Properties

Label 2.72.a
Level 2
Weight 72
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newforms 2
Sturm bound 18
Trace bound 1

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 72 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(18\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{72}(\Gamma_0(2))\).

Total New Old
Modular forms 19 5 14
Cusp forms 17 5 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.
\(+\)\(3\)
\(-\)\(2\)

Trace form

\(5q \) \(\mathstrut -\mathstrut 34359738368q^{2} \) \(\mathstrut -\mathstrut 70927982282139588q^{3} \) \(\mathstrut +\mathstrut 5902958103587056517120q^{4} \) \(\mathstrut +\mathstrut 8770870597804349527314150q^{5} \) \(\mathstrut -\mathstrut 2599421508451315417696174080q^{6} \) \(\mathstrut -\mathstrut 977029489621691011470940739624q^{7} \) \(\mathstrut -\mathstrut 40564819207303340847894502572032q^{8} \) \(\mathstrut -\mathstrut 10628802685401817991279833906512615q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut -\mathstrut 34359738368q^{2} \) \(\mathstrut -\mathstrut 70927982282139588q^{3} \) \(\mathstrut +\mathstrut 5902958103587056517120q^{4} \) \(\mathstrut +\mathstrut 8770870597804349527314150q^{5} \) \(\mathstrut -\mathstrut 2599421508451315417696174080q^{6} \) \(\mathstrut -\mathstrut 977029489621691011470940739624q^{7} \) \(\mathstrut -\mathstrut 40564819207303340847894502572032q^{8} \) \(\mathstrut -\mathstrut 10628802685401817991279833906512615q^{9} \) \(\mathstrut -\mathstrut 25280082439877112833087632519987200q^{10} \) \(\mathstrut +\mathstrut 12653540208550655455162359606709176660q^{11} \) \(\mathstrut -\mathstrut 83736981556687009481316699942590349312q^{12} \) \(\mathstrut +\mathstrut 1402114600026447424730787612645979920382q^{13} \) \(\mathstrut +\mathstrut 13556707247207600267061485252960961167360q^{14} \) \(\mathstrut -\mathstrut 463150161991131265454531829885637194022200q^{15} \) \(\mathstrut +\mathstrut 6968982874540819731729911960202612970618880q^{16} \) \(\mathstrut +\mathstrut 10181711179932788372373329307970525336341306q^{17} \) \(\mathstrut -\mathstrut 55594262306952284716516681362901274455965696q^{18} \) \(\mathstrut +\mathstrut 1325510285653528819282720455375706482906117100q^{19} \) \(\mathstrut +\mathstrut 10354816334164527159232103064349922641418649600q^{20} \) \(\mathstrut -\mathstrut 42823012539078891306604743063084645704384963040q^{21} \) \(\mathstrut +\mathstrut 161989115483968950704427146679034165000314814464q^{22} \) \(\mathstrut +\mathstrut 2576030858288230447152357060176894489164293220872q^{23} \) \(\mathstrut -\mathstrut 3068855251590236532635341250365968841950804049920q^{24} \) \(\mathstrut +\mathstrut 77470851888339292664441672086898049972906622581875q^{25} \) \(\mathstrut -\mathstrut 125659320313814536018994140772351903971115983175680q^{26} \) \(\mathstrut -\mathstrut 628306186559349752109366225672186417181006446589800q^{27} \) \(\mathstrut -\mathstrut 1153472828641177378005099339919048882393116443672576q^{28} \) \(\mathstrut -\mathstrut 6860435235110955235888980386324756040938899236397650q^{29} \) \(\mathstrut -\mathstrut 43906291162305836238335594921990591560880111380070400q^{30} \) \(\mathstrut -\mathstrut 101138229648341131683944451281012448339964583121499040q^{31} \) \(\mathstrut -\mathstrut 47890485652059026823698344598447161988085597568237568q^{32} \) \(\mathstrut -\mathstrut 2872718681843290704441919969985428791578040464860098576q^{33} \) \(\mathstrut -\mathstrut 2089174376329531075324499499506304385994387929631293440q^{34} \) \(\mathstrut -\mathstrut 26590843528869381245522388456150681325703999561842935600q^{35} \) \(\mathstrut -\mathstrut 12548275388644105841640834647323935802083744889448693760q^{36} \) \(\mathstrut -\mathstrut 179478717156161737952199117531851730305653000429723252154q^{37} \) \(\mathstrut -\mathstrut 149700690955753984862193540504327067209558169760793886720q^{38} \) \(\mathstrut +\mathstrut 94416102546770537282267371476315558928605691786370002920q^{39} \) \(\mathstrut -\mathstrut 29845453499564290134396131310707686627842346883796172800q^{40} \) \(\mathstrut +\mathstrut 3846369123867234867455010237081088489417820133981248732610q^{41} \) \(\mathstrut +\mathstrut 8690903445829590585189367602466647512114484579313630838784q^{42} \) \(\mathstrut +\mathstrut 17338922504730309176123187596751839749660440160795581433012q^{43} \) \(\mathstrut +\mathstrut 14938663542625748949816286011124914136888227642952478883840q^{44} \) \(\mathstrut +\mathstrut 37987004521035615334754948260762107212624595038820793319550q^{45} \) \(\mathstrut +\mathstrut 62889082130272532937079988402682372388244887148853272248320q^{46} \) \(\mathstrut -\mathstrut 429135394886087493770977961607764094246468597695160039852144q^{47} \) \(\mathstrut -\mathstrut 98859178769993095439877318533476707372172013910805381644288q^{48} \) \(\mathstrut -\mathstrut 1558097987331794956016239991867084432339286624611707370182035q^{49} \) \(\mathstrut -\mathstrut 2139017052573834010203754115251428811015502126501467258880000q^{50} \) \(\mathstrut -\mathstrut 10575181535498047247976526096408514112065655458660075650599240q^{51} \) \(\mathstrut +\mathstrut 1655324748076768470762542683337972394181722078994982363987968q^{52} \) \(\mathstrut +\mathstrut 27771480100294890260766648478005497510025062705378537144480502q^{53} \) \(\mathstrut +\mathstrut 43554539198535377139106282636115721830138588252581208575180800q^{54} \) \(\mathstrut +\mathstrut 138558736288553393791241265728689928701455253036545644497003800q^{55} \) \(\mathstrut +\mathstrut 16004934980572296291959343006311041571006711029663586205040640q^{56} \) \(\mathstrut +\mathstrut 552566598169568593712145239597492482576462242110041276545208080q^{57} \) \(\mathstrut -\mathstrut 430768800796615605993931812324062578087451922967420865945272320q^{58} \) \(\mathstrut -\mathstrut 1174522234873922120928153069161312202834475552876806608627334300q^{59} \) \(\mathstrut -\mathstrut 546791200380641247710709993664263053508874230184240483192012800q^{60} \) \(\mathstrut -\mathstrut 4863879043394431588707414533243661135597392802374498751561167090q^{61} \) \(\mathstrut +\mathstrut 715827820919445095111863683232549711182032799328549298644189184q^{62} \) \(\mathstrut +\mathstrut 1097403255444986437526353509940712168269796929769819540232937272q^{63} \) \(\mathstrut +\mathstrut 8227522786606030210774845912786752524913679328167899316743045120q^{64} \) \(\mathstrut +\mathstrut 64703442873238883952730087377637269598487910779977248208381371300q^{65} \) \(\mathstrut +\mathstrut 54280979636830909171702126269462964204393047322528230578312970240q^{66} \) \(\mathstrut +\mathstrut 160587947090088765174086426288352771987220715720185859562351491836q^{67} \) \(\mathstrut +\mathstrut 12020442903593436804339686791429503070247741817174352332790431744q^{68} \) \(\mathstrut +\mathstrut 206861013693649126634868127025776206951904728793061609744735975520q^{69} \) \(\mathstrut +\mathstrut 117995164260329317595196009846523553792560947455416245599128780800q^{70} \) \(\mathstrut -\mathstrut 540975001646035036849595506919675689377881219609882148412303342440q^{71} \) \(\mathstrut -\mathstrut 65634120239553687260700061362702410227631051770752626459991343104q^{72} \) \(\mathstrut -\mathstrut 493371973941352299010722149123744291895288018436097562205818780638q^{73} \) \(\mathstrut -\mathstrut 1481812414530170103203861927846532244801425966258053648728761303040q^{74} \) \(\mathstrut +\mathstrut 8355459350653696765097408822764273370397206988508516284275906182500q^{75} \) \(\mathstrut +\mathstrut 1564886336417298409206290858325907251950347510525226244133774950400q^{76} \) \(\mathstrut +\mathstrut 3611586259151978045873789678616930489070161307906038323413200819552q^{77} \) \(\mathstrut +\mathstrut 15689466188520597561055981542967906045955515089273553895606125068288q^{78} \) \(\mathstrut +\mathstrut 68407353017907947133105662345791767030213156146981156655342824585200q^{79} \) \(\mathstrut +\mathstrut 12224809398182422748460925484117190472447105178890074377997936230400q^{80} \) \(\mathstrut -\mathstrut 92110580265744318660232873064031880187735384501661563356167803540595q^{81} \) \(\mathstrut -\mathstrut 43263226932205879807538549281423394125232361098210197042043987951616q^{82} \) \(\mathstrut -\mathstrut 462431908164653495682410201119523266166808705767767883712370716520788q^{83} \) \(\mathstrut -\mathstrut 50556489777513174837671293461166995210134062551857783726792465448960q^{84} \) \(\mathstrut -\mathstrut 650648743460101540888363505415176392548070572704443583128819540574100q^{85} \) \(\mathstrut -\mathstrut 703663374002101694429668962592772073223408265270889950427032615649280q^{86} \) \(\mathstrut +\mathstrut 687295582331127203690300598059191003488341930862221453919507297367080q^{87} \) \(\mathstrut +\mathstrut 191242992387798810021694811771387247159324100604356524094954967924736q^{88} \) \(\mathstrut +\mathstrut 3715768144085451376614226040845977844148269767519963874570524425044050q^{89} \) \(\mathstrut +\mathstrut 1492233497358075465851610313531996223698852492419440484521371867545600q^{90} \) \(\mathstrut -\mathstrut 915815722483457918281503064193016464806477073013626088447027264966640q^{91} \) \(\mathstrut +\mathstrut 3041240446004566066241930447062732179186900059264170536705508641865728q^{92} \) \(\mathstrut -\mathstrut 9792641842112833965536483951253071508919251413753101313324589767158656q^{93} \) \(\mathstrut -\mathstrut 11723472961087541613965282430506838326173846879857508411173358943600640q^{94} \) \(\mathstrut -\mathstrut 5977957607192802395656169749882212142699617932843036690987662291887000q^{95} \) \(\mathstrut -\mathstrut 3623064795222056370310079435652767228100307337514111854737925162926080q^{96} \) \(\mathstrut -\mathstrut 16666024235652006649902412058319911035783942230801920498220735785536214q^{97} \) \(\mathstrut +\mathstrut 16974661415608528884927377182611327993795373631659676397521081525600256q^{98} \) \(\mathstrut +\mathstrut 123803061519529273406202571001432523025749676720081271360725002759840420q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{72}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.72.a.a \(2\) \(63.849\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(68719476736\) \(-7\!\cdots\!24\) \(40\!\cdots\!00\) \(-2\!\cdots\!52\) \(-\) \(q+2^{35}q^{2}+(-36645276287423412+\cdots)q^{3}+\cdots\)
2.72.a.b \(3\) \(63.849\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-103079215104\) \(23\!\cdots\!36\) \(47\!\cdots\!50\) \(-6\!\cdots\!72\) \(+\) \(q-2^{35}q^{2}+(787523430902412+\beta _{1}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{72}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{72}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{72}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)