Ramanujan's class invariants $G_n$, with Weber's $\mathfrak f(\sqrt{-n})$

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Numbers

$n$

convention

$G_n$ or $\mathfrak f(i\sqrt n)$

$G_n$:

$\mathfrak f(i\sqrt n)$:

1.189207115002721066717499970560475915292972092463817413019002224719466668226917159870781344538137674

$G_n$:

1.023807274909184034748664952421301398266836573959224042897868368355500754030548719031269015427436305

$\mathfrak f(i\sqrt n)$:

1.217518895713548480868409321542186128270770427496416662828070102569571075237846008749157491208572913

$G_n$:

1.059463094359295264561825294946341700779204317494185628559208431458761646063255722383768376863945569

$\mathfrak f(i\sqrt n)$:

1.259921049894873164767210607278228350570251464701507980081975112155299676513959483729396562436255094

$G_n$:

1.094588288578459130320957082252498229478023913160772989188705492843845419652414570169519693428444374

$\mathfrak f(i\sqrt n)$:

1.301692180776155281275810494127103187985637798553643888563039263015085924275580170090954315998461252

$G_n$:

1.127838485561682260264835483177042458436468335423655767290049186400476209539553612403996109293393067

$\mathfrak f(i\sqrt n)$:

1.341233551603846239042001605587290253445981548244120118800287006706080096468417964112692309432044359

$G_n$:

1.159287416192176064416754909845814112231606704790906857611078700657237474707582791869945271633129378

$\mathfrak f(i\sqrt n)$:

1.378632843668856481541320330743740452980424002187762797029736783442185100551146541257711564863959260

$G_n$:

1.189207115002721066717499970560475915292972092463817413019002224719466668226917159870781344538137674

$\mathfrak f(i\sqrt n)$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

$G_n$:

1.217855815652096293978713299458821102467272732345996841867457044017639413767180434851854965124016807

$\mathfrak f(i\sqrt n)$:

1.448282801020915144391302213193110660388445736120313511631803289407580167884474347933537540743048732

$G_n$:

1.245445108413552301377364874341648400757800564633019580694266561964799501770347871190061384491314176

$\mathfrak f(i\sqrt n)$:

1.481092184270731698455563868246081430655643995054625857304393098740749362747188178747915199145171498

$G_n$:

1.272142962269806534370186786345239532637330320472426116129544419548486973789089432973820206159163730

$\mathfrak f(i\sqrt n)$:

1.512841462031892066221052361565299056747656895155674137146748613298590500767155829815710386698019804

$G_n$:

1.298082557039312863788515964642505646858690843920203858872931594728899234940826443528624428193841196

$\mathfrak f(i\sqrt n)$:

1.543689012692076361570855971801747986525203297650983935240804037831168673927973866485157914576059125

$G_n$:

1.323370411329764949923802090511931770532358163647765022928671844033190145004186076998826343392241821

$\mathfrak f(i\sqrt n)$:

1.573761508937434068908197823251603868609915083857076387667756697723708085998064730825060787314332664

$G_n$:

1.348092734578734616835439723285445248486127988929798303596418871707203074029128016363351033770023258

$\mathfrak f(i\sqrt n)$:

1.603161471644505984309669419546610167598939286188408787124431061956350905643960569442726324079372939

$G_n$:

1.372320171553063074052411000147417546960527398097332202621215688692065736683736116065115496115295429

$\mathfrak f(i\sqrt n)$:

1.631972912072657382451465934095104506778757119550508280722764258004984571411907712791459206663189681

$G_n$:

1.396111310988422713966108174325423005520226218604564901200792342817349179842955940706272251116401929

$\mathfrak f(i\sqrt n)$:

1.660265504363208886098361557741394998149924900254897913395287105673759577693948174719238520336133374

$G_n$:

1.419515289323360485330852659900073398663404741223045782684474949833843869538836786978769453087270086

$\mathfrak f(i\sqrt n)$:

1.688097681918486420674063249691876356216566848550609966743567372493516757146263824511797572657659025

$G_n$:

1.442573740446059678174681402508318327108159234384216379759800395948420011870800339875782895487196044

$\mathfrak f(i\sqrt n)$:

1.715518956054542782392608550846686512169773250400408003734573119183705770495549508912795079306718755

$G_n$:

1.465322272243174793016687385887298351433456479500857615271344437934724231600985726003183370553825345

$\mathfrak f(i\sqrt n)$:

1.742571671923537713709509286885348121559704137568610055162882370132062740176676172326251690839845522

$G_n$:

1.487791598215070412726155231276027096345043597603051244663736428195447283102439602157782835513999399

$\mathfrak f(i\sqrt n)$:

1.769292354238631415240409464335033492670553045898857004233106130402673817350668329068741221494454818

$G_n$:

1.510008415083301592179540083715822516017409782848871066568820094758585214129978957197066505483628244

$\mathfrak f(i\sqrt n)$:

1.795712750931044404752165477481117174976217286727464006165706544493012873852245921642954774068931275

$G_n$:

1.531996091228447989044256781197750164851959319901586512208529009746371320717269972538915752167862292

$\mathfrak f(i\sqrt n)$:

1.821860651845228102554053992498295024399294112532360664983015140587247979659833614808129939536117067

$G_n$:

1.553775212614589265072873614350669941861774798209458238373632636235166320071812346068011132133766470

$\mathfrak f(i\sqrt n)$:

1.847760537956135232844804294943302587494768758673801689157719865473170378429585595577060102427429744

$G_n$:

1.575364020127322416384220749025769381553308552133179382068993947873027599246183932889827981520533303

$\mathfrak f(i\sqrt n)$:

1.873434101454701694066223299233722033945495873266348279245341001393913352713178846414591108454003293

$G_n$:

1.596778763268598989562220448215780667442447460996048418461882420506633470236920516916358298702661181

$\mathfrak f(i\sqrt n)$:

1.898900666364263516017144241431640453764423604715578853448309666500118196210565207137931587826904834

$G_n$:

1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137

$\mathfrak f(i\sqrt n)$:

1.924177531737607687421310699731866995296920658157885973453423544089791365356871432198637674177790432

$G_n$:

1.639142777915580803747755766131267168685060391569108559502384123613673641422331118310147150479374459

$\mathfrak f(i\sqrt n)$:

1.949280254002533778032466355059755569727393437484647217810152221401760919911225892669286495930988647

$G_n$:

1.660116943573778125864790193976759839935265951825088122055814973976560621887572445995004591887712435

$\mathfrak f(i\sqrt n)$:

1.974222881034507763672090410950360409028770763347375020691992114279325164924971214897902840384627683

$G_n$:

1.680967194344996071979248984268989770039472621058194306632797953558257968326051676573702927958408884

$\mathfrak f(i\sqrt n)$:

1.999018147601231117330516497986462473632983516967497717902368086675413903177921931506774966107296810

$G_n$:

1.701703272805981488130073655405932978597048612866896073072929330467232798166745459096151877921850648

$\mathfrak f(i\sqrt n)$:

2.023677639644289648381264765447731091298835593319339152150715331226642982556851802020753073996001460

$G_n$:

1.722334072330998414180991867466730743414125518313338136021241824438237760066223321384347292542246915

$\mathfrak f(i\sqrt n)$:

2.048211933227634535118388493272250998549055260660147544182935170249353584380089934368750454397142938

$G_n$:

1.742867736491155257605879741067303232575198817074817213065657375886901645908097611628018404605162934

$\mathfrak f(i\sqrt n)$:

2.072630712743969426305340645406994491246464974443411477746006440577006795803463660942523413761030477

$G_n$:

1.763311744076356468992530449773565060954511069760903734839359413930706863477202702656246643349525083

$\mathfrak f(i\sqrt n)$:

2.096942872023460305101577092180256546182846192181529841891695260808603913538757297920158102254371993

$G_n$:

1.783672982190861338057932354058738848801028866559429134624593425155119478600299094155090618441673439

$\mathfrak f(i\sqrt n)$:

2.121156601259494084383656908144514604287516384966162756813826914384106236401538085603845142357096648

$G_n$:

1.803957809396614565568859098045639191663852255856768778968045223331252856444926405651142212154565875

