
// Make newform 1.36.a.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1_36_a_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1_36_a_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [-2645665785, -12422194, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-198755104, 44712, 24], [324169574624, 84967848, -39144]];
    Rf_basisdens := [1, 979, 979];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1_a();"
function MakeCharacter_1_a()
    N := 1;
    order := 1;
    char_gens := [];
    v := [];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_1_a_Hecke(Kf)
    return MakeCharacter_1_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 36;
    raw_aps := [[46552, 1, 0], [-34958436, 1, -1], [297550684670, 7444916, -2484], [292807383115352, 723239370, -852426], [-385981809516662348, 583640593195, -3858182955], [-20713203516999550186, 54628153929972, -55424393460], [-1310878976731046574798, -2825301075479112, 6543407393352], [-10767798747644873516980, -84000151153806507, -9872745919317], [-171607356473791739074936, 2541836449593511358, 1160877406548546], [-12820133047994812532749370, -167162542839789844508, 30897215823693852], [34577198187908229910009952, 186942342299192193960, 125813443508636760], [821537724316296005673687902, 15156039498596862985764, -5309319421076092452], [7840995528177589376345695802, -39228367987312409807440, -24089178969404126640], [-15838765711814619394775091436, 17943293716363705085235, 66376501078524728013], [54782256280439649053243300752, 86799763531695964959772, 513097522165434502116], [-556826850370574494914819603186, -4793808545356933572226524, -4681503192076751946276], [1456842861030882233281747877060, 8362052198766408093261479, 13620914053734255305049], [7845692758838676315739693496102, 49041601588982484445785300, 10617827833982900544300], [-62948344461385639659105070640548, 320840674750454548819656753, -107623419866546146525233], [116258200765119403512873720668952, -584017934307415711475197350, -525446933143802590517850], [-95058321618274262205264072995686, 1786789981639487103639285528, -755642404117185982959384], [-142070954197020553967245543624720, -2625760784162711247787020348, 6218287018572069495754812], [4955729110554688439002961035255564, -4203954105400526589190325051, -2110524259087065705757893], [10194530282064583778401997472678090, -6722758070502638969648419944, 9140569715442095747986536], [-35388555681543650354442872900363998, 99290161366259802362440013592, 89027040356956206694959336], [25726462540703180983413645841987102, 36116929520540333067232206340, 306093610925198911255212540], [115146659582880222852093329781235064, -201300077870803307993730873534, 308554499757309948245326014], [-57157791107887934349790271662319148, 1961996425043287616432727356835, 349154391121339855087274589], [118124494649441487084516329725849910, -746322210191382580907742084156, -1038249231666732374265885636], [-435236404188654625657141424414432686, -15311507585474953966535627968, 687542908834973528280388800], [600930311318870247709582506334343552, -13554330632062333427367688882464, 2335462091799457107262851360], [13407994863688872686908556675611373852, 20700830689749645352860331073345, -21304168731856966330325098305], [4071563317262736784687108506933805402, 35039219196711759205236972346864, -58334014303166980123465850352], [4315340337976741793245879729605637940, -19831913232630101455110647803029, 57307343974332891986238794901], [-77452490489062355937089510933624957650, -31905055783730528705666189170060, -333682582061238385166366431860], [60181809862155879925808796711978553352, 445863403108413891003571372599390, -343553664936484782769097762910], [-36304867107788750812701631173561267898, -799746933791731923258402161560140, 54557994792913899282818636364], [217223801056582141334874670793092541564, -4010731179894368104375982543447223, -660844568459283881779898047305], [-451583921028551526722792966868234991048, 3057426306067795595656059086976338, 564382160369301736645651696302], [113804730470053148114388335021433165814, 1125727595021173041591486944002708, -3884686706105251808686336979604], [-269104706675481878998864509298763627220, -1535329714523708392383318973024323, 1838615119201394323027257067587], [-1392606181736613486098347608940717667698, -8499503454069015393887267589552300, -9030137499211558813439435241300], [-336852970839711316736819512215939405248, -32002755670326563615193584465134160, 5497450055151236273974061417040], [5257001876142716874991905194210559093314, 21709925193818678418192507679275480, 42456306182987955460705908090408], [-148423685969555302614826183610495106498, 12426004011796187209653414261777492, -28239961127174696074575248839764], [-911594982007749148604172478517890532200, -37500258281121801600239980641474630, -37949846329273318165803358873530], [448318418579873604860654243250880269452, 92688359709037404752033599300790565, 1399331811664113411608265463515], [55241546111393711683965563290638830908064, 420005569512009878680784325990236568, -120847875891241290793795650011544], [-84120368606618151531323823493215169610948, -109666487224112441449391909039106599, -138688081192076561706611996235225], [71318051999137541101261492546490561072030, -660013118474823008833635681474853548, -90614396956720607116159763330388], [53090362409645861840433397382491766920314, -569290483694654577551238116046569256, -50652575581379100105229005791448], [384547168195202724852742457792018699775440, -781516918527090433015712928440670004, -157622482742437671707921872005324], [-402604089291105033013522632112125374487598, 