Properties

Label 1.112.a
Level 1
Weight 112
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newforms 1
Sturm bound 9
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 112 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(9\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{112}(\Gamma_0(1))\).

Total New Old
Modular forms 10 10 0
Cusp forms 9 9 0
Eisenstein series 1 1 0

Trace form

\(9q \) \(\mathstrut +\mathstrut 7300500744476376q^{2} \) \(\mathstrut +\mathstrut 233675452617241539765416652q^{3} \) \(\mathstrut +\mathstrut 11020070169858572323722385321061952q^{4} \) \(\mathstrut +\mathstrut 814194067724935025770916308591172296830q^{5} \) \(\mathstrut -\mathstrut 21301527237484005705216688979545976184644832q^{6} \) \(\mathstrut +\mathstrut 78389901636433627730832560362319405491179083256q^{7} \) \(\mathstrut +\mathstrut 331917555337838634305814685721710111074446802593280q^{8} \) \(\mathstrut +\mathstrut 448549390735821931740487725597506918142357121619864013q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 7300500744476376q^{2} \) \(\mathstrut +\mathstrut 233675452617241539765416652q^{3} \) \(\mathstrut +\mathstrut 11020070169858572323722385321061952q^{4} \) \(\mathstrut +\mathstrut 814194067724935025770916308591172296830q^{5} \) \(\mathstrut -\mathstrut 21301527237484005705216688979545976184644832q^{6} \) \(\mathstrut +\mathstrut 78389901636433627730832560362319405491179083256q^{7} \) \(\mathstrut +\mathstrut 331917555337838634305814685721710111074446802593280q^{8} \) \(\mathstrut +\mathstrut 448549390735821931740487725597506918142357121619864013q^{9} \) \(\mathstrut +\mathstrut 70318383028880935426202467492101897354146229740372656080q^{10} \) \(\mathstrut +\mathstrut 946977307561400491686882997086862420104457212380759419908q^{11} \) \(\mathstrut +\mathstrut 621762406249451000840580799122879254067634743931860068057856q^{12} \) \(\mathstrut -\mathstrut 105889073807861007951354048376593318148850899094543430238180938q^{13} \) \(\mathstrut -\mathstrut 693200815745779812299859823081139405355406151517828024353685696q^{14} \) \(\mathstrut -\mathstrut 263011530334644917620915776775966612573140691671014743658639898840q^{15} \) \(\mathstrut +\mathstrut 27983078214052241229733802839129592879112204378169572191112640794624q^{16} \) \(\mathstrut -\mathstrut 222105261114216680375344353232960019775132216853502752977555776836734q^{17} \) \(\mathstrut +\mathstrut 13836898392348788400585588852374256049336540448485000158069800440043832q^{18} \) \(\mathstrut -\mathstrut 84745728104868082450386145180218866587906565903531918441867146236497220q^{19} \) \(\mathstrut +\mathstrut 7795554433199817270239288692578476214785961451581351341251140127219386240q^{20} \) \(\mathstrut +\mathstrut 53447991875461898796552150181251304409032073759002376573082504235219586208q^{21} \) \(\mathstrut +\mathstrut 1334480876839672633808741914071727189545841199132729059095736106759243606112q^{22} \) \(\mathstrut -\mathstrut 4280804965700905487867557627761776334077515300975222519458633688084497477528q^{23} \) \(\mathstrut -\mathstrut 129253468619699114441737651311892084919218132467191979241911494876891173877760q^{24} \) \(\mathstrut -\mathstrut 456274260699810946496432078768368872244545007142930133725494539638375760654225q^{25} \) \(\mathstrut +\mathstrut 9117596118629922830918453196718641926897783510547242670141134615123316525608208q^{26} \) \(\mathstrut +\mathstrut 36009290264725541742540148694316715091842126661227810228968619953124574644440120q^{27} \) \(\mathstrut +\mathstrut 389777110106676955938012042765885416118140162819674451900675730518240474989075968q^{28} \) \(\mathstrut -\mathstrut 1612630879112671269089242330901312038613162417445580611832712505926453350563056730q^{29} \) \(\mathstrut -\mathstrut 18167376599417385725382731299135244307087297190105602725235889269652594544837067840q^{30} \) \(\mathstrut +\mathstrut 67559679369166080772885534172593278837571532071620973699499831976537741546761534688q^{31} \) \(\mathstrut +\mathstrut 754093318949864663533121048615003094440304122496655835482200816722793265150805508096q^{32} \) \(\mathstrut -\mathstrut 3133890070812326529166672578753895355679540062945915391855288247367239239009432080976q^{33} \) \(\mathstrut -\mathstrut 24751022535879997806365605902203773919188021786642014954770492231640496515521010376656q^{34} \) \(\mathstrut +\mathstrut 57944164206332657023724654984518766320829384409054668126147883688641566073099799278480q^{35} \) \(\mathstrut +\mathstrut 