// Make newform 2736.3.o.r in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2736_o();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_o_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_2736_3_o_r();". // 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_2736_3_o_r();". // 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 := [36100, -329080, 3099194, -621764, 7434909, -6054124, 12843126, -6988600, 7652658, -2898276, 2838686, -837564, 597327, -93436, 70116, -10096, 4326, -152, 80, -4, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4269529384743869643540740187777234804446927982137838716, -534490497591433434456921928470657442365838658392622740, 5012460498053355272143122799802875032421783616611056332, -2969403699055313425590779880577920780114641081912893866, 10008711018291103782772765522774347682073723324536960626, -7183366877634472233112182199573662913687042388408986385, 6577776309982729242577910123707477099604349337976607401, -3445157862178527447769135330762124961553423863089009980, 2545063332062471226295911284130892081393721392601507965, -1177509515724863290839416980486734955127827549271460751, 581461112131295436163192257299778618173929893397912286, -178210434752887158754815769519072230463063887623750759, 69216077896065135575937729758920055947272833318300903, -20267469803255996022301528754691671638628732480549437, 5028742489907027849817901250181659288128053825765346, -852164320317524680923076331536987609691123088812367, 87734421639286034049665073405814073703943009127811, -16649312971260299731528869712996126671917106205624, 1446396638915256584879072815714823728247446429019, -166806236485515529815884032465889442546945004699], [-8141221969193371549371554422124195478238908997367378516, 2192612097451796046302623590687115643511773518291364200, -20560250074016852761336521693679238493078700157591303660, 26137658350959140146543803949033705802761015615940972064, -40727088376267893803963383617231248505468295217324162034, 28018816503184820943290250643160440029543715702559376422, -21427914688423680432187845211075286780525348882605425107, 11011346642437026385703642082113800393080607942441838164, -7880206065969231361352241703492308921033232954226816347, 2963560360634369329594865567759315880821165409022172014, -1433100352266341763631449377698736956373992460169888350, 235613754102479821752059324960586285643859410768549508, -180866110496302242649355012226559228321805501569766189, 16252435206907189564175438638880564028425512813446646, -8540008015504778477006536924603425810866854402382170, -983897077721064950772729717504798224364465852150528, -177789961923858510891189388874701077986391800187465, -16208873217271893266303348743941949448324602250840, -1276410756339469185713903432519289389930110967285, -389881742293104287858985196670785332798267092216], [1928619765202952528153918176197515811304402615640527208, -1040875737081318636330741655492626558232652108167398760, 9761582536559599451796751063752120545060194678905136268, -10797867518357384242263776659571094222801270674640444972, 19533241536810033966360342119887196522711212669622135970, -14200213027629554240267159681803811792488963214726930918, 13922648046764549111737467365649047494397332453097505779, -7154991193401501011441838240807136390944311776199264356, 5308010049656184646927356945098983675966718953704898963, -2376152806537385190144635517281726799189963325381868182, 1262512805942437170448744843684275443762023880837297294, -418990077278236991183844705399777339783890299814677628, 142633854071571083987032699861785340657518024641100053, -49514336748108294898051692238694594811565690894763302, 11345587753126080444714095598516767990935901526081514, -2556981195149459592080011809605865113424495973454076, 189127328099654534649222124668073853544801872120529, -49118906132977881587150927215181497520326389906136, 3554532868800298683079801626347055670664684546453, -556568228884371879547167568863830552719014657712], [2137397249847466328232840906066975310837966251950825212, -996832853466087956972485015428881773546047999164770760, 9348645693410782041382157225542866230256097473500028268, -9660316918113816279749191131333062901892231004768707368, 18723685030654720869464808986060674539498162194753732946, -13673375266972122896761856841292923435939604657444769038, 13629493495568889038872671632432214767355393991468697779, -7011753632887729199044296059522588207797115502683814476, 5214347815966428050595507042377798485974405526951684547, -2372331150685641366088101947057826977992310841224438558, 1257793675290189358488854908714657516514387679881246446, -426570306873482455339819553166959468455103124489412720, 141909909161536571228604836092833372813939389269243237, -50833656141252306158496641156534905749928176620902454, 11484592477665205564537456320810840207687451624914314, -2675760010511604172324279774659069693669470685422772, 190489281093003699268278417988741275071675795498961, -51324233457978645524411512502840170574237736645632, 3641604216405452916233349011602647301866952662709, -587320382914994856411893370103759117048775267516], [5960360032186397607340306035153957584908175823951740152, 299308085446748279873170338084547353443692717996578100, -2808430068198508980157024271419435435970780390584428380, -2984587769271271694149336370100940670605844655624547410, -5845345844031920907034407678437456207564968294753293162, 5085733199815623804520791138669632124073864508983025657, -8696357130359051981411330487950523873718097675819101637, 4215873758453834969824343323510976253650836840355542324, -3295396502062922271299496156660103981362721362240875209, 1972899844707951043376059353297519618006458257154739511, -1020916136788175852008407280622911574118240607940667446, 463174102823658690544779017628135455823877965578983379, -110872984501780693000198353995733960320577458661389843, 