622780839177717976351167333517183800, 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-11710986461450258461875577956293343], [14393581012688496971250724186210097628382814, -50644307851234371696024519338209528300, -19519594965845266280914946488509972], [-14000418501455798786557547888164378875768148, -93932706889681592538445139444468917635, -63900540825773667725620776998918781], [18116631508476387716748702218323609781835752, -3099566653934538269523127233231750490, 11150755210181725562937236087358810], [657825789235892022743372450799136421118314, 33653112371495536803070438646560541232, 137677548062102244951978040815810000], [34375760787907573285540574207438881838641702, -179421322680376638558008523343365614492, -62475391836598918587671934563222628], [90754124647956193419454221236238967587131252, 283081155379974904570911810339188783355, 102072302878928265097352313971296005], [-43611771344388550851558695497718865974691598, 208955552144492576992310281740552647904, -52600373269264798753464350620300512], [93609329612741954991454160897188682797865252, 726800017079195359018901828175997515407, -796256595044726210157907678865798799], [-54456972956711276458916139821666336404925050, 94868267619714517652512935028824954180, -605742108176561841115257494999172420], [234242231451421312983682399736188687418465314, -1876503430933553701280204242886730709664, 242147185226965075442931352289114784], [379747363995517355311154076676253306274877560, 2331737643157464824437971695886129722754, -1358023892502457054559749374950487426], [-636293203388900939794651459212213855356752048, -1653769363727405880663167874244416292692, -44413089167687216597634097073862828], [226156321709681701094506774979290394399236814, -3018800032635175837534963381382593280892, 697988710279594080414532760184572796], [-450770494530010804308521245356844146078756220, -5478276581019211683110662375662959748873, 1640256107201227210515442206682998537], [-50084592464025262594820849048973983234263936, 1664187372692997187958976242013629598944, 987980535704933447775027238503166752], [588268685619398714984339597148507144179352190, 3214997241012521737294327499031911037796, 6427013581201699904548791321430840476], [1894449700925145225476189255072339906027332502, 768842299290259072325242924176786847812, 7183656164202987044775400998849664956], [-2559538511690723273601149911778830006063429198, -4279248841764666115862194833573197027080, 3542310145659475778346439991483168520], [2273029472778921654503983019478827604143440010, 16525068025813775284826928942036972159984, 14737689539834012273772971257474546704], [-654108194618362780074518543914383407127451780, 14636788817347756061815412323911781122473, -12676803295289069175550823659592384937], [-5430677782317854015989107896357559693687972898, -21634704133354700724800778034743534064140, -6181423807017996003988946648764964340], [-282717221188344249347376593845525983133425648, -38855121297480272623711146704915487987780, -5652771408619027462226161033834293180], [-3574688086378326499751378315122724461800200686, -73629293916087258634821019809133462590216, 32723264868263959762174794048814576392], [17506334243136104011492598466872739868827361640, 54568723438995525747417851471613829120326, -12996574672034769941311426251081761094], [-23490828510620373775073047715159579798398837436, 2941052695575489786916702316109553065895, -19222002849514581710609147666546057127], [9467840489010768981236315229125820407549250, 79822677493321971818023852358314548903000, 72644456957876928943947836286397785000], [22628124819645567062990082666732900471524280602, 681299065550623577681428721624723520240, 78131064693849316273091266646119171344], [-22503020324860716164057415581142710935791568298, 101379064580294867034202422381309236130340, 21442853469456765645380989252784856540], [-3051556556020875885260543076726731962183297936, -30987562960965455864928339671215483323228, -129599657709903118558506017539408362660], [-26029839269927323447168317049440591580658194548, -160437344268288357767971554423119770964667, -102022578455245900104520141226057674053], [-2896679323514455728450896825417795820551105120, 511739978406801918575003064828538388267992, 74597506374513838133329152902821133352], [-52278446972193936732935484198303415851452003848, -248572006456467266926371067817946157380126, 228515597959522543614434050413590819358], [15102254847026653133980147732400938220329568852, -360133279746768289441363044806665856052445, -20650354322857475580045081567982293795], [-92130758457739246825478325600511884738905934100, -20092765224653640483095545328346759228915, -283186518355025997024265859934714532365], [145512332178044041458064562086407695687544641064, -284367192339958452635299100461437868505514, -262758594496676509637618311276512724566], [104159349459998989015175952881649857038138347110, -174411565997843833628182273877388002596876, 741564601318293649791677066620802476044], [-100470335577579433329884256491315209366664143398, -475609151149068449064176010127955945196240, 