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Critical values of symmetric-square L-functions of level-one cusp forms

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Numbers
$k$
$i$
$s$ 
$L(\operatorname{Sym}^2f_{k,i},s)$
12
1
1:
-10.64700013873556866381064613235751954963994629066090574279307417987239549998820307644429570053195948
12
1
3:
8.655979947619603661790489064417886505870990669471756291446066106928852104813208765439507345458284237
12
1
5:
-2.307166992698201131595024103437902625073770478776234456169903398615847238189935454873448990493583223
12
1
7:
0.5692706376295566502265366868953786254003980693454036898964265783424207024050486028285007589153168336
12
1
9:
-0.2293255506350864392112755646442107530414242744204102203374960709455464434346858264688577944321489639
12
1
11:
0.3520770499291561724325031811868759278582127323096649838101274135872131396453969062369046901847900792
12
1
12:
0.6317929457278832030111163264349193389373421402041367304133342436344486678433888516906717280431134615
12
1
14:
0.9017198031508855112530315939238999824903858062261610067477936654122167039152477460081836463027940653
12
1
16:
0.9759522932476373834191266099776808002275664535167743723371653000772411131391836423425590684503879336
12
1
18:
0.9941607924476083870681139401220098095255420608422467132555273413752955243678072598039914448726900788
12
1
20:
0.9985680684060846888738377399205788432625918756669900532532716276749354834192678158384739537503859545
12
1
22:
0.9996457111247713978357296220331531895497966586472549614937093413589918301344992671170017695706581921
16
1
1:
-11696840.57612226672550678334353275294453406548063023507713014530488964612223761750193610207238783932
16
1
3:
2435045.969539166769257264868034099200810702468811474491829318408675215775049134272808767564327594932
16
1
5:
-134759.9916304325991097006333693276592214580843246901294486320691707175880133193613577943953946409301
16
1
7:
5121.413055333427769663541870748754428644045531634887604569278706713176412707216354935211669574907230
16
1
9:
-199.0534422698161603075980543115920004178266155420222361706272098565016250760343190392420612521173933
16
1
11:
10.89369428035781000881714149980818253657919156292377611569518774158975454654349071749854467718693624
16
1
13:
-1.267028907566986371967155450870393281360618723321298155579762103073308899748660549925578540127846259
16
1
15:
0.7656167805569737256496502242816094099347482218462133341206866585918421840092314935838115976255818311
16
1
16:
1.007511299591063142930698025706096264548605248618220682762652569506569392618335014060411761755395872
16
1
18:
1.036176952183354156019341621057200580447853325190285672833078227553290640522570413273415123624105819
16
1
20:
1.011860201128456495753150688350726280182522483382255839736161137927652606170603450745125359086189578
16
1
22:
1.003185345139387906342167951636965487787479457555910468512166326340167917867848846004100733263892825
16
1
24:
1.000815287075349128268819003780595999195434295259201279189145598348180584559360770744054368393761060
16
1
26:
1.000205635384357022311209397050633029220491680646312961180045130818239576761497510411095605306545366
16
1
28:
1.000051595199374002894329213822464310028015585516530627514166501855463551208148029868868828412413718
16
1
30:
1.000012918700782179306169624867845596836978060981891819977030235502900390843187155459219035972498797
18
1
1:
44806971363.50051812176420531045229810702362130497263078161023971329909290332990302230050514991375634
18
1
3:
-5439147335.270878164952682130555783201098779457817865215387299173585069791466389364014127015521654141
18
1
5:
164785520.7231906908486216412022831829214016260851993159756250976425670088069338619535921469819654179
18
1
7:
-3154455.200398197780481068606596883219836681618769144082715693674567660319646844331794761779983649271
18
1
9:
54945.90697986000411127581670440202645261281158254084462324105105119961847702624018908521454814589969
18
1
11:
-1124.963805294978578024386642949960452541049059062628046069996987962631398243451587715492336218715330
18
1
13:
35.34455351318572085797418815052328126757153305172745471324130460578207640264219845571671754194366253
18
1
15:
-2.548559839989217078496137464378566499379934642980356013214064491192195361874725872927434968656974869
18
1
17:
1.067108738741698055784959395239965391791083080424785053524468325464379196544863481586793204660663357
18
1
18:
1.239051894626585611368065010879297635898091650119015727652846908994309633281999623507114609811402259
18
1
20:
1.109142946361634924214310616610071750673141146523450414670814144616373714517536233912877842270381357
18
1
22:
1.031792750056351384829483560083759347993834208765956188320597445431570336083455824553043911115830863
18
1
24:
1.008421343663506910344178318617202340773383056509797664596639741865559847278124721407496805822312331
18
1
26:
1.002158119291648204433080334373852125425024100614586828058856161964794621238130990140496182607930470
18
1
28:
1.000545466064543431237311369026720378349413982096288606482528879327045508785047654893277728747090285
18
1
30:
1.000137031791611812264535839656910613377631006298900207273584670244255357770347292922744569679143541
18
1
32:
1.000034332235924760559108732088263005294208401054815700642923205680398036196228807933781176810349982
18
1
34:
1.000008591335713395587779859542119067296948359459979151342406838623805853424541298851087389721621941
20
1
1:
-353211440318633.7401753401641918239650934524122599717661942301578790030062937301980068256044308789647
20
1
3:
26659796438041.75459922628505709848065883591354830481805447115386312894785602532500571607625274573046
20
1
5:
-478619760293.0545060966749543478341307282850975837298963710234347786502534801374288948648312264942054
20
1
7:
5107278958.990693368876231027600008471585358387257220452901662909816872426523306440592851638903896775
20
1
9:
-45701517.25549790936781456728884439848402854821452862788210989358653313164929881036407684428048459009
20
1
11:
427178.7685635022039908917381028339893478552933905392447795656184600202208489809320882112280058999165
20
1
13:
-5049.900176850194341739160780615300043650082436857757996461173510780762575710580698993329131482731799
20
1
15:
93.97020818708980476974496273872077704908626282782930580661737816555454702496440789170584371858605726
20
1
17:
-3.914556540204416433502472983967224049154277351625270479662673806382032455057985944144425217785996967
20
1
19:
0.9906442168593497753208954230319922331146411738913854730124721691729339499800877677604387262468745210
20
1
20:
1.029185949750395740997936645659692475901260439460765275939803131808004588275854356276453401182984820
20
1
22:
0.9735028065822470490118607569279356096479877241832688059142794868475015363138270684600690721721929586
20
1
24:
0.9866906942513397087089174408109246067077252227438959693635093881225387493032179916605273613669159308
20
1
26:
0.9958844555727935911801770057368885728106162619489375526784795892569084395318526037318176143877185654
20
1
28:
0.9988860766183455180029528363408436465660444593448200367294897779717298720987209367525054231136077266
20
1
30:
0.9997124040569169959528414423490868684830735045508481593696881145168113715326048082525980872815057134
20
1
32:
0.9999271146732426976208452166659279949064682391796852007587448442470541587380491853378304805477007834
20
1
34:
0.9999816709150071096024997204361935449200620947128615284724672395814124715308208893116156257045191560
20
1
36:
0.9999954058777174999130220664489152200956047218718670431803438515296640015605479268158363847882144877
20
1
38:
0.9999988501604831007613995010078249607752551131070115112932414111912232944506731834441576194529879826
22
1
1:
5301899611369328016.132963675182411611697611782058529462943473706572629796364710932763170127559644619
22
1
3:
-261729498891404466.2823671095140090659981505328376545116037396289562907935596648489772730456941771043
22
1
5:
2959229329215507.402923389251924928275549612832614510300891982983320687237690094163726470178112419480
22
1
7:
-18985042811258.22089535143546616128823549345287697393098452335397600164490429949424422035313253442052
22
1
9:
96304454823.49292016310898689720438350747910487476712312666518824194925709101681366922261882359728891
22
1
11:
-472165110.0216646632483521674769966736074536006913567543536372787178133031526242045739621655812940592
22
1
13:
2625889.325433166216397586593108172429398796237320665133947839943207387562497505485479081418356366442
22
1
15:
-19419.36082696673414574361060985163524775235211440688921804985544414011570517340277097310279967089631
22
1
17:
231.9309036009151037408092210242289514042122807514867843470864427931345696085463923845852248134526128
22
1
19:
-6.153540276665362039670064487952438494180711485756686039726264654659495619697476178045313716610879299
22
1
21:
1.001093541706435494248097254453347619278713513809610617979923480853918253824819204561305172048495001
22
1
22:
0.9409902119169494419299663653799137329189588578742307986803822836633800255985270482965708962129280388
22
1
24:
0.9254449606035450485760612921994299318594598265057639743413265107502927236611893978779561644713251058
22
1
26:
0.9741132586902896714532552545741497276252638603048133527821585572573993459268686466648393118437894639
22
1
28:
0.9928768105580782965727829974456231762271116948264109882754559066308446967602231138282628258354557749
22
1
30:
0.9981610520447368615428399572443091328227988014107317876090704746188336816062870629857145093848247849
22
1
32:
0.9995347912157085065306050988029328216528631360169286762989774258726495640259246888695749089022335797
22
1
34:
0.9998831538539650104651483264807361227426035399247050958512414411363264779517116337729775560632700872
22
1
36:
0.9999707320780518086108699729971818563961014168424646983463529841185994529650013279262137662478852711
22
1
38:
0.9999926770091193485096111153669466355816421631257496879531699086310581693631807825270714136040586878
22
1
40:
0.9999981686004098301144193896590221392012930902281467460152491850694691572976511920406854482163622397
22
1
42:
0.9999995420786704356713077680993716272335419125685875960232157729678159934028953917930355974163868558
24
1
1:
-142356548125531179238757.1982661107100335056451871260994091515604857541967476797623670521519288129771
24
1
3:
4787619648765159704360.175957856857638788308049811916899777609864693916510635381381386792060522903530
24
1
5:
-35769826679390089742.71423452357173598984685880005433215384975521703298644136306423170179682612908323
24
1
7:
146188029481033305.5974071969780340965172269975746956048565243478865010716659577077447590727370140864
24
1
9:
-451598759729740.5225173108876702292575686051694078464889321869757261206464959641482404264213159194071
24
1
11:
1273651670476.355842985933657625578518831209775022259830305185113971156871680819394618398090790070217
24
1
13:
-3780794856.057028206490451705931084393145635817907536433126750870480233131863031879093334740863895898
24
1
15:
13467367.44389404267735223330027625722716523013111278910987197830449041241321010152254168634701377763
24
1
17:
-66668.67467447111860261712027761306936436961336615744737230945585127093640998924254692040442460013891
24
1
19:
563.2940677877995418066108778782970436230305763644656575981472196515741572612202897944507003995249502
24
1
21:
-11.92948295039856756438285435532550612606302095370383636229000620672728333152874415857714847976832730
24
1
23:
1.835803719065742573194993621576915701190616345958745318491267754985517921640757281181736897255903087
24
1
24:
1.575535344802388295089157423240561097241279919861755716040181268927071874116716457889307599885475567
24
1
26:
1.136264698970243145104711099977565921613885836600994907094992240919587275453166836678794174215039477
24
1
28:
1.031265113956871718418988048383714680025569170796570574255884147494530416123264400570367482076561242
24
1
30:
1.007482405391612694235474953241303842759037548036849353268956095117235894445623223564538977312607989
24
1
32:
1.001833041553881108011313591625921440587647393613523625052678622452535308977305199597128356674220883
24
1
34:
1.000454056448863402498888004698687333461931660419760897639402655656083865063757981782296585409940785
24
1
36:
1.000113044736078798880256411047409244570828769686570749644639183069927215340574460468864115023810993
24
1
38:
1.000028208877358752095302613846810630092444272724623148040838623944535998524102442965408976425219831
24
1
40:
1.000007046397615009153597734540607147694422168182103214595657618399300764133229739137354860387269158
24
1
42:
1.000001760951925605178515184585392645891197588843704499114973546753735906259817167555591947732060528
24
1
44:
1.000000440166000784887488419823202396163053242204213437537803401849763168041080942795637352083768710
24
1
46:
1.000000110033499950683256553281172423462612157509254897849792352610549925643595392350312986405900587
24
2
1:
-142356568033026212099359.7876864302508484955733735323049797599124467894681328499757085876641405653313
24
2
3:
4787622326415866630813.080820220053293163760358521958792121812616468271653682646231864068242131658916
24
2
5:
-35769906674860669321.43224224168045913308625776432139055238793858520496057209135826384516924237080456
24
2
7:
146189336233998056.5072130636781052927647576559841940687617866592332683678981961985391913318383845996
24
2
9:
-451614879640156.6127049444852979419210046214353257158253790493379246193131440129716534505709716336428
24
2
11:
1273832840896.128483394226530814430670490319437615991906519140150610569938946271716052654065471782350
24
2
13:
-3782928147.569097299878295932956584187217190903461212982394328063767576700088384981395167061024617576
24
2
15:
13497206.15637656961568308335481650348905234433324490424744772223105322516524148902868622096498451818
24
2
17:
-67235.82019746269915955140374481605984923571807571775269924744243145518457057709430718876902616105294
24
2
19:
580.7542758811417259667892453346224407720382836802558290757918671438931239009675663787926802700014582
24
2
21:
-13.09734090674981796351300686335523625651025697141651656323251889428514512532791382393962417352642785