60534600337674057537094470523132753575740212426080877, -11605756676669099859159058917849244927505144845266730, 3855394687404192476757695769183305099314955213488647, -179605911528622260827657638844242597414867626435351, 72980853364232102577047514461895336387250269718528, -4253317256790401670880162155972287740944113692079, 909931800182497573366050329024413194281314615471], [-3257246256342481960513104552435719056949225241539665364, 1141561545151083218333752783929709258197168613822558420, -10702688213871019029390334783150918394603616212534268656, 21617723708435474086180880249021696919149711118812232102, -20922691966997472360873749729167721428109281442475498840, 13366856444521817240497755382533974431934573840049236693, -6251553215181943684884656566659427414772918399578560578, 3099409324037740261375766454424809868534947700526796976, -1937061127711042518636323588333369252088665203718166442, -101553426659655159148595818733690362424987369137137929, 59577263722633911144358343006047022826617746141272520, -294844988082085873707508759408376004602242115843918479, -4548503500451264062338032287074814361921869299640082, -47771031925331545168664482189677644004439119369605183, 5823700230905937098500893151659409127282482935119072, -4186752375986088004646463850461248459785910571093915, 64228826656426720236978666080847176223939037525830, -77855197301483160680621069928035456181472121736872, 3235650651601076770393143327140249952399637451662, -1077384078378390924255666032916739656546842692343], [2508316448743396310089679739273008056149508106505093140, -530648024694591707572803085450193419480835440245291080, 4976881782460324058288593609007663128993950793383396244, -2471094077351072826022165926140082223048165439798596256, 10008916670425479942257504853059881792639260254744727982, -7449032920487297433502583073585502267078458045894511126, 7844816440325299499285905490333755049843592069598287805, -4103386190560835637090655976142224584373614148316176772, 3086174823220216959246766328214614775215956154654157413, -1563715030964182788846266102375816892328896033642820910, 785295857543871374198759537472745472492923118516905458, -285399910564093321237438419212728322915016635800707868, 90897365935209108933371981403561441504909845055044659, -34784810972964398590051031135076643512281412701000238, 7627956084683707463515899313324949994012616074688502, -1840656522622678009812268262796091147292447560407256, 126518057152255977194826143978721313292372947722743, -35279103090965738857993643059772682134422385849456, 2458950829590264597306878049306097302312522925227, -405018529600763439316479463522234945816646892832], [504521100394146808519857341009549078397587939493673160, -9494643171446496179214803454765361389393555132185275276, 293082816788800868917350046354163595717939750564657080, -13141381924410410499174092564325907938918146087123532990, 30324977126736638714784154660840503183054028253002132752, -20534339805842743403946760888938011689422070399866279767, 22765826557089069173684289142335765959261729943102188603, -5209206320753106674710800737552481848014725273452524318, 9887780967239063971136552186116234794138157154577409597, -1241280663912244157010914230858739211262283997798993895, 2319402552621600718792610028392384737554784220384046920, 245544364921895862923601773236528645137710362729214647, 301166284463000997572947640854627942964498685889754759, 25758880662237633843933029146845574773021892847921075, 15451223518580919071443840231213662100428159559004500, 4463431531400694320111400614812762111588713882684495, 330146220931738840811141299013110815284657685807897, 73571652739651080833733916902693563999361848349770, 1711730005475383847244695829554470115263938892559, 1234680042057512920563380417639462370641346927497], [98339271865778929723524630482405291016770625696519880, -1442994091079612199201974886765610566588914151835447148, 2162305004605666166869624530835144966212614084511310936, 5687683375226678510094297077197506778575037963228258338, -5697814257458588294591586233060540735061751473973362528, 19584483696887784923816222211525464967588911911067495969, -11044192392993061480110988049648408156658668892962712317, 12584249348269999431053855691321491931835427153890176082, -4798608592790981853321215868644678568672730020592885891, 4953042366369318032796095213680524240135801771704067193, -1577123611997927960074448505397795177592952608212848968, 1067888912907892129240043616497666848235250350262874379, -177812270484862591584208134404343928597007222308980233, 129356026516397356816352097533256966241577888783290339, -19855867313748821129916096927298563873436851907080220, 8077552026690658131663656564616433321504172009694483, -300713390272663648521878657399819818794808500469719, 150268883486502430115061549863094115632315780502114, -7780858879887187832848279239770880709746705298745, 1880599743702100658963926622299438903621080054541], [185210450330505921356159379858765262385349355603723912, -30364151115570941086440620972408503731186118713340340, 285003672282627984243153479647099561316753709246227512, 523230750068029044817602651285638684759356058756879106, 608039443644579015737876645522483501101103471589139236, -582465425297015547196094208647715356728439858409642229, 1074324436363899648145568193870149847954893118349872940, -570763197408239682765013842079144336184330194079030168, 451295421282934609782916375119530917426442849953973972, -278371599403919148744974855449703302274241253667562287, 147061127995576275779297494407444833351439482789857900, -69726878304396887853854581646906865305596259872250749, 15882398574913858606897425787854176355713930945907640, -9219670060454991313166333758719958054408690266660353, 1726950925154977359124308501150805544084385321844316, -603238532806468093849221491076773800264925086138761, 26366941180389731161937037907053386209406387757240, -11397143734724605741174735749042061961655896660696, 