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

A nonlift newform in S₁₃(K(16))⁻

(Filename: eigs/eig-minus-13-16-5.html)

FORMULA FOR THE EIGENFORM f
with 3¹³⁻³λ₃ = 0

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	-42841669817732546105379356996067141904778362638570150213828487623924976076611809965583118958969727340335575366513601621759507852566374253253508824961096578712963362432005335833713432733893/19249061561020735460263141250733793847170035314917743784242848343821961819440170249939936681227578325883151493474080072398802580888700196008009380000000000000
b₂	46483673587976483395341812455080357069898432593006149414116417054351706881800090826053438370675951071995012484306931800867732345410249096807444630553614542231957339713681353735092726794311/129931165536889964356776203442453108468397738375694770543639226320798242281221149187094572598286153699711272580950040488691917420998726323054063315000000000000
b₃	1419159401456690749394548868156790515849509572474169324198816935245456523499848270487085982555748053036965404371508690858573508850556946870828241957427352760038247848186694745877121686369/3701742607888602973127527163602652662917314483638027650815932373811915734507725048065372438697611216515990671821938475461308188632442345386155650000000000000
b₄	20707047138314736779507429101538592520170501452295399549147463833502335078855372378014737561000029749049144473973809231575010964614070521222906043975563444326628117630000237259323263454207/24986762603248070068610808354317905474691872764556686643007543523230431207927144074441263961208875711482937034798084709363830273268985831356550637500000000000
b₅	71803614404047450781946883238334601706901177590919879740131476150558839459367402573785002582292271063633027930949849661025267554960436414043131635620852220062021731018529769330020178826229/39978820165196912109777293366908648759506996423290698628812069637168689932683430519106022337934201138372699255676935534982128437230377330170481020000000000000
b₆	-39295074907587866405192066479533677349164608573021837587300423053352881576525478968675707458127175762324040922583417952273302622732671334788959660684136324372460924869668998929087053659571/171866621080542280895206618310123159349732458168908426645025431641267516245001520088749434653817663623956709763161429217846451615077680321500083750000000000000
b₇	-1397640041478989476690802427048116680896491607960739704868483551081404870082469194712393455683134066942328260187559549335175641430171579233106465387751092567799070429238179975903270987/128441247070867896754424874893686346844995787243668219200908685568207040610143484763834097072248076017903590926206050305152152452549156112153087500000000000
b₈	-42148279570088993786142710701260058389092320863234543400954092256202400860550416828823242734294272044227392136469907623835403584515073255660224529614214566129432870503835272132533381368767/131243601552414105410885053982275867139795695328984616710746693253331557859819342613226841008369852221930578364596000493628199415150228609145518500000000000000
b₉	-188467202839796060964403070450512192619271811174748439583723963460302634490350812584460437117514884590458124947300660737577863039147854013056657563079611016291874720155299277457276684666943/601533173781897983133223164085431057724063603591179493257589010744436306857505320310623021288361822683848484171065002262462580652771881125250293125000000000000
b₁₀	-15933459060406209754178033931256617058545424785215403288544398380364283762412642666583158143023233906763827848514401783667584700602565509889657022161246971109325928812736773593526210445807/112787470084105871837479343266018323323261925673346154985797939514581807535782247558241816491567841753221590782074687924211733872394727710984429960937500000000
b₁₁	-1645910201735065009703629475796459682408182159200592756234020787047437997980793295739242203518793143643340294387456933377840247136022197923647018009356452450667168656989735100018476314283/40102211585459865542214877605695403848270906906078632883839267382962420457167021354041534752557454845589898944737666817497505376851458741683352875000000000000
b₁₂	-35677041571914370338196732353186351845378991430177851229675451918070984551360269614148033724043662815349503237889058654249275938394683340589834035970620517321468625573607093907234865650767/379915688704356620926246208896061720667829644373376522057424638364907141173161254933025066076860098537167463686988422481555314096487503868579132500000000000000
c_1,0	-4283424571392/281
c_1,1	323381857888792045537236896648265956557471404017911836961452537151488/2313271852808574933050569474311323070784104094675499345125
c_1,2	-28303269317293431471636146716327056397929088162252798647786184689828807585342707712/80715352210787299683703077791583405808949030873562498077996209489803125
c_1,3	-1705536113130861715926686299597467865997771186260164972219072742624157388266307594744/7667958460024793469951792390200423551850157932988437317409639901531296875
c_1,4	57479437260361937477085736868340493460107312006635909484219412133182790621142286176532727/30671833840099173879807169560801694207400631731953749269638559606125187500000
c_1,5	-70953025397838115309611523627258433875805497240226799826900420008399476661909317452281841623/49074934144158678207691471297282710731841010771125998831421695369800300000000000
c_1,6	-107647515356622064104196489919015788566094183192578152419946732765521385617206485923484963547/62815915704523108105845083260521869736756493787041278504219770073344384000000000
