
// Make newform 208.10.a.d in Magma, downloaded from the LMFDB on 20 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_d();".
// 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_d();".
// 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 := [1820, -376, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-2, 6, 0], [-2274, 45, 9]];
    Rf_basisdens := [1, 1, 8];
    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], [-52, 1, 1], [-424, -18, -1], [-5686, -51, -1], [-24658, -30, -400], [-28561, 0, 0], [124992, -306, 749], [-139446, 330, 3020], [-342056, 972, -6588], [-1025278, -28080, 6400], [-3095494, -64092, 11218], [-5882592, 67266, -11869], [-15752370, 64068, 40418], [20007920, -91701, -51421], [13608242, -59235, 96635], [2690682, -143388, 472322], [-46011770, 717594, 274544], [88067962, 474468, -30402], [-67722358, -609702, -1477932], [-41022370, -424617, 1051333], [-144337750, 1058736, 1083196], [-135472916, -5403300, 513120], [-35121870, 3004260, 2312590], [455125266, -3185616, -7356], [223230554, 1351920, -2810620], [99024994, -5795412, 420098], [1075575608, -2122872, 3465088], [-402313016, -1313208, 6229532], [432311224, -13280790, -4684725], [1032165542, -13932360, -8501380], [-630884296, -15684336, -10956296], [1371570556, 5501991, 4925531], [-2003054658, -12008964, 3649246], [-2381182012, 16753707, -5092893], [-3472537158, 47534856, 8132596], [501241554, 20892303, 43120773], [8579802650, -57332184, -17240644], [-5660249186, 45415200, -21772370], [-2405030730, -78272616, 11191794], [1261019450, -131697708, 23006922], [5144337744, -110368749, -89094169], [7508115870, -36621492, 32407938], [10451249180, 114809778, 32292478], [-5117672986, -174603036, -27582246], [2304774084, -251754402, 37067493], [-9706940600, -198262218, 118945182], [23369198212, -119731209, -105124709], [-23122551594, 341288835, 116499525], [13438712634, -39891036, 28379614], [11356695540, -379173630, -87514315], [1380346564, 429406194, -173022231], [-52926295794, 101118573, -309327997], [22674841578, 92355648, 516925768], [24996324660, -283190100, 53959620], [-45572307060, 186048918, 354052133], [12598573428, 615536388, 115384088], [92327752566, -190710756, -191139086], [42667771682, -494169243, 549739707], [-60366927438, -689353188, -28388358], [35903467834, 1062730656, 273390396], [9966188156, -402620208, 496751872], [13468622512, -500871918, -309099033], [22849242046, -902968038, -544781728], [47999508792, -262604646, -673360166], [-151869030796, 621340086, -201306669], [15063525882, 1251715080, -182366860], [-147735271406, -1744797312, -109432502], [-96702098572, -674569746, -778858181], [5568988284, 4152924363, 85678043], [-18756609728, -1526983554, 1072252831], [43773648334, 1445118204, -207933786], [-258456890994, -3067038336, 114959794], [424392280180, -1616048442, -455474622], [237892292486, 616069932, 1417874822], [233168295074, 2179430148, -746956082], [-43503718242, -4469756613, -2235910083], [17570329530, -229235304, -2490775304], [-26459350702, 4769451888, -1900626952], [59556506574, -6352726884, 968162346], [258416477854, -2822042316, -1347222746], [-367689594156, -1980184431, 77024129], [-61750320680, 1760550102, 2083550587], [185160669582, 6488650077, 1131919147], [411682977112, -2093805186, -2240389641], [-527744587928, 385419138, 2301249378], [462130054284, 13193740905, -1308953915], [63296025394, -7150117584, 1352744196], [-446518668110, 2136762144, -8269848776], [959399939424, 505582122, 3825970107], [-35325783466, -5074143840, -1644951310], [1158521140060, -6048584232, 439323968], [-271405005582, 14077995141, 1018436731], [303828876466, 7407822600, -2389565610], [479902455560, -3137981709, 358271131], [941338472762, 1557514044, 3160588374], [1138249172324, 9561502290, -1163924850], [-83199618926, 7416241968, 1056428048], [-1248220909116, 9601233678, -1391841317], [137224583856, 17739367176, 3882324596], [2859157104692, -5885998914, 3605937341], [2719018847228, 1879943697, 169134377], [-547900494732, -16440768666, 7074531389], [1588270429984, -7612338081, 