
// Make newform 36.16.a.c in Magma, downloaded from the LMFDB on 27 July 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_36_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_36_16_a_c();".
// 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_36_16_a_c();".
// 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 := [-2004, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-2880, 5760]];
    Rf_basisdens := [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_36_a();"
function MakeCharacter_36_a()
    N := 36;
    order := 1;
    char_gens := [19, 29];
    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_36_a_Hecke( : Kf := Kf)
    return MakeCharacter_36_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 := 16;
    raw_aps := [[0, 0], [0, 0], [-34830, -1], [1245752, -9], [7487964, -286], [5878790, -1314], [-2128038642, 1198], [4620844700, 5382], [-12127411368, 2002], [-73645129206, 79565], [-38338765216, -451413], [-485616578194, -307512], [1005351710310, 2742314], [349483151108, 8796114], [391922410032, 3658902], [2792937272994, -36041935], [7918081118604, 56238424], [908137030742, 19865196], [-1129033705492, 207154152], [98497659855240, -74494550], [-4012928613766, -525698208], [178193033072432, -306386217], [-180752178743388, 638552202], [92228154240150, 1605980116], [-907785730540126, 421648668], [755657650119762, 3377906761], [-1262861143980328, -2143882215], [-1375001387897796, 3345699500], [-1796417732601178, -2204478342], [-1690700157530514, -12578463900], [-697895426629888, 5163060609], [-6190467404330412, -26733881680], [-1044396556360506, 43354533374], [-2557132407459676, -3025147968], [20712375505689858, -51314168993], [-12676783695347800, -18210633705], [34470315737727734, -849282408], [-33127927975540, -20913555438], [1273929511089384, -133006353480], [28528318770810138, -219555215997], [-14604490737840636, 90768926024], [22810087101058910, 133817762034], [-142704879647056704, -116292471512], [-169471702645129918, 155020751424], [136654325582853042, -507767863043], [-53474505629767048, -866106223443], [74996380515965660, 1417198939128], [141866205516912800, 378599241429], [-424727952289090380, 31962599934], [-70894114561526866, 1471093211142], [450894927035548710, -380131396396], [200287333612662768, -2090286518292], [417537722066220818, -679617675468], [175738392623803212, 2709999725090], [1220326929844633854, -2828630710560], [211704664422746952, 5055422711056], [1350277075714411386, 4637542184045], [-638064460907011216, 5517894114333], [2403572617780496894, -1782992297790], [-1507454912406255786, 4737152330000], [706537092852998804, -3810817583136], [55265461902044946, -6898038772319], [-3499516399699528324, 11756695488264], [-3420414370058859144, 3209005954868], [3535899418515674762, 7033176519984], [6169301046960324138, -4082942530391], [1972448875269642212, -5939952393132], [2128362290842590386, 25290178735680], [-8608632246612540756, 17102121864954], [-2636783677467896266, 18024353241516], [-5949430704478536354, -21828010132732], [11716088438400116904, -8762590239990], [5393325675031146512, 81288515335347], [-15207709200976492642, 62002894325940], [-24709951895161840588, -26947933429914], [12522063343612404096, 104907587166272], [14203283745959991666, -108510764213249], [-18475351315645325434, 11671013384676], [-41693023860692604786, 5052007982810], [17529692897978550314, -5750443077948], [15116944049543662452, 137066512787094], [26765440998950544110, -57460403712942], [-7743276409800037968, -50625345458486], [-12109185263395450798, 59304956081544], [-59587387684855656568, -15353323377471], [23133541884767632908, 265938682259562], [6858778246638737598, 71245118569302], [53404467759570242810, 98411788017900], [-70140018056228840646, 167890764884987], [26519188815683471024, -251579165703291], [-35812849075160355036, 69529611870134], [63956922727843348704, 629307260670426], [81058139466245065304, -667880166481281], [-24391351537338850884, -605238259282196], [-49418113285512909124, 277763579626752], [-14180155885436757192, -793217325217278], [308226004302243161322, -180954066700351], [150079608821990171718, 287310106243138], [9499897058001072164, 1350797815401174], [-103594752357057689482, 