646325626958701120818053955039635719376243854840, -143575860869460849211244777171203230727569722525], [913443367761404031209067801751598031461862853090702160, -17944214979104788980528427004125459468587161833485917056, -1173675664665858890088112890373493215647159208338692464, -47126228424541193803761259703046208493643798508278340146, 38915717765226340410894361500744890532754565552709982056, -88702565776687546176442550668592661980099513104434335714, 48185725567571948923156130584745274067806164754416898440, -53914934872930627259135953428072429047855098209625287832, 20276904246740488258438614609959613717789719294697075424, -20207911262442713912130467989325059379116365778665928954, 5914106639359562218859333290165090003564697640987164816, -4282656947803344762123268594242278882959363385719140908, 659731165182384188976212171442532572652517449082357324, -503501060751612427090698504094000261677623868498769114, 71953772048458452054188893337967933890219433002902744, -31174808659494019879974425270547822216418224900428634, 1080219504282902120849591556397796191728453267823008, -576305751720448587058422740907539406999864800045560, 28654183735538721460765756067809270377712014806996, -7214661712976449257017006584964267659674749155684], [27128373738172048177741294310979916861339312548740, -527225548674347308529807046420152493251393426811184, -6280025135949294062128988226413392239724404713836, -1284041447240901162636777067515785119796530828196184, 1059636958415024058144614750634660126343981558443864, -2335489897822429002774400340055435695946668614508336, 1272589081945354439681551577858459518180152317895800, -1400223614010627996531573301801837394444230698792928, 533666519733926496100073581764557237052730338237496, -522590373831618914677700393576016319668991140648016, 155145331564664838966050656458090963054609766436924, -109994021191314624266357396758014624957993718264712, 17357849602382171815311126269957762072115868144016, -12935217835824153916223732580429561053554625454976, 1873034186159216465380881163306985317836099202216, -797307485518940297257624301718934356909326015856, 28243713683485261351890838538319124351743007032, -14751049693335048099386313170929310930286569440, 740357085590021774241041688758856276256338884, -184272489795434204559752438177859518665423336], [2013955413384305008691517315863810607437683307911859960, -38513781981654870399801731057086399769909600874684097036, 3165680496393898567513161151905028457351866261647520016, -85007709970334415413657498007388583730168447769550299431, 69686059307396133860436728468444091338175073670267004036, -148439657785764735030542366461268876430459707131302022664, 81967084114605656567240509735234183856144427053421739370, -86446037917821450623587653083216173597482353562891654982, 34387538584094859660016415374962047597706477055424572699, -32016676352362033849869870417830245509300454039435070344, 9946791761239755439527163239009039524743208565524829566, -6609695614525644543874607695252638161527197060383958353, 1123327929592687643673599105384532953373474503941274639, -778638299890507544329124267493840877861128292531964134, 117527585392142871123795384490230413624271520714318594, -47327330587082431242168734015612379155813039498815914, 1795682877035360005540741729783167000685197509185743, -877950761651526326642540731447159453587411390355870, 45408303818230174374381677508374176678874648909726, -10893032473550730702957897934397115203929550953699], [149386462189738937626980893480, -2902551308015028813357970788468, 22483154176508165188182643528, -7298723763043440916269489149558, 5687431451091777992503071581808, -13495817959110091324017587010717, 7220012841056819294873659827155, -8181180473639422515235180559426, 3032346600113285250089831095067, -3063426857283228628780585401437, 896485028349527988794130115028, -649453217913984091193050803409, 100032278348478581939611880237, -76489001891458174591084971587, 10968547040136510082156796632, -4742228507030942395130164877, 164667314028867329531666839, -87699295329507151964062430, 4368545710410150635558383, -1098257364735229985454387], [-16933560148350649481599389042795312738887538166719681820, 241854488874174126790690142501779714206303434733720378956, 37361868776678104065798216923046426282415182662207114684, 599974783479304045214703119446630439930497808667454669626, -420489835230589543341756462143541493702541671685068023066, 1074504379896022259175906787263681194755522914978408348859, -542073246393312433673253019532641293926412096357256143190, 652145357512307443229561232406215820357567689695269985912, -228983670700588617751286151981979636655294190997918369064, 243682538990593303568232460476803028002807438812937011949, -67501897557063811549523211561576023782628428798161371746, 51878591300202446345972485382618326238900690894362930543, -7511613492682665393851804364203440327959697114211886244, 6075706158518771762862072753993111859153408351388177819, -839335234746105544026912717311556833132756214917964314, 378351053647747673102149482017396166291152313262245099, -12492396149727720549217744328292696435013859008761138, 6980072219054765961684605459857737588880546128662900, -339848736882605053575702556255926802578237814656256, 87678756580181176886340231689738586294026906944919], [-4197932627181228481799322721996691704961603317447157800, 78670716852837425505085880651429140488069759846384504860, -16276026033820966846050550495606302285450859522869392776, 150860913925666668295823452113202106296930029465637238322, -96166208438676869902246667911549626484353944244257689464, 233888482819801490585682977950680881522525278614307049815, -115796101962405663953918925851500275197137541639694758689, 138858035304515820749358754161898841198682148187839461446, -47562868577288390489834269882910713604598785987693397191, 51163152476177628229039439708447727477040050746171869919, -14073484944562561148473289108995451002063225983831586912, 