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Properties

Label 9.38.a
Level $9$
Weight $38$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $4$
Sturm bound $38$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(38\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_0(9))\).

Total New Old
Modular forms 39 16 23
Cusp forms 35 15 20
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
\(+\)\(6\)
\(-\)\(9\)

Trace form

\( 15 q + 67746 q^{2} + 785430810132 q^{4} + 8198963257794 q^{5} + 1530671576429640 q^{7} - 13056161204613288 q^{8} + O(q^{10}) \) \( 15 q + 67746 q^{2} + 785430810132 q^{4} + 8198963257794 q^{5} + 1530671576429640 q^{7} - 13056161204613288 q^{8} + 701810811167570964 q^{10} - 16892975854486383540 q^{11} + 52320102428576639658 q^{13} - 2157518781494053266864 q^{14} + 60188323265818471794192 q^{16} + 35505852537394505978478 q^{17} + 340103894093608822057596 q^{19} + 5879902135370202309024456 q^{20} + 5423184383818460315790024 q^{22} + 49503812521744568187729960 q^{23} + 197740759565853839268042441 q^{25} + 28850718701503607618665212 q^{26} - 289768883906346168359703840 q^{28} - 3011713332589333357687728294 q^{29} - 781483822639604171690107920 q^{31} - 22636540887382406531598141600 q^{32} + 34944844065537508001921479068 q^{34} + 1478831265694675394820929280 q^{35} - 48698215244285736812332707510 q^{37} + 764165442780594139231937643144 q^{38} - 15584835515266358737879195152 q^{40} + 42713772542541291383087934678 q^{41} - 357193640373124911022591078236 q^{43} - 8456608498093768123832483825136 q^{44} - 6101690305449575035049134096368 q^{46} + 3241423397073527591847831069312 q^{47} + 6505178018778518629376164804119 q^{49} + 51936896861448473228503326404286 q^{50} + 201968765402745817321803572497752 q^{52} + 81355700301008161059189756616770 q^{53} - 396692024744375825342851041563304 q^{55} - 455988078172087222021525366964160 q^{56} - 603179502508704160403830986424668 q^{58} + 468992613634879660085979512345292 q^{59} - 1514039574412423112792120285935182 q^{61} + 1652905151731163411958039260712000 q^{62} + 3759236298641461088639243569404480 q^{64} + 10593915558478990030921473507246492 q^{65} - 266882168285961617964776700445908 q^{67} + 18057929447638014873272440437407832 q^{68} - 36774376798864258069820421993439200 q^{70} + 12003061736760778187081479098579672 q^{71} - 57831362673923186462758017850739970 q^{73} + 109748095358061943362910860700827756 q^{74} - 123611336921910076486872284521676400 q^{76} + 362439451050831948798822172417618560 q^{77} - 269095987265518808553895680263862624 q^{79} + 1823603543904832845123315675847913376 q^{80} - 886552596245968138461548193728960340 q^{82} + 1251081155754206345091629844210042228 q^{83} - 1975648459900377977331028992833131908 q^{85} + 4025042851501199044794491013596379000 q^{86} - 2423730904070903005805695363984983072 q^{88} + 3251871510831732053816609235628836438 q^{89} - 8162730110137418314902634761148155888 q^{91} + 24462997923659378849989105119795745440 q^{92} - 9326919510839415053010162109476526560 q^{94} + 29601754868160475604558651229979722456 q^{95} - 25066318508284790409476541519381690138 q^{97} + 54164241523199545963406990749090560978 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.38.a.a 9.a 1.a $2$ $78.043$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(194400\) \(0\) \(-55\!\cdots\!00\) \(-34\!\cdots\!00\) $-$ $\mathrm{SU}(2)$ \(q+(97200-\beta )q^{2}+(18860134912+\cdots)q^{4}+\cdots\)
9.38.a.b 9.a 1.a $3$ $78.043$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(310908\) \(0\) \(96\!\cdots\!90\) \(-46\!\cdots\!44\) $-$ $\mathrm{SU}(2)$ \(q+(103636+\beta _{1})q^{2}+(112825533616+\cdots)q^{4}+\cdots\)
9.38.a.c 9.a 1.a $4$ $78.043$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-437562\) \(0\) \(40\!\cdots\!04\) \(66\!\cdots\!84\) $-$ $\mathrm{SU}(2)$ \(q+(-109391+\beta _{1})q^{2}+(86524834843+\cdots)q^{4}+\cdots\)
9.38.a.d 9.a 1.a $6$ $78.043$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(29\!\cdots\!00\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(10522497328+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{38}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{38}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{38}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)