24775875045887211200171942360243620560], [-31241880467011967008028285398936536275125647436, -1234631903386141767160848463237396896688917, -360295717869951602183793884077045535979], [187854221469835304845491156381742857346315829702, 368617565386367901524706027088517472587220, -2485944130007257308189174399907140468180], [-416359605972593098315511203788494727346751871748, -318244247604609053498312488620194210853303, 761322802292129585208597371274094948791], [419478059708872442407016071708132786928053674102, -1400556065525029322213826784741960039066380, -153926022390877374260727473351884178676], [-386581407956488720381186752698317819417389732436, -7946553224010870046540219941039960023636803, 3803576737906415781103365840669061192515], [1311615579932983074183760728390956331242222859370, 161094350086203264910913776902395002985008, -1344702181247008710625414219765308578352], [-2077670339910090373549534703601185509214314540348, 3601669806227552701405646399221795718214855, -1775900608761153094519551787373662647495], [1280898817067908408854024800922808922407488388802, 2217136786535595034824177901162482560559456, 4227682007729853196525014850345647019680], [-363278955557112387227187530698729306037658494348, -3360336156618925008654672775299772881611141, 2300009624621639277006092947029562108293], [-421026458146410319345237235855296454498550940686, 18198122983050928668854276500342368820632000, 934084104071086755340332820354869323328], [104836445719138740052481798232637619659264848200, -3281357502133077833118075791655773690990770, 13985735369043208413928288828449769214130], [-359828346419941184464363912296318896641228658998, -14450340279163960571908506398750993709969000, -14602168654657790589548044733847902319000], [-66598625544887566328172743810786851831885543648, 5178833775727276962066255007851434436127800, 10398361624332062718036180291188564956104], [3503069789073074404976089055898516113798409718814, 9125103970552222270210796116441102169825652, -1773612975380181186615748277823968760180], [-3352934800314583962394225502640164519452301569798, 1454456858713319034007283557996562306327288, 636078134665723822581690089159191231752], [-4584353054301265451235604174934417904276033580, -32592528972838822238806199272969222438010397, -1061039093852584583785720524307866599907], [7554274291832271438897658908751471331992359690152, 5751715740645161651626200962750070334355190, -3700602675654486726001440242255672293110], [6230414247084410379637932716630642254219351472002, 10669523484080168039762763113052030924764440, -32592124817736094173523922344322892846360], [-2688122250526092596726010815447449206008381790436, 67190650568392210557557053609437582118442625, 33472157195557225145299332593004701624703], [11395156962837177853435208878296271366454055265752, 13448341145289881279783573992514740397940522, 50860707020165485742778171653650058289366], [-8944817653596404320430070975463458866135816782186, -65396334898781705378496505013685112050075484, -1813780613400936111493401153528173750436], [-4856052842833851923229808811094880589922405925940, -87939475205136261685388602317591991649758571, 28713246170585302418274998322205197659499], [-7363183002307555289374931301011306867077457548498, 105359461185514834374171449411936338680984660, 3949495130632895017934060868803377388460], [23876400992955813943033399141351594938955375929314, 41103976556198414773386420967021549825539288, -12012677265214536952359641232992639966424], [-21099009190872118970770458752252899623646951569698, 116519003979643620918293658440714684303976676, 50531105108863989665212742439556141590300], [-30913774673796745260918263752505834387450875907436, 92201660338257953561261068202514322655926419, -88318417233375798078561665226504169445523], [54807166603521078851866212973374730053443296574252, -227127211036152008787810591029464744061207235, -21624934807328968556615714928953215188285], [22562207229958925318747385730399619765408529844902, 64203395665075121286631593642651377910490260, 9658350966909890112867692491398063582060], [19863286517474171377345468889073922674960887912510, 42938472506495325760899539815570535218419684, -189153322994766605343319715026115345434596], [25189057740659676354417662759019007444855529969520, -225690769408180927881895416340926651070200732, -281575527219924614730372741096630595208292], [36576718297511072605913077225944383657699774372552, -327682990132997283153373205966007952605874194, 62973592155708779277877208337584445376530], [-156494621543231877529304783047014486367801560904186, 60566664880318791337788755582444395138750884, 92921096558683708573516753634234979366492], [40821469077771416476729423387790482379771934798140, 444652987563003453323808248934772170043079001, 265979861467915705582054319999206046385831], [96383791051080951426039682481836646837786665123064, 10646989961877506408800444394069144353651490, 53628844965592637043398659885024083281118], [-193091007967976619036876345090353228090833742924848, 429993993527018428550116422653758416055443660, -416734480250315146950140368567785664782540], [104159577446346243738571439199380957339620772705102, -183359033143096967683610474891073440818961100, 554892257640410243498647715278083677760204], [79619261231611916348148551909285120425257447148202, 1042421508230790040751875190355197252543720240, -723492140698857854882023589951436024736560], [-220645771164490802396533553149222857627330329048830, 756178007360671948003454839102980497452000728, 