
// Make newform 192.12.a.bc in Magma, downloaded from the LMFDB on 21 September 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_192_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_192_12_a_bc();".
// 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_192_12_a_bc();".
// 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 := [-7490, -1094, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-140032, 480, 192], [560128, 23424, -768]];
    Rf_basisdens := [1, 11, 11];
    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_192_a();"
function MakeCharacter_192_a()
    N := 192;
    order := 1;
    char_gens := [127, 133, 65];
    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_192_a_Hecke(Kf)
    return MakeCharacter_192_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 := 12;
    raw_aps := [[0, 0, 0], [243, 0, 0], [-1474, -1, 0], [13332, 1, 1], [252732, 22, 4], [435914, 34, 27], [-189750, -86, -54], [2480772, 1054, 46], [9924720, -3358, 314], [-51105186, 2477, -162], [-84007908, -9075, -2775], [-73077622, -2376, 6129], [271533218, -60834, -4590], [26599836, -94310, 16102], [-438287016, -95290, 5258], [1213664294, 276033, -44658], [3410280084, 301288, -33116], [4079867314, -331020, -44631], [4917343116, 207592, -28316], [-6738910944, 1113274, -178298], [6366080138, 1559008, 60426], [-13170554820, 3912945, 485421], [22656621108, -616466, -372152], [48221983834, 107932, 1265220], [23695849938, 3196140, -634608], [-38309748266, -10393143, 568242], [-63271471908, -11551201, 1066439], [166282152180, -570940, -1591072], [-24000620134, 18743014, -1413801], [-123985906686, 3448364, -4662144], [-173374436172, 21837767, -4657837], [253367949132, -37930992, -1787640], [-141216118510, -3903814, 7305174], [318519311844, 48525952, -3180164], [127890750030, 27607679, 16193304], [298652106084, -60067855, -2229319], [-66269702494, 12715272, 3309417], [57582256740, -49296742, -22248034], [1841447557512, -27790632, 5654304], [-991705505034, 113418307, -26774820], [108073191324, -241897544, 26161804], [899047607970, -138506898, -42941745], [4995787210176, 101137288, 145012], [-700558860014, -84570048, -376218], [2452956516678, 122657101, 36044946], [3103447453836, -494812885, -76725733], [3070812897660, 66955288, 80105836], [10681592780388, 42027139, 45109351], [3196656207636, -19559254, -51479464], [5944099565218, -386416486, -71903781], [1094307700202, 975853340, -7440552], [-587922285408, 442625548, -325427000], [-3074120256590, -1188078748, 207993366], [12729791517516, -564452010, -242428140], [5616374057474, 326099840, -354184272], [-899068455768, 1097158832, -42851404], [-23030599384402, 986477693, 3175578], [-13909901091468, -836934053, 165598807], [33887942350914, 717141534, 198414927], [1702685526954, -1274978992, 471265452], [8372774706612, -646514752, -269689072], [5589037495318, 1938438769, 263947950], [32300706166044, -360440056, 231785396], [-24021903841272, 2820775284, 326419560], [-39914494475782, 4091842224, 446119758], [-13614973335826, 2610644825, 642388374], [-22649551310892, -5374387036, 895491944], [-19239857508318, 4090986816, -758512566], [-68307017324052, 3431557214, 731649752], [60368296049554, 6466227732, -860022495], [58319153926994, 3454384812, -935583696], [11775993583680, 6622625850, 2937147990], [10909352922636, 2029490437, 1078241089], [89380713004714, 5117098860, -239132223], [-168497225278884, 7588222590, -573970218], [-5016224039904, -1480772544, -4431085476], [-64644973458850, -10130589873, 2240099388], [172848504068674, -199605924, 3810018573], [3141692920250, 18593406254, 5260709646], [-95101557621094, 28115349172, 110584656], [-70726050050700, -24730324174, 4658566616], [31735005523858, 17817567086, -4550993289], [-97752424887096, 8186083066, 34679134], [37845953919250, -21626186328, 