
// Make newform 208.10.a.e in Magma, downloaded from the LMFDB on 04 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_208_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_208_10_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_208_10_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 := [-66096, -6144, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-4092, -13, 1]];
    Rf_basisdens := [1, 1, 24];
    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_208_a();"
function MakeCharacter_208_a()
    N := 208;
    order := 1;
    char_gens := [79, 53, 145];
    v := [1, 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_208_a_Hecke( : Kf := Kf)
    return MakeCharacter_208_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]];
    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 := 10;
    raw_aps := [[0, 0, 0], [0, 0, 1], [79, -11, -9], [972, -40, 51], [10568, 380, -246], [28561, 0, 0], [302251, 1525, 1059], [-575952, -1540, -210], [-706512, 15720, 5208], [128502, 13080, 936], [494212, 37860, -40572], [-929769, 20365, -119877], [-2518892, -131870, 115638], [-12023940, -1780, -152121], [-2378488, -102580, 265575], [-4911188, 278110, -95478], [13270312, -475460, -926754], [76865436, -114170, -359142], [-67145768, 15780, 336906], [33256684, 604480, -2178279], [-117507622, -1579920, 201096], [-47076584, -2435160, 1809864], [453969672, -2560500, -1487136], [-236347886, 393400, 2877840], [-119261706, 4404580, 6250404], [410330864, 4147010, 7947006], [-493863024, 4059560, 4302132], [-565819164, 9382800, 1188768], [428543247, -232355, 11219619], [188717754, -3202200, -12115800], [-2405456224, 6752080, 1359876], [-2246152292, -7967660, -16643973], [-2662304252, -782990, 13420278], [-1757511016, 5486680, -18027069], [-5145511662, -15313620, 16365924], [-1638800008, -39134420, 46233669], [2367103646, 7117920, 83343312], [-353743136, -34125180, 7615188], [-7906871196, 54025020, 21871104], [-654367796, -3287450, 19289394], [-12853240536, -63203760, -8449413], [3310241264, -87024270, 73157454], [4177124872, -4734320, -133905306], [11081728248, 32673790, -126986118], [-20415556491, -19679085, -16504779], [-10876154136, -13942360, -156745614], [-38600205340, 62481100, 67927107], [26093267392, -137087220, 326037033], [-787836496, 125679860, 117955044], [13019689521, -222174545, -331569411], [42009973743, 48973665, -64114965], [10019780460, -405210720, -13518645], [35495916702, 23514940, -99757788], [-9348979844, -356427800, 258149340], [13014478119, 444352305, -416567985], [-56255981008, 205811240, -644088180], [-108165568660, -174481210, -229155174], [-14401488512, 134450820, 347787999], [-77720838432, 366522370, -468962754], [24653701922, -322958200, -363282192], [16940802652, 945466200, 241830288], [5222070067, 657835225, 192936711], [162776446896, -727933060, -348112722], [-150008462736, -31485600, 68117394], [4003555573, -273642145, 1834716309], [-119842998174, -1448088780, 259824252], [-1125121640, -1401769020, 722575908], [-97262405867, -968059225, 1630293765], [-169699031688, -1022009400, 36735075], [164417687167, -472711315, -1123596849], [188410405708, -2730590, 2726770134], [-22634573340, 1077480060, -1736429760], [171630051456, 644121800, 4256501442], [39010394176, -2605048030, -764074650], [-19339390744, -1806405740, 1315878096], [-305128745688, -1935878220, 2302884513], [58281799658, -561744940, -4725755676], [-265454440130, 1217963600, 3523899984], [180440922120, -1094939490, 2768715714], [-100936340836, -5575969350, 1556538102], [-130604999288, -2230013240, -900686439], [-486362766261, 2886607585, -915864693], [-385846255724, 5476307080, 3552436203], [-630266014163, -1202550625, -5048476923], [-224454749712, 5094673280, 3668985834], [-461520026836, -3019896460, 8531445], [596762416854, 387018660, 3277602228], [414023595022, -627766540, -3689761908], [-27849694017, 1789690605, -11360326701], [-56813239428, -6954968260, 2774020272], [-71009088508, 9050117240, 1237092288], [131964896336, -6725641180, -2992016145], [-53561407524, 2490748820, -10693758720], [1254339350964, 803830500, 1861787535], [-720827760256, 1253768500, -9848530356], [-987576943328, -8644528400, 8009354322], [939966308958, 