997373502/220315757724601081*e - 363812644937182669/440631515449202162, -9075088052358578/1542210304072207567*e^13 - 26736984526893581/1542210304072207567*e^12 - 178195025625226779/1542210304072207567*e^11 - 448353394303732198/1542210304072207567*e^10 - 321739465835937941/220315757724601081*e^9 - 5067612262281990252/1542210304072207567*e^8 - 13645935977611380659/1542210304072207567*e^7 - 17937762431898554428/1542210304072207567*e^6 - 33051047517693538639/1542210304072207567*e^5 - 4854323441832339276/220315757724601081*e^4 - 56110628552449874247/1542210304072207567*e^3 - 36672901311961886722/1542210304072207567*e^2 - 3317271400340878727/220315757724601081*e - 500020938049728949/220315757724601081, 4277365677175953/220315757724601081*e^13 + 12281445114992491/220315757724601081*e^12 + 84043386090875495/220315757724601081*e^11 + 207823872592735807/220315757724601081*e^10 + 1061722376422403474/220315757724601081*e^9 + 2350734066376089363/220315757724601081*e^8 + 6443906103743647803/220315757724601081*e^7 + 8402976471747301522/220315757724601081*e^6 + 15784865588682891109/220315757724601081*e^5 + 15657214060963845109/220315757724601081*e^4 + 25994167904263709880/220315757724601081*e^3 + 15502062925184370253/220315757724601081*e^2 + 10659932279284572715/220315757724601081*e + 290755997468506913/220315757724601081, -4976171541319268/1542210304072207567*e^13 - 27286717268538509/3084420608144415134*e^12 - 180563130066136773/3084420608144415134*e^11 - 447791967071794083/3084420608144415134*e^10 - 160252241275526998/220315757724601081*e^9 - 4966017721843534555/3084420608144415134*e^8 - 6196426840195734035/1542210304072207567*e^7 - 7921793055187807266/1542210304072207567*e^6 - 27046534461750091847/3084420608144415134*e^5 - 4405739248043402291/440631515449202162*e^4 - 20385424208895395877/1542210304072207567*e^3 - 24036481648365902381/3084420608144415134*e^2 - 137356961178605811/220315757724601081*e - 64164629429449347/440631515449202162, 12324466222521655/1542210304072207567*e^13 + 36651012297161994/1542210304072207567*e^12 + 246257932997524050/1542210304072207567*e^11 + 624126396310765024/1542210304072207567*e^10 + 3126831840156959912/1542210304072207567*e^9 + 7095675373307857442/1542210304072207567*e^8 + 19331067870042618897/1542210304072207567*e^7 + 26226090046301154199/1542210304072207567*e^6 + 6869480937979699074/220315757724601081*e^5 + 7162314486469666344/220315757724601081*e^4 + 79613329759735417162/1542210304072207567*e^3 + 7872748761590627386/220315757724601081*e^2 + 4600308805448449680/220315757724601081*e + 692646812097035481/220315757724601081, -10842842671992490/1542210304072207567*e^13 - 49949346647162881/3084420608144415134*e^12 - 389887918982555193/3084420608144415134*e^11 - 805280619646828639/3084420608144415134*e^10 - 2382338743639373011/1542210304072207567*e^9 - 8774305557675033555/3084420608144415134*e^8 - 12823521661235236589/1542210304072207567*e^7 - 11538208477637913710/1542210304072207567*e^6 - 93299873537368683/5308813439147014*e^5 - 4493460509339649095/440631515449202162*e^4 - 42423519114044116102/1542210304072207567*e^3 + 71516176638535279/440631515449202162*e^2 + 12360233331256391/220315757724601081*e + 4839667317862486371/440631515449202162, 2547995194623723/1542210304072207567*e^13 + 1741406853380707/3084420608144415134*e^12 + 66383978989647197/3084420608144415134*e^11 + 35241913416872301/3084420608144415134*e^10 + 365494599045300448/1542210304072207567*e^9 + 230402740915951013/3084420608144415134*e^8 + 987921644666329871/1542210304072207567*e^7 - 1092920687218136313/1542210304072207567*e^6 + 6968615046028443/5308813439147014*e^5 - 98537058657025111/440631515449202162*e^4 + 3279151693978758561/1542210304072207567*e^3 - 283788478377528873/440631515449202162*e^2 - 22263512356503320/220315757724601081*e + 643514910510982723/440631515449202162, -51054410743464028/1542210304072207567*e^13 - 167329934677592314/1542210304072207567*e^12 - 149933644906957012/220315757724601081*e^11 - 2846774891584886618/1542210304072207567*e^10 - 13422474040909572832/1542210304072207567*e^9 - 32544914262176131462/1542210304072207567*e^8 - 85089138911841801328/1542210304072207567*e^7 - 17740479323182784084/220315757724601081*e^6 - 210072543840756527661/1542210304072207567*e^5 - 34350023356406750790/220315757724601081*e^4 - 346714092700411022676/1542210304072207567*e^3 - 275102567457485958376/1542210304072207567*e^2 - 18821706282352312036/220315757724601081*e - 2824942101802751798/220315757724601081, -21713632128537864/1542210304072207567*e^13 - 62578015900673610/1542210304072207567*e^12 - 423948140613764590/1542210304072207567*e^11 - 1047301504357338515/1542210304072207567*e^10 - 761383323376029410/220315757724601081*e^9 - 11799399285858396620/1542210304072207567*e^8 - 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206233697230888939/3084420608144415134*e^12 - 1628949880747390249/3084420608144415134*e^11 - 3270956395306437357/3084420608144415134*e^10 - 9934825311028936543/1542210304072207567*e^9 - 35499233206796299597/3084420608144415134*e^8 - 52479329517677590687/1542210304072207567*e^7 - 43518862505325839271/1542210304072207567*e^6 - 381621757636802583/5308813439147014*e^5 - 17980971671976638401/440631515449202162*e^4 - 182459368326508269196/1542210304072207567*e^3 + 548098847683236465/440631515449202162*e^2 + 70144362105677482/220315757724601081*e + 8102521824304193221/440631515449202162, 62593929230507069/1542210304072207567*e^13 + 157316500754142123/1542210304072207567*e^12 + 1153381912732628737/1542210304072207567*e^11 + 2495245586190188656/1542210304072207567*e^10 + 14215847710683713177/1542210304072207567*e^9 + 27358520576516663902/1542210304072207567*e^8 + 78523058028646954872/1542210304072207567*e^7 + 73097885057846034285/1542210304072207567*e^6 + 285897634363008261/2654406719573507*e^5 + 14250990377422676946/220315757724601081*e^4 + 261720443069904908561/1542210304072207567*e^3 + 89816115529637546/220315757724601081*e^2 - 28392551617143651/220315757724601081*e - 4571272928069028644/220315757724601081, 60050096068507577/1542210304072207567*e^13 + 225409029154911648/1542210304072207567*e^12 + 183394026188016525/220315757724601081*e^11 + 3817437075544681710/1542210304072207567*e^10 + 16574417476714623532/1542210304072207567*e^9 + 43901129897020655036/1542210304072207567*e^8 + 108381798644179952863/1542210304072207567*e^7 + 24664377027584870455/220315757724601081*e^6 + 262924188199292288453/1542210304072207567*e^5 + 48397472910865590210/220315757724601081*e^4 + 434852496320886075870/1542210304072207567*e^3 + 399232932082533275348/1542210304072207567*e^2 + 21567051181359254736/220315757724601081*e + 3220962822061981801/220315757724601081, -280617424398881148/1542210304072207567*e^13 - 804123364544651322/1542210304072207567*e^12 - 5445710733083314086/1542210304072207567*e^11 - 13459133710125593435/1542210304072207567*e^10 - 9778390003380009510/220315757724601081*e^9 - 151538379494322867192/1542210304072207567*e^8 - 408090212256824888839/1542210304072207567*e^7 - 524045006139378943962/1542210304072207567*e^6 - 959296707537414549210/1542210304072207567*e^5 - 137082901777347082055/220315757724601081*e^4 - 1544892201643912512858/1542210304072207567*e^3 - 918752401732033644624/1542210304072207567*e^2 - 61116894405779897755/220315757724601081*e - 2459512777031281260/220315757724601081, -25697430535061449/220315757724601081*e^13 - 74287083154248868/220315757724601081*e^12 - 510504208294324956/220315757724601081*e^11 - 1261865893764888110/220315757724601081*e^10 - 6462298188271947364/220315757724601081*e^9 - 14308483723572762320/220315757724601081*e^8 - 39665623240194660486/220315757724601081*e^7 - 51812634831205466244/220315757724601081*e^6 - 98276035859164198564/220315757724601081*e^5 - 95748628403148848146/220315757724601081*e^4 - 163152862257429915588/220315757724601081*e^3 - 97395873560120027392/220315757724601081*e^2 - 70332956986936489433/220315757724601081*e - 1827336578014745448/220315757724601081, -5098436894526841/1542210304072207567*e^13 - 1285492223213642/1542210304072207567*e^12 - 74516577380539211/1542210304072207567*e^11 - 13985987233648641/1542210304072207567*e^10 - 115778377750527417/220315757724601081*e^9 + 18760066674013442/1542210304072207567*e^8 - 2765555677576288778/1542210304072207567*e^7 + 4714403642753655549/1542210304072207567*e^6 - 3907725597355410255/1542210304072207567*e^5 + 1732580689432009743/220315757724601081*e^4 - 14498310837851424239/1542210304072207567*e^3 + 9726811407611133288/1542210304072207567*e^2 - 2004695932000374569/220315757724601081*e - 310456014755196793/220315757724601081];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w}>] := -57463151712704/18580847037014549*e^13 - 329940423221169/37161694074029098*e^12 - 2267213063705377/37161694074029098*e^11 - 5604077071498749/37161694074029098*e^10 - 2049925008524500/2654406719573507*e^9 - 63543686338458539/37161694074029098*e^8 - 87621667763474479/18580847037014549*e^7 - 115025971620877732/18580847037014549*e^6 - 436286977162939755/37161694074029098*e^5 - 62406175242339603/5308813439147014*e^4 - 362105392051378503/18580847037014549*e^3 - 432318868008653613/37161694074029098*e^2 - 21451631341948629/2654406719573507*e - 6467542780550369/5308813439147014;

// 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;