4680415899], [1606179979336, -4258344270, 12760816695], [-2891328114048, -7580941203, -2501947343], [-1523124351222, -6445127760, -14085604280], [3423035295778, 16166125128, 4206784098], [1218943173098, -16193907864, -9801268304], [-1921349895988, 12492415158, -4644108862], [190309618960, 19130071446, 8428874061], [-1176007585956, 1433681982, -10802913798], [1502217009146, 39113899200, 4926594560], [-1539417189914, 25110265812, -22855921958], [1747606592246, 7010423280, -8022408530], [-521632272670, 40626039387, 24125781897], [-2079487068362, -28450215504, -1899739064], [2141771078146, -9125053686, -14394412916], [-2772682378312, -23594205534, -10078052894], [-2012763344106, -29456690244, 4296426626], [-5535566761368, 21163682664, 13271018084], [-883788598630, 47758631304, -1406576516], [-5719842233888, -11143591338, -17075038843], [503242896714, -7768520436, 27493346354], [-1568561026426, 96320280, 36990844990], [1214505167738, -22298098116, -29721576546], [-1802946641202, 548949900, -43251683210], [-156804348142, -20541745368, -16864284268], [-5772676536368, -10829586366, 19630082074], [10046400152504, 18284825130, 10967455130], [-756291084508, 351690906, -17513932279], [522220992558, 103114267536, -18335579794], [-3310556891338, -26066921343, -24239609653], [3255613880256, 37112289384, -47124725256], [-5147516765622, -13390333476, -18257433526], [3798538580826, 8581638624, -22210797636], [-1210486057650, -3611293572, 23881389298], [2324639984576, 5721141294, 62991202259], [-43935291842, -32036421228, 524882802], [6759841778538, 98009788056, 23946911916], [16461381811716, 50477324670, -10935713205], [-1894957106702, -86124322872, -5521828462], [4153414831160, -135768143610, -5893213065], [-6651938178384, 55677361662, -10225532218], [-1665092116890, 11469696234, -72868343256], [-1943264026178, -54330563088, 33256095212], [5754173850214, 181341995424, -20784980646], [-10867240435556, -25479863190, -8523665315], [13907789098546, -34322398728, -27061661988], [4857907131520, 34398488256, -41445035044], [659889659006, 23622475905, -36642760625], [3788153319396, 155918837010, 108043366605], [-3953377451244, 117478175718, -71360884367], [5032980526352, -150664426743, 83590998297], [5315826016868, 20054042136, 134944298916], [-2731174018176, 21664920519, 101785069999], [15686125913844, -145341122766, -24594478906], [-1272059408844, -142717964664, 143691286676], [-2984200460826, 236836591644, 144715906534], [-4411911513582, -126667922712, -62832983772], [-11152400836820, 43131468690, -13885441435], [6084909823878, -139137117678, -102053058648], [-1473592637512, -36201539574, -38678918929], [-18348828732938, 148006622259, -21149357751], [-10121256497340, 15368680137, 52076170357], [8833525276290, -51977334120, 86890238000], [-5236790272454, 162828151293, 252448999503], [-43553755807980, 74997585666, -10691179314], [-8293877889130, -3746358984, 108260967056], [-14661881626798, -72218521152, 12216803248], [-14130699683962, -269108272968, 164250149572], [-14200315063014, 75916559082, 60138215052], [13331912520996, 287684619894, 59318278609], [29088045999272, -113573933688, -63353003768], [2381364857590, 126794365596, -11156122574], [-8866764513648, -320935501662, -217808107822], [-38096877957248, 236899191294, -53170928561], [-6243041018114, 516907940256, -56818872034], [19144982111698, 134014937736, 283183068556], [-21451401821008, 67329750492, 82444063692], [1553106877070, -512192122884, -114547583754], [15668371314126, -501902873304, -297938712414], [-5434308633508, 17113074768, 309602895528], [-21913243901066, -184350514224, 107511836976], [-9861131035342, -378806579952, 28526099128], [-29240620713946, -244493211708, 120176914502], [14865569690866, 145430422800, 82451031820], [27599691650938, 132276583824, -114182822156], [5207432081322, 98804547558, 207856123448], [-472344240514, -438530532948, 28155460142], [72273037911174, 260480608008, 114160053498], [-44182514216672, 182160321330, -148338942315], [-7453320168558, -434732499216, 