13579507200, -6310664304], [-2025338591419, -901363585, 6196495245], [-905894693292, 7175028800, 1092886728], [634265005997, 5432608755, -14596552539], [-222374195384, 5916679080, 14905817793], [-1211255663475, 26319530595, 11408071677], [-432365966672, -13277181560, 10566941727], [-1922636897463, 8040932595, 2673941517], [-1743632960640, -12125180440, -115450995], [-868660857506, 8781639020, 2424431076], [1199504321032, 13535316340, -6857618604], [1746722191722, -1162260840, -22154565456], [1559095913640, -18233281080, -3579512874], [172448248211, -1756543755, -9813484485], [-6969618344, -19268386560, -1316201850], [2792300898822, -11428161920, 2653616064], [-644157029892, 12764419050, -30035651778], [-977392513496, 2733407460, -45291277584], [2013937785632, -19329472700, 15412520721], [675431299558, 25376319700, -1121503284], [537928473400, -9848219580, 16130691594], [3287551801008, 5317728720, -10623072678], [635607049972, 21978984430, 1162086882], [-2073371548836, -7366439280, 24980905128], [3189801424242, -15052260980, -12464266956], [4316803203795, 2590611085, -11899347981], [2276973687748, 14571270790, -23767725990], [-4865673766960, 30160700, -53853021276], [-4258877774952, -20101984180, -24566541204], [6302515874204, 28576728230, -11525972286], [2899606453042, -3714159460, 5948752596], [4748390231616, 204817560, 38888872062], [2746664983216, 27799909680, 12672813366], [-4076444577555, 68404968355, 8150776617], [-4882209089232, 6640775300, -13082746872], [12381887206704, 15543480540, -6141853965], [-798064624832, -41835673200, -8820297432], [-3024347129708, -96646718650, 55050794538], [-6959059488382, -20961928120, 3984512784], [-12000684084676, 4224169170, 32785850862], [-9171185471205, -52312110735, 11938955619], [8307731077584, 92910980, 11028028524], [-6289560686074, 49831462760, -77445382152], [11731474366573, 19956303655, 18726505461], [1112509294232, -24916095500, -118830010848], [-4637867850873, 5469499965, -29856289545], [-10543976704304, 82908472080, -895374450], [-15056469036608, -22220763020, -59355924090], [-7632208557810, -130157944880, 93553611672], [13177746847412, -42238768660, -5796050688], [6288166449, 91192771615, -46529704059], [4884607389918, -4539550860, 81135651900], [-540847417260, -52595290960, 33197780256], [-11575204457804, -10955951480, -29558893569], [5792178769281, -98042992745, 29184936789], [-1883895082733, -107521384355, -72038321829], [-5191264132356, 27591118940, -3066952107], [-11154990461256, -62344913040, 23738302824], [-20501092629972, 132664779500, 37391829171], [3628430605240, -140064002480, 24247455750], [19190334517864, -26556600960, -96373101672], [-13058665091920, 94178098550, 65491348674], [2799389187158, 200571777660, -13515470028], [-2681977760807, 34602824335, -87173738451], [-19847581934056, 35918663060, 62157410610], [-9534333506415, -126349679685, 173816364765], [-26527123776180, 87937792400, 152674528749], [-10644730253404, 157665990620, 127831171773], [8569129002114, -1497119280, -39273569592], [-17142055553220, -252781969800, 36440965419], [11401461884144, 98680521160, 275507677146], [936055145346, -121713541460, 251585285052], [1896870178734, 23631480460, 27807927636], [-7600831371914, 14452924000, -98564567760], [-32047468570560, -18487846620, -202561503870], [12134114917585, 192067776935, -314207049423], [-4809417177944, -125022112400, -155142449088], [17627622570244, -114051179950, -11518658994], [33804260892504, 85107062480, -115726378602], [-8731725553323, 66171946335, 12615938325], [10377310350688, -466317710100, -25890425928], [3050378953330, -154226710340, 118122704004], [15185409078608, 375996338440, -67613300052], [-32632641479436, -276238790930, 211792982106], [-5327491647636, -319814745700, 227892370992], [44740479113204, 232911365240, 44695606296], [14850540583874, -107697779220, 42490330596], [52999690313046, 49276569180, 129720701124], [15156388265004, 249252754140, -73051049532], [-20052535101970, 306806944760, -305477271024], [16487716884078, -106184051120, 100653209400], [-10242762101616, 101226454940, -253994141406], [-1791605238504, -500656466090, -104401285710], [19607028612972, -271559871780, -24879149184], [42378381402303, 