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{
"Title": "Critical L-values of higher-level newforms",
"Definition": "For an embedded normalized cuspidal newform $f(z)=\\sum_{n\\geq1}a_n e^{2\\pi inz}$ of weight $k$ and exact level $N>1$, this table gives the critical arithmetic values $L(f,s)=\\sum_{n\\geq1}a_n n^{-s}$ by analytic continuation, for integers $1\\leq s\\leq k-1$ CITE{PARI}.",
"Parameters": {
"form": {
"type": "Symbolic",
"title": "embedded normalized newform",
"display": "$f$",
"constraints": "a full LMFDB label $N.k.a.x.n.i$ of an embedded normalized newform with $N>1$, integer weight $k\\geq2$, and $\\chi(-1)=(-1)^k$"
},
"s": {
"type": "Z",
"title": "critical integer",
"display": "$s$",
"constraints": "$1\\leq s\\leq k-1$"
}
},
"Tags": [
"number theory",
"modular form",
"L-function",
"special values"
],
"Links": {
"LMFDB": {
"title": "LMFDB: classical modular forms",
"url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/"
},
"Labels": {
"title": "LMFDB: labels for classical modular forms",
"url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Labels"
},
"PARI": {
"title": "PARI/GP: modular forms, including lfunmf, mfembed and mfsturm",
"url": "https://pari.math.u-bordeaux.fr/dochtml/html/Modular_forms.html"
},
"PARILfun": {
"title": "PARI/GP: L-functions and numerical accuracy",
"url": "https://pari.math.u-bordeaux.fr/dochtml/html/_L_minusfunctions.html"
},
"Reliability": {
"title": "LMFDB: reliability of classical modular forms",
"url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Reliability"
},
"Completeness": {
"title": "LMFDB: completeness of classical modular forms",
"url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/Completeness"
}
},
"Data properties": {
"type": "C",
"rigour": "heuristic (agreement-checked)",
"complete": "no",
"complete-note": "it holds all $418$ critical values of every embedding in three disjoint slices: weight $4$ with trivial character and exact levels $2\\leq N\\leq30$ ($61$ embeddings), weight $6$ with trivial character and $2\\leq N\\leq15$ ($27$), and weight $3$ with primitive quadratic character $\\chi_D$ at $N=|D|$ for negative fundamental $D$ with $|D|\\leq40$ ($50$). These noncanonical level bounds balance low-conductor coverage across weights, characters and coefficient fields; the shorter weight-six range offsets its five critical integers per form. Every critical integer $1\\leq s\\leq k-1$ is included, including $17$ exact zeros. Empty newspaces contribute no entries",
"rigour details": "The nonzero values are recorded to 100 significant digits after fresh PARI/GP `lfunmf` and `lfun` computations at 160 and 220 decimal working digits, with explicit bit precision passed to every numerical call CITE{PARI}. The largest discrepancy, divided by the maximum of 1 and the two absolute values, was less than $3.4\\times10^{-170}$. These are agreement checks, not certified numerical error bounds CITE{PARILfun}.\n\nEvery reconstructed orbit matches 100 exact LMFDB Fourier coefficients under an exact coefficient-field isomorphism. Rational isolating rectangles identify all 138 embeddings, and the generator rechecks their Sturm-bound coefficients. Exact newspace dimensions, including empty spaces, agree with the 90-orbit catalogue selection CITE{Reliability} CITE{Completeness}. Catalogue floating-point coefficients are not numerical inputs to the L-value computation.\n\nAn independent implementation reconstructs coefficients from the catalogue's exact Hecke eigenvalues by Euler recurrences and evaluates the split Mellin identity CITE{formula-mellin}, using finite exponential sums for the integer incomplete gamma functions. Calculations with 450 terms at 150 working digits and 600 terms at 190 working digits agree throughout. The 600-term calculation differs from PARI by less than $3.2\\times10^{-199}$ on the same scale. For odd weight its root number is recovered separately from the Fourier value at $i/\\sqrt N$. Exact eta-product coefficients through index 100 for forms 3.6.a.a, 5.4.a.a and 7.3.b.a provide nonzero controls before the numerical comparisons.\n\nThe functional equation and conjugation identity are checked for every value. The 17 central zeros follow from exact Fricke eigenvalues, verified on Fourier coefficients through the Sturm bound. An exactly vanishing imaginary component is established by conjugation of the exact Fourier coefficients. Reproducible code, the full argument-selection explanation, the roster, source hashes and numerical evidence are attached."
