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

A nonlift newform in S₁₃(K(16))⁻

(Filename: eigs/eig-minus-13-16-6.html)

FORMULA FOR THE EIGENFORM f
with 3¹³⁻³λ₃ = 725184

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=1¹² b_i Grit(φ_i) + Σ_i=0⁴⁵ c_1,i T(3)ⁱ g₁ + Σ_i=0⁴⁵ c_3,i T(3)ⁱ g₃ + Σ_i=0⁴⁵ c_4,i T(3)ⁱ g₄
where g₁, g₃, g₄ are the Borcherds products that helped span the space S₁₃(K(16))⁻ and where φ_i are jacobi forms in J_13,16^cusp given by the following table.

φ₁	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3)\|W₂
φ₂	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,6)\|W₃
φ₃	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4)\|W₃
φ₄	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3)\|W₃
φ₅	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3)\|W₃
φ₆	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,10)\|W₅
φ₇	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,5,9)\|W₅
φ₈	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,6,7)\|W₅
φ₉	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,5,6)\|W₅
φ₁₀	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,7,7)\|W₅
φ₁₁	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5)\|W₅
φ₁₂	TB(13;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,9)\|W₅

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

b₁	69689643979591602559560768005870775668258041105503209161007985658978752665319936915488313800976146548119260244032977258617177382927121805388264575027767565131081615969848396256737509761587/562556906315339720288886582913622949952357024022478812260058192439457923868356215841290512480250469127267469606645117104544506197262712770757668325021718694383478622711888460000
b₂	-832682586074023554305848060466129433992417324766177559332426860746930472639577857024952287748771604195973027346224735266719020740388303033708851401208739151462020615123695824591966239840939/41769850293913974231449828781336504033962509033669051810309320788629750847225449026215820551658597332699609618293399945012429585146756423228756873132862613057973287736357718155000
b₃	-2974155350669938274469628300008529496919840637015826219066721954337101814636613345272210617374018932561185639301362037174337850932209702461655021769701552333012373378744074979415900181378177/139232834313046580771499429271121680113208363445563506034364402628765836157418163420719401838861991108998698727644666483374765283822521410762522910442875376859910959121192393850000
b₄	-4817371660119828617510169964659070217626238724579977607717947684134570313002751995583448960168906964236845688499492763673074750980593978596770255459969083163064243034318404317177848678734759/104424625734784935578624571953341260084906272584172629525773301971574377118063622565539551379146493331749024045733499862531073962866891058071892182832156532644933219340894295387500
b₅	-16692716459707957371906861803055175692825008967591741284949467379483998045771555080118580710400095414576445229886688804205399493531575334637263963381908202319851956742797649439109948607907373/167079401175655896925799315125346016135850036134676207241237283154519003388901796104863282206634389330798438473173599780049718340587025692915027492531450452231893150945430872620000
b₆	4918589406156792096343722125734847518774754049197472461931255320469669305415281081257183819943833368996812095050063998319132099100228813305121827800414679226294308850308417180663972454291153/386757873091796057698609525753115778092245454015454183428790007302127322659494898390887227330172197524996385354568518009374348010618115029895896973452431602388641553114423316250000
b₇	7014013355195064966699142213175615381009950742765944916186774499070202188885659630506279630284452841035243737741411657121206356262719670866078648733203181910666193491198545042029462509703/11602736192753881730958285772593473342767363620463625502863700219063819679784846951726616819905165925749891560637055540281230440318543450896876909203572948071659246593432699487500
b₈	8288961496343441690990243688995946846213600984537915053073094213533409762379275804029955720837679704363828200020834929193717662219847915646409129788159499687086492527178704288553331647231713/464109447710155269238331430903738933710694544818545020114548008762552787191393878069064672796206637029995662425482221611249217612741738035875076368142917922866369863737307979500000
b₉	3369946194978153102805546810367339407956490062351166408146544308216609447940490080673573523222918766681102636027232864842273841284679158841704418915346452137610303757510049383968193671667907/193378936545898028849304762876557889046122727007727091714395003651063661329747449195443613665086098762498192677284259004687174005309057514947948486726215801194320776557211658125000
