A nonlift newform <i>f</i> in S

A nonlift newform in S₁₄(K(16))⁺

(Filename: eigs/eig-plus-14-16-6.html)

FORMULA FOR THE EIGENFORM f
with 3¹⁴⁻³λ₃ = 632232 + 12096*α,
where throughout this wepage, α is taken to have minimal polynomial -3889 + x^2

Note that T(3) is the Hecke operator under the scalar-invariant slash and λ₃ is the corresponding eigenvalue.

The eigenform f has the formula
f = Σ_i=1¹⁴ b_i Grit(φ_i) + Σ_i=0⁵⁹ c_1,i T(3)ⁱ g₁ + Σ_i=0⁵⁹ c_2,i T(3)ⁱ g₂ + Σ_i=0⁵⁹ c_4,i T(3)ⁱ g₄
where g₁, g₂, g₄ are the Borcherds products that helped span the space S₁₄(K(16))⁺ and where φ_i are jacobi forms in J_14,16^cusp given by the following table.

φ₁	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,4)\|W₂
φ₂	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2)\|W₂
φ₃	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,6)\|W₃
φ₄	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4)\|W₃
φ₅	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,6)\|W₃
φ₆	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4)\|W₃
φ₇	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3)\|W₃
φ₈	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3)\|W₃
φ₉	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,10)\|W₅
φ₁₀	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,7,8)\|W₅
φ₁₁	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,4,4,5,8)\|W₅
φ₁₂	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5,5,8)\|W₅
φ₁₃	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,5,6,6)\|W₅
φ₁₄	TB(14;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,4,4,4,4,5,5)\|W₅

Here is a table of the coefficients b_i and c_j,i.