7113568068], [-143242191129780, -23319327945, 82113279], [1083358023564, 19694794990, -1912101728], [-336781030424710, -9546813054, -3796106526], [85890306745930, -31986016516, -10161627552], [-179735983290634, 47024957019, -2268314820], [-276227073446124, 7088749571, -4576977673], [6088950868548, -4740280046, -11093627816], [131618935079832, 51025591210, 2516066482], [-215943396330540, 35019604409, -2473381903], [318797990537556, -17760987164, 5460989668], [897907756416348, -18612270560, 821774176], [-302784958204944, 59193920626, 6250811470], [396045843090734, 21936493377, 3889608498], [-289819233421310, 15031724838, 11434721022], [9056181872844, -77007803634, -20367818946], [265047325395850, 18077244446, 3304067139], [1032576669698868, -3677809338, 3572247978], [-596910354833178, -92938763233, -14715663012], [958136581859892, 31569427454, 2251012244], [-1417920371979630, 2425268254, 11491415946], [-770208507524796, -13641734156, -12610184252], [514313539141394, -91847764080, -32639788674], [-468833294705580, -220549583172, 6782015364], [-1223722948135774, -37416226516, 17596243872], [1386818010201984, 188475753710, 3659085038], [2369277549942394, 29909983320, 11951921790], [2422867467368820, -23136410273, -18946914557], [-1725087896440454, 153161196084, -13145561631], [-1823019179302982, -115768433960, 8722310796], [-1230461754750972, -248778480264, 38925966360], [111828417647940, 125943777729, -32987369223], [-2287224222449734, -138531191638, -17226506682], [535036987722996, 130762525854, 35590242], [-11634960505440, 98654807090, -55417129558], [-1558545153432890, 264363147255, 28090482480], [165839339725068, 322282896940, -54222948836], [722382310839194, -212478880968, 36895124565], [2857084241541314, -212071423104, 42768040068], [-1197852795668114, -35836541541, -75356926668], [-1251268767065172, -206457649150, 87024332840], [2064078512523924, 358043490266, 54159884870], [-1144743856576658, -204419271671, 18860530158], [5093295751869634, 110010066266, -30654941805], [-370561108074408, -116270551098, -16346628594], [-5445178085169948, 29564416481, 76225791353], [839969268076890, -269085029094, 11380891095], [2809101184051548, -698436274940, -43572725036], [-6553805357053704, -147904681444, 79665932324], [-5472938974373892, 82625811177, -144134111499], [888584625236082, -331534708806, -97445486565], [-176779856832062, 770235539386, -57086696214], [1460424644474930, 158707114344, 160551700398], [-6863063964231738, -435066590491, -906121026], [4466786565448404, 342297180666, 56762390034], [-8067558656797578, 172351423543, -145593669780], [-4861172948052302, -320355778454, 175420155654], [-7629569021065284, 442019912298, -34901919606], [-4022774403360970, 665702742869, 44931502710], [-4280302186625484, 790588458069, 100447229013], [-6071783524125756, 1408206583372, 88609112896], [15177125798330922, 331121759526, 52833368583], [-9273317587173648, 202352328342, -29122837902], [206480521500026, -851879459808, 8941378293], [-4390709698562318, -510956967650, -211743733446], [-6934217922644868, 1494815171094, -11002355406], [9130978600558176, 492608121400, -257880643028], [3473936697313666, 1581543823212, -161917303671], [2708164055037490, -2155671737688, -166552176456], [-5709999541425228, 363078545634, -438134306274], [2003971493851992, -1644654724800, 28140386628], [-3185692666013652, 207400228174, 467641713250], [14174451136339680, -2096926867580, -178770482384], [5401333544023572, -321364629787, -95497819699], [9910489798357002, 466857551302, 259393491990], [6417551062080826, -336745480464, 5458632768], [-9121232631060810, -647577153085, -73474983900], [-1354036760251236, -358994664360, -376455314628], [14976734784167570, -325978032058, -59346924210], [-7035876114206052, 257074678775, -484299040129], [-1042923351918036, 943542423770, -513422990176], [7585199905225178, 