558131023412471084641987431705203920828968], [-295006588392112300653661205116542260077099679997186, -1460703850305968758679111948551020230888788572, 613124376080762178355134922496482798887516], [-142464522556046825006337920302527928707072389191348, 2227751977920341191371979929748459650648949749, 1092849280462181486162898989826219451497483], [400437396968130663097442186457917033328278150626502, 1409388187642721445345727024083116778966057332, 874495324898660148968495239465785141300876], [642947240414981000005511035291490968844999772561210, 3247521928601930891303681513428443336444642864, -1103741838947990209466771186679438490388016], [331595545197752157504677934414088699523144479660052, 2102493677748105082032257428314434031454713555, -2601488270360690414548285240017158893827795], [499303582903173287012668233459326147385277042124302, -5261939018239488568276082597161163526168296060, 110453562694809972931532283508632414358140], [-1321925364225788526098013126377591050447381696102936, -1430813424635598254687190425995206901242083802, 1341670773652578409847251373125428574452186], [71626953086577825106022505165915720224001034010052, 242188381617860315531081655895173450430975431, -1450848445588187010526080820932681170161095], [423868364554875556303099055227973078671196409096230, -5033643799802911571059058131380576661423894668, -1932607046306832241849124759244657800757108], [-513096321992268699685317926101293063044688143811560, 2388173279107536717848626684561869050887200346, 4490819720730241513243441457307286980911526], [-269487616422695274611798651682391098181654319959186, 2410477280656423384460960554970083658834163876, -7153144669007420667669865195240689618675876], [-351870998769757290185283549904502941388637395961398, -4659489079625353826994711001607492026886691920, -2833368427733091815771514146112838255748016], [463763276947580031666182610722919181809467803053860, -10311123342967576932927466086910323108358252401, -3765210732698982921320099290155806169231], [-301957339222779129086233002378296221785497949459936, 4692321784502792506620282370000922337369239352, 969542715399592696622989417506420994232520], [640091458733569885584777488241853945235020456821302, -444419227493908915633802771689163134638017244, -3263111462140052303782470938986271212234020], [-1466976267114146095346425966402357505721498368922798, 6984779360369236138751637518474782818077460960, 7563584178752544488725557051353531247113760], [-677171070257628870876714325052782725313539595788436, 16686745084524176234271933636932800941673210509, 1144361903576290745414488701841534776810867], [1574802452338163539539283711496417104965855120618152, 748464680258612488155303787661499845665647734, -6026579919780106852356704020924816053054582], [779260051626453371982949372959890236350055812044852, 17169832603292052457787255286969696604595314155, -4080313654413005026538835008020197934259691], [6943563956076854177842390645534246094112740863007152, 6266985462626364722378736620926691046148400020, 4307478493603842056019378008546798552128620], [-895474338022462398012292857503720816380054778066680, -11504530670408673524835649415113679704215748962, -985501272479703373862624323169233485643422], [-557231394688884997826666940702985600979362440131870, 23641015795240880808068198370479961835511733592, -14194894312280701873812979280910368795721048], [6016377417508770002294604605730185009866384433801402, -16749652746068371832680709358200319832207500456, -11418720911375973649561359749075768761482072], [1006915311350099336505683097755804268811486677395702, -15330890952116445778566820969865015418174761980, 30737990840610398388733059152233749691546620], [-7232205731892092194576854441713365035662964024289748, -21364690820723367575700408090856304599433251043, -20294418438138904293950252570442909988222749], [-1568383503167906152390194017421177433852118892625686, -27751873630342121151379596759615150750789949584, 60789298278103248610108427693119831480464], [13965209245278471488164766660937639776988848341996952, 17578129108808687814686220901898285169182736378, -39117353671072490489344340822608397090340858], [4356568564179692959208143210227189832957334461803252, -45260674413208006957623853165153202537936286405, 37115359500341472493918552723028672321909445], [9133564050667123260154041866913196627496757904498802, 29807360273482992786378156530485796146930473656, 66363361914714437372349497500134554138903880], [10031990408397316275842652123828195398699966647365064, 21651595577378885984812543557546497818127833774, 16862789205810942628535869332032570725049682], [2998646265125964631458975938606609805247038777678752, -2673468998762759487942074992472321946235910760, 51998526420177498483376612870976686077262440], [-33518602637694554214651470494555865141047351231558498, -40353876277847302702422542823581545995815680668, 33922918782546716571788101862217181673310876]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1_36_a_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1_36_a_a(:prec:=3)
    chi := MakeCharacter_1_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 36));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1_36_a_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_1_36_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,36,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;