$\mathfrak f(i\sqrt n)$:

2.145279462099176587842044545372242487868676322683251293136083363437445665665796819767962610184754488

$G_n$:

1.824172110505117927593152606263903370507337418274653277247992304492307363212112657988766510042858984

$\mathfrak f(i\sqrt n)$:

2.169318452802216177424719059453254976451340682085992420362652153849999063952848284602897079888479797

$G_n$:

1.844321344324062974834747388093763499821309761668387433750265790804236736369937154612485841825213478

$\mathfrak f(i\sqrt n)$:

2.193280065021559076807962554148024076127569483541687473682142754458088185332128953405587344340030902

$G_n$:

1.864410585430976405353832665974928944932525525303251110075703058973403255366462990454880200901928566

$\mathfrak f(i\sqrt n)$:

2.217170333480905668235672764299579682233691503395746294814106738221461761118695692964203980227242367

$G_n$:

1.884444560859056550889492985343969682204847112025201132297873643666304138150242028096925729653384365

$\mathfrak f(i\sqrt n)$:

2.240994879601768261880672063425564177099808310137828307911824463268526264293696883429840125622730187

$G_n$:

1.904427682429865483446160223509613505501608022404359469480083706792830625146058699038750541862066352

$\mathfrak f(i\sqrt n)$:

2.264758949953738596189633967657185420953156214379545970985793422839343659648126049315401099561597003

$G_n$:

1.924364075345721912983712335784804018494282040475383900627243508457875088024628953990004951968768507

$\mathfrak f(i\sqrt n)$:

2.288467450256764906769417438379625914545140014626991379489058400478883443591718866321796811665370057

$G_n$:

1.944257603555487839782609131435434645252232974157780446546168353665287054532331363731541013769130447

$\mathfrak f(i\sqrt n)$:

2.312124975546325891020739922754278926647764291916981663570011508937984834799519079269121108166292279

$G_n$:

1.964111892326305575758546419307429367345939298640682298505272064761420788031525598870290924966751785

$\mathfrak f(i\sqrt n)$:

2.335735837015900971855889919827580272285726220067962666719643909493456629746374472946800196732881990

$G_n$:

1.983930348387108341417740433627156181967482679229746566595203833964825093116732675216708594142169335

$\mathfrak f(i\sqrt n)$:

2.359304085971776420730660392800532555346481278067672283797141251583875588944652906082462811238040582

$G_n$:

2.003716177954573158710631368268449536089655952450782758108605597250907229631250221235332324551801966

$\mathfrak f(i\sqrt n)$:

2.382833535269636792529855486547661082755105279797348742832506956863369564708096374046125736760126264

$G_n$:

2.023472402906388536464318973845208227711065122284875276597879132991388136134537004256714504158305792

$\mathfrak f(i\sqrt n)$:

2.406327778547929929927133331870852586433944381067854567614177721584239938121517595723487403585571289

$G_n$:

2.043201875328515235835880400133775948303283012494550684812047488432638330810298190310881431601584365

$\mathfrak f(i\sqrt n)$:

2.429790207527572969462021891164167168334860492658818394177314209900017990537964569023559954740695461

$G_n$:

2.062907290631129605263112355692824398364920969750732729972836362964386840453395503701894007945036364

$\mathfrak f(i\sqrt n)$:

2.453224027609525475435859066041619029789836550359361292953024301302234505794661899744570233108213183

$G_n$:

2.082591199401039133241508201533787122572142677927701971579749746671473929129887559601171622002483863

$\mathfrak f(i\sqrt n)$:

2.476632271969766345243083078804718301549078008322086614178739029441999658272625490692208152720782601

$G_n$:

2.102256018135651980201975536878441931590558316923667563963834093072030279278601526055470641292032032

$\mathfrak f(i\sqrt n)$:

2.500017814324206748902641382625861921598163415685210724779258033995175625111262106404132243076853691

$G_n$:

2.121904038984342148307321231344364731418806034841055813091198298933620064099110107955930366568042019

$\mathfrak f(i\sqrt n)$:

2.523383380513190898819660668360825993405839503424408951170534643894486770247210559082207928125570038

$G_n$:

2.141537438606692283663785658679199805037045606502547161535080515129813162044778976705597220937325119

$\mathfrak f(i\sqrt n)$:

2.546731559035781416597044980036264504609037636805874761014196828647455485121128509875090248334414451

$G_n$:

2.161158286243138387125149474409246132393387036265175329125188474776980104205913478402682072016254792

$\mathfrak f(i\sqrt n)$:

2.570064810647427445782797628760467378451437811607708830474958991182237313847847945185569439423826731

$G_n$:

2.180768551081593462018421728385434620532810347399653679465484006696885368478074851027846001310081860

$\mathfrak f(i\sqrt n)$:

2.593385477120405907292708138862131958103712387550796517035923155455590767617222905647239434320135265

$G_n$:

2.200370108993368753432731793168934156930803328443113850434154072374829859841853564707465347490413029

$\mathfrak f(i\sqrt n)$:

2.616695789254226963270648960688526347766426474496897371695094924690205837658402505801344101105757264

$G_n$:

2.219964748702876502605275642454079679867340876627079666205806164424719494028404861135807215725643743

$\mathfrak f(i\sqrt n)$:

2.639997874212688430056228413735633947026841663249057214830318332547427446926821467221136449789317081

$G_n$:

2.239554177447967476875867616528956975939472979124566275824581011871825458616761908595125531279339314

$\mathfrak f(i\sqrt n)$:

2.663293762255189442123671428832267467580702577596628260389957956606294377090998286444467873277873712

$G_n$:

2.259140026181147500895251190790883935509757807186781329618855899520403644651156164998633025871469144

$\mathfrak f(i\sqrt n)$:

2.686585392922054157664515118383378811230911008108387052973241992890646619944621453196457961497394399

$G_n$:

2.278723854356177690700003398668430691791988419692329030666840914593876748676047901257512110868051975

$\mathfrak f(i\sqrt n)$:

2.709874620726790813623175217640813300373535481124628403911189228097972464187010283525428366903875189

$G_n$:

2.298307154339565814779492420966574166512079235930089995920838859695962896074188627259353540571793187

$\mathfrak f(i\sqrt n)$:

2.733163220402268640050810003672953598185474788444226250266050093836550236932928710358952555392406361

$G_n$:

2.317891355482094356768350208918799540716627778260016131706712676298982219937534089282481495804713706

$\mathfrak f(i\sqrt n)$:

2.756452891742608001192453695141751414240482947998706289009509604544293067025478744263274523963380239

$G_n$:

2.337477827881714493110382764100000837578107582468986732804017085055877235898307490524849921353508593

$\mathfrak f(i\sqrt n)$:

2.779745264078040686725828721460892801754520490734028996760225548484083183874529370513752588244289467

$G_n$:

2.357067885865788367934690849939944335866714254942791607929894208364343965343336191636635865293232645

$\mathfrak f(i\sqrt n)$:

2.803041900416017201207289630082908063166717028331214650467147031934966994445090915739355539634223676

$G_n$:

2.376662791217720408537792622813836596791584528410109674753553554681994859728409005152906178031959770

$\mathfrak f(i\sqrt n)$:

2.826344301278339681933752853776797008417683680350801922029879324741932781953054083204433863912778345

$G_n$:

2.396263756170427328403551697192212889840466872795364769443357463012623519279104480345824805140274805

$\mathfrak f(i\sqrt n)$:

2.849653908261017725081212084087511875035190496206659181336346106762542943397616879598477330021864052

$G_n$:

2.415871946186809362816339075281469555671777147068661130889389193937174657822300013953913991368541793

$\mathfrak f(i\sqrt n)$:

2.872972107340824562156317673449800800249094035004747996346160351198327764239829326194242142414672851

$G_n$:

2.435488482545362422908764880348829037174667123836701687190296874381562677813218667408195748403707611

$\mathfrak f(i\sqrt n)$:

2.896300231950125430227450885246281532660894712532648130202058186111713275702792992647188649877700520

$G_n$:

2.455114444747278202172679346166977183894024216019577244790954622338992524778187813562220224267002377

$\mathfrak f(i\sqrt n)$:

2.919639565839418145111413976212561732236473271543114541298868053809139888149851320806587760211476566

$G_n$:

2.474750872759787612018758536539649584753266322930346526590844659163979521346563023737065935277543230

$\mathfrak f(i\sqrt n)$:

2.942991345745133076335537827707883890373321996831056476858524692275515397508293837450431501126444164

$G_n$:

2.494398769109087125577877879872334472533068316141289272883577878812247577045333145904891996017241293

$\mathfrak f(i\sqrt n)$:

2.966356763878556046362906492015600512798761787830232626525983603809512092040321898990908062306025023

$G_n$:

2.514059100834925996137200124050951105643099919911963514811483077643057305183092152127096905535962970

$\mathfrak f(i\sqrt n)$:

2.989736970250237357649930723277234696762384672618614038676493748280785906671876380431095294244774288

$G_n$:

2.533732801317806153575519834727431713726278857635051200110328753264388103058622753887780021356968003

$\mathfrak f(i\sqrt n)$:

3.013133074842910910019494725154209091677210898092711903863788777539699902694976343349974536915055545

$G_n$:

2.553420771988739640294291067924059328647332175099323714469551386119476931720725926643168663935460149

$\mathfrak f(i\sqrt n)$:

3.036546149644749908398943272404725035286003475325317930334679764796688228858766098428586149615089336

$G_n$:

2.573123883930606639873828197268504992923639229106977152676924192171566742871287128861941482169243202

$\mathfrak f(i\sqrt n)$:

3.059977230553713224164767025535132331394731621511647512527883331807725404510055394075450386646573920

$G_n$:

2.592842979379348222794983628952988107672358785334798927654898506535175778287081622774112952684125672

$\mathfrak f(i\sqrt n)$:

3.083427319162774489345292324783240323631206778188880775493811952651605205568859652479329674270178256

$G_n$:

2.612578873132501203206488434354414381736735684859719538842558710698843713725941101262256846717218792

$\mathfrak f(i\sqrt n)$:

3.106897384434961770017741287377975235597406744982418977304871167574639186074707845387705419689872639

$G_n$:

2.632332353871928641263758311241488435006582405020587190840179654893272316080914319834083916471616736

$\mathfrak f(i\sqrt n)$:

3.130388364276358090920663235157428609213539766058278757372861507422350649730226242099976800630887363

$G_n$:

2.652104185407010375743852144953145480322606932561685231551261683914935349542229618010287067529893606

$\mathfrak f(i\sqrt n)$:

3.153901167014512462143665775334220670764614677388827442190903889269351060638942972155481017938679679

$G_n$:

2.671895107844026377539165983181927400172918228655185701686528016111187836188301987609786123851510374

$\mathfrak f(i\sqrt n)$:

3.177436672789078883267857753063711938623534830875377051240848206118960120567551097464956971342421208

$G_n$:

2.691705838686985394120847234879526491039323411007630186628857428500462840652737442295368850401446500

$\mathfrak f(i\sqrt n)$:

3.200995734860929599760542202160621061586643135406570913100724744295457659757721778488819295442483237

$G_n$:

2.711537073874716781946168175244708690104562119222163182295008323093231805699332088998538576765747836

$\mathfrak f(i\sqrt n)$:

3.224579180845472088985274491761305870562247166041482531500653353296483432593734836968263473187807003

$G_n$:

2.731389488758649728588750373781004142266714283817568529969063284245634897061128832346114759570388140

$\mathfrak f(i\sqrt n)$:

3.248187813875431068060951275215176195689175485981301860809665040354519106301788788092695889116867913

$G_n$:

2.751263739025346951211654258598676450164506455169894792809858184726195447167551393565432173493352025

$\mathfrak f(i\sqrt n)$:

3.271822413697932131915783511727532984401951492494827595907381177298047078836759746333664784375438678

$G_n$:

2.771160461567535623384230042017932898609441607683252769126715343961413037650562429542981061141658753

$\mathfrak f(i\sqrt n)$:

3.295483737710337928845310022246116052795311030045426769023571751151340742054679228870386099513907938

$G_n$:

2.791080275307083363993371754344339327498784620851244758717447465934866373771928825970534447810522517