377744374994], [16343539775156, 741566918553, -153046998287], [76751371158722, -121813643160, 247652675740], [-26161813999192, 198562888017, -127742800743], [-2913245240502, 364792112688, -254457724152], [-23126521028746, 591327200028, 125305369818], [-2713967619002, 23112949980, 52881098810], [78203182890182, 328207689924, 27207680094], [-898708033084, -444508923390, 20761360990], [-18426879874824, 56710193286, 513179717671], [30136321706788, -372328587324, -234252658384], [-21300785259364, 194926070214, -177172744451], [62795055585400, 1053010355682, -113685975143], [58589943920966, 275192249580, 269136519570], [-3408741162522, 460735706640, 275873790140], [-46731843052690, -48065880681, -373268636951], [-32727807553560, -328008498024, 53484357236], [-18746011732254, -537188188656, -43726407576], [-23735437864512, -887663466423, 178326057837], [-56185447978102, -531029509848, -429489676708], [45216857548782, -218993137800, 288085889980], [89498860991596, -219052638420, 11298702160], [-25331987771914, -1306663071330, -89235020900], [126672530893446, 551423818827, 126065972857], [-61711376102858, 31839501348, -954175983822], [40409764801604, -1304251363500, -672521590400], [83639644224188, -1239467109438, -139784627823], [-19761655265062, 510659448924, -396266415406], [136567712110764, 1004415389670, 1724468845], [31062008726122, 49869386628, -497421873702], [20783605416274, -567601206321, 576814744269], [-32237503598850, -322439671956, -513045580586], [-82607435947786, 247135333935, 827518462985], [207386037079344, -580010378115, -298902475395], [-38921456294938, 908126074512, 1164024315692], [51916736280464, -1673900695290, -848175780985], [-35131910253708, -102611138976, -834004520636], [97180253686400, -1153699115418, -90272481938], [-141889869894282, 684499906218, 311216648968], [-200057129826182, 955634961408, 618625724308], [-70881781813624, -112891144968, 644205125972], [-124695657414702, 1165437572136, 1030677208126], [90796704369244, 1624885746714, -892856486521], [20845293654772, 316229418585, -1203980447555], [-11111040471674, 973516847565, 1400563940595], [295215558039370, 185816181480, 4324547940], [188202714270364, -575127523878, -221348611343], [126930545858628, 259749936996, -495350620004], [53879123345140, 2132815219626, 786602295526], [-137776839810870, 2463385696158, 859060040728], [-61510334074216, -2060047230237, 431570217123], [-138583123818876, -1243935765528, 272629779892], [25945329321932, -2465469767382, -232720628307], [43403329465394, 565226945928, 1240282377308], [-127265882680248, -1540084698504, -87451237064], [54984887595010, -1551923856441, 174970260009], [146274207895766, 2527122090360, -16289060290], [-112880109576446, 1065054875598, -2301247124792], [-66540413325336, -1488651419478, 127824192882], [-132185794906576, -500667726642, -519784938277], [130554100073378, -2061334316208, -1282491125368], [207055777186946, -2069093351301, -95751864591], [-12342482476444, -995692524414, 671219780891], [329111515772734, -1136766901068, -93180364858], [127359156923510, -1348651433778, 137936145972], [-144208407651018, -2010497904636, -932868045706], [-28404464985830, 695343909474, -1011850929636], [40568233606590, -1193173961580, 119614755410], [-77577624876926, 140584035888, -2982388617792], [90655824233504, -1449859710132, 2067370136868], [183263383350108, -3370746508041, -96051751941], [-157799179186372, 3119552169270, 689045236185], [220754675042830, 427753103064, 1993151773704], [-155373016492314, -1229580338292, 658319903458], [119172630011540, 2555233031007, -1083722041573], [-168421662579800, 959260760250, -1595231523645], [-235353261954778, -258470228940, -291187315810], [266053758652542, 2361964269720, 643397660870], [33582770265762, 1709752711332, -850997462598], [193718043131122, 355459293780, 1310062358150], [-84526979203078, -3233691609414, -606557044444], [159981062219778, 3353882519232, -196416466288], [114210238431904, 3196824159924, 913766425164], [20714094870494, -329286682656, 