442175921437122], [350217645442036639244, -721648575897426], [-439523700283886261094, -1035207269110769], [-530691010170303892860, 366225560286330], [-38552374190065537674, -1959818241242582], [-123946701663722789260, -1553865593192316], [-774459511155523771966, 725373814422096], [85201871940992441628, -664791167179340], [-213342081045500253618, 3426400362262948], [542169224411303644056, -804400999152674], [205887453649610513258, 3606366989538360], [-549272490330968445664, 1785275445403593], [559978889471113585454, -817784932661748], [663915161104816776102, -2443098301890424], [-23078948749240152124, -5373136812635496], [-779354117341144911928, -4057977723707241], [471213691996018786686, 1528320368253230], [262436618540568033452, 3721587258991302], [-541852000138245207096, 1266859629796290], [-468452908930640305158, -5064800251999433], [515082459093858747108, 2350990878543172], [1039900065336445674878, -1507331800972440], [1393518026700776938850, -3764247324190272], [1044400616723910683538, 7195691144146363], [-341612248906991539332, -3162299815625754], [2298018317430807589628, -797723829906606], [223638542974047027114, -2192684696941543], [260782030274187617486, 10274863412864934], [-1742944912146065842992, 4731636790406166], [1614363295605586172264, 3805459652187783], [1454944025837570962358, 10525677891205446], [-2067931896423824285620, 11126027062894932], [-1441750015432077866712, 2469943627482780], [89461110710332602512, 1676713630795551], [-3582749681122207228642, 8864154098750502], [-4348981963974252358602, -2894734236978402], [1907664655563059940098, 3233995424858952], [313970558223766166514, -13546352477194587], [-2657907312000429206116, -1092871506983598], [5178407140747195237386, 18894091524560487], [-7451272291899558072474, -3365649820338002], [2715443449077521683460, 36814161766851234], [3037776730257131235234, 23206517012448853], [7547202718694631081224, -8691473016395325], [6916962888647236374156, 9574683728030212], [3981267291566933064278, -41034768351542982], [-2113434334436233042296, -46811434878486234], [-8800097241516284716354, -2890459524517632], [1664915987952289583766, 18194661230434890], [14755688480034385846100, 14822920584339390], [2709141363603939042144, 15727993638505048], [-10464149260483923895642, 3900890881502676], [1080890182749245321070, 31057381194999128], [376275893957852312636, -15750071401059558], [-9303396970641532992072, -17009971477828384], [5111571389765005220516, -12566116178935722], [-5154136493988950364528, 42674934157492548], [9146782636893708721832, -19024555449675789], [-20006871375581901155682, 33021454175144514], [4341506523259370929946, -92262069482313168], [-7708989078439678831590, -31774795053364349], [-4895396723699621236476, -81556885969209624], [1022515798240436347638, -30438081236751006], [-21229062907292455094920, 89385706561546737], [12974340978888249267612, 86956638153388398], [-7918394328377550241074, -67723204922358254], [43595312282342045110296, 20616102870688552], [19785778106923212249248, 48190839776787951], [-24624157552618586657362, 9801823087683540], [42491401727940188435474, 66564670903736436], [15782833918804742159202, 186160094710047179], [-28388040275206992333492, 53865818283169652], [2287707171247571182742, -109683590763159936], [-37534103528951964985272, 78817085694170204], [-18253046636543972815174, 12239342773087104], [8137018609734950489456, 121848509558220291], [62924221712738310564438, 17697773224399504], [-5341246162015051092076, -10934720887877568], [-40415056246668134930670, 21812553021982335], [-9198077207933217819880, -134611709583729951], [22619371392348659840486, -49566297740497416], [-31732069177077719177152, -146683523325784761], [71807678806947936296724, -133088206237428062], [-26476738915942260154546, 202692653229818430], [-39643643119342947796602, 124423280294623776], [-107239785084133508314800, -91363601626897792], [21153962977044239405250, -255482326819801469], [34799451543447564506678, -53122279799330658], [-77399117776486001674036, -9118031537591208], [-21659261276594735169958, -217466959070235420], [-37697477261707226657664, -459516700647782864], [50406100335291814866818, 168300501177911544], [-13349393356721911180452, 588171906659911296], [64273640564109490931996, 242481759986399472], [-51123002384410196927094, -60654123563813081], [-20683908888717284916876, 