},
"Display properties": {
"number-header": "$L(f,s)$",
"group parameters": [
[
"form"
],
[
"s"
]
]
},
"Numbers": {
"3.6.a.a.1.1": {
"1": "0.1178524725146029546964443363987177568867348323110446588788459428693034377551468781103452703516200788",
"2": "0.3351302899846574225582533474746834853903891613121037051691594825692163804730822531080955662237711119",
"3": "0.5600386910489645890343314763941015231677503853006209497922858574199182447081348598204262692902289580",
"4": "0.7350229744379839777779745236378771463081521009193329682206351541477319242523978416205262069407387992",
"5": "0.8503631276857206920542163035049044879358063675690591741944972104205579264543172997047772725642464268"
},
"4.6.a.a.1.1": {
"1": "0.2372550056342561811204793272201899477144301969252961789211385638328486685659980766849622255315768168",
"2": "0.5506574166704992687144129899621742698847443518696672478406329968492609895473066317396583328915805845",
"3": "0.7805376825961117937210607972807139466489105179797093059397325328877424287498884077729980264034539945",
"4": "0.9057951438439427225959494486851929814017881808666692687085096896728396651944304620943117658316813788",
"5": "0.9629497684208342292578060855601701864533533076889017966421375031326059379422869626758536931282851230"
},
"5.4.a.a.1.1": {
"1": "0.1613703351241217167582590553064809692741318360876510598175283664033840924442192244337832046665043666",
"2": "0.4118613283861991710154635203403986501988305072049907307406434553817911591541286434760481002744180423",
"3": "0.6370645478985185628978157970929321480716144311266447483707216454612172954794060216336591246391389269"
},
"5.6.a.a.1.1": {
"1": "0.3967318967572159305369572511355162209128100336043562808318219482832724640372531923397280053049421623",
"2": "0.7807297774317928553232791987347525435525638388697802253362531185752669947720838778494936245108636913",
"3": "0.9710655448233117529374157157645146593798877485038260259255714733827384883961703962652453112656834269",
"4": "1.027399206320311756908434297200049501731903461998204340625207127613284356592731960999000647158012141",
"5": "1.030541158595096077426808223198553629403774169316869308032666815545847023759069090178324778465861290"
},
"6.4.a.a.1.1": {
"1": "0.2216239155906735082467100442530338095614875988921917938713953739967396011777594087374808376908506514",
"2": "0.5097104233667223098633680858795421963250284450966691737039097935340986213530594014535543132059233905",
"3": "0.7291134575667892595145712159448379044293639045236670415770502101090630741407248851671769344372087330"
},
"6.6.a.a.1.1": {
"1": "0.6032956087515829906989918928382673258321131164997359201211283133156170624650450612618144779780366747",
"2": "1.036316412375335711701265107736594099108423437281685276222620068143007366654642624055178935779735859",
"3": "1.157778415751320856707368655073426800150873387762944257610824590028116664643349293987959488241882703",
"4": "1.136448113833416445260516457802184378939436030322027447267186893199900628073255281766284940480085714",
"5": "1.088268090246238240863815159854041578003787764568456494101588412172596681150404656488816717687160957"
},
"7.3.b.a.6.1": {
"1": "0.1967355870688958550489903975389425873775292307985145861005294477387848321639527067029568340172032788",
"2": "0.4672117688843731239421100514130912617241085499597108770877440238359901454261436221731572515501991028"
},
"7.4.a.a.1.1": {
"1": "0.2850516619814368206790532990666473751384807275386326475019152853438738717673292574561979830084667542",
"2": "0.5995661579686175665810611670752282076561559628744473357104439958045046085544866651766511487799053207",
"3": "0.8038134678942357281395913232443411484419209928521051914303167004364873159806914137600496887272863681"
},
"7.6.a.a.1.1": {
"1": "-0.5177803890674738779689240657842103361964640015695214861608483520944973691289657037276301099442099980",
"2": "-0.4408350004495845035842724891954411868026895108713281060308104469104116593414519698367684411310032618",