1112076164144], [-59165165014398, -369868718232, -2566824605242], [456322345386682, -1350780124470, -604658026820], [-410176729138110, -1653860345568, -972924219648], [194135511227422, 1179374190780, 3671507283270], [-277052750883384, -1089109921971, -2112996815051], [210887609850628, 3029983895832, -1990840082948], [-573824544270, 4742111153595, 1163456435625], [-149567640601940, -636261474750, 2197720466910], [-109167331709566, 2470980941256, 602471017196], [246307111269006, -4366870922616, -1110752453426], [-183479035455520, 3351830043906, 539390222186], [-335102440387324, 1769929012482, 1269167644917], [-535887044881564, -5537082965058, 189283879257], [240193713132106, 2438836704837, 1278439701487], [476858819768930, 61469170560, -330329755400], [509235218515602, 665492367588, -885938469622], [454158426359164, 1561136295219, 538958173319], [-290064568784226, -912378051012, -266246997862], [147961194796726, -5066793316830, 3230304589240], [-248351471569286, 2311232990256, 2817337559996], [-284411487311250, -1053592156164, 616571821406], [683077468440924, -1469779659906, -463514040006], [148334471329622, -5575606541580, 553234077870], [576974901396212, -5776316364099, -93830623599], [-461323850557634, 3799317410994, 1180000899824], [-547133107048696, -274329574614, 1239251852711], [151193013001784, 1863296034270, 1159520966955], [-158642891326920, 2773416446538, 5146622659818], [366588545808616, -1003369543584, 49202025436], [549490467954236, -2282584337811, 2525519004289], [-274700454188530, -1517498346108, 3978779445082], [491841284975936, 9823776296787, -841019993333], [-1159456986445374, 934588044156, 516803452426], [772449760602758, 1286569130172, 3320128943382], [73065208839990, 3012770443572, -4889180941138], [-520156046330388, -6305855049984, -85622939244], [-119975171121580, -2986891403778, 784814314977], [257977222850724, -4448343531654, 6130995686251], [119506501785824, -8784351357624, 4354814226476], [-212850903574598, -8221737988569, -2300322489299], [-4461265282912, 2820553141590, -475375770955], [878498406292242, -5577813575862, -3057905985132], [-210432742464134, -7777065083040, -4453276601090], [473443513034966, -5124290898156, -228671499086], [-367487141056714, 799215367200, -6163793383840], [248992379531332, 3397100315520, 201268145680], [1046406444472678, -7747834408692, -1526663319942], [-296576454255510, -2623800294804, 3498628595366], [-719169869128334, 3502374811887, 600144864777], [774523191003494, -752881057188, -3119723846998], [1157971386596674, -977173560096, 360064262304], [-273865498962106, -4529999241198, -2456548144768], [221207828165922, -4329782616012, -1595778060402], [106264312726832, 10943641207800, 1728176288640], [-837740020562026, 7678401670716, 5254789320006], [-544503115412750, 5328858043872, -3495797492328], [82714885015094, 7340727634860, 4948523363450], [-645605341765292, -1194966991164, 5536694257216], [-239109038370186, 7430887643532, -1537852588798], [-1845527020401974, 1618694707020, -1035529129450], [-633292124798126, 5331047649516, 6734068502406], [624129323381402, -12457967831952, -7428866206712], [-488922963176410, 3607491372588, 3352455172958], [-584778029369250, 11547551453028, 6060519079058], [-180612813499832, 6652154906598, 1779529402518], [432698107913376, 6677890654494, 5547330471759], [-900580752542004, 158940821262, -6263635650033], [-1191646895640128, -10031179840542, -816856704777], [196413991311752, 19331144450754, 1731874459954], [-187688193434572, 12056631122778, -678856178047], [1829906483367944, 1437973977387, -1712560769873], [-524553178365854, 15607245902064, 9736417652764], [-45643410098210, -20877500758500, 911953775250], [-1133357602552886, -3884520397173, -10040578384803], [973434884824146, -726043775112, 7869366478228], [-1732103966353030, 8716752709650, 2207058074340], [-373148784220502, -3125654692920, 47149194420], [-323068135291128, 11145235757754, 8454411756279], [1361953930793368, -12359960680680, -7205562962440], [-309885760219674, -10261099579872, -12418739938932], [717833730274782, 6666532565352, 