2538047677878, -344147053482], [-115482544536552, 2551309406920, -2835256724], [26635854214562028, 1655384518169, -352548106003], [-6871259500175174, -2796382991988, -48814186491], [-31251712136394782, -403302432028, -257618659248], [-2661808199775650, 1520243208123, -367235972208], [19335166321859748, -462075181380, 188540946324], [3253969530733922, 3175676630784, -418422135159], [3793845805746648, 2280843358460, 471878004428], [6282706914832538, 706436180032, -65478621024], [-13655852328740436, -1285541267451, -246024783879], [365144161390058, 3219928398864, -515807745660], [19771532170809012, 2292235967712, 167452728384], [5803077296510110, -2300176646081, -670332674808], [-26599052745727428, -597390158665, 372991409855], [-31039251551215726, -4007627522136, 446561726373], [26805689684988204, 1076652129505, 661676702773], [31739849463287652, 1833906571574, 715480227896], [-50748398175722110, 929081149762, 394883642079], [-4103984840103430, -5411813387712, -513184522212], [-24743436076818384, 1844204377984, -1043218585532], [16409434393635806, 3160482991571, -195692437944], [-24505586878553414, -2359318757470, 612100118655], [2903128376415372, -4711903731432, -1022622197256], [-822497730260454, -4884076926828, -301618202418], [27477876740536704, 3679873788496, 1501797683704], [1199301061814882, 23961431896, -402671979216], [6651465793378020, 9922756690656, 371096827164], [16223536376502780, 210567378832, 2283832961008], [99140243252389014, -1995947366041, 207582178248], [-44867690881924500, 2569721000304, -454826862096], [-13612610074255430, 6543608922984, -1021388087988], [7349609528742050, -10311321336480, -1020635731314], [-16208427346406118, -203689235646, -903500498097], [57719568842615874, 4221851218280, -1573591925340], [-8539531617124296, -5427750439092, -878141467632], [-6148471894558282, -1853789326621, -1197367168320], [-42834188872729068, 1843832107363, -1967609963129], [6175901795206242, 11403684489822, 349074434547], [10232486321690962, 292440982736, -1635063559470], [-50937098118154548, 8211832235986, -604783473692], [17946016649422126, -12694148687775, -723362398974], [-81192589275654828, 1623163286703, 104740759515], [-79110252068497788, -5123874687130, 1472121573236], [15733635538699306, -3290857847964, -374727688560], [-2538893160636948, -1675216631106, 670029515298], [52128456504187106, -4018344610968, 3350498756526], [2168430398379102, -718492706693, -804050551188], [54918685449975756, 694831153855, 2017460427007], [6868547527004820, -2673593557872, 1921343824380], [-9676535895000432, -4362645775434, -27059634870], [-45351396344315462, 5998775212440, -1917967796748], [44704460545426812, 2586005753689, 1662387207373], [94417231100842026, 2752029449930, -1754371737162], [-64442435904379656, -4514518236600, -3084058250496], [-2022156328722682, 15345616147331, -985852306368], [127763837529533266, -6138279412570, -1130830750557], [-42098538359843796, 18089428812767, -1676057017897], [30922920349573962, 7910468139286, -904029424614], [-72552556049616204, 8599631472043, 1504820342887], [-100828026516284748, -5972262836226, -6793157537160], [24411758306663066, 6148318330872, -5655029535963], [-80973622184599622, 12960498444708, 711930243492], [-150715164940798056, 13180079378794, -964062592358], [163231838378148972, 21111679386907, -1710078972437], [189866571525510996, -9370015100732, -1248209168708], [64711686523914458, -26425893211850, 1789947957519], [115523252672192676, -13888653354, 4028586508098], [-187428376316538972, 2588826621579, -2352763034553], [-883032268046182, 14761971123560, -5298958248084], [176748394944126828, -269795182418, -4087558348334], [42164297131244904, -8877961029054, -1055372302542], [-67468219517735422, -27558310849484, -1726892029080], [-7908566261475250, -24669292913553, 