Properties

Label 2.72.a
Level 2
Weight 72
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 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: \(\Q\)
Newform subspaces: \( 2 \)
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 and the Fricke involution.

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

Trace form

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

Decomposition of \(S_{72}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 34359738368 T )^{2} \))(\( ( 1 + 34359738368 T )^{3} \))
$3$ (\( 1 + 73290552574846824 T + \)\(13\!\cdots\!38\)\( T^{2} + \)\(55\!\cdots\!28\)\( T^{3} + \)\(56\!\cdots\!09\)\( T^{4} \))(\( 1 - 2362570292707236 T + \)\(13\!\cdots\!73\)\( T^{2} + \)\(28\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} - \)\(13\!\cdots\!24\)\( T^{5} + \)\(42\!\cdots\!23\)\( T^{6} \))
$5$ (\( 1 - \)\(40\!\cdots\!00\)\( T + \)\(46\!\cdots\!50\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!25\)\( T^{4} \))(\( 1 - \)\(47\!\cdots\!50\)\( T + \)\(39\!\cdots\!75\)\( T^{2} - \)\(39\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!75\)\( T^{4} - \)\(85\!\cdots\!50\)\( T^{5} + \)\(75\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 + \)\(29\!\cdots\!52\)\( T + \)\(13\!\cdots\!62\)\( T^{2} + \)\(29\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(68\!\cdots\!72\)\( T + \)\(22\!\cdots\!57\)\( T^{2} + \)\(13\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!51\)\( T^{4} + \)\(69\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!07\)\( T^{6} \))
$11$ (\( 1 - \)\(86\!\cdots\!04\)\( T + \)\(18\!\cdots\!26\)\( T^{2} - \)\(75\!\cdots\!44\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(39\!\cdots\!56\)\( T + \)\(83\!\cdots\!45\)\( T^{2} - \)\(71\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!95\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(65\!\cdots\!31\)\( T^{6} \))
$13$ (\( 1 + \)\(11\!\cdots\!64\)\( T + \)\(20\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!68\)\( T^{3} + \)\(15\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(25\!\cdots\!46\)\( T + \)\(13\!\cdots\!83\)\( T^{2} - \)\(43\!\cdots\!72\)\( T^{3} + \)\(16\!\cdots\!71\)\( T^{4} - \)\(38\!\cdots\!74\)\( T^{5} + \)\(18\!\cdots\!53\)\( T^{6} \))
$17$ (\( 1 + \)\(25\!\cdots\!12\)\( T + \)\(28\!\cdots\!02\)\( T^{2} + \)\(58\!\cdots\!96\)\( T^{3} + \)\(52\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(35\!\cdots\!18\)\( T + \)\(28\!\cdots\!07\)\( T^{2} - \)\(76\!\cdots\!04\)\( T^{3} + \)\(66\!\cdots\!31\)\( T^{4} - \)\(18\!\cdots\!02\)\( T^{5} + \)\(12\!\cdots\!37\)\( T^{6} \))
$19$ (\( 1 + \)\(15\!\cdots\!20\)\( T + \)\(64\!\cdots\!38\)\( T^{2} + \)\(93\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(28\!\cdots\!20\)\( T + \)\(18\!\cdots\!57\)\( T^{2} - \)\(34\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} - \)\(10\!\cdots\!20\)\( T^{5} + \)\(23\!\cdots\!59\)\( T^{6} \))
