Properties

Label 2.82.a
Level 2
Weight 82
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newforms 2
Sturm bound 20
Trace bound 1

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 82 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{82}(\Gamma_0(2))\).

Total New Old
Modular forms 21 7 14
Cusp forms 19 7 12
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators.

\(2\)Dim.
\(+\)\(3\)
\(-\)\(4\)

Trace form

\(7q \) \(\mathstrut +\mathstrut 1099511627776q^{2} \) \(\mathstrut +\mathstrut 5492623293753256284q^{3} \) \(\mathstrut +\mathstrut 8462480737302404222943232q^{4} \) \(\mathstrut +\mathstrut 3930103990742699543615489130q^{5} \) \(\mathstrut -\mathstrut 21678079061476108854899603668992q^{6} \) \(\mathstrut +\mathstrut 17840124989511368450810570225392568q^{7} \) \(\mathstrut +\mathstrut 1329227995784915872903807060280344576q^{8} \) \(\mathstrut +\mathstrut 895258546468549738568590790199579496731q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 1099511627776q^{2} \) \(\mathstrut +\mathstrut 5492623293753256284q^{3} \) \(\mathstrut +\mathstrut 8462480737302404222943232q^{4} \) \(\mathstrut +\mathstrut 3930103990742699543615489130q^{5} \) \(\mathstrut -\mathstrut 21678079061476108854899603668992q^{6} \) \(\mathstrut +\mathstrut 17840124989511368450810570225392568q^{7} \) \(\mathstrut +\mathstrut 1329227995784915872903807060280344576q^{8} \) \(\mathstrut +\mathstrut 895258546468549738568590790199579496731q^{9} \) \(\mathstrut +\mathstrut 50463440699837666565174903814070395207680q^{10} \) \(\mathstrut +\mathstrut 4574205974746866887301110535567689797851924q^{11} \) \(\mathstrut +\mathstrut 6640174117235059459063078612734728925609984q^{12} \) \(\mathstrut +\mathstrut 364156433571445284139262691120632802372144674q^{13} \) \(\mathstrut +\mathstrut 20830971491787782972327917197359525676676284416q^{14} \) \(\mathstrut -\mathstrut 288696232916218955954253101780543418545348888280q^{15} \) \(\mathstrut +\mathstrut 10230511461316320427425793829013981137591527800832q^{16} \) \(\mathstrut +\mathstrut 30797813462159183585571851099038617318785664498558q^{17} \) \(\mathstrut +\mathstrut 1243365571050882738535442992934696278613147598716928q^{18} \) \(\mathstrut +\mathstrut 10552219769880013876700985537471230307761596684415180q^{19} \) \(\mathstrut +\mathstrut 4751204188179343036375915600360129120490090271866880q^{20} \) \(\mathstrut -\mathstrut 534204395395054602762771158030159951278109283209553696q^{21} \) \(\mathstrut +\mathstrut 2213095756532053785240759362683971572702330104213143552q^{22} \) \(\mathstrut -\mathstrut 10513722214019385471431306716226633418248837618951393496q^{23} \) \(\mathstrut -\mathstrut 26207189497065736089112950693849540407165504893162094592q^{24} \) \(\mathstrut +\mathstrut 1014189466564068690368694254083309919300775340818247344025q^{25} \) \(\mathstrut +\mathstrut 846912971011690213566605075357303729752095555778833809408q^{26} \) \(\mathstrut -\mathstrut 41047056863228843015433175713593563710927499432759533085800q^{27} \) \(\mathstrut +\mathstrut 21567387724972458813165328982714265401134825135065654099968q^{28} \) \(\mathstrut +\mathstrut 111326085425145818041443197251224378309458208297450456511890q^{29} \) \(\mathstrut -\mathstrut 885918882743471050999671162877544194731846979137678323220480q^{30} \) \(\mathstrut +\mathstrut 569235729037298103337199475119942454463628025305311407367904q^{31} \) \(\mathstrut +\mathstrut 1606938044258990275541962092341162602522202993782792835301376q^{32} \) \(\mathstrut -\mathstrut 61496338392290018712116602303159393277177362460967281584608432q^{33} \) \(\mathstrut +\mathstrut 3718529020543483372117866554887703376895140544748422447497216q^{34} \) \(\mathstrut +\mathstrut 686063792178605252855004712321340223103017825503448653988751440q^{35} \) \(\mathstrut +\mathstrut 