b₁	57049884050163982957100640131868575697062146932925896755445178597829516414495824934332195726514944947483118906048661362437714309144697477779166466126086329572638363104560257075115588228850028832929359928244780500794750764189676714374740453960411278646008885333142217550462336292653914268981216404719098167108703977893827415496592899754111995979697709031374694562102783002819009846058663815357459029/102731402189125922618134456144737533609719677567447802195981905368371900675816216146927020445015594749268408994686791116373008816963651536445426412010948527932921535305264390593466371296326359604350766777423165710826193797171512415337442625131989415694943182576642397470604486395156729787072167650130161852014645793109836303616011477797903195718990739514653330136503418880000 - (2744460908891614011539653006375663205646033086041475067416403563153730878153476495790497844347794269533267673558249645475593403664017249683320299983520401714033973829903274832612040459357771909567114774089512384847900576353308969070669097519847478478788293454822490217884951105176698477998079549323393978690026237735833476731235436979824712223076950596183461744695773793984055983669680031149891639*α)/308194206567377767854403368434212600829159032702343406587945716105115702027448648440781061335046784247805226984060373349119026450890954609336279236032845583798764605915793171780399113888979078813052300332269497132478581391514537246012327875395968247084829547729927192411813459185470189361216502950390485556043937379329508910848034433393709587156972218543959990409510256640000
b₂	-171366269857808715194348145922010184717144449511716185244762256820822625872610422697908158585722099798318526322835346960423720230936068736804619887479213549994139309559208754752441505086278754244769901457251876796823572391461695867935787091607272409756134879096948186349985740544315068409723178321920379358398879736202406417810635612412266153554549149645719162371237644538976406727215140153913758393/256828505472814806545336140361843834024299193918619505489954763420929751689540540367317551112538986873171022486716977790932522042409128841113566030027371319832303838263160976483665928240815899010876916943557914277065484492928781038343606562829973539237357956441605993676511215987891824467680419125325404630036614482774590759040028694494757989297476848786633325341258547200000 + (8243803410954839334316469495259255249126467563600805233515843876697137520260582583369266463390967367442124012939796908031057434380514372492355736241262473041721120685112249517721689240808051453809228348334841531322291182834145540427715357171819855564475808213092282884777692791693202972978730795763728457097554902089103070366676040907872402104829862581305468585829481743412424269901187652586752963*α)/770485516418444419636008421085531502072897581755858516469864290262789255068621621101952653337616960619513067460150933372797566127227386523340698090082113959496911514789482929450997784722447697032630750830673742831196453478786343115030819688489920617712073869324817981029533647963675473403041257375976213890109843448323772277120086083484273967892430546359899976023775641600000
b₃	22951573282063404902025894650108492956054437888354786281403404354911006938828346745267440224304536028593968166775038202544422593797349674215714327583746240841463125897629465641167811420801364805336649922756008097136557600408082618849275032246584305765126412428991454157016542961554946508142065097985668149373520819619869830777712716625545305046195064428221864294394437866885786120310492525497195119/25682850547281480654533614036184383402429919391861950548995476342092975168954054036731755111253898687317102248671697779093252204240912884111356603002737131983230383826316097648366592824081589901087691694355791427706548449292878103834360656282997353923735795644160599367651121598789182446768041912532540463003661448277459075904002869449475798929747684878663332534125854720000 - (1104116103282459768202167456067468041142406488676244877489631957361986365783875022640354403080781682489741312025076468720324135528364684824853265973543546528507506325208933600006611168343894959486475445261223715646504567875283442552838529315058534626738631618038371929605759672996789292501533231527757844788235964399308011676619365692300341542382090140535537320469621934414363155734408469861933429*α)/77048551641844441963600842108553150207289758175585851646986429026278925506862162110195265333761696061951306746015093337279756612722738652334069809008211395949691151478948292945099778472244769703263075083067374283119645347878634311503081968848992061771207386932481798102953364796367547340304125737597621389010984344832377227712008608348427396789243054635989997602377564160000
b₄	4235369880106262845420366058160019769045334638643253242319366460947241107352980417983101725216054470394223727628948098211030891684220069612926643521729891970204153772886333704102612478278299209875866513570524566245656444009232409105346561838655232678536707091711764725982051662160103305361569331224628998855247941853041358412647309127115812160088319803674088092207407076839285984885299295273690231/4717258263786394405934745435217539808609577031158317447774679328139526051440540537358893795944593636445998372205005714527332037513637060346983865857645595670389254172180507731332639498300700185914065821412288221415488490686446998663453998092795432353339227771376436618548165191614339633079844432914140085041488837438716973125225016837658820211586309467509591689941483520000 - (203748127873977762616155754428900359774042282618675115393994992967855967682313952057943859007217383456552179209833145311113226633357423192825666442081179869973214545956641680801668803278830154903264284674692712377129770614999582926971330155297197849856757848110716944692278483226889151623329607625139405468321432442655523749803035683394760165414518767142767650433311488551296357576175297322668621*α)/14151774791359183217804236305652619425828731093474952343324037984418578154321621612076681387833780909337995116615017143581996112540911181040951597572936787011167762516541523193997918494902100557742197464236864664246465472059340995990361994278386297060017683314129309855644495574843018899239533298742420255124466512316150919375675050512976460634758928402528775069824450560000