$23$ (\( 1 - \)\(22\!\cdots\!56\)\( T + \)\(61\!\cdots\!38\)\( T^{2} - \)\(10\!\cdots\!12\)\( T^{3} + \)\(23\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(37\!\cdots\!16\)\( T + \)\(65\!\cdots\!33\)\( T^{2} + \)\(39\!\cdots\!88\)\( T^{3} + \)\(31\!\cdots\!91\)\( T^{4} - \)\(86\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!83\)\( T^{6} \))
$29$ (\( 1 + \)\(96\!\cdots\!20\)\( T + \)\(98\!\cdots\!58\)\( T^{2} + \)\(65\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(28\!\cdots\!70\)\( T + \)\(87\!\cdots\!87\)\( T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(59\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!70\)\( T^{5} + \)\(30\!\cdots\!89\)\( T^{6} \))
$31$ (\( 1 + \)\(40\!\cdots\!76\)\( T + \)\(39\!\cdots\!06\)\( T^{2} + \)\(30\!\cdots\!56\)\( T^{3} + \)\(59\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(60\!\cdots\!64\)\( T + \)\(23\!\cdots\!25\)\( T^{2} + \)\(91\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!75\)\( T^{4} + \)\(36\!\cdots\!04\)\( T^{5} + \)\(45\!\cdots\!91\)\( T^{6} \))
$37$ (\( 1 + \)\(11\!\cdots\!92\)\( T + \)\(73\!\cdots\!42\)\( T^{2} + \)\(24\!\cdots\!96\)\( T^{3} + \)\(48\!\cdots\!69\)\( T^{4} \))(\( 1 + \)\(68\!\cdots\!62\)\( T + \)\(77\!\cdots\!87\)\( T^{2} + \)\(30\!\cdots\!76\)\( T^{3} + \)\(16\!\cdots\!31\)\( T^{4} + \)\(32\!\cdots\!78\)\( T^{5} + \)\(10\!\cdots\!97\)\( T^{6} \))
$41$ (\( 1 - \)\(12\!\cdots\!24\)\( T + \)\(62\!\cdots\!26\)\( T^{2} - \)\(41\!\cdots\!84\)\( T^{3} + \)\(10\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(25\!\cdots\!86\)\( T + \)\(93\!\cdots\!55\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!55\)\( T^{4} - \)\(26\!\cdots\!66\)\( T^{5} + \)\(33\!\cdots\!21\)\( T^{6} \))
$43$ (\( 1 + \)\(15\!\cdots\!24\)\( T + \)\(71\!\cdots\!58\)\( T^{2} + \)\(14\!\cdots\!68\)\( T^{3} + \)\(89\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(18\!\cdots\!36\)\( T + \)\(38\!\cdots\!53\)\( T^{2} - \)\(37\!\cdots\!32\)\( T^{3} + \)\(36\!\cdots\!71\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(84\!\cdots\!43\)\( T^{6} \))
$47$ (\( 1 + \)\(38\!\cdots\!12\)\( T + \)\(13\!\cdots\!42\)\( T^{2} + \)\(20\!\cdots\!36\)\( T^{3} + \)\(27\!\cdots\!09\)\( T^{4} \))(\( 1 + \)\(43\!\cdots\!32\)\( T + \)\(60\!\cdots\!17\)\( T^{2} + \)\(99\!\cdots\!76\)\( T^{3} + \)\(31\!\cdots\!51\)\( T^{4} + \)\(12\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!27\)\( T^{6} \))
$53$ (\( 1 + \)\(30\!\cdots\!04\)\( T + \)\(39\!\cdots\!98\)\( T^{2} + \)\(82\!\cdots\!88\)\( T^{3} + \)\(70\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(30\!\cdots\!06\)\( T + \)\(74\!\cdots\!03\)\( T^{2} - \)\(15\!\cdots\!72\)\( T^{3} + \)\(19\!\cdots\!91\)\( T^{4} - \)\(21\!\cdots\!54\)\( T^{5} + \)\(18\!\cdots\!73\)\( T^{6} \))
$59$ (\( 1 + \)\(16\!\cdots\!40\)\( T + \)\(17\!\cdots\!18\)\( T^{2} + \)\(87\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(46\!\cdots\!40\)\( T + \)\(11\!\cdots\!77\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!43\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!79\)\( T^{6} \))