10900425552575721901324379562001373792489550681265787809, -1555144106497144843879163672174197916088144842627844517, 1275652979474665352752180395449274404655893146926853637, -175434628029540296130043747903669129741537607590684340, 79762217897446322370611086304080313063018281847494245, -2602553001468827985161776604705297351509837339786563, 1471176974582414763808940129174790715682931899017790, -71340148107837565486729521462226042856203421805829, 18511456648586485790829291792741202056269278586999], [-2751434641464910797535807551407748799117567299379797500, 82950155455623361298796479267570021728306871188155700652, -9419063119147296026616161270617277387146072057179915572, 209090248717075223603799275259156871671706930531208446002, -184403842066706349206549962985758963245305719977330667722, 392074450941944486096025236153931463843796091843825087723, -220499184447340231938043360916334789628997104675808223620, 237918220845662707170654980555635107112788028306773635344, -91916101470598364284235231717056275547651354495866022798, 89247940271638334295888665996489714186455439050099462413, -26703799721601748149299424626460938944830892409852485802, 18946808530551173387669530239999510215433777817205066331, -2971860923214692736544364588065718889042071084055426638, 2232729578464790488504309008253309356282374871192861603, -322649654545030791948060332333650839138010706972209738, 138311610890524741490660495682719659071892392862729983, -4850782178550122154137745033943150432769776771897216, 2558723668127106363865569950559040598193014467646060, -127971464880088097510995389817276054397475866272902, 32023679590419066275512989566122604485735521215403], [4317953741868386096135682016944897631024613452661990840, -83466474584800217409676464892674622561128647362884984724, 2315377403101398656907095004204320162907848908873382360, -199946520094109690676541099813210460371252293003408499518, 157264256250667299506605390154798545952837030480701754664, -360732873420826787006702462254707801500706676976434893149, 193493988957536361850003100764437635960642776022380432799, -216511158980806608192589508770581066769245229384149761074, 81093878687927630725192123447325082320063147070756349129, -80858547401949254655507777840880214196149956844809373869, 23772238473454978155837709024501414628707489694164454768, -17075557852571795460636753320468820191559133461762388463, 2655709801303155054176648996645427720323264239723828715, -2009150544830362384960364043910970846202185913069598591, 289132234805443714628999147001086461780198173042364924, -124222838890799662743420227287689465368892555062888883, 4349022092481835519441865436428509013641718538720717, -2297584704068277812081704025314338883529072278032370, 114790408456464069475367051140057573734269549447147, -28740584486151533669863468108164186482652119077681], [1910788265284959366349402535245428673974197980287976340, -36365841489776818361852729262927927512957880444022176724, 4253424316157546698069541932504723093350984396598372100, -78552694543213301304743644010404749867169766638964930208, 58111319563898433003158979828812779338122094905825905656, -133706715571344001204896521402269137252912979600484131787, 70524888306725308800950121279785147144057311338954355601, -79379328035499135067592105378717917194609337853551976674, 29411170809855210772984339120293946059869504378091869407, -29447823218257396518404435207484405501809523104312883355, 8616970276139555335014538913938498130675815779636214724, -6197339172801015390347477216244940872442621065908758967, 962701832489703207052479841521847812926111933936360285, -728151291943595494419518974869984719869163581993882117, 104711304010547752129889729318848496951615080647799036, -44974011240315257121103117754485008426582547631178801, 1575575598908148270928624890094143962006724898270995, -831750399938525041897125435916068477496167573562426, 41558787163073831721307056666809200000506235936049, -10401277785574669658094360467126782429635751568185]]; Rf_basisdens := [1, 271392329516404794256515171234763929778148545607150964, 271392329516404794256515171234763929778148545607150964, 271392329516404794256515171234763929778148545607150964, 271392329516404794256515171234763929778148545607150964, 271392329516404794256515171234763929778148545607150964, 271392329516404794256515171234763929778148545607150964, 90464109838801598085505057078254643259382848535716988, 271392329516404794256515171234763929778148545607150964, 271392329516404794256515171234763929778148545607150964, 14283806816652883908237640591303364725165712926692156, 339240411895505992820643964043454912222685682008938705, 7208219023341180817641118586648851799134897520535, 339240411895505992820643964043454912222685682008938705, 24246929119698667424763281595, 1356961647582023971282575856173819648890742728035754820, 271392329516404794256515171234763929778148545607150964, 452320549194007990427525285391273216296914242678584940, 271392329516404794256515171234763929778148545607150964, 67848082379101198564128792808690982444537136401787741]; 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_2736_o();" function MakeCharacter_2736_o() N := 2736; order := 2; char_gens := [1711, 2053, 1217, 1009]; v := [2, 2, 2, 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_o_Hecke();" function MakeCharacter_2736_o_Hecke(Kf) N := 2736; order := 2; char_gens := [1711, 2053, 1217, 1009]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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 := 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 := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 3; raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [3, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1, 1, 1, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, -2, -1, 0, 1, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 