1082301172056493072005884950823187296448920008772088480877510656q^{36} \) \(\mathstrut +\mathstrut 1905873955549544706440129014763384940073631587560832679296783178q^{37} \) \(\mathstrut +\mathstrut 1057413629544062188578721654616793655668978257175191520605634560q^{38} \) \(\mathstrut +\mathstrut 55341897122822588861691194763490464172205587304131594611109087752q^{39} \) \(\mathstrut +\mathstrut 61006556408625487129528265106360708571198708550074892262498631680q^{40} \) \(\mathstrut +\mathstrut 557515048551694176466040728107302945533850171681227845833793371814q^{41} \) \(\mathstrut +\mathstrut 555163309707595815805961702714236316200515286126475176926995546112q^{42} \) \(\mathstrut +\mathstrut 4816630411633048565398768161252558622962223074904367325846509636084q^{43} \) \(\mathstrut +\mathstrut 5529875707106989812609201886273374147571497497532759933118656282624q^{44} \) \(\mathstrut +\mathstrut 59280547321194031305904576843492222218648695345165516645244087774690q^{45} \) \(\mathstrut +\mathstrut 54763300416573635501069024199006675365149081237125318433109912322048q^{46} \) \(\mathstrut +\mathstrut 231600947615420523849173663516598760086361299122810432076865700336208q^{47} \) \(\mathstrut +\mathstrut 8027477937062241009643024641139128166341580904865027845844772061184q^{48} \) \(\mathstrut +\mathstrut 488884314080089494567910985806986572998097715062788175105646760287759q^{49} \) \(\mathstrut +\mathstrut 497356219041719558041226153518799564426131705030743706023953327718400q^{50} \) \(\mathstrut -\mathstrut 1837331169353117595512341888565294661065129266300924338435898332834696q^{51} \) \(\mathstrut +\mathstrut 440238114923299753371707677529614312190829786646038517437070913306624q^{52} \) \(\mathstrut -\mathstrut 21528641216322884743608350078749181085164032178710898763780234892918086q^{53} \) \(\mathstrut -\mathstrut 43755715708510428229418018240591061767395048738154899626547797043445760q^{54} \) \(\mathstrut -\mathstrut 89816572570227989101559828878826497696785313927220351593090382298274440q^{55} \) \(\mathstrut +\mathstrut 25183099284078520120341186590139951248547525837939632864247704207753216q^{56} \) \(\mathstrut -\mathstrut 325340069796986008297665253801465753346468478251645125296159579689898960q^{57} \) \(\mathstrut +\mathstrut 75286071641180057909408657391764386614076857088308740887661460911554560q^{58} \) \(\mathstrut +\mathstrut 662728117869435253821040267452447193179435605293938374690618618632935780q^{59} \) \(\mathstrut -\mathstrut 349012329997895887088397976680482613704259662419279599200034977250017280q^{60} \) \(\mathstrut +\mathstrut 5048975939077047323128803569069390709396201365602898856556981048321479794q^{61} \) \(\mathstrut +\mathstrut 9116117924911410298568489050005822874983045648944636378308346515042598912q^{62} \) \(\mathstrut +\mathstrut 35173215480755526937203547670059135126186051786127848366226864546062716504q^{63} \) \(\mathstrut +\mathstrut 12367929453448690307083082505200429610792387278129332706851243409048338432q^{64} \) \(\mathstrut -\mathstrut 46970532294923265265023129333973302067057379269613739355602364732812844980q^{65} \) \(\mathstrut -\mathstrut 85577854048600869442930812288143040550866591698175368757526131512174444544q^{66} \) \(\mathstrut -\mathstrut 184322092783437411769758942053951188704733055756695883309417066466667239652q^{67} \) \(\mathstrut +\mathstrut 37232271882079251195560462656419419341118428617840476746502613144025694208q^{68} \) \(\mathstrut -\mathstrut 1054123015060350205925859853673196722512269353610573887874785135858940609888q^{69} \) \(\mathstrut -\mathstrut 996287423890495461170521495492361754626623104274349693454298030309071912960q^{70} \) \(\mathstrut -\mathstrut 132669540113241156337773373156585489641837182225722090893045827548208576136q^{71} \) \(\mathstrut +\mathstrut 1503136742063299860150000575820714377378196845590270384949419035128997347328q^{72} \) \(\mathstrut +\mathstrut 7696425638326647416396579150109019018025911343704474407389216781457420473734q^{73} \) \(\mathstrut +\mathstrut 4212064098262262325478369202498743646631535719551325155846923504531753926656q^{74} \) \(\mathstrut +\mathstrut 18977872456506974957673383259339807360208907159415673322444899002577185656100q^{75} \) \(\mathstrut +\mathstrut 12756850934055889436101651315354615554023393824186865510409514056674894151680q^{76} \) \(\mathstrut +\mathstrut 79234255678237437380795543050254678320307999826725923179432696830135390143136q^{77} \) \(\mathstrut -\mathstrut 19194055024117071479861721778884928595303790396409596723840106893964354781184q^{78} \) \(\mathstrut -\mathstrut 212618578002421286534586833449108430493711472521556413568669436503059830318800q^{79} \) \(\mathstrut +\mathstrut 5743853417351171108173897102142362498933090902160685763748905880182985850880q^{80} \) \(\mathstrut +\mathstrut 79959423980560659225591806521539208152890161349457691318726618762317783721167q^{81} \) \(\mathstrut -\mathstrut 81302500671107676733587921195547926095566395205746092567558139477877051097088q^{82} \) \(\mathstrut -\mathstrut 1790164754511083345892373552571735287262201429345682698674912994270538116665716q^{83} \) \(\mathstrut -\mathstrut 645813486544703820860776922234667779771591725210663437335867182260916894826496q^{84} \) \(\mathstrut +\mathstrut 1564580034919142525696449604971603614529377387637357241796475674956634730426740q^{85} \) \(\mathstrut +\mathstrut 3294852804201453056359711287948686818251054091134890322093338475113528057397248q^{86} \) \(\mathstrut +\mathstrut 19247758453282545358950677651684018818005971882317880274189014839018404283268040q^{87} \) \(\mathstrut +\mathstrut 2675468601351170940457271556225519303767276818916460328318668092982006908977152q^{88} \) \(\mathstrut +\mathstrut 18806926689964364108023787783339140105636748008665375255555474899453722318915830q^{89} \) \(\mathstrut +\mathstrut 23999109123248994341661836060153247826889034621429882658172676529941076865187840q^{90} \) \(\mathstrut +\mathstrut 24928419343167225592864366889445673788456676771135660868978914924860750956057104q^{91} \) \(\mathstrut -\mathstrut 12710310244783919270419517984655261594177748110585961976304586108996071557431296q^{92} \) \(\mathstrut -\mathstrut 64439148604228135905283217235236807481608325977029514498260568870448376088280192q^{93} \) \(\mathstrut -\mathstrut 104782899901035220448955601374807674550698541121210045778060945597865908716437504q^{94} \) \(\mathstrut -\mathstrut 52275812910445671050882146644756147065705406087550783931933553231946215194105400q^{95} \) \(\mathstrut -\mathstrut 31682548042536096350318607947502050425635048441430143704736754878624462318600192q^{96} \) \(\mathstrut -\mathstrut 688354148445678317962501387361884373903301824763889876908335398557166488547019602q^{97} \) \(\mathstrut +\mathstrut 351029176762994484562468308831559055488776837904037672844652650080466774672801792q^{98} \) \(\mathstrut +\mathstrut 1787417826570478154481491653017470632071111476329584129222745030630914564983314692q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{82}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.82.a.a \(3\) \(83.100\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-3\!\cdots\!28\) \(12\!\cdots\!88\) \(-2\!\cdots\!50\) \(-5\!\cdots\!24\) \(+\) \(q-2^{40}q^{2}+(4201453557463404996+\cdots)q^{3}+\cdots\)
2.82.a.b \(4\) \(83.100\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(43\!\cdots\!04\) \(-7\!\cdots\!04\) \(24\!\cdots\!80\) \(18\!\cdots\!92\) \(-\) \(q+2^{40}q^{2}+(-1777934344659239676+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{82}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{82}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{82}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)