c_1,7	33938916039333628439022543416914611875825508429108455771037578605699026903511466980380969418527/17570885511754715554082540772173949576715103157214343637543991628907520000000000000
c_1,8	232013504660729488134694479930800187076067364895335237860811376865114083458739723613395804889529603/321617488407158313501926826293871973052193248189651345941605222775523246080000000000000
c_1,9	-169574702419917146459865060557941038478871179152784689157600378923649944282684337099264153232934051/161566085228085808980559787149197851454790956704377442231261650373889228800000000000000
c_1,10	-290191404114593740555115672641272463512406199532275850883725552623094157927809063606534647980075815777/1646681540644650565129865350624624502027229430731014891221018740610679019929600000000000000
c_1,11	33645735993801744128814453419824863846236000669139760365511320579356073635041792639371802794284705290269/105387618601257636168311382439975968129742683566784953038145199399083457275494400000000000000
c_1,12	46936991061530983955366155698704165247689315111587864570652288830016647588138394456407822243138352455341/1686201897620122178692982119039615490075882937068559248610323190385335316407910400000000000000
c_1,13	-262110552807436813774419321613819605404037251336069180506270776174045079777549876820299757529450835654509/4316676857907512777454034224741415654594260318895511676442427367386458410004250624000000000000
c_1,14	-5195427961117621712253101832765223711590215228495486733762357913190237614449835584728543218076792163368187/1726670743163005110981613689896566261837704127558204670576970946954583364001700249600000000000000
c_1,15	839103585475173926461844494521155713671826390775695811822269544779623810161473875812494952848265699919330279/110506927562432327102823276153380240757613064163725098916926140605093335296108815974400000000000000
c_1,16	49682180788208500482289683341662587024727612084642603956625259643438125811126467129382818589247849244247669/221013855124864654205646552306760481515226128327450197833852281210186670592217631948800000000000000
c_1,17	-1129762131636006206094252000978302441199147578128389834987482531744537536506267629838894158855592568468766979/1768110840998917233645172418454083852121809026619601582670818249681493364737741055590400000000000000
c_1,18	-155673738149547069732925065490535547196629107505280311794015852516523582095684033195853901293194167072212027/14144886727991337869161379347632670816974472212956812661366545997451946917901928444723200000000000000
c_1,19	79320202088246252695723538904403299470296515642432593906266150088364073461149380061879859294812102604529851/2176136419690975056794058361174257048765303417377971178671776307300299525831065914572800000000000000
c_1,20	24370334864071823170395114462404910563406308209677605615741475792217081613314000881735643190726687969989227/82297522781040511238757116204408266571487838329930546393405358530629509340520310951116800000000000000
c_1,21	-10159684412029825525024427601546840548188534247941549446867993201914345838429310802323229049716601592853388051/7242182004731564989010626225987927458290929773033888082619671550695396821965787363698278400000000000000
c_1,22	-64346761278258943285887964112633242371849001925848312237593437799210357019546709889957622219298630564656323/57937456037852519912085009807903419666327438184271104660957372405563174575726298909586227200000000000000
c_1,23	214054514587022380175716851809112872743602118329452322379130472088345950702827261539974855050818084121994009/5980640623262195603828130044686804610717671038376372094034309409606521246526585693892771840000000000000
c_1,24	-643223942391021833014515862002956336982276604651398478105587311497818630359113371937132367957247534563176449/3707997186422561274373440627705818858644956043793350698301271833956043172846483130213518540800000000000000
c_1,25	-1766040566839802691519049412682129749058585138091939771686503120319497574256239937995130968708292213699263/3019849077815381267941313755639473772692624284877003520962045675623368154613851678887116800000000000000
c_1,26	103227929005594403538853484637863369670189311841152225870642370474473889852650231354205571480313998836563709/21573801811913083778172745470288400632116107891161313153752854306653342096561356393969562419200000000000000
c_1,27	5465072349852000444568796262682565922560485060926698195712822534399363675377566779759551370329067742477907673/949247279724175686239600800692689627813108747211097778765125589492747052248699681334660746444800000000000000
c_1,28	-741393520443499718274520935657902179459878476526267502249013905338248006908863714649070645318611689558322797/15187956475586810979833612811083034045009739955377564460242009431883952835979194901354571943116800000000000000
c_1,29	-15093399407925715416132433344800631696762855076814485778210207273639144660663145881263872108151821268889121/495933272672222399341505724443527642286032325073553125232392144714578051787075751880965614469120000000000000
c_1,30	159398762954619011368548950126663570491227253625589350424446084665134914520635130289828154750146540201621461/972029214437555902709351219909314178880623357144164125455488603640572981502668473686692604359475200000000000000
c_1,31	2025465365549235636389379590279612045176776005882811610332067323733925450296747107485272801773488928849912987/31104934862001788886699239037098053724179947428613252014575635316498335408085391157974163339503206400000000000000
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 77