1, 1, 1, -2, 0, 0, 0, -1, 0], [-2, -2, 0, 2, -2, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0], [3, -1, -1, 3, 1, 1, 2, 1, 0, 0, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, -1, 1, -1, -1, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, -2, 1, 1, 1, -2, 1], [9, 3, 0, -3, -5, -2, -2, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0, -1, 2, 4, -3, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 5, 0, 0, -1, 1, -1, 0, 0], [8, 1, -2, -4, 3, 1, -3, -2, 0, 0, 1, 0, 0, 0, 0, 2, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, -4, -2, -1, 2, -2, 0, -2, 2, -1], [9, 0, 3, 0, -9, -2, 0, 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, -3, -1, 0, -1, -1, 1, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, -1, -2, -3, -3, -1, -3, 2, 0], [16, 2, 1, -8, 7, -2, 2, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 0, -3, -2, 3, 3, -1, 0, -1, 1, -3], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 1, 1, 4, -2, -2, -4, -2, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, -3, 0, -1, -5, -3, 5, -2, -1, -2, -1, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 3, 0, 1, 2, 1, 2, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 5, 5, -2, 1, 2, -1, 2, -2, -1], [-23, -1, -2, 2, 1, 4, -2, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0], [-16, 6, -3, 4, 6, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9, 0, 2, -1, 10, -2, 4, 4, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -2, 2, -18, -1, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 0, -1, -8, 0, 7, 1, -1, 1, 6, 1], [-21, 2, 6, -13, 4, -1, -1, -2, 0, 0, -2, 0, 0, 0, 0, -2, 0, 2, 0, 0], [-43, 0, 2, 3, -2, -1, 3, -4, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, -3, 0, -3, 7, 3, 3, -6, -2, -6, -2, -2], [0, 0, 0, 0, 0, 0, 0, 0, -1, -2, 0, 9, -5, -8, 4, -3, -2, -3, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 4, 5, 0, -2, -2, 2, -2, -3, 1, -3, 10, -2], [0, 0, 0, 0, 0, 0, 0, 0, -6, 1, 0, -4, 9, 5, 1, -5, -5, -5, 2, -2], [53, -5, 1, -5, -3, -3, 0, -1, 0, 0, -3, 0, 0, 0, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -6, 0, 7, -2, -6, 4, -1, 0, -1, 1, 2], [4, -2, -1, -8, 5, 2, -2, -2, 0, 0, 2, 0, 0, 0, 0, 6, 0, -6, 0, 0], [-23, 6, 2, -3, -10, 2, 0, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -8, 4, 5, 6, -3, 1, -3, -1, 4], [0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, -6, -6, 0, -14, -6, 1, -6, -5, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 5, 0, -5, 0, -3, -3, -8, -2, -8, 0, 4], [-56, -11, -4, -10, 1, -5, 3, -2, 0, 0, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0], [-12, -8, -1, 10, -2, 2, -2, 4, 0, 0, -4, 0, 0, 0, 0, 4, 0, -4, 0, 0], [-13, 3, 1, -11, -9, -1, 0, 7, 0, 0, 1, 0, 0, 0, 0, 3, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, -1, 9, -6, 3, 4, 5, 4, -3, 4], [-7, -3, 0, 8, 1, 4, 2, 8, 0, 0, 8, 0, 0, 0, 0, 4, 0, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 6, 0, -12, -8, 6, 0, -1, 2, -1, 4, -11], [-73, 9, -3, -11, -17, 1, 0, -3, 0, 0, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, -5, 0, 8, -2, -4, -5, -1, 1, -1, -1, 6], [-39, -16, 4, -3, 20, 5, 1, 6, 0, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0], [-44, 5, 4, -6, 5, -1, -5, 10, 0, 0, -1, 0, 0, 0, 0, 3, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -2, 0, 2, -10, 13, -13, -4, -3, -4, -2, 6], [49, 6, 2, -3, -8, 2, 0, 0, 0, 0, -6, 0, 0, 0, 0, 4, 0, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 4, 0, -5, 11, 22, 8, -1, -2, -1, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, -8, -6, 0, 2, -1, -5, -8, -2, -1, -2, 1, -2], [-79, -1, -1, -29, 25, 1, 2, 3, 0, 0, -3, 0, 0, 0, 0, 5, 0, -5, 0, 0], [-62, 2, -2, -2, -32, -4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -11, -11, -17, 9, -11, -7, -11, 1, 2], [0, 0, 0, 0, 0, 0, 0, 0, -2, -12, 0, 14, -9, -1, -6, -2, 1, -2, -9, 4], [0, 0, 0, 0, 0, 0, 0, 0, -6, 5, 0, -4, -3, 8, 9, 3, 5, 3, -6, -4], [55, 7, 2, 2, 11, 0, -12, -10, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-34, -17, -4, 4, -19, 3, 11, 2, 0, 0, -11, 0, 0, 0, 0, 11, 0, -11, 0, 0], [-14, 4, 2, -14, 32, 2, -4, -14, 0, 0, 2, 0, 0, 0, 0, -2, 0, 2, 0, 0], [3, 11, 5, 19, -35, -7, 4, 1, 0, 0, 3, 0, 0, 0, 0, -3, 0, 3, 0, 0], [-77, 20, 4, -1, -28, -1, -9, 10, 0, 0, 8, 0, 0, 0, 0, 2, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 3, 0, 4, 21, -5, 15, -3, 5, -3, -4, 2], [0, 0, 0, 0, 0, 0, 0, 0, -5, -10, 0, 3, -13, -5, 7, -3, 5, -3, -12, 14], [0, 0, 0, 0, 0, 0, 0, 0, -5, -6, 0, 3, 28, -19, 3, 1, -1, 1, 6, -2], [124, 2, -8, -12, 15, 6, -6, -8, 0, 0, -6, 0, 0, 0, 0, 8, 0, -8, 0, 0], [138, 11, 12, 20, -43, -9, 3, 2, 0, 0, 5, 0, 0, 0, 0, -5, 0, 5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 9, 13, 0, 1, 5, 12, 15, 12, 11, 12, 2, -4], [0, 0, 0, 0, 0, 0, 0, 0, 9, 5, 0, -7, 11, 0, 9, 2, 1, 2, 3, -6], [131, -4, -9, -6, 21, 8, 4, -3, 0, 0, 13, 0, 0, 0, 0, 5, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 12, 0, -3, -28, 21, -12, 1, 0, 1, -5, -4], [0, 0, 0, 0, 0, 0, 0, 0, -2, -7, 0, -6, -11, -16, -4, -11, 7, -11, -2, 8], [0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 0, 7, 6, 14, -4, 11, 2, 11, 3, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, -10, -10, -9, 2, -8, -10, -8, 2, -5], [57, 3, 2, -4, 27, 2, 2, -14, 0, 0, -6, 0, 0, 0, 0, -2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, -6, 0, -2, -36, 6, 12, -3, -10, -3, 12, -3], [-165, 5, -4, 19, -35, 10, -10, -2, 0, 0, 3, 0, 0, 0, 0, -3, 0, 3, 0, 0], [154, 6, 8, -8, 9, -2, 4, 18, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [139, -14, 6, -41, -26, 4, -2, -10, 0, 0, 10, 0, 0, 0, 0, 4, 0, -4, 0, 0], [105, -7, 4, -16, 25, 4, -2, 20, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-67, 0, -12, 35, -9, -3, -4, 16, 0, 0, -16, 0, 0, 0, 0, 2, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 12, 12, 0, -4, 13, -11, 0, 13, 4, 13, 6, 5], [33, 