b₁₀	569997185246293797499402450560634888580433884002610273407570621262625134325745880566604622663221876701966782907920476157691386634334005623389735457483629613990645584651375486560877214843686/72517101204711760818489286078709208392296022627897659392898126369148872998655293448291355124407287035936822253981597126757690251990896568105480682522330925447870291208954371796875
b₁₁	88207238149912840281539694474694511883086756010843951672493019351308175830543580114328327473687089566160488621543533074205706087706527676791592059193858912215213959630628817598555769048701/38675787309179605769860952575311577809224545401545418342879000730212732265949489839088722733017219752499638535456851800937434801061811502989589697345243160238864155311442331625000
b₁₂	12121900477727473936422318072117996612662553819976185027846020598377211454281981552144731518781220160899487264249190724690002826979465176205453721959864755456032986579534542033781139264108777/2320547238550776346191657154518694668553472724092725100572740043812763935956969390345323363981033185149978312127411108056246088063708690179375381840714589614331849318686539897500000
c_1,0	0
c_1,1	-6280202095022716448118838913605702742689554832116568593602173653208793088000000000/232367842943031729258423711359982295041913265132018900634482738607872423852276344607291382929
c_1,2	883542819760157298073414187207562896648897555773570388734261294871200008264941568000000/3596657816555463644205781740815500075098451727497250902520710440742828162392548637227128757734747
c_1,3	-404698357922463499836930115930997333083127679962999959085934301772828608953718483630080000/672575011695871701466481185532498514043410473041985918771372852418908866367406595161473077696397689
c_1,4	-46823288319915805312497089685598761723712908902112720753977933105369579099732132305272000/105610952249764977916224318389400593114254537089237458319471770214539408768435746347669326249847571
c_1,5	1631015907870471374557468103293497531441823122152137821675129691501338509040567649989466137/496268940668798537004393884470581427837856271370785726472080939648903629552649526526912173834235188
c_1,6	-93882488805185318484897763071786022416395911810154940924951309992053438401667275150034159847607/40892560711108999449162056080375909653839356760952743861299469427069659075138320985817563123940979491200
c_1,7	-2110289132359526109860824943079434132061331189266211000783392588601095409207160366809046726057757/654280971377743991186592897286014554461429708175243901780791510833114545202213135773081009983055671859200
c_1,8	13240673002857249605571211144058880090631489448626957611314864111375622839717909963765486778835317273/4187398216817561543594194542630493148553150132321560971397065669331933089294164068947718463891556299898880000
c_1,9	823554037852511523948241951068047100944013200807208829851934757265673215484154961669831197089716080613/535986971752647877580056901456703123014803216937159804338824405674487435429653000825307963378119206387056640000
c_1,10	-37214426657169801010958322493796979528297796711378245961756539640184373897172816831771167570141950412957/21439478870105915103202276058268124920592128677486392173552976226979497417186120033012318535124768255482265600000
c_1,11	-1245325636659117405390193570335708753249245695032669433516008938077432748976879379246500333483446365850067/2744253295373557133209891335458319989835792470718258198214780957053375669399823364225576772495970336701729996800000
c_1,12	1055983135730499310547367260851175474231751787278402588323663959467912499410827601696751056516691772355573/1995820578453496096879920971242414538062394524158733235065295241493364123199871537618601289087978426692167270400000
c_1,13	259755685500182164410321128848693925564209942545560564536027374451396001958834529866773031949755478932390987/2810115374462522504406928727509319669591851490015496394971935700022656685465419124966990615035873624782571516723200000
c_1,14	-2251051871423479415746111928686800289375539785765951834157059940364253554260200691502587742214973051445737519/22480922995700180035255429820074557356734811920123971159775485600181253483723352999735924920286988998260572133785600000
c_1,15	-38813142712000847415481322927699981079240938894002311139595511755096002077385384420002389601152117481905350593/2877558143449623044512695016969543341662055925775868308451262156823200445916589183966198389796734591777353233124556800000