[8 -89 992] 15 15039

[4 -25 160] 15 14823

[12 -109 992] 23 101593

[6 -13 32] 23 101593

[6 -19 64] 23 247391

[14 -42 128] 28 197120

[14 -70 352] 28 197120

[8 -42 224] 28 197120

[8 -106 1408] 28 -197120

[16 -113 800] 31 -455926

[8 -113 1600] 31 -455926

[14 -47 160] 31 362398

[20 -101 512] 39 5953761

[10 -91 832] 39 615879

[10 -101 1024] 39 -615879

[8 -27 96] 39 4089591

[24 -73 224] 47 10797382

[12 -73 448] 47 5640814

[4 9 32] 47 -10797382

[8 -73 672] 47 5640814

[16 -55 192] 47 15652394

[2 4 32] 48 7667712

[26 76 224] 48 7667712

[6 12 32] 48 7667712

[14 36 96] 48 7667712

[2 3 32] 55 -16080845

[14 -29 64] 55 12199099

[14 -99 704] 55 -12199099

[8 -35 160] 55 -41395387

[2 2 32] 60 38831616

[30 -210 1472] 60 38831616

[16 -50 160] 60 37615104

[16 -178 1984] 60 -37615104

[10 -30 96] 60 37615104

[10 -50 256] 60 37615104

[8 -50 320] 60 -38831616

[8 -114 1632] 60 38831616

[2 1 32] 63 -29245293

[16 -225 3168] 63 -38862747

[4 33 288] 63 38862747

[12 -159 2112] 63 0

[8 -97 1184] 63 8588349

[2 0 32] 64 0

[2 8 64] 64 71368704

[2 16 160] 64 0

[4 8 32] 64 71368704

[20 56 160] 64 71368704

[10 -24 64] 64 -71368704

[10 -64 416] 64 0

[10 -144 2080] 64 0

[36 -469 6112] 71 106753777

[18 -181 1824] 71 -46812719

[18 -235 3072] 71 -52691425

[12 -85 608] 71 -46812719

[120 277 640] 71 -106753777

[10 -43 192] 71 -56591305

[10 -53 288] 71 52691425

[40 -441 4864] 79 162678186

[20 -121 736] 79 -36843246

[190 441 1024] 79 -162678186

[10 -71 512] 79 36843246

[10 -121 1472] 79 -65516814

[44 -397 3584] 87 -353650077

[22 -45 96] 87 429317955

[22 -243 2688] 87 925201629

[16 -115 832] 87 -429317955

[12 -51 224] 87 -678177819

[134 307 704] 87 -353650077

[46 -138 416] 92 454641152

[46 -230 1152] 92 454641152

[4 6 32] 92 -454641152

[24 -218 1984] 92 454641152

[26 70 192] 92 -454641152

[162 374 864] 92 454641152

[6 26 128] 92 -796117504

[26 -86 288] 92 796117504

[12 -26 64] 92 454641152

[12 -38 128] 92 796117504

[12 -122 1248] 92 -454641152

[12 -134 1504] 92 -796117504

[48 -337 2368] 95 1280455847

[24 -337 4736] 95 -97971049

[4 17 96] 95 -95591375

[16 -81 416] 95 95591375

[6 49 416] 95 97971049

[12 -145 1760] 95 -1739798137

[60 145 352] 95 842320369

[10 15 32] 95 1280455847

[52 -261 1312] 103 238993197

[4 5 32] 103 -238993197

[26 -261 2624] 103 -300894771

[14 -59 256] 103 -757572405

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[16 68 320] 496 3533441597440

[56 -188 640] 496 -3533441597440

[20 28 64] 496 3410852577280

[28 -92 320] 496 -3533441597440

[28 -148 800] 496 -4596871200768

[28 -260 2432] 496 -3410852577280

[28 -316 3584] 496 3533441597440

[28 -372 4960] 496 4596871200768

[28 -428 6560] 496 3910372687872

A nonlift newform in S13(K(16))−

A nonlift newform in S₁₃(K(16))⁻