24
2
23:
2.195966712350937994325739605303527848082871917355232029742999298144673240423215974431765393285181855
24
2
24:
1.884636759031699759799793951629984342333206076517341712797328358867106880550010774217687526816524978
24
2
26:
1.247501353126249258595048293112459054531516043503645156311252054748243471316694688853926541684011850
24
2
28:
1.063230839354630323953834626402746679134777309916188640117117241934854308373885649313657934834324390
24
2
30:
1.016052984280432859315092401308875545945706776444417614264982023554320991345887410158165935434228831
24
2
32:
1.004052733576613811119083595337447805298232606939510521394780385190071375586113548511348846328455460
24
2
34:
1.001018556832537554819596928165143149821206126123612386880971942303678071973216573289553165758976495
24
2
36:
1.000255305688845548721329424397505235755216846697775395236456562667883836956514041368252993446971756
24
2
38:
1.000063905089523536136870675502757870077098173250223785868104920930998740525474741681409319719292124
24
2
40:
1.000015985311268137132016570530193186727181932964038060407924121107765289246768070306726620022306512
24
2
42:
1.000003997351116598425945010816510230740640113403258175475564069461629305488740678471458580478988335
24
2
44:
1.000000999452678085555221591012386625211572232830566445032094657887579500877644483393932703416193694
24
2
46:
1.000000249876011485855919810013688314184269438627920477557529502203426420213512891363884808213896580
26
1
1:
6494268833813833376482190646.413430859312976007523575448835348663437115353485550833527045227942408194
26
1
3:
-153887393245823079613744145.7507689587404242475908343780080435599504614411335084607845822512968504876
26
1
5:
789956517607407078267850.0642110178031320150911459559960781428350191927347022351453435427700140671511
26
1
7:
-2153335665530687944110.353216675357670764305088699663509652534175574134134071473390529868021356176371
26
1
9:
4281158004603019978.123590159846995504388619218319315007870003247390673721908882295439538343059650524
26
1
11:
-7435889081653446.134157278617479041685774903380968272774301906940726560149768293040823293053491371394
26
1
13:
12849274596767.28502877913320790055155321792997363012241804015893202215863478958436281711022318457809
26
1
15:
-24697177114.59177698819309568715711221136957828973072893486140454256980784067176912671587083039202119
26
1
17:
59030093.47471766924038078182769435580607698491510453378624536489661722804154124472260760684690897093
26
1
19:
-199484.0510090306810556803693140776746272731101876998687052999540722558027845925423484718321049990236
26
1
21:
1128.814958451685917340947972839568865648261961774102841112097230716040678713416826704631218620690625
26
1
23:
-14.02585373423369703180712087277930104329394851465638479143152974021309928911749890161876209374987307
26
1
25:
1.000007630764705449470618532695544934320062050869073721053690076116596528297385626096140035085046846
26
1
26:
0.7895743770974623360218388416245196788825393559149423950533432513047141092062694839199849055067613443
26
1
28:
0.8792088735894675484872086954314395368549710385249646745174160318604840087554592243700238162181605848
26
1
30:
0.9671313739667744291971653612942281061575472481895166474081613395530423526339480799901218791830169890
26
1
32:
0.9918872451052130722546741489898523124426580386056850120107141849986540368556349081620345876869009769
26
1
34:
0.9980066519270045420151765112034703842006925137186925365720263940072229394719556833894221116986474765
26
1
36:
0.9995068593394989476544751834938020526219695741682555376813093614992672252279710495427211774469519518
26
1
38:
0.9998773697903389254937609782777744111494430592917230011460280244075840849613498463806719548307838688
26
1
40:
0.9999694198713617841887344040441004138362112527913647683941030804338108337961704114496792728584957189
26
1
42:
0.9999923638537724522957763594014815490177001183126418800125941064346350104363334092405466921338817071
26
1
44:
0.9999980919681961339226479948673788259247672887190301243480023546460464618751129021260706215333720644
26
1
46:
0.9999995231047668505511353145179081205329414880418400906877169439273657323493710867019609468056502706
26
1
48:
0.9999998807887829174227015931851405349880958904802772877096402388453063702574769774286194179265634904
26
1
50:
0.9999999701985989238583154280409821341513732466110513072524888374295728140867401556005658111076571873
Definition
For a normalized level-one Hecke eigenform $f_{k,i}(z)=\sum_{n\geq1}a_n e^{2\pi inz}$ of even weight $k$, this table gives the uncompleted arithmetic symmetric-square value $L(\operatorname{Sym}^2 f_{k,i},s)$ at its critical integers [2] [1].
