
// Make newform 75.22.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_75_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_75_22_a_e();".
// 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_75_22_a_e();".
// 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 := [-2792983104, -6395796, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4263646, -656, 1]];
    Rf_basisdens := [1, 1, 1];
    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_75_a();"
function MakeCharacter_75_a()
    N := 75;
    order := 1;
    char_gens := [26, 52];
    v := [1, 1];
    // 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_75_a_Hecke(Kf)
    return MakeCharacter_75_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 := 22;
    raw_aps := [[-268, 1, 0], [-59049, 0, 0], [0, 0, 0], [525814764, 153760, -264], [28162718864, 9117600, -7808], [-355198062282, 104131392, -89144], [-4616822158622, -1409034048, 729992], [8957650249812, -14111812896, -9362104], [-25265117784328, 18765594464, 10535432], [-303392922896886, -993243207040, 732279424], [-4923704756606944, 477897067264, -479108872], [11294499909514134, -6894023111104, 1056894152], [1451864625405546, 47659581755648, 26907294352], [25526910116705860, -12561144913856, 53393843840], [-59685136446539480, 178318406875104, 68756675576], [-1337869814576843578, -13520998705664, -411233665296], [-1492246460343526352, 434925331623008, -1595336337024], [4557750515913238550, -437849824268416, 961515250192], [-3339354295109713212, -6230202201438656, -536812296576], [10513525059014066432, 7945981682866048, 8510370755328], [-26877162613795978258, -6166519714978048, 2078961676560], [-33883127166957598320, -10734583048382272, -28313946671352], [32907085470311429076, -77743160475758592, 11083286184960], [16564107621445798242, 65549330041667328, 120822099984240], [-255822414225813915202, -497730817948697216, 227359737881344], [25170518928513051874, 159596427519511168, -439942032096800], [906496265918103363660, -421308909560934688, -175730032557832], [-339755542121478985700, 234495714542803456, 45533145436528], [1569557270853664979782, 435473584885966592, -9722314220416], [-2873924021140691953574, -910302789779632320, 1177997964016232], [2544319425057545478868, -2382995427479968928, 1433865126339992], [9986350960256539256640, -2773973628868416672, 730188706284816], [-31835092671084660454318, -718639807972548160, 5232270885597464], [-9612721752121186307932, -13226089467475259232, 11080160023582776], [11094471658232420937066, -7555736920610063104, -19726655545804928], [44268346425959028556352, -19018713189350219968, 8173358613352312], [14813544222924086185062, 48254546662394878400, 1455390511072184], [820158703396042547684, -19413334802307073280, 45015362954371840], [-211076466582564780900136, -28377810097471043680, 2970591658758536], [226396644878271529969886, 70794087377629025792, 27978420782292224], [218557883544972671539480, -208331614767478604896, -88812373588632336], [163030240399347226939326, 126499502576231773312, 64247004164050816], [-414072824765185760539096, 216686961726046240576, 22093459248161568], [-392993947065204499657594, -166904138301348051968, -181532080880139536], [-94637088836010268781434, -205851380201084333056, -545550490603517968], [185963604442433844143456, 871996047632588583680, -51686047564369544], [388138341981110338486396, 390495808993219673888, 894581902683302072], [-1033765097728229742963932, 1139993347345632703776, 1235512003282344840], [-6955312522063280965138996, 83412084664845946944, 1011936468302752000], [6742171978995847156524278, -2089901494515804276992, -724163502914266864], [4815333049864084644656330, -1389122945859532733632, 751293548482861304], [3291012102542900974543592, 3199928176586260263936, 1333933935476079568], [2626030529744041702011714, 3974227455067092334848, 1403178674191556480], [-6781882608723732281295288, -8235050007816719328096, 2997022356528823824], [804414714729509868797850, 9515872478363751506496, -2551260450443457432], [-1524576363715658675444536, 1547309286063891217248, -3065645162514621032], [26945212885362916489105802, -14138565919430321015808, -4937894674471120256], [1688764900983453656914344, -15590009837642722489728, 2890616682489405464], [-32826351626895223751069114, -15508362346820408098624, 13493930779729411384], [-3718272879478950630845662, -18208388881756985546368, -1337272244055050512], [6004820018113341442028988, -21088945872503929779392, -331924015345315488], [22172877541114052452914374, 