541692723550179664], [-15491255834265058888218, -369796142916821384], [116117893160548638426962, -165612692889291744], [141510657619856316422870, -79969761823351842], [38509752740231062043838, -475381245629762984], [59554764229852850851080, -692665825522472660], [84429570011563574947386, 512837549755006323], [-86216202225798027816016, -497374487137560987], [4113845453450331159614, 310177568932243074], [-55476034958995115068126, 427621102269167952], [328327582342565493100476, -45934168181991466], [-186678316347798372776982, 270483558325111201], [114579068638171732182656, -952511857153414263], [71874442283819799159252, 36087445792258034], [60852972508776746107974, -102164119789746356], [183124184365768373533988, -552489485067874458], [62210249275180363467506, -1315107857800703736], [-94000500570400005924990, -79799735959921925], [192340124924928367456808, 47315647150035993], [-236894781497504491226772, -875813408062736240], [240111459196504341740136, 894448752374443974], [166562259386027847007130, 655495633068793272], [235883006705232802020944, 644193923056968159], [-100652833647744089915826, 242119155738710382], [-146899938556669318182504, -838064640819137080], [348824443184549298677322, 461202978260478835], [-237526294500656756459218, -751334528538441990], [1786905516128167293896, -1418956741106085015], [195773587805936274154110, -735845774000325166], [-37704954574400749869328, 1729946079294502797], [-278958602181526147941660, 268193576929960698], [414805492986648531545726, -1453461160184584632], [59298954276923323639254, 2568206909949111372], [31268733381193064010912, 483408891552310394], [8817525631682387719064, 325271253542269005], [-16431370437017794464900, -1489658277408148276], [296822682188841898294118, 406819377372738762], [-730391474600956393637380, -1152485926423562178], [-169212968679324302140480, -1195725626993753379], [259407592204570149547782, -2601336271519318120], [543913165859501514172004, 126148540910456214], [-724451276633321800711728, -994427669249099950], [45440341433369382789170, -2005112760263312508], [663320260813486355368002, -1109082112905246609], [-118681809007435243603668, -3790464053247046578], [521737760635173035426088, -69357107093711086], [-125452250297126999306172, 1375109863515751786], [-216193816440853410829132, 1043199532355921028], [-470098707556535482183624, -1322447107395537], [944516478555945080304902, 2561723442188339040], [-198437311877645573805042, -2080710787466917520], [512619758597344832848728, -891412648500486658], [287199946504383645983840, -378513438192151533], [-30143692940182704824076, -621828232879862932], [347993505545606256080900, 229404086588989620], [-894446465130248679927504, -2906442211187005788], [-901281560822259239619754, -3189792829688596278], [961253424241545275693118, -2602786699608225564], [-129504432295419636793464, 487961331477143098], [-537029036946174780856198, 6192025029972256680], [313668040246983842880186, -1586095328122998435], [440018231342954066418852, -923199395462642900], [-1021342458285092091657538, -3181826627522474958], [71281272755415221358932, -6004835024821239420], [-287511871740421748442030, 3171928269245828949], [-904884882750709322109622, 5626371195114592824], [964723512233404694198528, -5912884617613620255], [-1407030830625019226723244, -4985439341510518392], [1985090369733522102330638, 2129616890272225242], [1577341476393670513071350, 3497940172760971968], [-537570760302033718726242, -1188605282363146294], [617619331557781510497932, -1160538344178187860], [-1051782414530413842969318, 5582138422873413223], [-284624261482594223506698, -8177620685530902660], [948460738105739576503028, -3215100919846087206], [1616505697888358080941234, 2375747076290937039], [964955061025826877715526, -1177310736358453404], [-2173902021856157279032612, 7183052098493043708], [-1903745194179972860994838, 633357650666824704], [2813868006129445515077024, 775021393299989007], [-3481961329776884271417070, -3181919296768446816], [-2549404291933621480300984, 2551822372623298113], [-3542101452024804617684148, 891064874035577054], [-3149731881174049467070570, 1902857349727942380], [2234056909963671062212538, -4422695340135622104], [246203129290696045676388, 