2958659234572], [-249661858829502, 12610154619015, -3828982173295], [-582637349349894, 1204657567122, -3068080202828], [2162912926393248, 2559992053398, 4607439739563], [-2324880902627082, -1663341067125, -80994563215], [645087909295474, -23497436569176, -9249889609356], [113666600434206, -3715394074848, -12244309345208], [1410733800480844, 9862811920938, 8046451489358], [-2083037218483526, -807180655968, 128034954882], [-1009638864252068, -12291098185008, 3612156724792], [-117598313971400, -20754745724106, -11209208049031], [-237471097170204, -6605592226794, -2341840020719], [498195718605950, -4545586758912, 3004108516658], [-1069950075288696, -8173038939078, -268105807703], [-741890630031444, 1234565897217, -499965989103], [948547536458312, -10610129337399, -13787122974539], [-1409347695583142, -5595658589511, 8526391790119], [324318484327590, -14493094186296, -11040396890776], [-1198408325008716, -10775910001944, 5025716251716], [588833243637342, 17074987775016, -13166084849824], [55400920659042, -22346974475754, -10292816222784], [-567877486129536, 20020732577322, -7311478666438], [1279870300887878, -8002714929372, 10192983903078], [-1650457386954142, -14436589806336, 1852623905384], [-1336795319696210, 1248064288272, -17485962468008], [-659478025230102, 12741422090256, -16079610173304], [2179382669619026, 8781357159528, 3804878006628], [2223718438767342, 4009688155596, -6139817987474], [-1359983020609478, -13681193041944, 6982115919336], [-512419551871114, 9712670352180, 17610586821970], [-345049416212142, 22788133408641, 11506715835331], [583776423315966, 14898763545297, -10710424452553], [10660895296922, -14235142004364, 458964825286], [366073459433026, 30903913117350, -4718358020480], [1139234523689548, -4823335835628, 20574753368712], [475380348753786, -2199776206176, 4785352520464], [-1072714184983052, -19592424150738, -16508520533623], [-2959905110648034, -6345892957518, 9613442542792], [1471073690053996, 16025119646175, 4448920716795], [-734273927263806, -21531913152651, -4805734004961], [-810205450521788, 26295847665078, -8382698009467], [967276992235110, -29641797296181, 9017469078109], [312548718466304, 8915559813786, 3866423571261], [1010404462084992, -16510250442624, -6224887276164], [2528632539440760, -17904311018046, -8763528655341], [1710568207650016, -18882871947462, 11121317019763], [2163828125003534, -28238132372376, -3116524325276], [236440529785058, -2770551678369, -18318500181359], [-1891555738636418, 20961609953316, 8678652178826], [2906360972978412, -5170709607030, -13801134370775], [-280811095829348, 5693799184386, -22688330335874], [-953921605705748, 7587439289898, 15147941760133], [-3565264548644538, 8676879610572, -981207386398], [2170304448583574, 2493687903570, 20917433767960], [-1902662815910898, 3697581672372, -4121277478318], [-1621600058051876, -4538039315394, -20112476355539], [1259880975924102, 5190384241104, 8275439008264], [1164318691741948, -12572529851031, -17016414810411], [-529406862673192, -23723699072913, -30869545987293], [-1404694176174340, -33268140329622, 15986124256573], [1830261805068834, 36972309604140, -12018744059910], [295513421496438, -3457960370841, -13676707102271], [5640770740691012, -2661912325608, 1087476232292], [2227182228809682, 22382272994352, 5088900880792], [-1382263258985376, -15960194768898, -7924742973498], [-2483281757337142, 20828461627596, 5359334942586], [1232315696322490, 28160057792184, -4303794191756], [451696515928938, -44397153999048, -10811946316058], [-1064719222539996, -130721174643, -5757313121063], [-4535133335270578, -4129676776188, 6976170495222], [-2056270616298374, -966294470760, 11027128162580], [2423805528289240, -10787642196021, -20192307573281], [-1385083250124618, -2510439975540, -9775345043150], [-352441255191898, 38434947067500, -115057117470], [-345067528483636, -10033485947514, -12670738071054]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_208_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<3,R![-3817152, -38493, 156, 1]>],Snew);
    return Vf;
end function;