21706072455308218547840, -23513296346775924880], [-95789544592736814165866820, -7621698182769911172672, -17500887496774102640], [11392289920761637813362064, 35733332076410092582912, 50828773804127079184], [151862771742024964771868654, 4176653307508981110400, 5697429871007783440], [156842638121881570029205230, -447930822754440051200, 61789423451262535952], [-194808946764138682294099028, 90844039223526150631392, -5219186708917457288], [296672054475579827122465526, -89562126442868792924800, 21056143024325004048], [-293172584869794844415941692, -8157329881658670065920, -98388809956908142160], [-22782133309065757551950666, -165614516090768725027328, -75242390962348391184], [107857112768890061509255538, 132749737510389139987904, 184965814743995141528], [-234738227150410089368713432, 48734470222681761086912, -264435922224808015840], [346649067569223520480535476, 355679934627661188582368, -232355233343952962744], [-900985197491643659154113970, -241763053971958340440896, 23268459674053028440], [795945744572404210415137460, -357497392270164614575072, -110413007932503158072], [-682221208890209485505168056, 276276634304241766796192, -226838904050829061768], [-306823516905053671871634262, -40241693694164373103360, -207310206511815876352], [604640715940082974347625390, 42351936904540797762112, 281887598008515674024], [2055336044898896931043095610, -483947210001608581178240, -244078514105590431872], [1669835180055081587142988858, 26475209617377761502464, 594549283776259581632], [552850523008311912933934272, 1710514942007935535571488, 125686497715166806000], [3009602620680369539550345398, -305081289710095573322368, -693542434270311114992], [-1596225747306999513473769664, 56643700235796148417088, -328508010742743647008], [-503867579944892125150570186, -732899237709394294177792, -1312451656287178954992], [-3572058730503093660270325120, 1581886084710983496179840, -791454839494818513320], [-4351820255086735968725953292, -769646811490104506104640, 1751745895424968175104], [-1751628626699814020437211174, 3967326747889558727194880, -242125685425231513824], [-5395224074578368752404062586, 1364244939360749214629120, -1751001276650947898304], [2854827847273388822513308882, 2863067399542964233164160, 2861296861245266027680], [1315414772503769780556060460, 426270027947761203631136, -448359362402575440008], [-3108686237169754513788807660, -1765551133671884807684288, 3202825450295130088304], [5141948161692824273316481960, 1853478611354152115783680, 5553777119676291688960], [14096006518822032969047353108, 7720035951925814996676000, -3533506203703043506680], [-21379791172480285397494635800, -1279185971115669805786464, 1169388341364009172544], [7955128787118242102185159012, 4315842170563039189436512, 2819610268305866665944], [-11811851416830361806481561232, -3389800562811946323928480, 1171025500317782931864], [2238131647931118680541008778, 12223415887847542868994688, -1941775631535399076864], [-22063394899650039221309338414, 6515114946049513936165888, 3332047526264493872352], [-9789932540793862730690237220, -1870383565627107060557760, 11640267545305354908016], [-3834057808295507714026976954, -22794909375769397639948160, -2023993208748399310976], [3653144263765004941255709564, 23829814454741538099993536, 15949239751536662121312], [-28884476228280343076833933442, 19179441285734576813493760, 1099502249047243824160], [1581676264863052439671825276, -46669108184160257603350784, 4414787824448684861008], [-5100219144010592130712630318, 32680984410313671089341568, -1056134368935588633760], [-46852851839192432372843412676, 12880978684834642656801120, -23548471289400854637288], [13332748719608328433825795038, 22534595352758872424724224, -10802524508612061837856], [5197809905366478083294424380, 5217396362698968834756224, -31281100165602713374384], [-37558254827673801200402226230, 34771489286121116169858368, -16144606326363162333816], [-51236323900324398913559016024, -40506536045621377385329216, -15894835473018610216160], [60119249358010979676561689674, 58061400710431774237668608, -8013567905837407720960], [36247872697235527538360398108, 24993124723986887318356960, 18556435421948163188360], [-20063143547263018774231245834, 119303227231983147032007744, 24989900472794690067496], [-141314218225633335883084656038, 64203503985791079068289856, -6627926863641844519880], [19740263937698165711315069908, 105081692628642222198160928, 69774003448284390363896], [-171294599624809811862636787608, -11555735644767199064221632, -30694315251099996159352], [338147940570478386912811372306, 11606787149188934416448896, -40722189363383596659920], [234383137114842433799019368580, 