18552089532803849356], [763824563356644295292448, -4392000318217209958], [-1394313152681382051868648, 16805079547740015201], [-3369831176921342038076808, 5683085127065948822], [1862656126811915613411662, -6152849881025338476], [704497728372754258342628, 5752584469530158850], [2599899100787823144044112, -4964757590419335540], [622217572825618768628402, 14334654530336675004], [-592666502510921117593662, -7092326927148093005], [1379736665706659625091304, 21531988522272511071], [1606759105990761398854494, 11362574768960454852], [3299520674919890970555354, 3852686129458379223], [-4604623847155501271751996, 11451681952406040800], [2195983849476410498745270, -12897162113847431444], [-248615047737294420228516, -25515675065377885824], [-960009092890433690415226, -14479328739941885016], [7475974533600578771993226, 1396186722052819147], [-4971479898899623453611040, 3174670868926620609], [3164948659312576675867170, 10787737529067968617], [790758879792276567363084, 26635568469125633046], [4357302378544803332305772, 13535057286316674234], [3728531669918528995139834, -11638885992685793928], [1359954468353684720369466, -8034404218879788825], [-469536394270778880086608, 37077694445911164939], [-2763622482594898469520348, 16240047367912583576], [2276970675477069070378196, -3684347088393864654], [2293456663547980980593186, 17282839860091847412], [-6569737137284789574572868, 13561913876915404418], [-3648957358750145215166170, -19464708471339275154], [5825391286837822014070584, 18443873171446026790], [3223125376061523604586126, 6529562658636178572], [3926077965724327847397648, -27509547487232906824], [8416457649375738954218370, -11674851421178734071], [-4974323167774718922627810, -9799434253534850154], [-3420486219166091911662580, 38050251892943346054], [-7015240706886938941532568, -3644322862512770636], [3371297503250938216181402, -28331446815877252248], [-7050442989403803447695484, 16051122803761416620], [4031661059439414946189632, 1346871619288509358], [10985363725563745066519490, -10929787812069219360], [12141935099840771652983886, -11171242391799190836], [7870464259838255579520860, 13541814844665642072], [-9064645160021418046757014, -29126605198345699176], [-9599415000228085490858934, -21972447977894113987], [2468300547677419622929952, 12274467504184756395], [-1356889829207304414617946, -13084906611112284566], [12453965816397528143033234, -7414036146738901764], [-3543643739853033014652244, -46062438020130673386], [17915170357865703726567956, -8375057045875416006], [-4479141345891806922818400, 22152555192200325840], [-262259894583166142362734, 19322703417974217311], [12211688775766841249349734, -4185242766112416894], [12555531163990339892881578, -39990462831692472485], [-2271187322357791755928384, 55954445328811966293], [3530443222276860383186580, -50757670313328428844], [-8826226419950404338816412, 2951942881482885528], [21656426733713964740091948, 20622123361481895110], [1085134525664882089021622, -70964941367302415424], [-1601077245827021090027298, -11885541069451746676], [-17076145419605902463121958, -8986260132842542632], [-1072477475749047516252784, 79425696253084766073], [3451769040698843061258782, -76656307469723174448], [-4946398221734761018143690, -33144731969198502214], [-2660599031242318812341262, -27526611093664144335], [14569059345862044440587832, 1700357885379322317], [-11129509543484836193220342, -23565047911899442535], [-12603481418600764493565708, 47507475441435480146], [3384834838590083547647342, 29171491714092118950], [5475472540071621904340228, -39500651932548944652], [6515530065792153483793200, -36440830999830449114], [-2748396723883882036316766, -82493665688113071235], [-362537348112892261030420, -23176864745670656358], [-14565269830058024468570758, 93466118384108978940], [-22843123422587651656303686, -10906170262723127351], [4547530595944626103874864, -50312816897056425927], [-2470096325677704090337090, 39739492915025995008], [19305195928872042987289494, 56474241316404479576], [5587128100321976438489952, 142649630609258033220], [26395198597858412918214972, 15815024158038251188], [686196041504821969454190, 64359726709558715158], [5869409466211852019498424, -79669850045118080452], [-6118956907048308212314354, -67616325044327228976], [-17338487897608630392793914, 