Parameters
$k$
—   weight ($k$ is even and $\dim S_k(\mathrm{SL}_2(\mathbb Z))>0$)
$i$
—   embedded eigenform ($1\leq i\leq\dim S_k(\mathrm{SL}_2(\mathbb Z))$; forms are ordered lexicographically by their real Fourier coefficients $(a_2,a_3,\ldots)$)
$s$
—   critical integer ($s$ is odd with $1\leq s\leq k-1$, or even with $k\leq s\leq2k-2$)
Formulas
(1)
For $\operatorname{Re}s>k$, $L(\operatorname{Sym}^2f,s)=\prod_p\bigl((1-\alpha_p^2p^{-s})(1-p^{k-1-s})(1-\beta_p^2p^{-s})\bigr)^{-1}$, where $\alpha_p,\beta_p$ are the unordered roots of $X^2-a_pX+p^{k-1}$.
(2)
Put $\Gamma_{\mathbb R}(s)=\pi^{-s/2}\Gamma(s/2)$ and $\Gamma_{\mathbb C}(s)=2(2\pi)^{-s}\Gamma(s)$. Then $\Lambda_2(f,s)=\Gamma_{\mathbb C}(s)\Gamma_{\mathbb R}(s-k+2)L(\operatorname{Sym}^2f,s)$ satisfies $\Lambda_2(f,s)=\Lambda_2(f,2k-1-s)$, with conductor and root number both $1$ [2] [1].
(3)
With $\langle f,f\rangle=\int_{\mathrm{SL}_2(\mathbb Z)\backslash\mathbb H}|f(z)|^2y^k\,dx\,dy/y^2$, the Petersson self-product satisfies $\langle f,f\rangle=(k-1)!L(\operatorname{Sym}^2f,k)/(2^{2k-1}\pi^{k+1})$.
(4)
$L(\operatorname{Ad}f,t)=L(\operatorname{Sym}^2f,t+k-1)$.
(5)
$L(f\otimes f,s)=\zeta(s-k+1)L(\operatorname{Sym}^2f,s)$ and $\operatorname{Res}_{s=k}L(f\otimes f,s)=L(\operatorname{Sym}^2f,k)$, where $\zeta$ is the Riemann zeta function.
(6)
For $\operatorname{Re}s>k$, $L(\operatorname{Sym}^2f,s)=\dfrac{\zeta(2s-2k+2)}{\zeta(s-k+1)}\sum_{n\geq1}\dfrac{a_n^2}{n^s}$.
Comments
(7)
The Fourier normalization is $a_1=1$, with $q=e^{2\pi iz}$. Every finite Euler factor is included; level one has no ramified finite factors. The values retain their signs. The arithmetic reflection is $s\mapsto2k-1-s$, so the centre is $k-1/2$. In the analytic normalization of [1], replace the argument $s$ by $s-k+1$; that paper writes the weight as $2k$. The completion in (2) is $\pi^{-k}$ times the completion in that paper after this argument shift.
(8)
The index $i$ counts all real embeddings of all normalized eigenforms of weight $k$, in lexicographic order of $(a_2,a_3,\ldots)$. In the selected weights the $a_2$ values are distinct, so this agrees with the increasing-$a_2$ convention of the ordinary critical-value table, the normalized Hecke-eigenvalue table and the Petersson table. The roots of the Hecke polynomial $\chi_{k,2}$ give these $a_2$ values.
(9)
An integer is critical when the gamma factors in (2) are regular both at $s$ and at $2k-1-s$. For even $k$ this gives every odd $s$ from $1$ to $k-1$ and every even $s$ from $k$ to $2k-2$. These sets are exchanged by the reflection.
(10)
Write $\Delta$ for the modular discriminant, and $E_4$ and $E_6$ for the constant-one weight-four and weight-six Eisenstein series. The selected eigenforms are $\Delta$, $\Delta E_4$, $\Delta E_6$, $\Delta E_4^2$, $\Delta E_4E_6$, $\Delta E_4^3+(a_2-696)\Delta^2$, and $\Delta E_4^2E_6$ at weights $12,16,18,20,22,24,26$, respectively. At weight $24$, $a_2$ runs through both roots of $X^2-1080X-20468736$; $a_2$ lies in $(-4017,-4016)$ for $i=1$ and in $(5096,5097)$ for $i=2$.