2556307590025860700768000, -20093328165977529234144], [-23226215725231332574875740192, 62634433114200847606506912, -138682036049420855377048], [-303075701633244216286855854946, -35994099652227621712354176, -103973377698138164806016], [-399203226338264923251809504000, -4928231887532687302866400, -19459395324314516621760], [302686212054737875486777590382, -46094531191691089749664640, 6607599860697174013680], [75143715328033102760924906902, 2384152038122484415921664, -46543860768639813076688], [37772835497292718512357144726, 121013325721962529079069568, 166129575287979337351296], [-81413444513354906487640929636, 154837500966016916579457216, 104387907501466782240336], [-323302701863887026805114726476, -281785915219496996919095136, 249239529598840247795096], [122775201954885835889900185770, 175631600647627401134898944, -555007350571935515168], [-23580441726378516777602024058, 148262860405270675662128640, -135917489565276925001488], [683628820313795380151575193136, 185540902137502822024296576, -103348207825738652004048], [815194253389502612887998905332, 147299944900063545718805664, 204853779036265440364008], [462347623546090143531139628830, -668757268485658396512714048, 72914721338664144613848], [275543146181186171504614879188, -416891124259991198678820768, -222982309667633488988680], [-561650784580968076983697186000, -472422886249584651841314464, 417053112561053415422952], [-235385329854232314029236294096, 779153292902359703864681600, -85401232694603740876808], [-730964186894087939395384460378, 320403407345742832295280064, 104131816896383754758168], [-342429167110308811755965108782, 72214938958106054733546752, 616297657200808504210208], [153934536292616713238134891746, -999742337646739875885592832, -611520946201953698477216], [-1169731051171054594764354344266, 790367432014596729830155520, 535799999951041296096080], [1484914345545420488877527815220, -830599614622111712149048384, -177365496347220995517072], [-1880159901128694359988623342130, -247381357685493250036044416, -681117162925483127422144], [-101278150211929120713034043110, 2318552572000078890227452288, -434850071101437426255312], [-1928799174894586917829630928828, -1266563068290178635074449824, -23337219308577432686024], [2134765455851318570119991234978, 676018377594610956985546496, -362419088404193901390688], [2252561423045302774299613716756, 840169019372015341231490528, -724172228243708272655080], [-1267369755637714589527929516492, -1971121924610976073838143360, -312141426707782137133280], [-1443101490083315659095839102410, -2837517427867089374861755904, 1492593991602306723783296], [471739830248669938756432036920, -1759034956109547067931038912, -511089492267125176008512], [-6253608099142272773028642293946, -1284050959712076736905378752, 513924709271046445845768], [-6558457322655903818427225467110, -717473842929411023342491328, -666174430896658365867576], [2282078888501038497647661828500, 1302598336524417670979391392, 2428991195729536046446312], [-784360674666150059745058836800, 3643569789816021222769289120, 125299501079683631949576], [5098034740884269388209597095974, -1246373333264613481511155520, -2106976125842271502040904], [-4094990147752046119616304555302, -2324030845249479928065146368, 2115912010547236243082448], [-2671884013576665485305639990460, -961320278293932534422823168, -1769099371097108510913312], [11474070140597011370609102750520, -945093257363538043280410464, 1630235322412689964927848], [-4669763493163227807252429108860, 3071008447035137306049775616, -1549288122092086622969072], [3389486831488171874276351674752, -1273334581507707395018667968, 4410206188675606864289552], [-7213620236813017577848751194360, -49676968824880606108053376, -6087313040394166583738696], [-3801669744469722492016623730758, -8163619350782258075125366528, 3075289668528220760399104], [-8894879629738872765015718074234, -1321589081138115856593128576, -1265963603923770371977728], [-9047986954343840064326136374334, 374711788814866853418788480, -1211045436546705469795616], [5991604557880638489420637555372, -113610413607155060748538304, 2466704382799652802201040], [-8632397580744276805288069169262, -1772876163375273033750907456, 340736353185268535033368], [1185368951763147098599979152348, 3517423342064692003596963360, -8211482323711999515323160], [11133131955044506032378200863256, -7851440069723921137696876384, -5987429987459458569478480], [-807567559577579759152251831662, 1222577430841457661225425088, 2632431909295178238034568], [17506921695883129160631761560752, -2916869669485864691759716512, 8407716020214932401493736], 