328041564753120514775451575146434843390262564311310021231853773527241323710816435244864q^{36} \) \(\mathstrut +\mathstrut 3043820559995771428343491307335152699983691109501186564620625804804431981535340669576286q^{37} \) \(\mathstrut -\mathstrut 21382754992423793021134163907198673444659659051000850342253103089169961740533890647606880q^{38} \) \(\mathstrut +\mathstrut 3273459216524867162345405450876825490686869325852879816870160180171296350982671401366216q^{39} \) \(\mathstrut +\mathstrut 123460999717991511987771461057366625032200502337474727630106509260305540512417484610278400q^{40} \) \(\mathstrut +\mathstrut 104629960197719783697927100274525225542843896456497561309270816521996960904985588920195178q^{41} \) \(\mathstrut +\mathstrut 2027994299853353493651484516191980857294428743343203061454578749195409490693400034208902912q^{42} \) \(\mathstrut -\mathstrut 19157176231150752403174922289745163210049080257481323543299910438470109245483512805859143708q^{43} \) \(\mathstrut +\mathstrut 54900509500333144241574805152152293176467261284516409174290950395775810400803069025987309824q^{44} \) \(\mathstrut +\mathstrut 15479509256451142996000421697851689107817887106618468041729343908392688500639929264156928310q^{45} \) \(\mathstrut -\mathstrut 398788118873285310214184399746055204854113651152175668288210597675002274814165601481840895552q^{46} \) \(\mathstrut +\mathstrut 1456966582071114763048986859064421525928331803997603240440970059620025856397045514594291849296q^{47} \) \(\mathstrut -\mathstrut 7452969450778497532129812087184007504558526604092287637802268612794467934087599186261483798528q^{48} \) \(\mathstrut +\mathstrut 17298008315064903937184133864291881308696588708445105876848488522049133450344930890256623872337q^{49} \) \(\mathstrut +\mathstrut 20891593843500014301021428427705110541710363262225372510717283195064501890651418995923815351400q^{50} \) \(\mathstrut -\mathstrut 85813999871822828588324201494589306593218994449334105811559220660851252439965047034531679254312q^{51} \) \(\mathstrut -\mathstrut 228851992565399673815270028508808722169419264169266009042427250901076945058351740642305444481664q^{52} \) \(\mathstrut -\mathstrut 330560931516346297292158383237634604339355604111254380988674341402404197811701576314208409413298q^{53} \) \(\mathstrut +\mathstrut 4500007476747002196644470511723672131082224104684813911982619633059661906955824254260654375081280q^{54} \) \(\mathstrut +\mathstrut 5411788085074080207797655602098900123199579033664526066719620291468503584226809836679744568059960q^{55} \) \(\mathstrut -\mathstrut 42315784638223426280810136290704778028064917513845974684816911806737536885555338292253409336340480q^{56} \) \(\mathstrut +\mathstrut 9918498856896672609870803756688673833744336383910153405185505298972528406991914653827866717244240q^{57} \) \(\mathstrut +\mathstrut 32435179551725169564404757555461461839261719646775117357648454983301156661453863733221818065697680q^{58} \) \(\mathstrut +\mathstrut 322553637419248081900645358050759529139017004186411501272056030884685696264538328889393003620694740q^{59} \) \(\mathstrut -\mathstrut 132467389966865887348037310509899613119681515385419367203561785040397906770826770994985507300267520q^{60} \) \(\mathstrut -\mathstrut 1759437984456387651106179880992119801754779277040263539338974358093077280629950964970586556177810042q^{61} \) \(\mathstrut -\mathstrut 1712844297363572226123727147322557976426042183124057300794490386880408520919151071285053014838119168q^{62} \) \(\mathstrut +\mathstrut 12759876033417698231199447475315412125628460028323317128859307435477632649190559962838360311186507992q^{63} \) \(\mathstrut +\mathstrut 14281879241185987264094876187662890300953975181530786481128692077543427681889775277266607957835251712q^{64} \) \(\mathstrut -\mathstrut 18446575904127982541910334362372985348294257469022062289060943594932692744923327761688031698818070540q^{65} \) \(\mathstrut -\mathstrut 105227969740255769667487712999247235831860856852926275817422436766136475148421006520466387572795840384q^{66} \) \(\mathstrut +\mathstrut 137686280746299018306956215858337887331602769875959175009115726791437961931237555780269025002728818316q^{67} \) \(\mathstrut +\mathstrut 1157398196638034088725243365559060772268690235343345029534596768981482256873945115267239799529371269248q^{68} \) \(\mathstrut +\mathstrut 1095507431560680024171724297772352455505639098885833012944706047494456930144134797052286618395241366496q^{69} \) \(\mathstrut -\mathstrut 3640686416490340897700513163360856154742313115593190855720847062189976425143796505336061966373300603520q^{70} \) \(\mathstrut +\mathstrut 9881612323140206421591424247232489623122284569259126072100461213939705350610294394975802208678001259448q^{71} \) \(\mathstrut +\mathstrut 26396053480919173194166495118665950061506035792323405768129609103560026391739244345645460880788768488960q^{72} \) \(\mathstrut +\mathstrut 68010594140875249475223137869343627059525985931897139097425407128220372890956239808513785565566231240522q^{73} \) \(\mathstrut +\mathstrut 220837506033562453797347134849380708116539322544975559087316116631000736843141442638912881210212941372624q^{74} \) \(\mathstrut +\mathstrut 181912260264814567208746612550869983753545991059523226326816406967698797635966067835830629689080707385300q^{75} \) \(\mathstrut +\mathstrut 761174863647605635882918277985364633628165523124208915131122931858702438047845125322045601860264208695040q^{76} \) \(\mathstrut +\mathstrut 1384642526997449454787538522373154719967650987275787604770256389537561344719927104096300078005408626536672q^{77} \) \(\mathstrut +\mathstrut 10154718868595236540854386892056000355936883967938908970289471652675003137655678023153495552339085989663424q^{78} \) \(\mathstrut +\mathstrut 7526673880001773966576204583811316437192125118919660383683770302134291678384367816740764114717599435401520q^{79} \) \(\mathstrut +\mathstrut 32601244901804291407696890633217962008785625969393161121607747994402795735664787250217837405420588998778880q^{80} \) \(\mathstrut +\mathstrut 48874282116782895668905156724559809826804317530845116719941792255495466449412888893127682785590511826600049q^{81} \) \(\mathstrut +\mathstrut 113239522172902215034601434016259598176780723606704841332886416969022724124423260666831983947062879210014192q^{82} \) \(\mathstrut +\mathstrut 199958416124075286987530689011944805825753524406236269102187579359597825528337021515595830260563921442487932q^{83} \) \(\mathstrut +\mathstrut 880467563819420401586770374036347249339328631628161679582704747941404166343896577833615976614908144379930624q^{84} \) \(\mathstrut +\mathstrut 707266002569821004378084548147696321676161909242361424333261618590644246794997074911164911557278987628650780q^{85} \) \(\mathstrut +\mathstrut 749430950077361833338178064514980710590134374932604590973028338058871330476392166543416304708654721880933728q^{86} \) \(\mathstrut +\mathstrut 2802132957491922284048190008280399903628415414482410070524683686927111831188200973776028254889144423439332360q^{87} \) \(\mathstrut +\mathstrut 7195654272713601174999226937725203391076863795283476240434396643890296393716359696024184526976625256798607360q^{88} \) \(\mathstrut +\mathstrut 3700827869341881331782801900641952675754040149647113243871021736652252631453067855946419168721111605247226810q^{89} \) \(\mathstrut +\mathstrut 8997316939698708062605864942661951051594004217268128464090536667598399664816028357347925999912163977799180560q^{90} \) \(\mathstrut -\mathstrut 254313076672373460593401973466735362362295247774148430702156053605292605984743534435031357382528251918280752q^{91} \) \(\mathstrut -\mathstrut 16891756811302405862414651288981793929299127803233629387746389497252488943048833672780023478534318077838493184q^{92} \) \(\mathstrut -\mathstrut 55378555126167982657281578020507978450769506755610812937817924402648049087367995838404448550854542227572273536q^{93} \) \(\mathstrut -\mathstrut 71374169795251900964177147542915102589642305140612249128066111463360036313665875245965949313319551156299307136q^{94} \) \(\mathstrut -\mathstrut 167271491201189176415157992539596873672401782137242927001530501790921136440405267833568144254517552969527081400q^{95} \) \(\mathstrut -\mathstrut 1187167956876210917054814422873111467722801274363871607741943257050230342552379215783989919197651861627909046272q^{96} \) \(\mathstrut -\mathstrut 631372797244867221191218982738135503959422268139699621925372110836288083138321626431181260783380505262640568654q^{97} \) \(\mathstrut -\mathstrut 1898458058047689123148702915382067648076866113471255418148529297211872922066143741813213498341870918619707529832q^{98} \) \(\mathstrut -\mathstrut 3056373683854811033724902889292841037638123003448050519838010864270837225210323694704228642303518682358171259244q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{112}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.112.a.a \(9\) \(78.026\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(73\!\cdots\!76\) \(23\!\cdots\!52\) \(81\!\cdots\!30\) \(78\!\cdots\!56\) \(+\) \(q+(811166749386264+\beta _{1})q^{2}+\cdots\)