(11)
For these trivial-character forms, the automorphically normalized adjoint function is the argument shift in (4). It supplies the same values in a different argument convention. The tensor square has one additional zeta factor, as in (5); its residue at $s=k$ equals the symmetric-square value at $k$.
Programs
(P1)
Sage
from sage.modular.modform.constructor import Newforms
from sage.rings.real_mpfr import RealField
k = 24; i = 1
f = Newforms(1, k, names='a')[0]
embeddings = sorted(f.base_ring().embeddings(RealField(750)),
                    key=lambda phi: phi(f[2]))
L = f.symsquare_lseries(embedding=embeddings[i-1], prec=750)
critical = list(range(1, k, 2)) + list(range(k, 2*k-1, 2))
[(s, L(s)) for s in critical]
References
[1]
O. Balkanova and D. Frolenkov, The mean value of symmetric square L-functions, Algebra & Number Theory 12 (2018), 35–59, section 2, equations (2-3)–(2-5). (doi)
Links
Similar tables
Petersson norms of level-one cusp forms —   the same eigenforms and embeddings, related to the $s=k$ values by (3)
Ordinary critical L-values of level-one cusp forms —   degree-two values of the same eigenforms, with the same ordering and Fourier normalization
Normalized Hecke eigenvalues —   stores $a_p/p^{(k-1)/2}$ for the same eigenforms; multiplying by $p^{(k-1)/2}$ recovers the coefficients used in (1)
Hecke polynomials of level-one cusp forms —   the roots of $\chi_{k,2}$ identify and order the embedded forms
The modular discriminant —   its Fourier coefficients define the weight-twelve form $\Delta$
Critical L-values of higher-level newforms —   ordinary degree-two critical values with additional level, character and coefficient-field coverage
Petersson norms of higher-level newforms —   Petersson self-products beyond level one, where ramified factors must be specified separately
Critical values of Rankin-Selberg L-functions of level-one cusp forms —   The degree-four tensor construction pairs each higher-weight eigenform in the same roster with $\Delta=f_{12,1}$. For equal factors the tensor construction specializes to $L(f\otimes f,s)=\zeta(s-k+1)L(\operatorname{Sym}^2f,s)$; the linked table instead records the critical integers for distinct weights $k>12$.
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds all $162$ critical values of all eight embedded eigenforms in every nonempty even-weight level-one cusp space with $k\leq26$. This noncanonical cost bound includes the minimal cusp form, all intervening spaces, both embeddings of the first quadratic coefficient field at weight $24$, and the next one-dimensional space. Every critical integer is included for each form; no values are selected by their decimal size)
How they were obtained:

Values are recorded to $100$ significant digits after fresh PARI/GP evaluations at $160$ and $220$ working decimal digits, with explicit bit precision and freshly refined quadratic embeddings [3]. Exact Fourier coefficients come from the divisor-sum expansions of $E_4$ and $E_6$ and the identity $\Delta=(E_4^3-E_6^2)/1728$. Reciprocal cubic Euler factors and multiplicativity give the symmetric-square coefficients. The two working precisions agree to at least $120$ relative decimal digits throughout.

more

An independent calculation uses Sage modular-symbol eigenforms and its Dokchitser symmetric-square evaluator at $700$ bits [2]. Its coefficients use the distinct Dirichlet convolution in (6). Every value agrees with the higher-precision PARI calculation to at least $171$ relative decimal digits. Dokchitser reports a maximum loss of $9$ decimal digits from cancellation. The requested working precision is therefore not taken as an accuracy guarantee. Neither evaluator supplies a certified analytic error bound.

The exact Fourier coefficients through index $650$ agree with independent Sage and PARI eigenform constructions, beyond the Sturm bound in every selected weight. All dimensions and embeddings through weight $26$ are included. Exact convolution coefficients, gamma-pole critical sets, and functional equations are checked throughout. Nonzero controls include $\zeta(2)=\pi^2/6$, known initial coefficients of $\Delta$, and Sage's published high-precision regression value at $(k,i,s)=(12,1,22)$.

PARI's independent Haberland period pairing [4] checks (3) for every embedded form, with relative differences less than $10^{-150}$. The attached generator, exact roster, argument-selection rationale and numerical evidence reproduce the calculations and record the software versions and source hashes.



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