"3": "0",
"4": "0.4143682914848998668543631546634890695462180336293224812594078631577890283440169619291622807091279668",
"5": "0.6862111163849627659962483841040660417721788468280211781201686106350409627933591468871069207803884778"
},
"7.6.a.b.1.1": {
"1": "0.8025199270624613668310708244260478293052341861889141451484703662697495839309501522514143855017597377",
"2": "1.201502229107919151887816015132123833886455440697652729745252633899391567911338306671102088606059846",
"3": "1.214062261927596625577806728491261464114035313075185963484560982266661195291762682055456573572588132",
"4": "1.129366827459256547156554497715848684908396903512485571405582878000232123078986425708384101058551173",
"5": "1.063574648052085651926070419027660096375764260580486337176072619120626902173628086580070035703800997"
},
"7.6.a.b.1.2": {
"1": "0.8901671627150316383461966402147953746398505689883712231351509999305753449982165724637045674293740225",
"2": "1.377153780309742960949051015678646572560537378841944268621692259461413392971466750737027895150074985",
"3": "1.411116560073870027403459482986777420139453975077987818542766553238628067953473748750300410520332096",
"4": "1.294472667725893983837400163729049920235724912810466124662875535403023666075422111827624868286411710",
"5": "1.179732982154941183778255651256508495684033051570099289334836199467498693571487947425287736057705599"
},
"8.3.d.a.3.1": {
"1": "0.2380208810160280648641285918559743141066037083062833382606840844916544581506062671590192116570780997",
"2": "0.5287494555957668337759580638598658713411539297330525367243832947211294834920535857738431390794928794"
},
"8.4.a.a.1.1": {
"1": "0.3545006837309647187655598914949323644154302800304152530026834597078810578234301999188187313777903141",
"2": "0.6900311631233975251191054202183187040456219271857341158693739871906502602439104359044727941870360455",
"3": "0.8746953770850790449445947283566370007122192855441703635420319759089211078173230524925146366120660984"
},
"8.6.a.a.1.1": {
"1": "1.044885995210620120241402179565850005801548824412939087422980191369839167979142713432803031393287990",
"2": "1.391304368376811756590303614254293845934000905967725684216564248592854845306068757740094809904399898",
"3": "1.289076427120921353755651965702721082779993916534729740020839407878138251486744347866047721812259341",
"4": "1.144301976448885964660728923373689112604986706363105235101264115489861956982509337403005725268334576",
"5": "1.060222864871099270108440537247257112920766731362609739002998869352207872073558024167325331923858115"
},
"9.4.a.a.1.1": {
"1": "0.4189910775022040675514095346065758603044153972758781792250219546093954605640124037757785215696620431",
"2": "0.7599657499933076359136227474262080196061237300276900939175306460912019326559284082054146601652510736",
"3": "0.9189502627850946258433477704757323821417759411345570189877669347049866498941369358072092731731613067"
},
"9.6.a.a.1.1": {
"1": "1.478136100797008284788349773389101448205983092258173927530296535851837940724795451748840675750634935",
"2": "1.852593077576951430417662676570949129168846820965713272598075469298190879895720956865464324275215492",
"3": "1.620957618426135197288575434282088495505075300164021496415214232759159244340579153302026659964721121",
"4": "1.354397095694900697221335128812159321503525167373347410773699656686488540629106819203922556695472382",
"5": "1.185052625540585864485343162465121597398400834571757835141346478706021175135856286785335677489068523"
},
"10.4.a.a.1.1": {
"1": "0.5155982898017706462773964532180964166033718325912743375412018466880247432010424596392098781754573246",
"2": "0.8824450529181852676121956311868236797285093690830353312507774298668380280140405696662966084856709349",
"3": "1.017750230044340427828423630559708624681032442151307460492682490006139338954702385880651848687534017"
},
"10.6.a.a.1.1": {
"1": "-1.214557132442393485781740630376306109386291385959124860046419408779632452414080939266798125983667452",
