↑

Properties

Label 9.90.a
Level $9$
Weight $90$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $4$
Sturm bound $90$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 90 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(90\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{90}(\Gamma_0(9))\).

Total New Old
Modular forms 91 37 54
Cusp forms 87 36 51
Eisenstein series 4 1 3

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

\(3\)Dim
\(+\)\(14\)
\(-\)\(22\)

Trace form

\( 36 q - 13815144306990 q^{2} + 9494254941648978191253428052 q^{4} + 8378288596333374877629303041844 q^{5} - 22141006204118239644392005856890047360 q^{7} + 31906320320401034235282184071821556336120 q^{8} + O(q^{10}) \) \( 36 q - 13815144306990 q^{2} + 9494254941648978191253428052 q^{4} + 8378288596333374877629303041844 q^{5} - 22141006204118239644392005856890047360 q^{7} + 31906320320401034235282184071821556336120 q^{8} - 134038824668325674393648794052448340317724236 q^{10} + 6509639905902455149255015416959460954658399824 q^{11} + 13971971499476806189828829969885313684546957957600 q^{13} + 160863021054237332691202710522443977525553369805744 q^{14} + 2489613658335614921428740416367092978709618079888084752 q^{16} + 838459091653533362788447762101347006496458185402830180 q^{17} - 1672064984005446341410389286570052508600500675989867787232 q^{19} + 26112865566263594307753685540162317834179347722912682361736 q^{20} - 119254586282802139640134008757942649634208839960142818811320 q^{22} - 8962749613517740893960933756633480926099382713735133447554160 q^{23} + 1218111300149993219901490978423187347518188808974742686561576196 q^{25} + 2332397522602045637918391522135787948944237040149062315067445116 q^{26} - 6429467764521716914299716453318228887721588353548022591650745440 q^{28} + 71638218169044553674138000559279450054643297987083976529868564116 q^{29} - 4173444102746117207116928046794182040462450980437839340334253636640 q^{31} + 52702908634101277357268615544353320464624398863159192435985469807840 q^{32} + 183377506332226202571017616559982306314930967067792513095470029889020 q^{34} - 731756947721216374032789029324049775340148830726863692550775171843280 q^{35} + 4419173627818762190286115792939263521543117592034887367404494520121560 q^{37} - 15552718067922022346410355142392585280060168315972982977713778085910520 q^{38} + 363427221340565749908754799311072387727058741124797071655114798878922288 q^{40} + 1941573294930497660166939068775529069619813841945404456751838640012854452 q^{41} + 214539555787108416740969710978230808277715789596551538392278934712926080 q^{43} - 24207082866886693668071321149739255834207530668845187013813164446533035888 q^{44} + 131506405800281669795802709855269954471930784779788768241561377892639205968 q^{46} + 316975679771727745479460197573319274572312312685420208064258432085633167920 q^{47} + 7143641309673518741877636270133953744631765677589329741244041878944465031844 q^{49} + 7232632159406320670481565950540436320489625308393438715465047119397844165806 q^{50} + 35158335055098743348131097851274441733184148215536513291806910561480416080920 q^{52} - 25350402070315611342151876227095113903632047291205570905700989953791062377500 q^{53} - 24075316431541153412323953361586658537159926841887966405244046142971548556224 q^{55} - 251451773778168441553353639395543738276475197789892356731102634893006478080320 q^{56} - 509445432112141526112221768870579200075847882314517529371222008400133194202300 q^{58} + 1702013909769262149460247192817068893706706767126322534840893085369457513579392 q^{59} + 27145056353584836777635266369371004179892200392473622098706393993564693782683480 q^{61} + 18443839744084581317924047634878318222771441923045538375689116912276650148589600 q^{62} + 571117690573045023464111232381706027100510567661320022289589088551063526703858752 q^{64} + 921767802254428866439157560369554443097880854942303464586342627315981604445160312 q^{65} + 2747356811924453029321345856616977788678596561783219570888766303693071712243840800 q^{67} - 641762985096982852422362731162613154772660694257972915628791390882650370484415720 q^{68} - 10174022431787029910932853947549220264036648226328722142063434408969445596834692640 q^{70} - 14961281715796686451292853068413668932488330853308657453689383377304437550413161264 q^{71} - 125414711371796425465298418739704130048784556995838746682338050102853366075784153840 q^{73} - 53728200480408930083022038596493565317126381484974241352594544662989288075451757076 q^{74} - 356974242895738942167665221444190975600803251945968974377497133906740400521998954224 q^{76} - 48121656327526369989851907753494698057008110684686333860953596717990833009944230560 q^{77} + 2819781643357627866775583815109709775781412612091532479320843259851120061944614879424 q^{79} + 10931726145428123903823862165703831993922311600303122678141276168417969375593865784736 q^{80} + 12491410764187220469042598329667341868481502891540815652007331478960697807638054274380 q^{82} + 23111555386594531264298181710780229350965182925437535456899317852742188260922214239760 q^{83} - 184039900313339886738771584809835286201051273812033690944671960807846156645569423665608 q^{85} - 148100807548304045982158701016784270668147892086629301897909255422240901589677779170696 q^{86} - 345161799060482845914567216592377137039645407171611713104446897374295304848849670273440 q^{88} + 325265390775569708815845634213676499421821832428689035847555438508279004781992465142628 q^{89} + 3860250106672241255270534942151902723656401725582540175826444162732114995365849298441888 q^{91} + 3794209583113715897167905338121233768205429358624356237536976643236458582012707761864480 q^{92} + 2108213347880518284056060019772238301215013363838463819961347459211252148249233498642912 q^{94} - 16447808554826651620924269020628392807878186143018013635576760186138865374041584231956224 q^{95} + 3548443491053517432505305229847968356952548698985696584409665009639612311492838045331600 q^{97} + 77713316052871858704077238702235692130560366004734609213766208095415994683039055536231490 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{90}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
9.90.a.a 9.a 1.a $7$ $451.462$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-25\!\cdots\!92\) \(0\) \(23\!\cdots\!14\) \(-43\!\cdots\!84\) $-$ $\mathrm{SU}(2)$ \(q+(-3598993697727-\beta _{1})q^{2}+\cdots\)
9.90.a.b 9.a 1.a $7$ $451.462$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(31\!\cdots\!08\) \(0\) \(-10\!\cdots\!50\) \(38\!\cdots\!92\) $-$ $\mathrm{SU}(2)$ \(q+(4486761478773+\beta _{1})q^{2}+\cdots\)
9.90.a.c 9.a 1.a $8$ $451.462$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-20\!\cdots\!06\) \(0\) \(16\!\cdots\!80\) \(76\!\cdots\!72\) $-$ $\mathrm{SU}(2)$ \(q+(-2503689846788-\beta _{1})q^{2}+\cdots\)
9.90.a.d 9.a 1.a $14$ $451.462$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(0\) \(0\) \(-24\!\cdots\!40\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(158511273851367030316695397+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{90}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{90}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{90}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{90}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)