656232515784], [400829690850430620, -1237877821190, 995171048656], [48616052958547968, 167100190978, -4707397572206], [91817455351357620, 15831499017102, -2150244733836], [30648263212095204, -21690037438604, -3300641522216], [172911917449928244, -21058253206631, 2578678810417], [54882387459775730, -7395233116032, -9158459541615], [-18216833611800302, 32619374327184, -5264673570756], [327224265977843616, -7215336152114, -801643289054], [293256532832579004, 15131419292821, 7049965801297], [-243356522669189124, -18636495315068, 2681737406872], [105070957502554356, -15147841295836, 347445054584], [-48585562258010208, 1369726128868, -7271699447156], [412348425817665130, -25419051231466, 160768040247], [-249015126827501070, -7042025736020, -850322740896], [420905920308078720, -11574502497494, -849398406254], [326811167009031674, -7324632228920, 2242193017680], [-450925036899721074, -1567961331779, -3615354349650], [-251243963275092660, -37136151896700, -2561553280488], [113653850226325922, 1600265977294, -675377846397], [19257082035319716, 4886531010260, 2083129169300], [-125896701954051578, 2644610948293, -201715007238], [-157834406043245846, -29962370269800, -11637835913412], [-160873632026253324, 11635466990631, -4242891270117], [-572441181347162292, -3786498499400, 2036867193964], [-204589312352065166, -19054962556282, 5930362256931], [-337406691564248254, 24622408758240, -10457520105231], [-72021896213212342, -49496358743394, 10603767109050], [-113602810403596644, -21350986404228, -4525264310544], [141135617346758790, 33270404217079, 12848655424788], [341379713189314874, 29761056741620, 8262160619976], [224877049330675596, 66983912240450, 3403400032058], [-383705666976772610, -16953301535537, 188832452904], [284238211229000290, 23468543777916, -9510766550271], [-348210487526629044, 40999966574444, 1050362698796], [219178160095530186, -52107565351104, -3729259614312], [-732113782602414036, -10205791779671, 6726050605045], [344673595943009618, -69398644808672, -14134229567568], [-741445842598720500, 56143764551, -6354051558001], [-115062764302419876, 29380441739562, 5824141490988], [-249222042075166574, 52349957626068, 5216137629525], [426035353233859850, -39646064585016, -5367484121994], [118183767474678924, -53652209693660, 14676738634792], [-796340118745823592, -55548616047478, -6338600168902], [-828611434575143172, -47419791256233, 4031185507623], [97142677572698304, -34239856128826, -6389877553330], [-685323519565187462, -64710520821396, -12439172343303], [-483002048412339876, -109357870571766, 1104896357154], [-220225777825035360, -124340057269204, 13889942884724], [239654108894836274, -72005692220660, 9322680360222], [1101365137745200766, -9002622642381, -8877406770180], [-495885368801928636, -148905215605671, 11868153251745], [408860333412850002, 81319480164716, 13926638578896], [-508414800459591578, -4170273421353, 1688838493464], [-649670035681732740, 12898115168704, -24393632516516], [-1228001727496634598, 59315859913796, 14476074583932], [568589067433662372, -15979861302432, 29292431988324], [-70994279516730574, -97470641992392, 5574672881109], [-607720483321712826, 12808713715787, 20533892491116], [262453399211524500, 86125683029751, 9181107585819], [-91836138295265754, 63431299476073, 34437039868026], [-458491179060675204, -70442654944046, -21311436933500], [-1120437051416752380, 50000623588546, 2987228936266], [116171940426993802, -108779486607720, 6583022917686], [-150429177032723578, 63860123044743, -35267764261320], [-315411528339628020, -43589642405475, -17298064320015], [-1407062092997028, 109678188230984, -13470872207416], [-586447920131567892, 20871558257146, -18355801190906], [464131443460345410, 172776748960932, 5842497524238], [-1531716840603127524, -58811018299018, 2614038498524], [162374609315589450, 113359781507538, 