"2": "-0.8008325275854027251525369083162444813366254021211495657873503078998761052366787893276217895828281845",
"3": "0",
"4": "0.5269266825861603952070425111869395117522003862232018397737885053455305392459666166525919201962883502",
"5": "0.7887260418671870798684280819756412617235773853730910331345736523359107004972565447132485991656350352"
},
"10.6.a.b.1.1": {
"1": "1.507082489621681173208855585221881569480694049925940480482054603208378665296444923102401239132762176",
"2": "1.581093100367651107060843256245387797140493444518727729966085337356070417847791554675895180409119238",
"3": "1.239525664364571011821933660184509229057369694768578838661841317833782636077177350100984110795057016",
"4": "1.040317561461372558665187822244457472562710032237336692930144807708930140568758129436400889367202189",
"5": "0.9786902361820624982505841924555265436029372026283760580603076061267410239459158273521622783782948484"
},
"10.6.a.c.1.1": {
"1": "1.776860829136941545462207576335838226990165016832088918155526662632674990477271842530723898284180793",
"2": "2.011552603420606710737691374035522130732373929218083133076823267447270945293409691841122634972761087",
"3": "1.636778589541502879577124233552452123351701683625886974293820336868063423541862539708843149645144066",
"4": "1.323548561849518480582762851565713948130884189233324003890341789966078983730540717430560609181740762",
"5": "1.153882655067689261894846664794102192594424329277298922505868396593739822837664061571626567685560302"
},
"11.3.b.a.10.1": {
"1": "0.3689122286920574381658931950275207124290446092070461789356584457260156879037538579615124680790551206",
"2": "0.6988863804242464992183227193400899508043808043220733131221996202810940996843218666667536516127682798"
},
"11.4.a.a.1.1": {
"1": "0.5348944609676166586195140338635358218355559379381301995805350623046633025464272634432113772224531658",
"2": "0.8560195510902055721043474835217754372541165710182774369344052783268343589476796577987968581936232305",
"3": "0.9598539501971471859357011043690754661558604329160349711371623611950834473292888102899345065736751545"
},
"11.4.a.a.1.2": {
"1": "0.6043704120931276166698390848693060657610498284919420580942438077511836964884140844596094994071685035",
"2": "0.9790655727628448861228878601811118219704684545698714263021304554284831963007533965134607035430513179",
"3": "1.084526705287731209568792515744422146896775254635315452813577357220126183620026598911515288654748925"
},
"11.6.a.a.1.1": {
"1": "-1.498109551321394341462022512194525143368620225160238954672881474121199299381317297970005201302598456",
"2": "-0.9180990932207589220579687426322889734218764777046924188254710973288827955971517346331623352013590259",
"3": "0",
"4": "0.5491681727931970659773787815683347989563216052895967456151885782132976867934108676313706188803362035",
"5": "0.8040192267965523286307257438054062796746026617090093439277685853670475268729698405517757152313805161"
},
"11.6.a.b.1.1": {
"1": "1.511448059847108983420869889937136637692230836250456089797563657771941667000329622766097387240787737",
"2": "1.278155454329647392795226159802421704597636814812023227403943047457499917256407512374278637625594639",
"3": "0.8699849810884206509076034060111386987396693193995144932999338056007359482165405582100602734598432207",
"4": "0.7645387089289852496220974805517878056570849281515753988036294488956891913306663088528413356927193430",
"5": "0.8111778603570986382592387953768430932428763020475986480285899506372994186356980804149331172706313338"
},
"11.6.a.b.1.2": {
"1": "2.098500958616069997806137721391170046891282880581942950324614634193096702448453297894413950778457044",
"2": "2.162704249621651850784848114466854127496714755555165637091048681291838855938394245794492496825609338",
"3": "1.645807385054622089885096713107395513100960281219529487880300249549081652037844471327269057346837316",
"4": "1.293638507898212967035733776886367570718828752969561335800849755082975117160840173737729905697004028",
