Properties

Label 9.74.a
Level $9$
Weight $74$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $4$
Sturm bound $74$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 74 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(74\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{74}(\Gamma_0(9))\).

Total New Old
Modular forms 75 31 44
Cusp forms 71 30 41
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
\(+\)\(12\)
\(-\)\(18\)

Trace form

\( 30 q - 23369856750 q^{2} + 146334199690942151938644 q^{4} + 14390748999542377719422244 q^{5} - 8761908508171710478503105837360 q^{7} - 1704090635258843910856136812481160 q^{8} + O(q^{10}) \) \( 30 q - 23369856750 q^{2} + 146334199690942151938644 q^{4} + 14390748999542377719422244 q^{5} - 8761908508171710478503105837360 q^{7} - 1704090635258843910856136812481160 q^{8} + 2814935259330323878569614456669126964 q^{10} + 139125474186027087679626225093630348600 q^{11} - 49094325732096197424363524487029679319500 q^{13} + 1952628903334357435373709765972347266274352 q^{14} + 825716191268928231990745531142676305773538064 q^{16} - 824023217957467950724723432777198579080380580 q^{17} - 112919306642652100133390016372623444036450777448 q^{19} + 774335284095359134456588121414461071731680578696 q^{20} - 18604743580701928620445485100959646642753651924920 q^{22} + 100745233947787733204991999701814462283447479411600 q^{23} + 6727872316842451098536904493766129457690625415208306 q^{25} - 17898384262997390732342646175313561448196609750132356 q^{26} - 65843337531362226461152404106373720413088966049203040 q^{28} + 98018265953181929188359411333254532628627224473260692 q^{29} + 3564727052795299984339294318479281672782881181470899040 q^{31} - 6775572173869419759685933432929070768058930470254201120 q^{32} - 127547057125118955051932009821343769734678013025623059844 q^{34} + 229272902223641210409272421321984075600446736909195379200 q^{35} - 1370745735867298148851026346524539492240042949235441603020 q^{37} - 18932868357889054442024889865690077926067124974044858137080 q^{38} + 86047107283742460765356633347665845266907591174283349182768 q^{40} + 102982949984475362418095305611061267346144504571565535620716 q^{41} - 675232728624313426854332905648999087898126240954052758974680 q^{43} + 367246953584975903028628053784419098391590209451728793839248 q^{44} + 4938864612920687600883209477188662945308520519915474097855824 q^{46} + 1153884722930524703402946932923544379070130138574388214127360 q^{47} + 152594307485252444212059639944328811127227412709036119332235118 q^{49} + 140620204117602181510629073107724842396186997667820951130097646 q^{50} + 1006298612290540542931996580332106496531824675271249146689203480 q^{52} - 106424988802563098074317885365939008770286913839980972609603420 q^{53} - 5172506256875754978997472475478683726776787429731630192149394064 q^{55} + 3602265481776389260603645344014133732283302292852140721422425280 q^{56} - 83529227013630801500006894760141841387896718576010287954894045500 q^{58} - 49984396630229824435111431513397844626345660251719235901793169096 q^{59} - 123169208072529914222388671074767427883138549457609422200299940284 q^{61} + 555686760763584790416873594778826974751940683302515521381627748000 q^{62} + 3207309826365786381683045867046449394584452467128502349115009248320 q^{64} - 689190540488371610400476881600694254048410665782109610267982594248 q^{65} + 2614993700422390459507424898782798433910928201492955050860914804600 q^{67} + 14225880228434803401462147987839914605433934271721010447358779513880 q^{68} - 60748475224255963614769611759324103360762995014574562086231408556320 q^{70} + 128062254369082888926645866892895909655495135483231510561414962583664 q^{71} - 13220052289112410192691656515596078649240054749331877062599037563460 q^{73} + 194096296326621869697503017391786759391601211749309862894908837532652 q^{74} - 1824635067839255269227452427727008036294635235562048130282166241244400 q^{76} + 3530983837600232657060192275000666699588607460189174099795153897196800 q^{77} - 3650121983097869417651735867683540329672698768483471714001487240133568 q^{79} + 24389616527041600642144497852492740180314641704786117190232176120202656 q^{80} - 7721249799873782847473618167038859124404610666729744191971101145467700 q^{82} + 41474232258935659291619018672190962545342732086194560710959275846916040 q^{83} - 99568558626310339044059823226961309351274435232278830569929532241070408 q^{85} + 280718552781544448134542140346054570217080280203377932911933000268300920 q^{86} - 268798596627176104123227944893341869900193387236079002781276093967051680 q^{88} + 340493802614077054772144510678601157968972034668677965546609198054766316 q^{89} - 389500299615486721283649374245840742231432933832434796409200037491732576 q^{91} + 1526143867696300349330686472362360419008785621025159836973088828219344160 q^{92} + 1501060085454864277632560209698242677068504367114639704260914912150406880 q^{94} - 2649882625283732762195274397352298215155973081829473225351901433936381584 q^{95} + 6880327527427864608453171215509160734787960274977247235790036258780745420 q^{97} - 25957220874051698526791109302156968000188855104784869050150467917988447870 q^{98} + O(q^{100}) \)

Decomposition of \(S_{74}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.74.a.a 9.a 1.a $5$ $303.736$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(92089333488\) \(0\) \(-23\!\cdots\!50\) \(-43\!\cdots\!08\) $-$ $\mathrm{SU}(2)$ \(q+(18417866698+\beta _{1})q^{2}+\cdots\)
9.74.a.b 9.a 1.a $6$ $303.736$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-71256829788\) \(0\) \(15\!\cdots\!84\) \(-91\!\cdots\!84\) $-$ $\mathrm{SU}(2)$ \(q+(-11876138298-\beta _{1})q^{2}+\cdots\)
9.74.a.c 9.a 1.a $7$ $303.736$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-44202360450\) \(0\) \(22\!\cdots\!10\) \(10\!\cdots\!52\) $-$ $\mathrm{SU}(2)$ \(q+(-6314622921+\beta _{1})q^{2}+\cdots\)
9.74.a.d 9.a 1.a $12$ $303.736$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-13\!\cdots\!20\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4752100926420020816128+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{74}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{74}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{74}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{74}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)