3960806075775], [132530266084042272, -96628137258314, -23414198940974], [-334797868645737446, -2664211724748, 28601288794449], [1396696176798041616, -50331691271176, 12883299924908], [1684620987930155014, 90347984570409, -22893733526742], [1755615004323957114, -57238450921246, -26121725308602], [784139774393389236, 78863627386622, -10573152699850], [770118897490352184, 6138643814068, -21524345775824], [-335539179771330918, 113858301942984, 12953829192336], [1275020611519602636, 36990777764004, 694041115020], [713492018680929384, -172288788291842, 14015344681318], [-884090585974467982, 183142088979616, -17744153782086], [720036588654158818, -84418860403644, -4522591969452], [605815573868327196, -221050313478504, -25993644349896], [-607987589605870022, -166737059797320, 28094687816022], [1077584121978131806, 24764904191677, 42647133511422], [839870838381741660, 26131418870061, -27212087519511], [1977523014118526050, -11558978084226, 31778728879122], [-779413036886235326, -161009442466420, 29608316658792], [1841773511648028756, -63548103002930, -10931427114434], [620571581123425164, -188319724276638, 23206373220174], [1315561152150058080, 114818109561136, -5676000196112], [2077520078782918078, 134040142409199, -14975901750924], [1979374155606450618, -12184683854946, 17834785445511], [-1715416636370353530, -108268545072725, -4152016059984], [-1185930098513635836, 130303001763203, 12552622837583], [3014166157575502332, -41727302981828, -12324076570304], [-83632046015932476, 208139886876280, -70823042479436], [-1747420289462310372, 215015035937442, 2744964376692], [1881379950499733762, -193938164116128, 19136028202209], [2501032350914732178, -180416571423260, -32086409803452], [-180461119774934838, 8791115866680, -85749162843510], [2765965201706791428, 14996873200799, -10063019247253], [-1126261653471092326, -1266659174736, 19422107278605], [2496970035337200818, -376057307109138, -30274316097498], [1657219346263421814, -104649698146751, -30142801732266], [-2848091324887007412, -245960035645253, 35903284146427], [-4398778724537919122, 188738414008249, 5186696984154], [1176370035161671332, -124165415105082, -43984139567652], [-2857559783505116222, -139651639518694, -56197019335041], [-3038189851857491916, 28977695155492, 8193547633552], [1980843971041953912, 98035988751574, 9155767917022], [4432243189361327734, 86645221976653, -30113352552222], [1647394512994025988, -313658964943774, -23443099182358], [-938232523716469046, 219321830528588, 26642413734696], [1850166280034697182, 397493789990729, 18191006521938], [605850384535613988, -48317712197569, 44916882818483], [-160936986350862982, -112726804942656, 5136431066037], [-1359816693312536214, 182371542968104, 26797830347160], [-5167134275397668688, -4527270379612, -34869065108032], [-1550407339243971084, -384935357522500, 86528234415692], [1332718449954933226, 144839379660258, 25001555530374], [767931364341508008, 523562287295260, 6402493543600], [-1182018125254633078, -52822811673072, 39392550447705], [2117227506395676970, -2263703633252, -51501361372308], [3603385301258851536, -76887681369560, -59223458888324], [-108536990989300836, -370799554129838, 18186238483300], [-6435118313005893156, -167905749823788, 4778815853220], [2137009779116631914, -82462216040712, 18796378471902], [-3279062682115871698, 91310240579877, 70424040802770], [-4625707288779127356, 304875160703125, 7821197468773], [3478661210782376426, -59559635844144, -33869921045508], [-1068661913794912428, 647765982363490, -18575827001876], [1293060862348865964, 542296721963598, -1984438643754], [609936845472186768, 561477874272456, -7033365331296], [-371334197136845882, 239991904478473, 13845988851030], [3095926299300619908, 147568634592801, -90872980750263], [405333546668007418, 155544463795466, 28501430959215], [1975003342269879444, -54660461248546, 