$61$ (\( 1 + \)\(48\!\cdots\!56\)\( T + \)\(15\!\cdots\!06\)\( T^{2} + \)\(27\!\cdots\!16\)\( T^{3} + \)\(32\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(16\!\cdots\!66\)\( T + \)\(47\!\cdots\!35\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!35\)\( T^{4} - \)\(55\!\cdots\!86\)\( T^{5} + \)\(18\!\cdots\!81\)\( T^{6} \))
$67$ (\( 1 + \)\(13\!\cdots\!72\)\( T + \)\(12\!\cdots\!62\)\( T^{2} + \)\(60\!\cdots\!76\)\( T^{3} + \)\(20\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(29\!\cdots\!08\)\( T + \)\(41\!\cdots\!37\)\( T^{2} - \)\(35\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!71\)\( T^{4} - \)\(59\!\cdots\!12\)\( T^{5} + \)\(89\!\cdots\!87\)\( T^{6} \))
$71$ (\( 1 + \)\(14\!\cdots\!36\)\( T + \)\(10\!\cdots\!66\)\( T^{2} + \)\(40\!\cdots\!56\)\( T^{3} + \)\(75\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(91\!\cdots\!96\)\( T + \)\(80\!\cdots\!85\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!35\)\( T^{4} - \)\(69\!\cdots\!36\)\( T^{5} + \)\(20\!\cdots\!11\)\( T^{6} \))
$73$ (\( 1 + \)\(32\!\cdots\!24\)\( T + \)\(56\!\cdots\!98\)\( T^{2} + \)\(65\!\cdots\!48\)\( T^{3} + \)\(39\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(28\!\cdots\!86\)\( T + \)\(59\!\cdots\!63\)\( T^{2} - \)\(84\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!51\)\( T^{4} - \)\(10\!\cdots\!94\)\( T^{5} + \)\(77\!\cdots\!33\)\( T^{6} \))
$79$ (\( 1 - \)\(32\!\cdots\!60\)\( T + \)\(74\!\cdots\!58\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(35\!\cdots\!40\)\( T + \)\(17\!\cdots\!37\)\( T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(91\!\cdots\!23\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!39\)\( T^{6} \))
$83$ (\( 1 - \)\(29\!\cdots\!76\)\( T + \)\(35\!\cdots\!78\)\( T^{2} - \)\(52\!\cdots\!92\)\( T^{3} + \)\(32\!\cdots\!89\)\( T^{4} \))(\( 1 + \)\(49\!\cdots\!64\)\( T + \)\(13\!\cdots\!33\)\( T^{2} + \)\(21\!\cdots\!48\)\( T^{3} + \)\(23\!\cdots\!11\)\( T^{4} + \)\(15\!\cdots\!96\)\( T^{5} + \)\(58\!\cdots\!63\)\( T^{6} \))
$89$ (\( 1 - \)\(38\!\cdots\!20\)\( T + \)\(87\!\cdots\!78\)\( T^{2} - \)\(99\!\cdots\!80\)\( T^{3} + \)\(65\!\cdots\!21\)\( T^{4} \))(\( 1 + \)\(17\!\cdots\!70\)\( T + \)\(72\!\cdots\!67\)\( T^{2} + \)\(90\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!63\)\( T^{4} + \)\(11\!\cdots\!70\)\( T^{5} + \)\(16\!\cdots\!69\)\( T^{6} \))
$97$ (\( 1 + \)\(57\!\cdots\!72\)\( T + \)\(21\!\cdots\!02\)\( T^{2} + \)\(66\!\cdots\!16\)\( T^{3} + \)\(13\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(41\!\cdots\!58\)\( T + \)\(31\!\cdots\!47\)\( T^{2} - \)\(74\!\cdots\!04\)\( T^{3} + \)\(35\!\cdots\!91\)\( T^{4} - \)\(54\!\cdots\!22\)\( T^{5} + \)\(15\!\cdots\!77\)\( T^{6} \))
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