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50482168298516707097125211147008, -84523014660258936350209336704], [-135314216270843056084164682413401894, 7785668002211850448682419881984, 35429887766251113467185458848], [258317407067489549311470793876346548, 21053617047198165550686877296160, 47693815344126161197670112104], [-206094212532421955913634916297062898, 123511408776120123231224955916800, 41171653103694609624278387152], [143884315517314041114243514806968986, -108872650009451378863229474434496, -1862006593661460115052356904], [169633921857011654845347189176293232, 80516544462051299580529978664320, 49793199780398115537605296992], [-345601680888315233282825455707908112, -15663171487605611057656998121824, 25310118778337408570822348112], [-147699193640615921470730317971068614, -115080281012519041682674046339008, -53547592061106527832978745000], [-105922863686988561043352517660362024, -118978345729327266359131199800992, 36197887594592583582989744120], [-266285431651566477164936376624367762, -121714633405359557696116319454784, -23548114390431313322539407144], [-7372689561389772282553476185338742, -71229317120222229631695535558528, -112093770502072993933424637312], [111506657535611151366263852048871736, -95913521691339392058268384965920, -15302122172594939835126863672], [-60631473422369226684983032234272352, -87414873239398554162644699044896, -27249338048308920390426499088], [114721333349097547743390486983489380, -195898822616934063233985970440000, -17492827294401088745361526848], [-385184586453409921896296729254451018, 80407882858204042534212153447040, -113482714263098420141846900992], [-249295885853464723629616050093008642, -170576552226880950950444340892160, 8463210203818805427066595120], [451629762578463326492510374541112236, 201017451569013910453252173294816, -85478950131909406189311835672], [404802929022899905256060114250703514, 139758342526238154160101984263424, -95716986077830612112936537184], [92526100962453823253705585298242408, 126665717183438161110456677981024, -87948253839247105584805199920], [-488132189531653430182375443653505148, -36320932018723125601618383343712, -105336644262921760200895806040], [-5227268182988396788108898838193128, -133294884543960802404881937126112, -32867348984008570207360852904], [239028097624802832720769782677140962, 81320205212051773121182462687488, 3085922519638873936277165760], [-540915585869984579870076878822118352, -41626847352328729924103508689600, 16407222849695815400316574616], [-100890431836134743967843448690709986, -319647211851876200013609456515392, 33089178222733816077058898904], [-43178983012114346347083026472317688, -240496280974874827739598141916256, 332745495853360754973999481376], [-168510402700622508138503915120358720, -353993446822273865020758798153024, 1978514680002585337991279664], [-48136063071715397635297871355580722, 85965341403315831870177899072000, 110556603806564635264047781664], [429119035733466850778275238002238506, 78307779318179709725314754216704, -195050913317188050337579723520], [-253950904225687739467485406711433210, 29837803497457919661876626697472, 176330648104608911657980331264], [-635229607700305190049649097037904958, -13125908172500334831859221148800, -29755475294936738460315270880], [221190704755591610004720314746225170, 137148172859351207704715170815936, 167759071520563167511919107032], [629779845801419696631004155377586956, 165117643731312955887683774665760, -170246653880196658208937148872], [-630139437011654982477012518934780382, 135586874703641417258577449050560, 59215627025992909539136564120], [-4805927182204718511910554096192916, 316709123642558831753018117644768, 144947771208103588691048274392], [538295758376373907248247850226319960, 246534398767371166816464493890144, -133052619594439275416391212680], [-598614431819810151800689748212880672, 300234252929158778939880803127296, -234583998777712186129852258888], [-395372052796083792307689277558322650, 7757556638574300935196618759616, -337485163425285105559067970632], [871717444114618369581034770644159860, -358877603427070379031358304385856, 118953498246500733921799277200], [-583129297302838974257762077956086800, 622721809855841860319138188563680, 184125710653400058685602097832], [1086794615724961448455744628548480578, -385847754337993025691090883382272, -166696759374400329882214576704], [-10202900176906587588583723908189194, 248360520300403765785267409946112, -381429970201917994068466437856], [-876770867857083887521065338005004896, 319839767202379668123376761913056, 174381141435283552529942416480], [732607416796779180751483777475084964, 134327341249759610622288846396608, 