8, 11, -70, 55, -4, -6, -9, 0, 0, 3, 0, 0, 0, 0, -3, 0, 3, 0, 0], [138, 0, 8, 4, 20, -11, 3, 20, 0, 0, 0, 0, 0, 0, 0, -7, 0, 7, 0, 0], [163, 4, 14, 1, 5, -5, 2, 16, 0, 0, -4, 0, 0, 0, 0, -4, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 8, 0, 5, 7, 17, -7, -5, -3, -5, 5, 4], [0, 0, 0, 0, 0, 0, 0, 0, -6, 1, 0, 5, 59, -7, 8, 7, 8, 7, -16, 5], [0, 0, 0, 0, 0, 0, 0, 0, 6, -4, 0, -10, 6, -14, 24, -9, -2, -9, -4, 2], [-75, -11, 8, 13, -33, 4, 8, 20, 0, 0, -1, 0, 0, 0, 0, -5, 0, 5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 12, 0, 13, 13, 16, 1, 23, -2, 23, -9, -6], [76, 8, -18, 32, -39, -4, 4, 12, 0, 0, 0, 0, 0, 0, 0, 10, 0, -10, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, -8, 0, -14, -37, -15, -9, -7, 9, -7, -2, 3], [0, 0, 0, 0, 0, 0, 0, 0, -3, -4, 0, 0, 16, -23, -7, 5, 1, 5, 1, -1], [-96, 4, -2, -32, -42, -6, 2, 10, 0, 0, 10, 0, 0, 0, 0, -7, 0, 7, 0, 0], [-163, -1, -6, -15, -53, -2, -14, 10, 0, 0, -7, 0, 0, 0, 0, -17, 0, 17, 0, 0], [-319, -3, 12, 22, -41, -6, 6, 2, 0, 0, -6, 0, 0, 0, 0, -10, 0, 10, 0, 0], [-80, 20, 2, 44, 68, -4, -2, -14, 0, 0, -6, 0, 0, 0, 0, -18, 0, 18, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -15, 0, 9, -51, -29, -5, -8, -2, -8, -6, 14], [0, 0, 0, 0, 0, 0, 0, 0, -2, 11, 0, 4, 49, 16, 23, 9, -13, 9, 8, -12], [0, 0, 0, 0, 0, 0, 0, 0, 6, -4, 0, 18, -37, -11, 20, 1, -10, 1, 2, 5], [131, 11, -8, 7, 27, 16, -8, -20, 0, 0, 21, 0, 0, 0, 0, 1, 0, -1, 0, 0], [72, 6, -9, 8, 0, -14, 0, -26, 0, 0, -16, 0, 0, 0, 0, -10, 0, 10, 0, 0], [-138, 0, -8, -32, 60, 2, -10, -20, 0, 0, 0, 0, 0, 0, 0, -3, 0, 3, 0, 0], [-93, 16, -8, 7, 56, -2, -6, -20, 0, 0, -8, 0, 0, 0, 0, -18, 0, 18, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -24, 18, 3, 27, 1, -1, 1, -1, 3], [21, -10, 10, -11, -16, 18, -4, 16, 0, 0, 14, 0, 0, 0, 0, 12, 0, -12, 0, 0], [-75, 9, 1, 29, -69, -3, -8, -9, 0, 0, -19, 0, 0, 0, 0, -11, 0, 11, 0, 0], [-102, 12, 2, -8, -77, 0, -2, -14, 0, 0, -18, 0, 0, 0, 0, 2, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 24, 0, 15, -13, 17, -7, 15, -1, 15, -6, -2], [0, 0, 0, 0, 0, 0, 0, 0, 6, 11, 0, -16, -45, 15, -27, -3, -3, -3, 8, -8], [0, 0, 0, 0, 0, 0, 0, 0, -20, 15, 0, -12, -47, -4, 21, -9, -9, -9, 10, -4], [0, 0, 0, 0, 0, 0, 0, 0, -8, -23, 0, 42, -30, -7, -2, 17, 6, 17, -30, 22], [0, 0, 0, 0, 0, 0, 0, 0, -23, 5, 0, 27, 56, -5, -1, 8, -8, 8, 0, 6], [-94, -18, 0, -42, 2, 17, 3, -12, 0, 0, 8, 0, 0, 0, 0, 1, 0, -1, 0, 0], [-40, -6, 7, 38, 27, 0, -18, -4, 0, 0, 12, 0, 0, 0, 0, 14, 0, -14, 0, 0], [-11, 10, 4, -19, 50, 5, -3, -26, 0, 0, 14, 0, 0, 0, 0, 6, 0, -6, 0, 0], [-73, 1, -1, -9, -43, -5, 14, -17, 0, 0, 13, 0, 0, 0, 0, -1, 0, 1, 0, 0], [-37, 6, 2, 27, -54, -12, 2, 22, 0, 0, 2, 0, 0, 0, 0, -16, 0, 16, 0, 0], [-1, 34, -12, -21, 42, -13, -1, 26, 0, 0, -2, 0, 0, 0, 0, 4, 0, -4, 0, 0], [37, -8, 10, 59, -122, -10, 0, -8, 0, 0, -2, 0, 0, 0, 0, -12, 0, 12, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 13, -9, -27, 1, 16, -4, 16, 4, -2], [0, 0, 0, 0, 0, 0, 0, 0, -1, 8, 0, 3, 34, -5, 27, -3, 10, -3, 13, 1], [-20, 9, 8, -62, -7, 7, -1, -18, 0, 0, 3, 0, 0, 0, 0, -7, 0, 7, 0, 0], [-132, 0, 7, 92, -64, 4, 10, 14, 0, 0, -22, 0, 0, 0, 0, -18, 0, 18, 0, 0], [-66, 35, -8, -14, 25, -23, 1, 18, 0, 0, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -17, -24, 0, 27, 29, 13, -27, 3, 9, 3, -13, 2], [0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 0, -18, -60, 2, -32, -1, -2, -1, -10, -2], [0, 0, 0, 0, 0, 0, 0, 0, 6, 25, 0, -24, 50, 15, -22, -5, -6, -5, 18, -10], [-214, 36, 5, 12, 28, -12, -8, -20, 0, 0, -4, 0, 0, 0, 0, -34, 0, 34, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 13, -4, 0, -3, 3, 40, -5, 17, -6, 17, -3, 4], [-24, -20, -6, -14, 107, 6, 4, -26, 0, 0, -10, 0, 0, 0, 0, 12, 0, -12, 0, 0], [-187, -8, 12, 41, -16, 16, 32, 32, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -16, 29, 0, -22, -26, 18, -18, -9, -3, -9, 10, 10], [0, 0, 0, 0, 0, 0, 0, 0, -30, -17, 0, 32, 63, -7, -7, 13, -13, 13, -8, 14], [0, 0, 0, 0, 0, 0, 0, 0, 10, 36, 0, -14, 62, 8, 2, 22, -4, 22, 10, -4], [206, 12, -12, 36, -40, -12, 8, 8, 0, 0, 20, 0, 0, 0, 0, 6, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, -36, 0, 42, 12, -41, -17, 21, -5, 21, 1, 5], [204, -8, 28, 18, -28, 7, -11, -28, 0, 0, -16, 0, 0, 0, 0, 7, 0, -7, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, -23, 28, -11, 33, 7, 11, 7, -2, 8], [0, 0, 0, 0, 0, 0, 0, 0, -10, -31, 0, -12, -73, -17, -13, -33, -17, -33, 28, -4], [-4, -22, -27, -58, 52, 20, -4, -4, 0, 0, 6, 0, 0, 0, 0, -10, 0, 10, 0, 0], [124, 0, 12, 46, -48, -5, -3, -8, 0, 0, 12, 0, 0, 0, 0, -25, 0, 25, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 18, -9, 0, -14, -67, -30, 32, 11, 1, 11, 14, -6], [0, 0, 0, 0, 0, 0, 0, 0, -9, 50, 0, -25, -18, -1, -35, 6, 19, 6, -2, -2], [0, 0, 0, 0, 0, 0, 0, 0, 12, 14, 0, -6, 48, 40, 32, 0, 0, 0, 2, -24], [81, -4, -36, 57, 26, 1, 7, -2, 0, 0, -36, 0, 0, 0, 0, -8, 0, 8, 0, 0], [-260, 16, 10, -34, -78, 2, -34, -8, 0, 0, 24, 0, 0, 0, 0, -26, 0, 26, 0, 0], [-81, 11, -10, 13, 101, 2, -18, -24, 0, 0, 7, 0, 0, 0, 0, -20, 0, 20, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 2, 0, 11, -67, -6, 4, -11, -14, -11, -10, -8], [165, -36, 5, 40, -55, 26, 16, -11, 0, 0, -19, 0, 0, 0, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, -7, 0, -33, 47, 4, 11, -10, -5, -10, 4, 12], [-321, -48, -16, 27, -54, -13, 9, -2, 0, 0, -24, 0, 0, 0, 0, 6, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 8, 0, 21, -15, 23, -13, 13, 9, 13, -34, 6], [0, 0, 0, 0, 0, 0, 0, 0, 21, 31, 0, 5, 71, -15, 18, 42, 8, 42, 23, -14], [0, 0, 0, 0, 0, 0, 0, 0, -6, 30, 0, -30, 46, 80, -38, 6, 2, 6, -2, -8], [0, 0, 0, 0, 0, 0, 0, 0, -9, -20, 0, 9, -35, -7, 33, 18, 4, 18, -13, -15], [185, -13, 8, -35, 35, 18, -22, 20, 0, 0, 9, 0, 0, 0, 0, 7, 0, -7, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -19, -18, 0, 1, 2, 2, 28, 19, -2, 19, 1, 18], [124, -10, 30, 54, 