c_1,16	2846391293334139978829121068666682902646330244735937988609596000203943140828476329497658095197862549659472413/230204651475969843561015601357563467332964474062069464676100972545856035673327134717295871183738767342188258649964544000
c_1,17	64704265962172012108926346755509155230701170239270659174630577462731392058054691786119709146351138647674795549/46040930295193968712203120271512693466592894812413892935220194509171207134665426943459174236747753468437651729992908800000
c_1,18	-22066350102868668222701735051747854425224167981924569927255102732893523964374706790378398139427971374280916291/21666320138914808805742644833653032219573126970547714322456562121962921004548436208686670229057766338088306696467251200000
c_1,19	-75465010448649641519981915969328096485064893041399833082732263807073611133887226947922456944436865880022204707/736654884723103499395249924344203095465486316998622286963523112146739314154646831095346787787964055495002427679886540800000
c_1,20	1326802522074090662773783897563416521634920428348930709974159103435594348569831010220826346944211849644274675353/23572956311139311980647997579014499054895562143955913182832739588695658052948698595051097209214849775840077685756369305600000
c_1,21	963305080656909527335796098474952865389268093941054276350791633404566188674600289479577839820893918058819757101/188583650489114495845183980632115992439164497151647305462661916709565264423589588760408777673718798206720621486050954444800000
c_1,22	-3124966766392455732631194360689179474242459475821572453337298284329087996738574963801301030461497048587099420983/1508669203912915966761471845056927939513315977213178443701295333676522115388716710083270221389750385653764971888407635558400000
c_1,23	-4118801132729201060235273093572990466882474435840480567250402363527660326849794439804356402269024249495897527349/24138707262606655468183549520910847032213055635410855099220725338824353846219467361332323542236006170460239550214522168934400000
c_1,24	4785706185671665123605608594516915104141879176418709727295672303090320051426425998312896427121475351750573222879/96554829050426621872734198083643388128852222541643420396882901355297415384877869445329294168944024681840958200858088675737600000
c_1,25	2879881777006970377111092803722645767225921649038457854504222199732685091565929109259474038823175801600971756223/772438632403412974981873584669147105030817780333147363175063210842379323079022955562634353351552197454727665606864709405900800000
c_1,26	-906284397472247610371305207355275096008848999508959078830168311480324087702963525411064325041984016389090245861/1235901811845460759970997735470635368049308448533035781080101137347806916926436728900214965362483515927564264970983535049441280000
c_1,27	-1266207010278003450733548255218607625219488205777403768651550211157069899365204626579848452308177629306743047643/24718036236909215199419954709412707360986168970660715621602022746956138338528734578004299307249670318551285299419670700988825600000
c_1,28	2387402781634213291813302120723731927342177337319712427456692992081928119005003716049440921961754806399894953523/395488579790547443190719275350603317775778703530571449945632363951298213416459753248068788915994725096820564790714731215821209600000
c_1,29	67439181331340536929530367930372929801281638351497953382510844702256821284197228854158817666797453494919518473/166521507280230502396092326463411923274012085697082715766582047979493984596404106630765805859366200040766553596090413143503667200000
c_1,30	-531130364436884131631142620873280490249832690826578759761192024426541002544687335127856247744067851996894261089/25311269106595036364206033622438612337649837025956572796520471292883085658653424207876402490623662406196516146605742797812557414400000
c_1,31	-1148438862266416605832778227688540029615231995335554183058282172557135730318255609944149678605568222657900663629/809960611411041163654593075918035594804794784830610329488655081372258741076909574652044879699957196998288516691383769530001837260800000
c_1,32	0
c_1,33	0
c_1,34	0
c_1,35	0
c_1,36	0
c_1,37	0
c_1,38	0
c_1,39	0
c_1,40	0
c_1,41	0
c_1,42	0
c_1,43	0
c_1,44	0
c_1,45	0
c_3,0	0
c_3,1	0
c_3,2	0
c_3,3	0
c_3,4	0
c_3,5	0
c_3,6	0
c_3,7	0
c_3,8	0
c_3,9	0
c_3,10	0
c_3,11	0
c_3,12	0
c_3,13	0
c_3,14	0
c_3,15	0
c_3,16	0
c_3,17	0
c_3,18	0
c_3,19	0
c_3,20	0
c_3,21	0
c_3,22	0
c_3,23	0
c_3,24	0
c_3,25	0
c_3,26	0
c_3,27	0
c_3,28	0
c_3,29	0
c_3,30	0
c_3,31	0
c_3,32	0
c_3,33	0
c_3,34	0
c_3,35	0
c_3,36	0
c_3,37	0
c_3,38	0
c_3,39	0
c_3,40	0
c_3,41	0
c_3,42	0
c_3,43	0
c_3,44	0
c_3,45	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