112236367705, 218882960859], [-3878659259136, 321741531900, -306260023200], [17110721957296, 336998616000, -506101148559], [-5217896959862, 243072806620, -689841976092], [7279966396540, -305304555380, 398406769317], [-31738359857910, 152188649760, -389441456736], [-8719290690092, -285797970790, 137692701078], [47850180212728, 16303801970, 82058267454], [-9120329543796, -230378674590, 213368396286], [-2353855593952, -289363086880, -550107370326], [-8539263201921, 78938823645, 350032489623], [32051293237880, -445187355320, 385633952328], [-35730386904147, 224739744955, 650444443941], [6452509991027, 626059722045, -684231867585], [24697572787600, -261047640260, -833274807564], [81386843585626, -113522920340, 280649928492], [10794556168024, -9486636780, 402276676437], [-19604341956484, -270401887840, 459333664008], [-1564131997782, 515988418080, -94327722144], [-3914969342576, -162268749480, -333015663447], [-84859256071218, -56860905140, -726092160396], [-62498683812282, 789625021200, 469873403952], [39779826940248, 294798024560, 257141107464], [21242538976272, 334639674420, 570368676594], [-75706244259196, 638080583240, 496538047221], [62041383093856, 197208564350, 1182310417914], [21845184357320, 607720900840, -1213393301844], [56959400594367, -110392120215, 221745543459], [7552169273644, -740399580380, 239470672956], [7434674231461, 345249626155, 54888696117], [31652855287744, -364095912910, 804307942518], [-78833559364848, -528948635260, -1400872566387], [-174400097979448, 62367348110, -246848379558], [-9147726409352, 1148676732420, -345579147099], [45903021048828, 579651013260, 619726910577], [50376818176338, -320760447860, 417216023724], [157496407154333, -231973612825, 304026884853], [-5509972869600, -878666308080, 86471440548], [44805671625840, -409685336800, 378175724142], [-53639668145136, 787358929860, 953985530994], [48053314803118, 61417523840, -1183842445944], [23244795217916, -1493955737840, -1071893497416], [-97341905541308, 1224994632180, -299155613256], [-61271439526083, -522778154265, -948336481839], [48199834789836, 242070483620, 974855875749], [40841506480404, 1885416507240, -90762221109], [60527081105994, 1315698766360, 1103824789176], [-90288879124155, 179443398915, -1915071797127], [150632461349068, 1350700592440, -644127755244], [-60974551952448, -379498738800, 1379419125582], [95942104735592, 87206020580, 1891960886346], [-142483856684772, 480331640540, -842316872097], [-204192858078568, -240112724960, 1146516253464], [162071153690937, 744946399855, -563582743467], [-231239386816658, -1443167791700, 971252780484], [-123692456415336, 605398524000, 1533327424836], [83705167282364, 588814439560, 633860926761], [-163730314503360, -1163345419980, -619474177776], [-70517762401984, 110006308900, -842838236466], [-118134504844872, -284566321920, -3324446434086], [103559519948075, 2093183187945, 610646212539], [-190737093336758, 1016659892200, 702996220728], [311042463354904, -1250839458740, 235732227441], [29044987041713, -347593461045, 1283691780189], [40392477599988, -2181134558610, -248651819790], [194072611177800, 1644209894580, 116037566430], [85933649470804, 98530608230, -1201404461430], [-44794461616040, 289131235020, -528610363146], [-187281809448672, -1723192529070, 554673054294], [-60157532297770, -1230305176980, 2706611479524], [194487793165320, -230364315880, -317692287156], [-80963411418132, 12694886820, -924770252133], [-3743714742279, 1122309298375, -2672061175455], [61377131656178, 1291781063940, -728402668524], [185257500660660, 513778276170, -2399754558186], [-112186447156852, 926717545540, 3140188563], [414744637879983, -195654716355, -112085234493], [40012124277636, 3323669362530, -2289067670850], [-197952460157696, -691040497620, 775621238760], [-214838845964928, 1171150343250, -1514905411770], [167774100008544, 579698476330, -1348702996794], [111917788516568, 1332156453220, -4236210387330], [-338091931594614, 2685881468080, 486390269280], [227572204217288, -2368167932120, -970883890800], [363517392761190, 766694677540, -1981782225924], [138571385924924, -3455301007100, 1386374574288], [33756798435120, 1128939873060, 118465381134], [97743561741822, -776956515680, 