67, 4, 14, 38, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 26, 10, 0, -16, -83, -24, -6, -6, 10, -6, 2, -14], [0, 0, 0, 0, 0, 0, 0, 0, -39, 14, 0, -19, 48, 37, -2, -17, 8, -17, -29, -10], [529, 28, 2, -69, 21, -11, -20, -12, 0, 0, 12, 0, 0, 0, 0, -30, 0, 30, 0, 0], [149, -48, 6, -37, 74, 9, 17, 16, 0, 0, 10, 0, 0, 0, 0, -3, 0, 3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -36, 18, 0, -20, 87, 43, -46, -9, 0, -9, -24, 11], [144, -24, 13, 72, -6, 8, 6, -30, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [269, -48, -2, -37, 142, 3, 19, 32, 0, 0, 6, 0, 0, 0, 0, -4, 0, 4, 0, 0], [310, -42, 4, -102, 71, -8, 4, 0, 0, 0, -10, 0, 0, 0, 0, 20, 0, -20, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -46, 10, 0, 8, 5, 60, -4, 1, -7, 1, -9, -5], [186, 75, -4, -40, -19, -1, 11, -10, 0, 0, -23, 0, 0, 0, 0, 5, 0, -5, 0, 0], [-249, -6, -18, 43, 134, -4, 26, -18, 0, 0, 2, 0, 0, 0, 0, 12, 0, -12, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -20, 13, 0, -1, 112, 14, -34, 15, 3, 15, 4, -25], [0, 0, 0, 0, 0, 0, 0, 0, 16, -15, 0, 6, 77, -35, -31, -7, -23, -7, 6, 4], [0, 0, 0, 0, 0, 0, 0, 0, -38, -10, 0, 10, 94, 20, 6, 2, -34, 2, -7, -2], [-35, -55, 43, -77, 69, 27, 14, 5, 0, 0, 23, 0, 0, 0, 0, 15, 0, -15, 0, 0], [94, 28, -14, 18, 29, -2, -6, -4, 0, 0, 16, 0, 0, 0, 0, 14, 0, -14, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -26, -47, 0, 20, -11, 33, -85, -5, -11, -5, -40, 14], [0, 0, 0, 0, 0, 0, 0, 0, 24, 2, 0, 14, 12, -15, -16, 11, -5, 11, 41, -40], [0, 0, 0, 0, 0, 0, 0, 0, -23, -22, 0, 25, -72, -14, 24, -29, -16, -29, -9, 8], [-379, -4, 10, 17, 125, 15, -14, 26, 0, 0, 30, 0, 0, 0, 0, 8, 0, -8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -25, 26, 0, -37, 120, 54, -14, -3, 0, -3, 13, -30], [71, -23, 24, 72, -115, 4, 22, 14, 0, 0, 2, 0, 0, 0, 0, 12, 0, -12, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 20, 47, -74, -64, 14, -10, 14, 0, 12], [0, 0, 0, 0, 0, 0, 0, 0, 28, -9, 0, 8, 76, 0, 33, 11, 7, 11, -3, -26], [291, 16, 27, 56, -63, -6, -6, 19, 0, 0, 7, 0, 0, 0, 0, -29, 0, 29, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -24, 20, 0, -24, 36, -6, -38, -2, 18, -2, -15, -14], [-135, 31, 22, 31, -71, -12, -8, 12, 0, 0, -25, 0, 0, 0, 0, 6, 0, -6, 0, 0], [-71, -58, -36, -67, 14, 9, -19, -14, 0, 0, -26, 0, 0, 0, 0, 22, 0, -22, 0, 0], [-328, 104, -8, 46, -36, -14, -38, -12, 0, 0, 24, 0, 0, 0, 0, -12, 0, 12, 0, 0], [-199, 49, -29, 19, -27, -21, 14, 17, 0, 0, -21, 0, 0, 0, 0, -33, 0, 33, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 60, 0, 5, -31, 40, -14, 7, 8, 7, -2, -6], [0, 0, 0, 0, 0, 0, 0, 0, -19, -52, 0, 35, -17, -45, 11, -14, -4, -14, -55, 51], [0, 0, 0, 0, 0, 0, 0, 0, 9, -25, 0, 51, 69, -53, 47, -8, -7, -8, -5, 26], [0, 0, 0, 0, 0, 0, 0, 0, -11, -38, 0, 81, -40, -10, 50, 39, -2, 39, -43, 8], [9, -28, 15, -92, 25, 34, 6, -15, 0, 0, -3, 0, 0, 0, 0, 31, 0, -31, 0, 0], [404, 58, -4, 10, -23, -12, -10, 6, 0, 0, 12, 0, 0, 0, 0, -18, 0, 18, 0, 0], [-219, -26, 50, -19, 38, -46, -24, 10, 0, 0, -2, 0, 0, 0, 0, -30, 0, 30, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -13, -36, 0, -3, -95, -11, -9, -5, -5, -5, 0, -10], [726, 22, 17, 20, -4, 22, 14, 12, 0, 0, 2, 0, 0, 0, 0, -6, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 25, -8, 0, 1, -109, -43, 103, -11, -27, -11, 24, -20], [13, -10, 28, 9, 38, -27, 13, -6, 0, 0, -2, 0, 0, 0, 0, -44, 0, 44, 0, 0], [166, -6, -17, 42, -55, -4, -4, -18, 0, 0, -22, 0, 0, 0, 0, 8, 0, -8, 0, 0], [-233, 50, 10, -69, 50, -8, -42, -22, 0, 0, -22, 0, 0, 0, 0, -12, 0, 12, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 18, 0, -15, -11, 10, 8, 8, 26, 8, -36, 29], [0, 0, 0, 0, 0, 0, 0, 0, -9, 19, 0, -33, 63, -21, -31, 4, -6, 4, 26, -8], [0, 0, 0, 0, 0, 0, 0, 0, 43, 71, 0, -29, 5, -20, 35, 26, 27, 26, 43, -18], [483, -4, -16, 3, -188, -1, 27, 10, 0, 0, -4, 0, 0, 0, 0, 10, 0, -10, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -18, 4, 0, 10, -126, -19, 25, 0, -5, 0, 30, -12], [0, 0, 0, 0, 0, 0, 0, 0, 25, 4, 0, -47, 74, 25, 89, 7, 19, 7, -18, -12], [152, -14, -22, -88, -23, 30, -10, 8, 0, 0, -26, 0, 0, 0, 0, 26, 0, -26, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 34, 0, 8, 0, -1, 22, -38, -38, -38, 4, -8], [0, 0, 0, 0, 0, 0, 0, 0, -12, -44, 0, 4, 90, 8, -72, -34, -9, -34, -19, 11], [0, 0, 0, 0, 0, 0, 0, 0, -2, -3, 0, -1, 93, 53, -40, 3, -32, 3, -22, -13], [-669, -8, -16, -41, 100, 42, 10, 8, 0, 0, 16, 0, 0, 0, 0, -2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -16, -15, 0, 13, -26, 50, -70, 13, -5, 13, 4, 5], [-125, -5, -48, -13, 71, -12, 40, 16, 0, 0, -7, 0, 0, 0, 0, 9, 0, -9, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, -68, 0, 32, 136, -25, 19, -5, 19, -5, -33, 25], [0, 0, 0, 0, 0, 0, 0, 0, 25, 7, 0, 11, 17, 29, -1, -10, -20, -10, 30, -2], [0, 0, 0, 0, 0, 0, 0, 0, -6, -10, 0, -14, -113, 3, -2, -43, -18, -43, 24, -13], [397, -31, -6, 40, 85, -28, 32, 4, 0, 0, -24, 0, 0, 0, 0, 28, 0, -28, 0, 0], [382, -59, 4, 0, -67, 13, 25, -18, 0, 0, 15, 0, 0, 0, 0, 11, 0, -11, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -16, 24, 0, -56, -21, 8, 78, -28, -18, -28, 50, -12], [0, 0, 0, 0, 0, 0, 0, 0, -7, -14, 0, -57, -83, -35, -19, -50, 0, -50, -1, -11], [0, 0, 0, 0, 0, 0, 0, 0, 7, 4, 0, 22, 56, 95, -37, 31, 23, 31, -19, -1], [493, 24, -26, -17, 30, -18, -48, -4, 0, 0, 42, 0, 0, 0, 0, 8, 0, -8, 0, 0], [927, -11, 7, 41, -71, -19, 0, 5, 0, 0, -21, 0, 0, 0, 0, 21, 0, -21, 0, 0], [-52, -79, -26, -48, -61, -3, 33, -2, 0, 0, -7, 0, 0, 0, 0, 42, 0, -42, 0, 0], [-384, 82, 40, -36, -93, -30, 18, 16, 0, 0, 6, 0, 0, 0, 0, -12, 0, 12, 0, 0], [-445, -12, -11, -164, 81, 14, 24, 7, 0, 0, 47, 0, 0, 0, 0, 31, 0, -31, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 25, -31, 0, -5, -149, -87, 95, 14, -4, 14, 0, 16], [0, 0, 0, 0, 0, 0, 0, 0, -6, 24, 0, -31, 67, 72, 18, -2, 18, -2, 8, 31], [-352, 50, 16, -104, -30, 15, -55, 20, 0, 0, 36, 0, 0, 