b₅	322767557831905427636937346769488033944183586214056962555405235568666919007469510539778767706751554205147145142642168837426204127811273588289138058718817189883627664781992600832574374603393652230890486391258352370900568889180773709812605919966134057625041338408138875893690955056156496106162426417675948304403193196873677762818663346001358134578581472921366495658138260079418656118654815057013920397/277374785910639991068963031590791340746243129432109065929151144494604131824703783596702955201542105823024704285654336014207123805801859148402651312429561025418888145324213854602359202500081170931747070299042547419230723252363083521411095087856371422376346592956934473170632113266923170425094852655351437000439543641396558019763230990054338628441274996689563991368559230976000 - (15527164688874022869340434291788598906124744812432135066696158853699077141481932790626877330851703004507782091843066169899326594045244638676031213188195961094603356096952536441535145898629642869606217023339279159504015231718530588458523770946589526134849574640702811519928345338760258091491345063586420226948214609076673047146093315861832502088235164327074558368740284364841408539858841276808662047*α)/832124357731919973206889094772374022238729388296327197787453433483812395474111350790108865604626317469074112856963008042621371417405577445207953937288683076256664435972641563807077607500243512795241210897127642257692169757089250564233285263569114267129039778870803419511896339800769511275284557966054311001318630924189674059289692970163015885323824990068691974105677692928000
b₆	10333316458431990498023354248176515698990135022368140251401978786561647841415771425920931159234368934053116266651570976103232447296310504587811074495439161374262063173565270145609184408728047885934342326457345074925877997207815841213390760540445448356759367097124398947843687242387703022380410306749760889639714235070100399683894053438056422299377303217811244729790649269593587139064615144051833173531/13868739295531999553448151579539567037312156471605453296457557224730206591235189179835147760077105291151235214282716800710356190290092957420132565621478051270944407266210692730117960125004058546587353514952127370961536162618154176070554754392818571118817329647846723658531605663346158521254742632767571850021977182069827900988161549502716931422063749834478199568427961548800000 - (497097996806374895961062932765040017606318327774182194525471469649338526085292323631711266176178077317179358078218360137628586222725592716760203836503352218894888030146455090991412263878244276850189898673150434689057891273803821800462887287532983997847859016155351527521459918314387335979574411686883204043320021677619320405470212717164652952547355795495599728347937816862290104241965213411066701721*α)/41606217886595998660344454738618701111936469414816359889372671674190619773705567539505443280231315873453705642848150402131068570870278872260397696864434153812833221798632078190353880375012175639762060544856382112884608487854462528211664263178455713356451988943540170975594816990038475563764227898302715550065931546209483702964484648508150794266191249503434598705283884646400000
b₇	6467578848901185137816572470123359840520811928749117959518407268840011390190825666914507284750896001366402835353477397119302658321868478425856542313562451847684200734782873362849555763167772565210074861846768638918749259491429790367234233297188884453757048328526320759908531271071659197694144715895727502945667804738528819836312898574143338988728435369078785207421800348160948329231796207117975107231/2971872706185428475738889624187050079424033529629739992098047976727901412407540538536103091445093990960978974489153600152219183633591348018599835490316725272345230128473719870739562883929441117125861467489741579491757749132461609157976018798461122382603713495967155069685344070717033968840301992735908253576137967586391693068891760607725056733299374964531042764663134617600000 - (311131523253854292234681529937894956979425635554940878558189177594425964135732938584595241613683066327965654233876528852485685940468441798991422855445395026499609955645786439331400665637692485600210631856853850335894430047586598180839906581846948069316876422452076237095294442784023054251609717958635017761782094529802611994184866125145852526361205398742441161854505966024835420201647130463514060421*α)/8915618118556285427216668872561150238272100588889219976294143930183704237222621615608309274335281972882936923467460800456657550900774044055799506470950175817035690385421159612218688651788323351377584402469224738475273247397384827473928056395383367147811140487901465209056032212151101906520905978207724760728413902759175079206675281823175170199898124893593128293989403852800000
b₈	30185919753139735213423131007420609246136944658784051245610826786615231971456277157483897611860186742098247633241481439088036903947585288512392958898849364427310778870162469279873043410430987363340896692343809321971607618304813241401475037655066864590068869878393822379472161389037533098277720235212751780739620949706106493938380550193825885867056390730770792161954998882021503347398873181235391701/60298866502313041536731093824085074075270245528719362158511118368392202570587779042761512000335240396309718322968333916131983436043882423565793763571643701178019162027003011870078087500017645854727623978052727699832765924426757287263281540838341613560075346294985755037093937666722428353281489707685095000095552965520990873861571954359638832269842390584687824210556354560000 - (1452134007644695826120465211886186583461760162215538601107231467436772602082477153710747084358876263402912428928054951676762666165157800621674238029703525237070313690429782613171286028426593122123338382898512455549948484186902612947434867543922985149687267306138275146571385916399111094437597221389272166233525734622165560126145070236823304413728354626065014769165461634672385687290305447945585591*α)/180896599506939124610193281472255222225810736586158086475533355105176607711763337128284536001005721188929154968905001748395950308131647270697381290714931103534057486081009035610234262500052937564182871934158183099498297773280271861789844622515024840680226038884957265111281813000167285059844469123055285000286658896562972621584715863078916496809527171754063472631669063680000
