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Properties

Label 9.16.a
Level $9$
Weight $16$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $5$
Sturm bound $16$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(16\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(9))\).

Total New Old
Modular forms 17 7 10
Cusp forms 13 6 7
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
\(+\)\(3\)
\(-\)\(3\)

Trace form

\( 6 q + 90 q^{2} + 149748 q^{4} - 111330 q^{5} - 4637508 q^{7} - 2789784 q^{8} + O(q^{10}) \) \( 6 q + 90 q^{2} + 149748 q^{4} - 111330 q^{5} - 4637508 q^{7} - 2789784 q^{8} + 72765540 q^{10} - 91787220 q^{11} + 155922528 q^{13} - 1085740992 q^{14} + 6338575248 q^{16} - 5398648326 q^{17} + 142099284 q^{19} - 13005535320 q^{20} + 19969914312 q^{22} - 2268016344 q^{23} - 914723490 q^{25} + 110758903164 q^{26} - 423973368384 q^{28} + 201269476278 q^{29} + 256723352868 q^{31} + 115812052128 q^{32} - 66406867956 q^{34} - 237548797920 q^{35} - 742162256988 q^{37} - 1258670491128 q^{38} + 5578833690000 q^{40} - 2870975534814 q^{41} + 3774514173276 q^{43} - 8906559408 q^{44} - 13698912405456 q^{46} - 186132202848 q^{47} + 964196339814 q^{49} + 18637799721750 q^{50} - 13081373394504 q^{52} + 2605172709726 q^{53} + 14532457260600 q^{55} + 24312633327360 q^{56} - 67113774788364 q^{58} - 40346133868404 q^{59} + 57964344662388 q^{61} + 11039482068336 q^{62} + 170352596954688 q^{64} - 159905667699180 q^{65} + 1735689827676 q^{67} + 18210068335800 q^{68} - 200566302007680 q^{70} - 50889630307176 q^{71} + 229110272621184 q^{73} + 220679668039788 q^{74} - 326909774547888 q^{76} + 68507717213376 q^{77} + 317743497713172 q^{79} + 568465455515040 q^{80} - 1415898324911556 q^{82} - 154016576449740 q^{83} - 88713966524220 q^{85} + 100990225245240 q^{86} + 1585183587204384 q^{88} - 572646020260446 q^{89} + 436763017451544 q^{91} - 888807497693472 q^{92} + 1410397675781760 q^{94} - 414257174849160 q^{95} - 240276113030544 q^{97} - 1374140508526422 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.16.a.a 9.a 1.a $1$ $12.842$ \(\Q\) None \(-216\) \(0\) \(-52110\) \(2822456\) $-$ $\mathrm{SU}(2)$ \(q-6^{3}q^{2}+13888q^{4}-52110q^{5}+\cdots\)
9.16.a.b 9.a 1.a $1$ $12.842$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1244900\) $+$ $N(\mathrm{U}(1))$ \(q-2^{15}q^{4}+1244900q^{7}+397771850q^{13}+\cdots\)
9.16.a.c 9.a 1.a $1$ $12.842$ \(\Q\) None \(72\) \(0\) \(221490\) \(-2149000\) $-$ $\mathrm{SU}(2)$ \(q+72q^{2}-27584q^{4}+221490q^{5}+\cdots\)
9.16.a.d 9.a 1.a $1$ $12.842$ \(\Q\) None \(234\) \(0\) \(-280710\) \(-1373344\) $-$ $\mathrm{SU}(2)$ \(q+234q^{2}+21988q^{4}-280710q^{5}+\cdots\)
9.16.a.e 9.a 1.a $2$ $12.842$ \(\Q(\sqrt{370}) \) None \(0\) \(0\) \(0\) \(-5182520\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+87112q^{4}+464\beta q^{5}-2591260q^{7}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)