72205942157005092884], [-4637162580413648003956080, -31158866068868893080], [2461319303119702987204332, -159034846203939795434], [11032890957437295273404300, -92844211622598495612], [28585274888914397650070810, 2306149332992586648], [16132307847866895688572186, 83782111906784515093], [-20043702329117002366982536, 147312730590749609571], [52523620306500714402519530, 30250225472557991184], [3687024152568540154763700, 186810874507084267814], [36669837595403265078164804, 33227975814416204022], [-40032748445202619836052752, -6019294037603807320], [24191710097545982973597282, -31330749591878319527], [-6913744418582197382748088, 100417237680384879687], [-9213580944553036090266730, -132228332627975497386], [-42109952463276298167782556, -24591525805922984742], [49807048993116820679672352, -62641824529456994616], [-18961701794750336059073566, -46723806845769177216], [-29672345991140773117117650, -24852641577636896318], [14714113147229639932581770, 215468348448813494076], [29042352203789771808904746, -71332201788838284133], [34435895881557585803304582, 109404469748730019822], [-6637436255351950372171288, -67743820143663382377], [-3080369991520855923386274, 57995583083788887376], [42843065909216673396000716, -126365750670327760530], [-54959446842910430842581336, -8695315297972270544], [-17967946686847025258501872, -195124744184620892049], [-47587092599747227874300002, 98827229405650289490], [-27741066354286852635869644, 195360634319356537632], [-8734375508343008477106816, -164692830053980041030], [-2867753303330637357864574, 4086261531334003320], [-27844656928655646299191182, -234809873851830249533], [-34089457571819026063853220, 93238318770727312240], [15078218879489431981382684, 267546783870934315758], [-31868330956498605676916520, -37876618496247475044], [17840006697632872208409386, -230765438934680354244], [-4535312481844722590597920, -236207775337789901895], [-66723165759335173818238746, 83628448613223596394], [-40078356008062049309862268, -215887807686832783002], [-75026826077216220902058462, 107591223236139821231], [-1900780211622317308166698, 144616134781363661892], [14858197479343203116537790, 247763580297797446474], [2027246384209191338377136, 379796397991300799361], [69081307187308782977136150, 127605663844122294066], [12160319939676969068604114, 72357251564840082945], [15365527049134818806884184, -48134929850205685389], [17390688577558273989777446, -277591538760851983446], [4970077617372765948287982, -305496814583484316678], [-6390937841449620691459396, -105469978420531960938], [69154900562337683859722196, -192623297150984323790], [-6507267738693275337036046, -377984795912861861232], [437197743879618377184642, -352000937068067827645], [59658722106131159254540908, -303224208496063284934], [49584018031554123077289740, -193628258148299707494], [-21716272503607359642661990, 192841024721819799672], [63103806428824350385082010, 219144834691734662893], [-24425427564018800200126656, -124401346800743141320], [9385633755684934811683064, -146921603459086691541], [145342018471788502221900366, 33143436148126488674], [17179214522503212852355224, 95876308452380068070], [-88177484478131984349774774, 76142395321003968587], [107966933402432870090616878, 12734529100316508744], [107968191694353962332736112, -204561039075746950196], [-32947634180010698451623988, -311287232180991291594], [4074093656757928293270842, 117515076601385926296], [44123363526033204220792314, -556241369389861991843], [-83328903130576055916191004, -107159575696938771850], [22570006129687658561878998, 220983556879663583148], [-104364564018829947439568620, 161599904118942268086], [-62184705882793609279336200, -230701891789259311118]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_36_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_36_16_a_c();".
// 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_36_16_a_c(:prec:=2)
    chi := MakeCharacter_36_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 16));
    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_36_16_a_c();".
// 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_36_16_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_36_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,16,sign)));
    Vf := Kernel([<5,R![-65283075900, 69660, 1]>],Snew);
    return Vf;
end function;