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c_1,38	20844981465927229683280533413452927132888981532740499153872064615672709889745506873507020112599253584562128959781356392922753979506878728557255978449202729538715505595179732542296695330629900077679555449459595886655287512523703391431376033726500851681606588973329690071924877943/82016541039869278755898164434354236076438834725634946350936691661473776785005668076276633786822043901106625349220373385413053794961614106027692870881879320107931833503380079940035868412275892526493839223006232978365708245377803833267791136267246950944730111722422846416261178419940478156800000 - (334258562965269273517545766866881964960096656423380150756034715187547185357518380427258114692374218317297782272555778391382922373335549243990747784533734567774439701318835718258841798257760848180248508389665563256161783893502876419543968718664813801466317316005075026758078831*α)/82016541039869278755898164434354236076438834725634946350936691661473776785005668076276633786822043901106625349220373385413053794961614106027692870881879320107931833503380079940035868412275892526493839223006232978365708245377803833267791136267246950944730111722422846416261178419940478156800000
c_1,39	-280421605967686606507180172227407485985999144443459777018166093638526704451017292186467908586557235739134925094172924156360299547340798451836566949652593551002752349481169906906973197486424475470404075045554081946569030837460312723071031237729721649667115523695165656237689559387/13122646566379084600943706309496677772230213556101591416149870665835804285600906892204261405891527024177060055875259741666088607193858256964430859341100691217269093360540812790405738945964142804239014275680997276538513319260448613322846581802759512151156817875587655426601788547190476505088000000 + (4496685362295319949755388437335586703299891824387206641023630195656236176465721431115316986457354478152141427884839187462329213554601035737830370262590698115198260731375871787856297231038513904832649623955447867828287699955293248100802950990220458491934289904369628845724711539*α)/13122646566379084600943706309496677772230213556101591416149870665835804285600906892204261405891527024177060055875259741666088607193858256964430859341100691217269093360540812790405738945964142804239014275680997276538513319260448613322846581802759512151156817875587655426601788547190476505088000000
c_1,40	-3463259700926539690512161789435242159302845032169365115598714284754792455755661966390245018241180230425087371651132364100502888897193268085681637453492741431086874989931800755272544656947860401629259115269139743878404289262531769183199450828516366600098930896752097599367600469/7498655180788048343396403605426673012702979174915195094942783237620459591771946795545292231938015442386891460500148423809193489825061861122531919623486109267010910491737607308803279397693795888136579586103427015164864753863113493327340903887291149800661038786050088815201022026965986574336000000 + (55534911973870248588848262584005178306087616802063989998211923835111381290424671925479172964695006850864430737473184794429157764324983822884845843412696461530551408000828961629857390424450258617254165208033623698305627919228205148422840340406264755171402393380385693268226493*α)/7498655180788048343396403605426673012702979174915195094942783237620459591771946795545292231938015442386891460500148423809193489825061861122531919623486109267010910491737607308803279397693795888136579586103427015164864753863113493327340903887291149800661038786050088815201022026965986574336000000
c_1,41	66450411537377087379593352792032981358685671961084780324420814000530098061017535555974179437820880893738277875951911748636058672235202805903067084000446649038370430735766444821523824134620991389948162521158744803838119670168445738076161083581291646764947716099087073314248527769/1679698760496522828920794407615574754845467335181003701267183445226982948556916082202145459954115459094663687152033246933259341720813856891447149995660888475810443950149224037171934585083410278942593827287167651396929704865337422505324362470753217555348072688075219894605028934040380992651264000000 - (1065561948579169685979652535583799544419348127491330178805545390552404188603565633002600743995651971073864331892542034501687100115290188875330974268884027454847499917031791994114588344117741931494206790786864470226170265442186323541610946166847200964883417410196842701861397793*α)/1679698760496522828920794407615574754845467335181003701267183445226982948556916082202145459954115459094663687152033246933259341720813856891447149995660888475810443950149224037171934585083410278942593827287167651396929704865337422505324362470753217555348072688075219894605028934040380992651264000000
c_1,42	0
c_1,43	0
c_1,44	0
c_1,45	0
c_1,46	0
c_1,47	0
c_1,48	0
c_1,49	0
c_1,50	0
c_1,51	0
c_1,52	0
c_1,53	0
c_1,54	0
c_1,55	0
c_1,56	0
c_1,57	0
c_1,58	0
c_1,59	0
c_2,0	0
c_2,1	0
c_2,2	0
c_2,3	0
c_2,4	0
c_2,5	0
c_2,6	0
c_2,7	0
c_2,8	0
c_2,9	0
c_2,10	0
c_2,11	0
c_2,12	0
c_2,13	0
c_2,14	0
c_2,15	0
c_2,16	0
c_2,17	0
c_2,18	0
c_2,19	0
c_2,20	0
c_2,21	0
c_2,22	0
c_2,23	0
c_2,24	0
c_2,25	0
c_2,26	0
c_2,27	0
c_2,28	0
c_2,29	0
c_2,30	0
c_2,31	0
c_2,32	0
c_2,33	0
c_2,34	0
c_2,35	0
c_2,36	0
c_2,37	0
c_2,38	0
c_2,39	0
c_2,40	0
c_2,41	0
c_2,42	0
c_2,43	0
c_2,44	0
c_2,45	0
c_2,46	0
c_2,47	0
c_2,48	0
c_2,49	0
c_2,50	0
c_2,51	0
c_2,52	0
c_2,53	0
c_2,54	0
c_2,55	0
c_2,56	0
c_2,57	0
c_2,58	0
c_2,59	0
c_4,0	0
c_4,1	0
c_4,2	0
c_4,3	0
c_4,4	0
c_4,5	0
c_4,6	0
c_4,7	0
c_4,8	0
c_4,9	0
c_4,10	0
c_4,11	0
c_4,12	0
c_4,13	0
c_4,14	0
c_4,15	0
c_4,16	0
c_4,17	0
c_4,18	0
c_4,19	0
c_4,20	0
c_4,21	0
c_4,22	0
c_4,23	0
c_4,24	0
c_4,25	0
c_4,26	0
c_4,27	0
c_4,28	0
c_4,29	0
c_4,30	0
c_4,31	0
c_4,32	0
c_4,33	0
c_4,34	0
c_4,35	0
c_4,36	0
c_4,37	0
c_4,38	0
c_4,39	0
c_4,40	0
c_4,41	0
c_4,42	0
c_4,43	0
c_4,44	0
c_4,45	0
c_4,46	0
c_4,47	0
c_4,48	0
c_4,49	0
c_4,50	0
c_4,51	0
c_4,52	0
c_4,53	0
c_4,54	0
c_4,55	0
c_4,56	0
c_4,57	0
c_4,58	0
c_4,59	0