1983575861760], [243426053855936, 61175516670, 1189078302186], [-62273905719636, 3576282812700, 559252306281], [302781618896784, 2962557579840, -3748906171020], [805444379810224, -222650745660, -272565748863], [211737860052520, -1855208439320, 1451129463558], [-21470487585246, 808198285900, -2481200181132], [-246478901438072, 2504005678260, 3408043392540], [331827490892352, -323994937560, 510412248090], [4284645497563, 963812314805, 626455898223], [266199338098073, 111263390135, 7464938234745], [563960343192340, -445250492400, 1541223942651], [367298167758378, 3650628522600, -3572607479064], [66649546952648, 62122585010, -3513324295290], [-516081252883316, -2036939730140, 1323090076959], [-449688219919636, -4598144602390, 3030433742934], [-62995536148008, 2726042408460, -20909057670], [320009459889722, -1848897695940, 3566591230716], [13853842267200, 367376299290, -1467712135650], [528855762712248, 892734512720, -729231389958], [-428504139634560, 3358415003010, -1522651911858], [-99996227790516, -1691058650460, -2210332848351], [-348181698288056, -5202427033740, 4053469327722], [-212594102093671, 2559005634195, -6556580119179], [-153417522512229, -2191935825015, -5463989904213], [-22724127341264, -1708419658680, -6027534165186], [214748204950332, 1951982176560, 9784360873992], [-472595119921628, -1228772815620, 2122518936393], [130952217718928, 3482634135510, -6045904439982], [-417880712894484, 3383057583900, -4824633756945], [688872769799024, 3429431873370, -5365146212010], [-201568586076156, 5473085924220, 6442016747388], [-276128443347832, -132099035430, -3870168238890], [725583205272536, -3591900317320, 3401237590452], [-637021688587631, 3922333633615, -3845910713103], [-586955514904743, -4529529349125, -251236021515], [1429024969484620, -461465603280, 1339245959184], [-117794157520080, -1962321282060, -1461040219947], [-625078969905093, 3512029408045, 3306075261555], [-447157047513648, -4625518566180, -612059355402], [192560142132420, 123437300340, 7893645854952], [-183666492030364, -8059826141190, 6066954415710], [21436050343718, -521912668540, 9097235366076], [-1208907853290780, 94813354520, 477947455872], [-245047175901668, 1199695629050, 5836706258550], [-252689823487344, -3337699008150, -4675736664834], [-288414839073644, 3187080622560, 7033538476953], [-403376430948036, 3337849053450, 3735441488406], [-206960987320998, 7179593394760, 559445112408], [-507262535632560, -987540970660, -5074762511982], [237266849177512, -8069886314460, -5757287182380], [-1010931699272872, 7583252145800, -22153812276], [746755427208588, 3590581037070, -5669141331990], [825278475331534, -7675186037440, -3955842632208], [352761224332148, 3312766658750, 10370242640226], [717994680832776, 5547932109800, -7151990729664], [678482528706016, 413134509460, 39783466392], [-4347868181808, -1748079433060, 5941238384796], [-303778239547720, -6960498618700, -10472871249228], [293943156272318, -2941626144960, -9744364081728], [-274940171285696, 6050320194310, -5432294437662], [-195649038002448, 12435464272270, -6963551375046], [-265059139581984, -2580743643640, -6325672067454], [-1569694582751193, 5489650745725, 3585734992479], [-268721045945631, 1168367482395, 14380161410409], [-1142502955991469, -4739148857895, 684232417527], [-1182869262962480, -9599840321040, 10827396215226], [185723472035589, -5741992817445, -3261770019615], [-215563782297568, -5045743649800, 9894154687467], [642998184414686, -5873479841640, -3717727322832], [580380172069224, 1209878430020, 12724404073368], [-727833375852860, -10995455911760, -4010879731227], [273729679618498, -14162352712580, -6061641486924], [648439129209344, 14257076181340, -3812622354618], [-359630696958978, -4045803514460, 6040601816124], [471125681755011, 607766113665, 4690226776071], [-1167309071746472, 1947167346600, 3877784930304], [-827626153453394, 18387900485240, 8741040100080], [1612891655986122, -3375924719760, -1059388367184], [318893828413524, 2469544381440, 6533099409393], [1567539399798768, -406994586300, -8430871479438], [-109575467824911, -12251192450925, 5941833402993], [-1615781577190884, -562116851760, -9348056112819], [85351161140946, -1212941495300, -13582908630060], [-764452334184122, -5844334718900, 