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 -21

[8 -89 992] 15 3049

[4 -25 160] 15 -143

[12 -109 992] 23 -26801

[6 -13 32] 23 -26801

[6 -19 64] 23 -2487

[14 -42 128] 28 -107520

[14 -70 352] 28 -107520

[8 -42 224] 28 -107520

[8 -106 1408] 28 107520

[16 -113 800] 31 -260010

[8 -113 1600] 31 -260010

[14 -47 160] 31 -499854

[20 -101 512] 39 500551

[10 -91 832] 39 1061121

[10 -101 1024] 39 -1061121

[8 -27 96] 39 -1209679

[24 -73 224] 47 1320554

[12 -73 448] 47 -1890766

[4 9 32] 47 -1320554

[8 -73 672] 47 -1890766

[16 -55 192] 47 -1910906

[2 4 32] 48 -2883584

[26 76 224] 48 -2883584

[6 12 32] 48 -2883584

[14 36 96] 48 -2883584

[2 3 32] 55 -183579

[14 -29 64] 55 2561037

[14 -99 704] 55 -2561037

[8 -35 160] 55 2466867

[2 2 32] 60 12342272

[30 -210 1472] 60 12342272

[16 -50 160] 60 2536448

[16 -178 1984] 60 -2536448

[10 -30 96] 60 2536448

[10 -50 256] 60 2536448

[8 -50 320] 60 -12342272

[8 -114 1632] 60 12342272

[2 1 32] 63 -10763179

[16 -225 3168] 63 9121091

[4 33 288] 63 -9121091

[12 -159 2112] 63 -15228864

[8 -97 1184] 63 -1008581

[2 0 32] 64 0

[2 8 64] 64 22020096

[2 16 160] 64 0

[4 8 32] 64 22020096

[20 56 160] 64 22020096

[10 -24 64] 64 -22020096

[10 -64 416] 64 0

[10 -144 2080] 64 0

[36 -469 6112] 71 87097687

[18 -181 1824] 71 27402359

[18 -235 3072] 71 60704121

[12 -85 608] 71 27402359

[120 277 640] 71 -87097687

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[62 -372 2240] 496 -856845975552

[8 4 64] 496 1320910061568

[64 -196 608] 496 -2570537926656

[64 -452 3200] 496 1320910061568

[64 -708 7840] 496 2570537926656

[10 12 64] 496 1713691951104

[50 -252 1280] 496 -2570537926656

[50 -452 4096] 496 -1713691951104

[50 -652 8512] 496 2570537926656

[14 20 64] 496 1713691951104

[38 -196 1024] 496 -2570537926656

[38 -348 3200] 496 -1713691951104

[38 -500 6592] 496 2570537926656

[16 4 32] 496 856845975552

[32 -68 160] 496 -1713691951104

[32 -196 1216] 496 -1320910061568

[32 -324 3296] 496 1713691951104

[32 -452 6400] 496 1320910061568

[16 68 320] 496 -2662965706752

[56 -188 640] 496 2662965706752

[20 28 64] 496 1320910061568

[28 -92 320] 496 2662965706752

[28 -148 800] 496 -2570537926656

[28 -260 2432] 496 -1320910061568

[28 -316 3584] 496 -2662965706752

[28 -372 4960] 496 2570537926656

[28 -428 6560] 496 -856845975552

A nonlift newform in S13(K(16))−

A nonlift newform in S₁₃(K(16))⁻