$\mathfrak f(i\sqrt n)$:

3.319172521938937062002061262572005802753617955852975145036696482514853239452908524766699017890071121

$G_n$:

2.811023781973098636841548739127350476609662114255742437257986168725105532710270402692503100761371624

$\mathfrak f(i\sqrt n)$:

3.342889481964266620783998780615375903396770165387502089419115144614977708044921078412650692740090909

$G_n$:

2.830991566838090208005457166630336411429949489483729525297853741663040074797398481689589599121738403

$\mathfrak f(i\sqrt n)$:

3.366635313796558245301965842017758521338110466124299480242202042133429681845535512493458956806252288

$G_n$:

2.850984199414897033746355662161839629254202035231049349250784971279170907968944747443610017467798663

$\mathfrak f(i\sqrt n)$:

3.390410694704532107740707434123994647533240133570949082232573558652694061608999360458631534454519718

$G_n$:

2.871002234116896005973487121858349504422362100959529826268794890644447576957494103013607596239539527

$\mathfrak f(i\sqrt n)$:

3.414216284000520660643876930880144971278455455854131134059380437394124522328119168696140125512442164

$G_n$:

2.891046210883808493121534900508693326238288707320832020533369064578073385383541972218951228538417718

$\mathfrak f(i\sqrt n)$:

3.438052723784682227940748195845007503901004583432763985327699970145437233806511878200257032463232410

$G_n$:

2.911116655775255910079152641839488785779752108166742225692606270232518955311059428486040760193908108

$\mathfrak f(i\sqrt n)$:

3.461920639650861511854718402161639405968878887115277978175968982184979245593405122112256004894949746

$G_n$:

2.931214081534058136512594580563828634354612062926663891048072706267305796383889770591661698371294002

$\mathfrak f(i\sqrt n)$:

3.485820641356468079843548332961372665493686870052225887881414592755059984664425552691458865129402733

$G_n$:

2.951338988121125139916674512315155540403528494050796184631575802633987864940479533552394606879427820

$\mathfrak f(i\sqrt n)$:

3.509753323458573288486916271775202661791554874886227159723438311452750233609708801910185847365756036

$G_n$:

2.971491863223660447821970830243121443339780458330932359363816002197679443968515914098829802763968601

$\mathfrak f(i\sqrt n)$:

3.533719265918269468504313841871360435536253377168935509981760545362269880144671193200908777886227253

$G_n$:

2.991673182738274075446684709320859411765340501731670188119211703657396675843966225641111331453205731

$\mathfrak f(i\sqrt n)$:

3.557719034675191255668430584512729171298757644501762231620379217098938688194856339168631323923611723

$G_n$:

3.011883411230491182450649600210977832926850067440498837601271639759892789373105976780814585045571067

$\mathfrak f(i\sqrt n)$:

3.581753182193966554822365207552129797487673280607007622870478297503047797818302630345234707610204144

$G_n$:

3.032123002372040234095784547376540338206368302108276416856357579679731456678084782954817657367477718

$\mathfrak f(i\sqrt n)$:

3.605822247984242732438093218216547146192573447392943014890038706982625521474457492996089526063322942

$G_n$:

3.052392399357209993206777813519171179441193228411246838913561846560963063594335578990587134280862595

$\mathfrak f(i\sqrt n)$:

3.629926759095821313837242624706335362844936824407749724840126023559405916598731897033677220121152795

$G_n$:

3.072692035299477562055795263627321140456963716815328698568499335392659407931207373368922361129848991

$\mathfrak f(i\sqrt n)$:

3.654067230590330872609986407569567895904759184232542697406078338766097757513186114749278346166874209

$G_n$:

3.093022333609529288631641532688074318890844270618653424625237191273180171317333863626002814699575590

$\mathfrak f(i\sqrt n)$:

3.678244165990772181970105705533194581731285869245764191497278523608263784859398372687312118333296998

100

$G_n$:

3.113383708355722072127648242976038402217695205175897107613660884731164606562735179536148943681165989

100

$\mathfrak f(i\sqrt n)$:

3.702458057710181363924355869088965653403263217854565824980305751589633851337680575782946055741657348

Definition

For $n\in\mathbb Z_{>0}$, let $q=e^{-\pi\sqrt n}$. Ramanujan's class invariant is $G_n=2^{-1/4}q^{-1/24}\prod_{m\geq1}(1+q^{2m-1})$. Weber's value is $\mathfrak f(i\sqrt n)=2^{1/4}G_n$, written $\mathfrak f(\sqrt{-n})$ in Weber's notation. Both normalisations are stored [1] [2].