118769494094588], [1723094983451812512, 696268983179944, 26594064969652], [711220327984320226, 977968475712, -86999302536048], [-3076094568889877414, 186477913827846, -122340789434286], [120505790769489898, -579261108136180, -42624940072914], [-4751579269172840026, 519862762872539, 28059948851436], [-2169601722856108494, -310266427121878, 95262325168770], [8810779969132964388, 84498908124657, 55058972270745], [4470729820598865954, -288768749204624, -41069501579592], [-102366958666036236, -37913526202010, -97366332265430], [-6034946411922635544, -279937406082032, 54002147283064], [-3197878248237815748, 24581691776921, -58075216298347], [499031008515517730, -335421313784018, 201030726497031], [6541198578348734772, -412027603768992, -23262612589452], [226489374786952920, -708655963608278, -112408840754882], [1365381575608200674, 577793261314648, 29434844321856], [510314003890850318, -96843141515341, 47658626986032], [7261843057971009732, -113781122986608, 33877573314984], [-2214419056594761564, 498326799517734, -101465129964354], [1990626425720392872, -522359728776356, 51224216565172], [3629131244454444794, 839301648009100, -9049883210064], [-4699440180918962700, 576672517828783, 65015366136355], [-1532814238435309182, -336015192072706, -37683500094522], [5605167557460181548, 474280721005150, 3456734594374], [1025749675935774798, 834415388285983, -171886233051036], [3512959637706977842, -130212751193604, 98545458194973], [-6703785663061778134, -714991509569058, -23395284768150], [6829211936034878532, 129282930166583, 129072048266315], [-5828725492459733934, 692274447705974, 59467154147226], [7093432780510787766, 256238029830977, -70257628287918], [-3573441776438279772, -189636322428779, 119256349351333], [-5394843944298565094, 178082447555030, 91715995128447], [4095731197157630298, -915492045699314, -76111011027666], [1353759057632412708, -713241894235282, -205186461416026], [4172057541345483396, 45145893808422, -174322796456556], [2960913943408344386, -807150297429936, -170104474675818], [-10697373453852786306, -208278345888157, -9598400041572], [8017808508980318604, 925150353640414, -118927330591988], [3831811710702492612, -1022278770934238, 57900824317078], [7202518833779455690, 245720111827736, -85201283094606], [-5728553547753098226, -862458655558723, 51382740155778], [-6846052980759543936, 973437424786008, 189148098705012], [1763386267917542244, -993124652689075, -146689686381235], [-2617567116314479334, 280770994353574, 182733485321142], [6844959512183716416, -1739196431420970, -192245500976358], [-9880275997009033306, 750054418132443, 180910478767176], [2453235145287617450, -537955931855880, 29027222514693], [10578360645909424320, -55232617620724, 66260181649556], [7186835942229081948, 1013838660728690, -37602181916608], [-1165720043991054710, -759128600093256, 231349840448238], [-8950669898809651602, 622298360694125, 155874951253386], [-2920296283427335548, -194600933065198, -26850028945096], [-12874136981898072198, -1090930814776220, 72230762190204], [-1355060643658540644, 1348378876637582, -61748427515566], [5578703540830336464, -973184732254878, -205472549760558]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_192_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_192_12_a_bc();".
// 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_192_12_a_bc(:prec:=3)
    chi := MakeCharacter_192_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 12));
    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_192_12_a_bc();".
// 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_192_12_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_192_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,12,sign)));
    Vf := Kernel([<5,R![-285604670200, -134222580, 4422, 1]>,<7,R![112744965783744, -3170001744, -39996, 1]>],Snew);
    return Vf;
end function;