COEFFICIENTS OF f (return to top)

Use the notation

[a b c ]

a	b
b	c

Here is a list of some Fourier coefficients of f.

2u det(2u) a(u, f)

[4 -53 704] 7 -81025/3425723482112 - (609*α)/3425723482112

[8 -89 992] 15 1562625/3425723482112 + (11745*α)/3425723482112

[4 -25 160] 15 -6279807/3425723482112 - (263967*α)/3425723482112

[12 -109 992] 23 50274987/3425723482112 + (2039755*α)/3425723482112

[6 -13 32] 23 50274987/3425723482112 + (2039755*α)/3425723482112

[6 -19 64] 23 58303275/3425723482112 + (438219*α)/3425723482112

[14 -42 128] 28 0

[14 -70 352] 28 -81025/418179136 - (609*α)/418179136

[8 -42 224] 28 0

[8 -106 1408] 28 81025/418179136 + (609*α)/418179136

[16 -113 800] 31 223128811/1712861741056 - (1466037*α)/1712861741056

[8 -113 1600] 31 223128811/1712861741056 - (1466037*α)/1712861741056

[14 -47 160] 31 -195050325/1712861741056 - (1466037*α)/1712861741056

[20 -101 512] 39 103254675/3425723482112 + (27871923*α)/3425723482112

[10 -91 832] 39 -2661402093/3425723482112 - (79397709*α)/3425723482112

[10 -101 1024] 39 -2661402093/3425723482112 - (79397709*α)/3425723482112

[8 -27 96] 39 1843168275/3425723482112 + (13853619*α)/3425723482112

[24 -73 224] 47 -2664100767/1712861741056 - (15905343*α)/1712861741056

[12 -73 448] 47 8217107361/1712861741056 + (67650241*α)/1712861741056

[4 9 32] 47 -2664100767/1712861741056 - (15905343*α)/1712861741056

[8 -73 672] 47 8217107361/1712861741056 + (67650241*α)/1712861741056

[16 -55 192] 47 -5155516575/1712861741056 - (38749887*α)/1712861741056

[2 4 32] 48 -120537/104544784 + (2481*α)/104544784

[26 76 224] 48 -34725/6534049 - (261*α)/6534049

[6 12 32] 48 -34725/6534049 - (261*α)/6534049

[14 36 96] 48 -990663/104544784 - (10833*α)/104544784

[2 3 32] 55 -9832517119/3425723482112 - (413576607*α)/3425723482112

[14 -29 64] 55 -14707647871/3425723482112 - (29908383*α)/3425723482112

[14 -99 704] 55 -14707647871/3425723482112 - (29908383*α)/3425723482112

[8 -35 160] 55 31418890625/3425723482112 + (236150625*α)/3425723482112

[2 2 32] 60 1562625/418179136 + (11745*α)/418179136

[30 -210 1472] 60 0

[16 -50 160] 60 0

[16 -178 1984] 60 6279807/418179136 + (263967*α)/418179136

[10 -30 96] 60 -6279807/418179136 - (263967*α)/418179136

[10 -50 256] 60 0

[8 -50 320] 60 -1562625/418179136 - (11745*α)/418179136

[8 -114 1632] 60 0

[2 1 32] 63 -20615468181/3425723482112 + (157627659*α)/3425723482112

[16 -225 3168] 63 -8678638101/3425723482112 - (974133237*α)/3425723482112

[4 33 288] 63 -8678638101/3425723482112 - (974133237*α)/3425723482112

[12 -159 2112] 63 -9984347037/428215435264 - (170638461*α)/428215435264

[8 -97 1184] 63 -27194987925/3425723482112 - (204402933*α)/3425723482112

[2 0 32] 64 676137/52272392 + (1695*α)/52272392

[2 8 64] 64 1046537/104544784 + (4479*α)/104544784

[2 16 160] 64 -46300/6534049 - (348*α)/6534049

[4 8 32] 64 1046537/104544784 + (4479*α)/104544784

[20 56 160] 64 -1046537/104544784 - (4479*α)/104544784

[10 -24 64] 64 -1046537/104544784 - (4479*α)/104544784

[10 -64 416] 64 -1416937/52272392 - (7263*α)/52272392

[10 -144 2080] 64 -46300/6534049 - (348*α)/6534049

[36 -469 6112] 71 49302471915/3425723482112 + (1814594827*α)/3425723482112

[18 -181 1824] 71 -17010206229/3425723482112 + (566451723*α)/3425723482112

[18 -235 3072] 71 -52237976085/3425723482112 + (816566027*α)/3425723482112

[12 -85 608] 71 -17010206229/3425723482112 + (566451723*α)/3425723482112

[120 277 640] 71 49302471915/3425723482112 + (1814594827*α)/3425723482112

[10 -43 192] 71 -53161509525/3425723482112 - (399572469*α)/3425723482112

[10 -53 288] 71 -52237976085/3425723482112 + (816566027*α)/3425723482112

[40 -441 4864] 79 -41006840289/1712861741056 - (1227198081*α)/1712861741056

[20 -121 736] 79 -18500062241/1712861741056 + (3483555327*α)/1712861741056

[190 441 1024] 79 -41006840289/1712861741056 - (1227198081*α)/1712861741056

[10 -71 512] 79 -18500062241/1712861741056 + (3483555327*α)/1712861741056

[10 -121 1472] 79 70915591775/1712861741056 + (533015679*α)/1712861741056

[44 -397 3584] 87 -70212215415/3425723482112 + (3385994217*α)/3425723482112

[22 -45 96] 87 239712637065/3425723482112 - (822233367*α)/3425723482112

[22 -243 2688] 87 -117555799671/3425723482112 - (1957218327*α)/3425723482112

[16 -115 832] 87 239712637065/3425723482112 - (822233367*α)/3425723482112

[12 -51 224] 87 -272167709175/3425723482112 - (2045666583*α)/3425723482112

[134 307 704] 87 -70212215415/3425723482112 + (3385994217*α)/3425723482112

[46 -138 416] 92 54289131/418179136 + (1238987*α)/418179136

[46 -230 1152] 92 62721/6534049 - (12512*α)/6534049

[4 6 32] 92 62721/6534049 - (12512*α)/6534049

[24 -218 1984] 92 -54289131/418179136 - (1238987*α)/418179136

[26 70 192] 92 -62721/6534049 + (12512*α)/6534049

[162 374 864] 92 54289131/418179136 + (1238987*α)/418179136

[6 26 128] 92 0

[26 -86 288] 92 50274987/418179136 + (2039755*α)/418179136

[12 -26 64] 92 -54289131/418179136 - (1238987*α)/418179136

[12 -38 128] 92 -50274987/418179136 - (2039755*α)/418179136

[12 -122 1248] 92 -62721/6534049 + (12512*α)/6534049

[12 -134 1504] 92 0

[48 -337 2368] 95 -850935124791/3425723482112 - (10374478423*α)/3425723482112