"5": "1.126242814946414014405623480728752907751498621103918426767222587686321413281270153274374638892912378"
},
"11.6.a.b.1.3": {
"1": "2.341751098112947645231142526215691399065301642374555112521811709911914237079412310326564347557597790",
"2": "2.524126077812215000508757288129604184627363523341757461949070124717679463542407400909998824786010585",
"3": "1.961193867291087052023283548753575032836794276005453022266444130166754466373403230286183273334944169",
"4": "1.509825808877567153979204161237338262186567183722199081111144378898819882571727698573988088233701438",
"5": "1.256792539366717852283784067621764785604727713469079795070569713898689228046315595058949213861506395"
},
"12.4.a.a.1.1": {
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"2": "0.9344401381419144428104289823001602595950770395124332831581817614802993401022671860290980269161648683",
"3": "1.010941976484379270317945520601225863420443549935621377874902918112044142491887597586463952258797581"
},
"13.4.a.a.1.1": {
"1": "-0.2552019412596977930931895492653394489443954813305394937576694751898959843170776514223670019623484959",
"2": "0",
"3": "0.3874988004035786639442120731656919855015829832648016839269538193779045468037265439253700525723703889"
},
"13.4.a.b.1.1": {
"1": "0.6354458199293281607438678388239528239024632794463064963389740291851264573455101973558981290453664994",
"2": "0.9114306732800435241761308370532268147515384798721311216744628176895711440647829293890893793980703728",
"3": "0.9648613632351281977021459839341143199808450873874035348151105491172115346599092755860120307206485376"
},
"13.4.a.b.1.2": {
"1": "0.7567964575022277818152599565492465104937788029477761672700672214432948247450560602084560569941225607",
"2": "1.107518455078538737312991324032872913944229986234495007473834094139204717039710489195218680846640438",
"3": "1.149120253491203549059937242330075765545508573013039805996046658224049165148543606712980339727250298"
},
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"39.3.d.c.38.1": {
"1": "-0.08110199934534890379061262997467722513658271941104539059203025670068790163573796265438200890454397827 + i * -0.4826433262652738118329755707405254672986875884462946261630754863717046419580375186654133970879908122",
"2": "0.4134900581959818662262983889144427758841281090071003919185727619526555360543575323463540391594324086 + i * -0.2673692811791359319588216066014164472162046757469921951655477099566278949458008299866227221200018368"
},
"39.3.d.c.38.2": {
"1": "-0.08110199934534890379061262997467722513658271941104539059203025670068790163573796265438200890454397827 + i * 0.4826433262652738118329755707405254672986875884462946261630754863717046419580375186654133970879908122",
"2": "0.4134900581959818662262983889144427758841281090071003919185727619526555360543575323463540391594324086 + i * 0.2673692811791359319588216066014164472162046757469921951655477099566278949458008299866227221200018368"
},
"39.3.d.c.38.3": {
"1": "1.291563348967326290487394476636939900549234037659021750237669511394116419946661307804451491015561569 + i * -0.4163372251284021623315078709618770932984669546818344768148940772280434431461018924210692320264615197",
"2": "1.342361812321934219264208722980306658285957953174019854715585968708955077800457364626388353581634294 + i * -0.2492526743232178137251788868356377380126626153389084244988059295567935047168897517099742268309982917"
},
"39.3.d.c.38.4": {
"1": "1.291563348967326290487394476636939900549234037659021750237669511394116419946661307804451491015561569 + i * 0.4163372251284021623315078709618770932984669546818344768148940772280434431461018924210692320264615197",
"2": "1.342361812321934219264208722980306658285957953174019854715585968708955077800457364626388353581634294 + i * 0.2492526743232178137251788868356377380126626153389084244988059295567935047168897517099742268309982917"
},
"40.3.e.a.19.1": {
"1": "0.8288454392391412846720725861425002850424553919324615182653848099701913610044253740775539511233028225",