599718505172314116733288614704], [-85497150975038502405766368498194920, 222475468968755802598630222718592, -57903105947460389371901124800], [-163198703238040398307261762379700746, -262750213623276592267170973826304, 264474466442246121096041290176], [-112598048949584504535295946475993720, -317049536413899129120151015747136, -139694719127759358748403312216], [-889957938736896048279652230563394342, 322012726626995188248378109837440, 35873387368373664397587082128], [-1141582892088519987424183826538967124, -260377443332866121289338285538848, -157924397561883373345206666744], [-1415277141009347210335795370398998022, -16737057606743396920619754968320, -81248439830562637898704579264], [-273072471318049180246511068685403554, 14860533265670999519567653590016, -336304736549115190440424936128], [560417776701122077868371473218167258, -519802998352015212703166270034368, -385142724141279358697176281704], [298417623912385876696022734048958404, -352803482400574463227701239118752, 235281247148555610704181608024], [-1938634550945285313923400635675219646, -33018346823760024889179021668672, 49799168152860082790180753144], [-362546946855467852106174782096951918, 266045388007808006154234597058560, 41983506356901226937690340992], [689554691753793395117202119398002296, 166757933079997305484808056576448, 564676247601255528478260278152], [940920642305557502971024929328858558, 554236230801930504344210067916224, 170064893078921583640913781144], [102402054034450754109647577527632954, -483075585783540578896221484437120, -398772948367181969059072371328], [1743940350850236164249170411654103204, -298769824426313124431632968397824, -18423071424860315902375374784], [-5240075263898762673019162776294832, -243941014851550209275337138681568, -344935289603748519792416432080], [-1298950203292130369791524883826064642, 104038641720278622324422469065728, -147678235032297320833262710848], [247550614502266705075910166170925030, 618988488040338674327196995706624, 120861996337178010140516224416], [670023987336932083796245332194091212, 85415456063888377937761756022848, -705305217817030837715998366704], [-543747195949106248341355900990949676, -510683958951139951480792023689056, 357213829109932941668495522616], [-1247935412125846049709538712381557970, -568661534046008311392173035342080, -488533110856051067530875870768], [-1328395140641944037906841445008379222, 999452502554459736626935149573504, 421566442094579872169932252000], [-845417637754355012484029851124416608, 655258418696623928642000899997696, 560685357913956079340141049088], [639062345083697542710143884759561420, 811835590192113901681420707050272, 342436302910700923060982856824], [-1734568503106825853235310637763676566, 841780285469184681368407143161664, 371961874477494546457770070680], [-2088364116609802892885239283500987112, -386179420038101843222838789803296, 505113269933277325711766221128], [-1823413854940966593915864722484222334, 606924553294293130807581138636544, 513029903131523783808126612480], [-380289167917043019667806887744277554, 400544095590139638543076460241856, 666896218330382711074792117784], [-989305598923184367764133598567954104, -1333592891910346378049693141558688, 725244745193267063225219055720], [1317237426133658448415662571654796168, -559772413852697194017664762274208, 594045574177725283211035296160], [-2212550729385973676092983903019365682, 1254307492806616028453893360140160, 484210768036270069139624054704], [-825084683442070922969068191084288850, -20145413609998773916415274253952, -1274226150044545102297513904400], [128649134574790952269840317534647092, -656872485488043912897701684669056, -269965691425226578972289491008], [1112635687160306266201581724673665210, 833035834647145774170165342516736, 186822122325457246941924075616], [-728362860997883237670517630569601396, 1265148212622368064776354394257376, -185455291508004712370962581912], [-896487753665354855029779161509159512, 627509023956619824260080560932416, -533570074004557333788250753696]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_75_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_75_22_a_e();".
// 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_75_22_a_e(:prec:=3)
    chi := MakeCharacter_75_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 22));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_75_22_a_e();".
// 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_75_22_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_75_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,22,sign)));
    Vf := Kernel([<2,R![-4487879424, -6180860, 803, 1]>],Snew);
    return Vf;
end function;