[24 -337 4736] 95 -1013361069879/3425723482112 - (25620255575*α)/3425723482112

[4 17 96] 95 355332514377/3425723482112 + (8446354601*α)/3425723482112

[16 -81 416] 95 355332514377/3425723482112 + (8446354601*α)/3425723482112

[6 49 416] 95 -1013361069879/3425723482112 - (25620255575*α)/3425723482112

[12 -145 1760] 95 580849415625/3425723482112 + (4365779625*α)/3425723482112

[60 145 352] 95 -1522628651703/3425723482112 - (24514473559*α)/3425723482112

[10 15 32] 95 -850935124791/3425723482112 - (10374478423*α)/3425723482112

[52 -261 1312] 103 406694408271/3425723482112 - (4280107281*α)/3425723482112

[4 5 32] 103 406694408271/3425723482112 - (4280107281*α)/3425723482112

[26 -261 2624] 103 818292606287/3425723482112 + (21246301167*α)/3425723482112

[14 -59 256] 103 -419973861425/3425723482112 - (3156606993*α)/3425723482112

[14 -101 736] 103 818292606287/3425723482112 + (21246301167*α)/3425723482112

[56 -169 512] 111 -1744499975961/3425723482112 - (4001953401*α)/3425723482112

[28 -169 1024] 111 2054640482919/3425723482112 + (25922023047*α)/3425723482112

[120 297 736] 111 -520719115545/3425723482112 + (17386753671*α)/3425723482112

[6 9 32] 111 -1744499975961/3425723482112 - (4001953401*α)/3425723482112

[14 -169 2048] 111 -844119920025/3425723482112 - (6344573049*α)/3425723482112

[14 -183 2400] 111 -850609809177/3425723482112 - (70190183289*α)/3425723482112

[12 -87 640] 111 2054640482919/3425723482112 + (25922023047*α)/3425723482112

[10 23 64] 111 -520719115545/3425723482112 + (17386753671*α)/3425723482112

[2 4 64] 112 -4732959/6534049 - (11865*α)/6534049

[58 172 512] 112 -4732959/6534049 - (11865*α)/6534049

[4 4 32] 112 -4732959/6534049 - (11865*α)/6534049

[28 -84 256] 112 -2592800/6534049 - (19488*α)/6534049

[28 -140 704] 112 -2592800/6534049 - (19488*α)/6534049

[28 -196 1376] 112 9918559/6534049 + (50841*α)/6534049

[14 -28 64] 112 9918559/6534049 + (50841*α)/6534049

[14 -84 512] 112 9918559/6534049 + (50841*α)/6534049

[8 12 32] 112 -4732959/6534049 - (11865*α)/6534049

[16 -84 448] 112 -2592800/6534049 - (19488*α)/6534049

[16 -148 1376] 112 9918559/6534049 + (50841*α)/6534049

[16 -212 2816] 112 -2592800/6534049 - (19488*α)/6534049

[2 3 64] 119 740698102221/1712861741056 + (16188550637*α)/1712861741056

[30 -61 128] 119 -781412995251/1712861741056 - (32722812563*α)/1712861741056

[30 -451 6784] 119 740698102221/1712861741056 + (16188550637*α)/1712861741056

[20 -61 192] 119 927631676685/1712861741056 + (7172050669*α)/1712861741056

[90 221 544] 119 826397372493/1712861741056 + (2409603693*α)/1712861741056

[16 -67 288] 119 1375481453325/1712861741056 + (10338391917*α)/1712861741056

[8 67 576] 119 -781412995251/1712861741056 - (32722812563*α)/1712861741056

[12 -61 320] 119 927631676685/1712861741056 + (7172050669*α)/1712861741056

[24 61 160] 119 -3960897805683/1712861741056 - (23170854035*α)/1712861741056

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[30 -395 5216] 455 -188335130632383/53526929408 - (1613311285971*α)/53526929408

[28 -427 6528] 455 89931569777511/53526929408 - (1035752291567*α)/53526929408

[18 43 128] 455 156933214359789/53526929408 + (2519415155117*α)/53526929408

[24 -341 4864] 455 195599607834687/53526929408 + (4249414310421*α)/53526929408

[20 85 384] 455 64265554629129/53526929408 + (2857704476249*α)/53526929408

[24 -299 3744] 455 64265554629129/53526929408 + (2857704476249*α)/53526929408

[232 -2553 28096] 463 -734435391671533/1712861741056 - (20549196745869*α)/1712861741056

[116 -697 4192] 463 -7327103705566381/1712861741056 - (139458739462029*α)/1712861741056

[1054 2457 5728] 463 1730184272909395/1712861741056 + (172216626219891*α)/1712861741056

[58 -697 8384] 463 3544614932494675/1712861741056 + (26642030162163*α)/1712861741056

[58 -871 13088] 463 1730184272909395/1712861741056 + (172216626219891*α)/1712861741056

[16 7 32] 463 -734435391671533/1712861741056 - (20549196745869*α)/1712861741056

[236 -2125 19136] 471 -10551058374765651/3425723482112 + (172772681865741*α)/3425723482112

[118 -237 480] 471 -9991246888407891/3425723482112 - (200626182808563*α)/3425723482112

[118 -1299 14304] 471 7765302304750509/3425723482112 - (102383864174835*α)/3425723482112

[80 -563 3968] 471 5712740417691693/3425723482112 + (114880539405837*α)/3425723482112

[60 -243 992] 471 11349377241194925/3425723482112 + (85304174512653*α)/3425723482112

[74 211 608] 471 4946526592915245/3425723482112 + (138068809292301*α)/3425723482112

[10 13 64] 471 7765302304750509/3425723482112 - (102383864174835*α)/3425723482112

[10 83 736] 471 179800709036973/3425723482112 + (166997581377549*α)/3425723482112

[40 -563 7936] 471 179800709036973/3425723482112 + (166997581377549*α)/3425723482112

[12 51 256] 471 -12955124948456403/3425723482112 - (33356049449715*α)/3425723482112

[30 -333 3712] 471 -12955124948456403/3425723482112 - (33356049449715*α)/3425723482112

[24 -51 128] 471 4946526592915245/3425723482112 + (138068809292301*α)/3425723482112

[50 -173 608] 471 3367070583961389/3425723482112 - (200499488231667*α)/3425723482112

[26 -371 5312] 471 5712740417691693/3425723482112 + (114880539405837*α)/3425723482112

[26 -397 6080] 471 -10551058374765651/3425723482112 + (172772681865741*α)/3425723482112