"2": "0.8234238174823788412713095581121362783623477513189646688788124610204171756905571808992242861489527796"
},
"40.3.e.b.19.1": {
"1": "1.482683795983513185665056863395831203842663804658974149543269961536893923161369937319889967505277279",
"2": "1.472985304146383062803793778003493047956792862723583612549249832083964340920353641814096016979082453"
},
"40.3.e.c.19.1": {
"1": "0.4973646584989966635719642698496373027720508330761557908704564738166363495690633887820372094536809162 + i * 0.5559232353073140180264064630315980829234406251345708854158349757215407220633305074350194856436899245",
"2": "0.7044610105625110085183175170252831916216921437166596321629033776096192921186439459772806209275555631 + i * 0.2300031096748946189752741850055102675470138737972066367716666666502693239885099346851014113020771701"
},
"40.3.e.c.19.2": {
"1": "0.4973646584989966635719642698496373027720508330761557908704564738166363495690633887820372094536809162 + i * -0.5559232353073140180264064630315980829234406251345708854158349757215407220633305074350194856436899245",
"2": "0.7044610105625110085183175170252831916216921437166596321629033776096192921186439459772806209275555631 + i * -0.2300031096748946189752741850055102675470138737972066367716666666502693239885099346851014113020771701"
},
"40.3.e.c.19.3": {
"1": "0.1980678825086382501700402304753642034988273386367203400109766091463333757745245337399417829084486372 + i * -0.8269946671167701058985813721145041302100352425234389689928848033553904130230421252971726911062312811",
"2": "0.6312554157353222373027081476546184338610395768993858923813595261053472051183201320723635941708516265 + i * -0.5614606789408411038633715957342993413439835222719396557629124577141775884939514460828697388081980049"
},
"40.3.e.c.19.4": {
"1": "0.1980678825086382501700402304753642034988273386367203400109766091463333757745245337399417829084486372 + i * 0.8269946671167701058985813721145041302100352425234389689928848033553904130230421252971726911062312811",
"2": "0.6312554157353222373027081476546184338610395768993858923813595261053472051183201320723635941708516265 + i * 0.5614606789408411038633715957342993413439835222719396557629124577141775884939514460828697388081980049"
},
"40.3.e.c.19.5": {
"1": "0.7234531663447768702521120360510903003265814370925349738245196579412232430795164946461471789079485860 + i * -0.9592779828158811103168417871659430964690063547303516444324587512454120089130526816100820756996808148",
"2": "1.006063141234042685942616515986763578258039788640051607511580374773525186634326514240997435881982723 + i * -0.6423486560525901552259226757208455405426452494329422134424040891211812554354679232791890035239847624"
},
"40.3.e.c.19.6": {
"1": "0.7234531663447768702521120360510903003265814370925349738245196579412232430795164946461471789079485860 + i * 0.9592779828158811103168417871659430964690063547303516444324587512454120089130526816100820756996808148",
"2": "1.006063141234042685942616515986763578258039788640051607511580374773525186634326514240997435881982723 + i * 0.6423486560525901552259226757208455405426452494329422134424040891211812554354679232791890035239847624"
},
"40.3.e.c.19.7": {
"1": "1.358371274029354792022787381390192832396320550585924803976210201583635180017896096272006515495923303 + i * -0.08457744724849438301239304187761081439996549032822421694414297144363132069336670451035964767638986166",
"2": "1.349228062995807878913277291899449106163112514643205547081989145126220071972266683824058361140841971 + i * -0.08806805715726683544134166885731053408006702230192924235586309070822078265172503572612786742289418534"
},
"40.3.e.c.19.8": {
"1": "1.358371274029354792022787381390192832396320550585924803976210201583635180017896096272006515495923303 + i * 0.08457744724849438301239304187761081439996549032822421694414297144363132069336670451035964767638986166",