0, 0, -25, 0, 25, 0, 0], [-507, -44, -20, -79, 220, 21, -19, 10, 0, 0, 12, 0, 0, 0, 0, 32, 0, -32, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -38, 26, 0, 14, 162, 51, -94, -4, -6, -4, -16, 44], [0, 0, 0, 0, 0, 0, 0, 0, -30, 23, 0, -37, 131, 119, -8, -23, 0, -23, 8, -29], [-233, 49, 5, 33, 105, -27, 22, -3, 0, 0, 7, 0, 0, 0, 0, 1, 0, -1, 0, 0], [471, -48, -4, -5, 128, 14, 10, -20, 0, 0, -16, 0, 0, 0, 0, 2, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -21, -31, 0, 2, -24, 55, 9, 16, 26, 16, -20, 7], [0, 0, 0, 0, 0, 0, 0, 0, -35, -2, 0, -9, 87, 35, -51, -33, -9, -33, -32, -18], [-69, -85, -45, -95, 15, -25, 14, -1, 0, 0, -11, 0, 0, 0, 0, 47, 0, -47, 0, 0], [150, 4, -10, -128, 102, -8, -8, 12, 0, 0, 4, 0, 0, 0, 0, 8, 0, -8, 0, 0], [18, -10, 45, -56, -88, 18, 10, -12, 0, 0, -38, 0, 0, 0, 0, -18, 0, 18, 0, 0], [-1427, -12, -6, 27, 83, -3, 6, -24, 0, 0, -12, 0, 0, 0, 0, -10, 0, 10, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, 14, 0, 39, 162, 30, -34, 33, 8, 33, -39, 18], [-534, -48, -26, -18, 5, 38, 50, -12, 0, 0, -36, 0, 0, 0, 0, 40, 0, -40, 0, 0], [209, 16, -45, -78, 141, 4, -28, 7, 0, 0, -9, 0, 0, 0, 0, 13, 0, -13, 0, 0], [231, -79, -4, 115, 39, 6, 30, 28, 0, 0, -5, 0, 0, 0, 0, -11, 0, 11, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 34, 14, 0, -24, 69, 71, -50, 16, 15, 16, 19, -18], [0, 0, 0, 0, 0, 0, 0, 0, -24, -7, 0, 10, 156, 34, -67, 25, 31, 25, -29, 24], [-252, 58, -42, 4, -28, -10, -26, -34, 0, 0, 8, 0, 0, 0, 0, -13, 0, 13, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 12, -36, 0, 56, 30, -28, 22, -2, -37, -2, 9, 5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, -13, 44, 22, 2, 31, 1, 31, 40, -33], [-107, -31, 64, -123, 31, -4, -24, 8, 0, 0, 59, 0, 0, 0, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 34, 37, 0, -68, 121, 39, 77, 31, 33, 31, -6, -36], [-466, -80, 19, -42, 148, 14, 44, -30, 0, 0, 6, 0, 0, 0, 0, 32, 0, -32, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -17, -48, 0, 5, -58, 57, -71, 0, -3, 0, -22, 14], [0, 0, 0, 0, 0, 0, 0, 0, 16, -31, 0, 26, 64, -4, -35, -43, -9, -43, 25, 0], [-194, -26, 23, 222, -40, 0, 38, 42, 0, 0, 4, 0, 0, 0, 0, 16, 0, -16, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 41, 12, 0, -19, 21, 18, 41, 9, 40, 9, -17, -14], [264, -44, -10, -116, 153, -16, 16, 16, 0, 0, -52, 0, 0, 0, 0, 26, 0, -26, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -21, 64, 0, -1, 46, 27, 110, 19, -10, 19, -35, -44], [0, 0, 0, 0, 0, 0, 0, 0, -27, 24, 0, 37, -20, -35, 83, 32, -3, 32, -20, 4], [615, -15, 8, 17, -21, 32, -28, 12, 0, 0, 69, 0, 0, 0, 0, 0, 0, 0, 0, 0], [635, 20, 24, 63, 160, -6, 22, 60, 0, 0, 28, 0, 0, 0, 0, 14, 0, -14, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -17, -16, 0, 25, -2, -28, -70, -25, 8, -25, -21, -16], [605, -56, -16, 65, -6, 5, -21, -2, 0, 0, 36, 0, 0, 0, 0, -24, 0, 24, 0, 0], [-769, -49, -24, -20, 75, 6, -48, 12, 0, 0, -4, 0, 0, 0, 0, 34, 0, -34, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -29, -44, 0, 87, -50, 77, -52, 27, -4, 27, -29, 24], [0, 0, 0, 0, 0, 0, 0, 0, 42, 17, 0, -42, 7, 138, -41, 31, 7, 31, 2, -30], [-547, -49, 28, -52, 137, 6, 42, 34, 0, 0, 6, 0, 0, 0, 0, -10, 0, 10, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -76, -15, 0, -64, 145, 64, -6, -69, 5, -69, -26, -6], [883, 28, 16, -25, 218, -13, 5, 2, 0, 0, 28, 0, 0, 0, 0, 34, 0, -34, 0, 0], [967, -27, -1, 33, 13, -11, 28, -35, 0, 0, 3, 0, 0, 0, 0, -11, 0, 11, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 20, 14, 0, 28, -70, 9, -122, -10, -20, -10, 20, -10], [857, -46, 58, -107, 136, -6, 32, -28, 0, 0, -18, 0, 0, 0, 0, 16, 0, -16, 0, 0], [273, -31, 43, -59, 53, -11, 4, -5, 0, 0, -27, 0, 0, 0, 0, 33, 0, -33, 0, 0], [-62, -23, -2, 108, 153, 9, -23, -16, 0, 0, -17, 0, 0, 0, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 45, -20, 0, -41, -11, -33, -13, -3, -25, -3, 1, -32], [-997, -24, -4, 87, -76, -30, -22, -28, 0, 0, 16, 0, 0, 0, 0, 14, 0, -14, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 38, 26, 0, 46, 20, -22, 80, 10, -16, 10, 33, -2], [-574, 6, 17, -32, 173, -2, 52, 24, 0, 0, 32, 0, 0, 0, 0, 6, 0, -6, 0, 0], [259, -9, -20, 66, -105, 12, -6, -18, 0, 0, -10, 0, 0, 0, 0, -28, 0, 28, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -38, -6, 0, 42, -106, 16, -62, -22, -14, -22, 31, 10], [0, 0, 0, 0, 0, 0, 0, 0, 19, -21, 0, 35, -154, -45, -3, 0, 2, 0, 26, 40], [0, 0, 0, 0, 0, 0, 0, 0, 10, -74, 0, 10, -32, -57, 116, -24, -14, -24, -64, 26], [-1012, -54, -21, -218, 31, 20, -10, 36, 0, 0, 0, 0, 0, 0, 0, 16, 0, -16, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -6, 0, -1, 12, -115, -1, -33, -8, -33, 41, -5], [942, 34, -8, -72, -207, 26, -4, -26, 0, 0, 4, 0, 0, 0, 0, 26, 0, -26, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 39, 74, 0, -8, 53, 15, 75, 1, 21, 1, 56, -11], [-271, -108, 22, 71, -150, 2, 92, 44, 0, 0, -14, 0, 0, 0, 0, 24, 0, -24, 0, 0], [9, -125, 34, -267, 95, -2, 42, 22, 0, 0, 45, 0, 0, 0, 0, 27, 0, -27, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -16, 0, 7, -101, -48, 92, -7, -22, -7, 66, -34], [-655, 50, -42, 17, -50, -54, 0, 14, 0, 0, -30, 0, 0, 0, 0, -18, 0, 18, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2736_o_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2736_3_o_r();". // 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_2736_3_o_r(:prec:=20) chi := MakeCharacter_2736_o(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 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_2736_3_o_r();". // 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_2736_3_o_r( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2736_o(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![2944, -253056, 370656, -7072, -96156, 3716, 6433, -60, -142, 0, 1]>,<7,R![774016, 223616, -4274352, 2619680, -51568, -140592, 12441, 1976, -218, -8, 1]>],Snew); return Vf; end function;