[238 -714 2144] 476 1058089777773/209089568 + (13263471277*α)/209089568

[238 -1190 5952] 476 9918489861/6534049 - (91408730*α)/6534049

[4 6 128] 476 20357446515/209089568 + (8267130963*α)/209089568

[120 -1082 9760] 476 23782985898/6534049 + (764240050*α)/6534049

[90 262 768] 476 0

[738 1718 4000] 476 927631676685/209089568 + (7172050669*α)/209089568

[6 26 192] 476 1581786003/6534049 + (74413234*α)/6534049

[90 -278 864] 476 877014524589/209089568 + (4790827181*α)/209089568

[60 -122 256] 476 -1058089777773/209089568 - (13263471277*α)/209089568

[60 -422 2976] 476 -23782985898/6534049 - (764240050*α)/6534049

[60 -602 6048] 476 -9918489861/6534049 + (91408730*α)/6534049

[60 -902 13568] 476 20357446515/209089568 + (8267130963*α)/209089568

[10 22 96] 476 877014524589/209089568 + (4790827181*α)/209089568

[210 518 1280] 476 -1581786003/6534049 - (74413234*α)/6534049

[10 42 224] 476 -1567250216595/209089568 - (10380625171*α)/209089568

[62 -198 640] 476 74801487159/6534049 + (399694652*α)/6534049

[40 -122 384] 476 1567250216595/209089568 + (10380625171*α)/209089568

[40 -442 4896] 476 74801487159/6534049 + (399694652*α)/6534049

[12 26 96] 476 1581786003/6534049 + (74413234*α)/6534049

[180 442 1088] 476 -877014524589/209089568 - (4790827181*α)/209089568

[34 -102 320] 476 -74801487159/6534049 - (399694652*α)/6534049

[34 -170 864] 476 -1567250216595/209089568 - (10380625171*α)/209089568

[30 -122 512] 476 0

[30 -182 1120] 476 -20357446515/209089568 - (8267130963*α)/209089568

[30 -362 4384] 476 -781412995251/209089568 - (32722812563*α)/209089568

[30 -422 5952] 476 -23782985898/6534049 - (764240050*α)/6534049

[16 6 32] 476 9918489861/6534049 - (91408730*α)/6534049

[32 -134 576] 476 781412995251/209089568 + (32722812563*α)/209089568

[32 -390 4768] 476 0

[16 134 1152] 476 -1058089777773/209089568 - (13263471277*α)/209089568

[18 26 64] 476 23782985898/6534049 + (764240050*α)/6534049

[30 -218 1600] 476 -9918489861/6534049 + (91408730*α)/6534049

[30 -278 2592] 476 -20357446515/209089568 - (8267130963*α)/209089568

[30 -458 7008] 476 1058089777773/209089568 + (13263471277*α)/209089568

[24 -122 640] 476 1567250216595/209089568 + (10380625171*α)/209089568

[24 -314 4128] 476 -74801487159/6534049 - (399694652*α)/6534049

[48 122 320] 476 -927631676685/209089568 - (7172050669*α)/209089568

[248 570 1312] 476 0

[24 -58 160] 476 -1581786003/6534049 - (74413234*α)/6534049

[24 -250 2624] 476 -877014524589/209089568 - (4790827181*α)/209089568

[240 -1681 11776] 479 -211290566443983/428215435264 - (10709508118431*α)/428215435264

[120 -1681 23552] 479 1220381901787953/428215435264 + (74316991624449*α)/428215435264

[4 17 192] 479 13349577379761/428215435264 - (15258517891103*α)/428215435264

[80 -401 2016] 479 301424690863281/428215435264 - (30163717280191*α)/428215435264

[6 49 480] 479 -3375392811587439/428215435264 + (16584388575905*α)/428215435264

[60 -721 8672] 479 374800230960225/428215435264 + (2817073010241*α)/428215435264

[8 17 96] 479 301424690863281/428215435264 - (30163717280191*α)/428215435264

[48 -145 448] 479 787035937542129/428215435264 - (17343082976863*α)/428215435264

[10 81 704] 479 -999462011080575/428215435264 - (3219101869663*α)/428215435264

[40 -401 4032] 479 4523061534303393/428215435264 + (45676343055841*α)/428215435264

[54 -175 576] 479 -1014067163832543/428215435264 - (15482245505663*α)/428215435264

[12 17 64] 479 13349577379761/428215435264 - (15258517891103*α)/428215435264

[42 -593 8384] 479 -999462011080575/428215435264 - (3219101869663*α)/428215435264

[36 -401 4480] 479 -1014067163832543/428215435264 - (15482245505663*α)/428215435264

[178 433 1056] 479 -2374111919613711/428215435264 - (50335779628767*α)/428215435264

[30 -271 2464] 479 4523061534303393/428215435264 + (45676343055841*α)/428215435264

[28 -145 768] 479 787035937542129/428215435264 - (17343082976863*α)/428215435264

[56 145 384] 479 -598721362040943/428215435264 - (22118706548415*α)/428215435264

[24 -145 896] 479 -598721362040943/428215435264 - (22118706548415*α)/428215435264

[20 81 352] 479 188372927309409/428215435264 - (13773894699071*α)/428215435264

[28 -401 5760] 479 -3375392811587439/428215435264 + (16584388575905*α)/428215435264

[168 401 960] 479 -2374111919613711/428215435264 - (50335779628767*α)/428215435264

[24 -367 5632] 479 -211290566443983/428215435264 - (10709508118431*α)/428215435264