6325670427780], [1453302753364704, 18393902316600, -15155149941546], [-1022923845405560, 8484844046500, 12452790222012], [-597051446009532, 8181398980520, 12182279124696], [74301460406775, 5269517352985, 11712048359583], [-260213561166855, -5995974183105, -15784105141479], [40577759011732, -10548053008500, -20960692208784], [1160242277429163, -8598558429155, 14989813344903], [2175310710948592, -9451120383920, 1255176278529], [551401613732432, 12076971854680, -2711922142335], [932158279575376, 1000164908260, 22744166245971], [-1273222604215498, 5308430259200, -7444336522248], [2258015165823352, 3699166179120, 9242024594184], [-1259133795497326, 6787906444740, -13453186189068], [172728269268336, 4229331487260, 24973759239798], [377871722771256, -17338646956080, 142852329618], [-2160669512690028, -7692963735830, 5374427844630], [-629162601975918, -10786915263840, 25962367510992], [-138712510101810, 20915238575620, -12469375466340], [743048428892862, -7301358303840, 23051584116192], [2744633948308118, -5833290493540, 13818757281108], [-1729749516830172, -2947572968260, -13306769941044], [339237910162082, -5185707117280, -17606393642232], [-24839872265272, 3837192664540, -20579137899864], [-1371061027459820, -8156678272400, -6414722529897], [-2018129530791940, -7241763864320, 3015237229383], [-971424574013072, -6878781997390, 23516056545630], [-70411502899456, -5752515906620, 18093128484522], [-3002360752989368, 2412620920360, -16124014358880], [-2261964292998018, -11273232751220, 3633818335572], [111751697827737, -25306383332985, 6548738831865], [-1150552921757312, 9632028758980, -30291979885674], [1388296683262452, 1525206827420, 1789752699987], [1623308776557868, 9113314510600, 20664313445571], [3711780278122559, 14651211565425, -10226908684305], [-1040843966051648, -15395541362100, -6234419071719], [-594806713128113, -10057766085695, 6029068618491], [-62123701894892, -14851416651360, 15199438127112], [-611458925442737, 16896020965525, -14307243733389], [-659648396509793, 14026549268765, -3805407422589], [-1855616879017702, 25290192453560, 17867855820504], [-470847673031572, 15898183537480, 12667868408145], [-1181441673201832, 10782590439590, -18576038296878], [2657492946595293, 7353664121835, -3075684199215], [-3015468591306712, 4045035222280, -31323902486262], [1502352299969617, -2194609210105, -13527375577491], [619244740428396, 12117974644050, 7438770277566], [-657261910928456, -5169770732940, 28410680658546], [-1558721230517504, -10226577435740, -24653811057540], [-1176979102629353, 8691869680625, 12965020116855], [-1338336363520290, -5803071717240, -16506626818104], [-695147590037524, -28678177607980, 9071385853125], [896578316686256, -16305049655240, 11904524122143], [-637712133012797, -20200629598555, -33858048066681], [821604165822116, 6818119632830, 15758055850146], [-191195872530248, 18081039137140, -8508487975947], [-1437183714366424, 27129692636080, 3119193849576], [-1096616509702078, 17943481082480, -9493735119504], [1444005564433824, 38514577859520, -14371244754786], [-302424983289292, 12818795194910, 4288196997186], [-4195331450390350, -2349944135940, 20625294895236], [-501489130017628, -27013235690620, 8501394093288], [1189145130736628, -28469325623380, -17931129244407], [-2476156749970436, -17338736002390, -13129653082890], [-154902563735846, 32825340363580, 13731206689092], [4511850165985472, -7667454684280, 2545664333835], [1754633853869508, -17153701461630, -1168622495730], [126899964921184, -10847120694660, 619017760980], [785115566710152, 6967031554680, -1252146785478]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_208_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_208_10_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_208_10_a_e(:prec:=3)
    chi := MakeCharacter_208_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 10));
    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_208_10_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_208_10_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_208_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<3,R![1012908, -19093, 0, 1]>],Snew);
    return Vf;
end function;