"2": "1.349228062995807878913277291899449106163112514643205547081989145126220071972266683824058361140841971 + i * 0.08806805715726683544134166885731053408006702230192924235586309070822078265172503572612786742289418534"
}
},
"Comments": {
"comment-labels": "The embedded form is identified by its full LMFDB label $N.k.a.x.n.i$ CITE{Labels}. Here $N$ is the exact level, $k$ is the weight, $a$ labels the character orbit, $x$ labels the newform orbit, $n$ is the Conrey character index and $i$ distinguishes embeddings with that character. The attached roster fixes an exact coefficient-field polynomial, a rational isolating rectangle and Fourier coefficients through the Sturm bound for every embedding. Software list positions do not define these labels.",
"comment-characters": "The weight-three slice uses the primitive Kronecker character $\\chi_D(m)=(D/m)$ for negative fundamental discriminants $D$, with $N=|D|$; its value is zero when $m$ is not coprime to $N$. The weight-four and weight-six slices have trivial character. Each newform occurs at its exact level, with all coefficient-field embeddings and both members of every conjugate pair.",
"comment-normalization": "The form belongs to $S_k^{\\mathrm{new}}(\\Gamma_0(N),\\chi)$ with $a_1=1$. The stored function is uncompleted and has arithmetic centre $s=k/2$. The completion in CITE{formula-completion} is half the convention using $\\Gamma_{\\mathbb C}(s)=2(2\\pi)^{-s}\\Gamma(s)$; the uncompleted $L$-values are the same in both conventions. HREF{Special_values_of_the_L-functions_of_level_one_cusp_forms}[The level-one critical-value table] uses the same arithmetic normalization with a different form index.",
"comment-zeros": "For a form with real Fourier coefficients and root number $\\varepsilon_f=-1$, the functional equation forces $L(f,k/2)=0$. Such central critical values are included as exact zeros. No derivative is substituted for a vanishing value."
},
"Formulas": {
"formula-completion": "With $\\bar f(z)=\\sum_{n\\geq1}\\overline{a_n}e^{2\\pi inz}$, the completed function $\\Lambda(f,s)=N^{s/2}(2\\pi)^{-s}\\Gamma(s)L(f,s)$ satisfies $\\Lambda(f,s)=\\varepsilon_f\\Lambda(\\bar f,k-s)$, where $|\\varepsilon_f|=1$ CITE{PARI}. For trivial character and even weight, $\\varepsilon_f=(-1)^{k/2}w_N$, where $w_N$ is the Fricke eigenvalue.",
"formula-mellin": "The Mellin identity is $\\int_0^\\infty f(iy)y^{s-1}\\,dy=(2\\pi)^{-s}\\Gamma(s)L(f,s)$ CITE{BC}. If $I_f(s,T)=\\int_T^\\infty f(iy)y^{s-1}\\,dy$ and $T=N^{-1/2}$, then $(2\\pi)^{-s}\\Gamma(s)L(f,s)=I_f(s,T)+\\varepsilon_f N^{k/2-s}I_{\\bar f}(k-s,T)$.",
"formula-conjugation": "For real $s$, $L(\\bar f,s)=\\overline{L(f,s)}$."
},
"References": {
"BC": {
"bib": "K. Belabas and H. Cohen, Modular Forms in Pari/GP (2018), sections 2, 6 and 9.2.",
"arxiv": "1810.00547"
}
},
"Similar tables": [
{
"table": "HREF{Special_values_of_the_L-functions_of_level_one_cusp_forms}[Critical values of level-one cusp forms]",
"relation": "uses the same arithmetic L-function normalization for level one, with embeddings ordered by the coefficient of the second Fourier term"
},
{
"table": "HREF{Hecke_polynomials_of_weight_2_newforms}[Hecke polynomials of weight-two newforms]",
"relation": "records characteristic polynomials of Hecke eigenvalues in weight two, rather than the higher-weight critical periods stored here"
},
{
"table": "HREF{Special_value_of_L-function_of_elliptic_curves_over_Q_of_rank_1}[Leading L-values of rank-one elliptic curves]",
"relation": "records leading central Taylor coefficients in the rational weight-two case"
}
],
"Keywords": [
"newform",
"higher level",
"critical L-value",
"modular L-function",
"Mellin period",
"quadratic nebentypus",
"LMFDB embedded newform"
],
"Programs": {
"program-pari": {
"language": "PARI/GP",
"code": "\\\\ Example: all critical values for the unique newform 5.4.a.a.1.1.\ndefault(realprecision, 160);\nmf = mfinit([5, 4], 0);\nf = mfeigenbasis(mf)[1];\nL = lfunmf(mf, f);\nvector(3, s, lfun(L, s))"
}
}
}