[60 -209 736] 479 188372927309409/428215435264 - (13773894699071*α)/428215435264

[244 -1221 6112] 487 17000501986770839/3425723482112 + (246846879552951*α)/3425723482112

[4 5 128] 487 -3350201139002345/3425723482112 + (79900081122999*α)/3425723482112

[122 -1221 12224] 487 5887706515454871/3425723482112 + (89498549366967*α)/3425723482112

[62 -933 14048] 487 -3350201139002345/3425723482112 + (79900081122999*α)/3425723482112

[16 5 32] 487 17000501986770839/3425723482112 + (246846879552951*α)/3425723482112

[248 -745 2240] 495 -11287727131444203/3425723482112 - (376865380258059*α)/3425723482112

[124 -745 4480] 495 20466356932909077/3425723482112 - (19042546198539*α)/3425723482112

[504 1257 3136] 495 -42512708307448299/3425723482112 - (483323203800075*α)/3425723482112

[6 9 96] 495 -3208953925427427/428215435264 - (47551311558531*α)/428215435264

[62 -311 1568] 495 -39599191123221867/3425723482112 - (89919816240651*α)/3425723482112

[52 -681 8928] 495 27161900421225621/3425723482112 + (392135996225013*α)/3425723482112

[42 -87 192] 495 -1338197020245603/428215435264 - (24601593181059*α)/428215435264

[42 -297 2112] 495 -1338197020245603/428215435264 - (24601593181059*α)/428215435264

[36 -471 6176] 495 18894061885024917/3425723482112 + (288822965599989*α)/3425723482112

[56 151 416] 495 -1210804330564203/3425723482112 - (139858127396619*α)/3425723482112

[16 23 64] 495 -42512708307448299/3425723482112 - (483323203800075*α)/3425723482112

[28 -87 288] 495 6042879636132885/3425723482112 + (100073323356405*α)/3425723482112

[142 343 832] 495 -22825768419889131/3425723482112 + (374744745536757*α)/3425723482112

[18 9 32] 495 -11287727131444203/3425723482112 - (376865380258059*α)/3425723482112

[26 -265 2720] 495 -1210804330564203/3425723482112 - (139858127396619*α)/3425723482112

[26 -343 4544] 495 18894061885024917/3425723482112 + (288822965599989*α)/3425723482112

[28 -151 832] 495 27161900421225621/3425723482112 + (392135996225013*α)/3425723482112

[24 -105 480] 495 3871608855508125/428215435264 + (66168110368125*α)/428215435264

[2 4 256] 496 -22787169174/6534049 - (57124890*α)/6534049

[250 748 2240] 496 -22787169174/6534049 - (57124890*α)/6534049

[4 4 128] 496 14280243904/6534049 - (93826368*α)/6534049

[124 -372 1120] 496 -69950174553/13068098 - (1724855343*α)/13068098

[124 -620 3104] 496 12829198937/13068098 + (2100160815*α)/13068098

[124 -868 6080] 496 14280243904/6534049 - (93826368*α)/6534049

[62 -124 256] 496 47753610774/6534049 + (244777626*α)/6534049

[62 -372 2240] 496 47753610774/6534049 + (244777626*α)/6534049

[8 4 64] 496 14280243904/6534049 - (93826368*α)/6534049

[64 -196 608] 496 -69950174553/13068098 - (1724855343*α)/13068098

[64 -452 3200] 496 14280243904/6534049 - (93826368*α)/6534049

[64 -708 7840] 496 12829198937/13068098 + (2100160815*α)/13068098

[10 12 64] 496 12829198937/13068098 + (2100160815*α)/13068098

[50 -252 1280] 496 12829198937/13068098 + (2100160815*α)/13068098

[50 -452 4096] 496 12829198937/13068098 + (2100160815*α)/13068098

[50 -652 8512] 496 12829198937/13068098 + (2100160815*α)/13068098

[14 20 64] 496 -69950174553/13068098 - (1724855343*α)/13068098

[38 -196 1024] 496 -69950174553/13068098 - (1724855343*α)/13068098

[38 -348 3200] 496 -69950174553/13068098 - (1724855343*α)/13068098

[38 -500 6592] 496 -69950174553/13068098 - (1724855343*α)/13068098

[16 4 32] 496 -22787169174/6534049 - (57124890*α)/6534049

[32 -68 160] 496 12829198937/13068098 + (2100160815*α)/13068098

[32 -196 1216] 496 14280243904/6534049 - (93826368*α)/6534049

[32 -324 3296] 496 -69950174553/13068098 - (1724855343*α)/13068098

[32 -452 6400] 496 14280243904/6534049 - (93826368*α)/6534049

[16 68 320] 496 -12483220800/6534049 - (93826368*α)/6534049

[56 -188 640] 496 -12483220800/6534049 - (93826368*α)/6534049

[20 28 64] 496 14280243904/6534049 - (93826368*α)/6534049

[28 -92 320] 496 -12483220800/6534049 - (93826368*α)/6534049

[28 -148 800] 496 12829198937/13068098 + (2100160815*α)/13068098

[28 -260 2432] 496 14280243904/6534049 - (93826368*α)/6534049

[28 -316 3584] 496 -12483220800/6534049 - (93826368*α)/6534049

[28 -372 4960] 496 -69950174553/13068098 - (1724855343*α)/13068098

[28 -428 6560] 496 -22787169174/6534049 - (57124890*α)/6534049

A nonlift newform in S14(K(16))+

A nonlift newform in S₁₄(K(16))⁺