Parameters

$n$

— index, with the CM point $\tau=i\sqrt n$ (positive integer)

convention

— whether the row holds Ramanujan's $G_n$ or Weber's $\mathfrak f(i\sqrt n)$

Formulas

(1)

Weber's function is $\mathfrak f(\tau)=\eta(\tau)^2/(\eta(\tau/2)\eta(2\tau))$, equivalently $\mathfrak f(\tau)=q^{-1/24}\prod_{m\geq1}(1+q^{2m-1})$ for $q=e^{\pi i\tau}$ [2] [4].

(2)

If $m_n=\lambda(i\sqrt n)$, then $G_n^{-24}=4m_n(1-m_n)$ [5].

(3)

$G_{1/x}=G_x$ for every real $x>0$, where $G_x$ is defined by the same product [2].

(4)

The Weber value satisfies $j(i\sqrt n)=(\mathfrak f(i\sqrt n)^{24}-16)^3/ \mathfrak f(i\sqrt n)^{24}$ [6].

Comments

(5)

The index $n$ is the $r$ of the singular moduli $k_r$ and $m_r$ and of $K(k_r)$. The point is $\tau=i\sqrt n$, and the order is $\mathbb{Z}[\sqrt{-n}]$, of discriminant $-4n$.

(6)

The companion invariant $g_n$ used at even indices is built from Weber's $\mathfrak f_1$; it is a different function and is not stored here [3].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.rational_field import QQ

n = 2
CBF = ComplexBallField(numberdb.bits(100, losing=256))
tau = CBF.gen(0) * CBF(n).sqrt()
f = tau.modular_eta()**2 / ((tau / 2).modular_eta() * (2 * tau).modular_eta())
G = f / CBF(2)**(QQ(1) / QQ(4))
G.real(), f.real()

References

[1]

Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley-Interscience, 1987.

[2]

W. Duke, Continued fractions and modular functions, Bulletin of the American Mathematical Society 42 (2005), no. 2, 137-162. (doi)

[3]

Bruce C. Berndt, Heng Huat Chan and Liang-Cheng Zhang, Ramanujan's class invariants, Kronecker's limit formula, and modular equations, Transactions of the American Mathematical Society 349 (1997), no. 6, 2125-2173. (doi)

Links

[4]

DLMF 23.15: Dedekind's eta function

[5]

DLMF 23.15: modular functions and the elliptic modular function $\lambda$

[6]

Wikipedia: Weber modular function

Similar tables

Singular values $k_r$ of the elliptic modulus — holds $m_r=\lambda(i\sqrt r)$, and the entries here satisfy $G_r^{-24}=4m_r(1-m_r)$ by (2)

Modular polynomials for Weber's $f$-function — those polynomials give modular equations for the same Weber function

Rational singular moduli — Formula (4) gives the rational singular moduli in that table for the orders with $\Delta=-4n$, namely $n=1,2,3,4,7$

Hilbert class polynomials $H_\Delta$ — those polynomials record the algebraic $j$-values of imaginary quadratic orders, while this table records Weber and Ramanujan class invariants at $\tau=i\sqrt n$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds Ramanujan's $G_n$ and Weber's $\mathfrak f(i\sqrt n)$ for every integer $1\leq n\leq100$, matching the range of the singular moduli and $K(k_r)$)

How they were obtained:

The generator computes Weber's $\mathfrak f$ in Sage's arb-backed ComplexBallField from the eta quotient in (1); the imaginary part is required to contain zero before the real part is stored. The $G_n$ entries divide the same real ball by $2^{1/4}$, except that $G_1$ is returned exactly as $1$.

Before the values are sent, the generator checks (1), (2), (3), and (4) on the computed range.