Properties

Label 9.22.a
Level $9$
Weight $22$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $6$
Sturm bound $22$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(22\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(9))\).

Total New Old
Modular forms 23 9 14
Cusp forms 19 8 11
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.
\(+\)\(3\)
\(-\)\(5\)

Trace form

\( 8 q + 738 q^{2} + 7749908 q^{4} + 15764994 q^{5} - 100527428 q^{7} + 9551476440 q^{8} + O(q^{10}) \) \( 8 q + 738 q^{2} + 7749908 q^{4} + 15764994 q^{5} - 100527428 q^{7} + 9551476440 q^{8} - 16900144236 q^{10} - 75612986748 q^{11} + 113810666956 q^{13} + 592920650448 q^{14} + 3910337823248 q^{16} + 8628605985798 q^{17} - 10662351659300 q^{19} + 14714389148616 q^{20} - 171534735404856 q^{22} + 609437868332904 q^{23} - 664483930375924 q^{25} + 2989931563865340 q^{26} - 2252900947835936 q^{28} + 5675325104176074 q^{29} - 10391139615445580 q^{31} + 50294786793537888 q^{32} - 16837742531975652 q^{34} + 50227680602579760 q^{35} - 28530856099631648 q^{37} + 2687179498625160 q^{38} - 10740114847398672 q^{40} + 22700756653984494 q^{41} + 11499738512971156 q^{43} - 458077673064011760 q^{44} + 824931070322758416 q^{46} - 1201161753249386832 q^{47} + 285465997666025016 q^{49} - 3553060248038159874 q^{50} + 1040821712107144792 q^{52} - 258396493840894926 q^{53} + 2963017709798311656 q^{55} + 4026414482735883840 q^{56} - 10703738500402761180 q^{58} + 2430508798668060468 q^{59} - 10346467496967980192 q^{61} + 23991502848168805056 q^{62} + 24106700803027558976 q^{64} + 29197660799078311932 q^{65} - 50050446811785975548 q^{67} + 51210264797751150936 q^{68} - 17553160186729786080 q^{70} - 17816961785052742920 q^{71} + 118493973361426701076 q^{73} - 168827260634054945172 q^{74} - 72749295691041449072 q^{76} - 88961926910059121184 q^{77} + 42169984472805393508 q^{79} - 323207258606355191904 q^{80} + 454740848769958082604 q^{82} - 28836035863347453156 q^{83} - 188834896121087115108 q^{85} - 561142243596068882568 q^{86} - 24966377872189224480 q^{88} + 548363532687143292702 q^{89} - 596833249332639026344 q^{91} + 1331501228265777010848 q^{92} + 7945648195379471136 q^{94} + 2381010806096571520296 q^{95} - 802385631083615464748 q^{97} + 135856630542555134226 q^{98} + O(q^{100}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.22.a.a 9.a 1.a $1$ $25.153$ \(\Q\) None \(-1728\) \(0\) \(41512770\) \(538429808\) $-$ $\mathrm{SU}(2)$ \(q-12^{3}q^{2}+888832q^{4}+41512770q^{5}+\cdots\)
9.22.a.b 9.a 1.a $1$ $25.153$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1123983020\) $+$ $N(\mathrm{U}(1))$ \(q-2^{21}q^{4}+1123983020q^{7}-370076825230q^{13}+\cdots\)
9.22.a.c 9.a 1.a $1$ $25.153$ \(\Q\) None \(288\) \(0\) \(-21640950\) \(-768078808\) $-$ $\mathrm{SU}(2)$ \(q+288q^{2}-2014208q^{4}-21640950q^{5}+\cdots\)
9.22.a.d 9.a 1.a $1$ $25.153$ \(\Q\) None \(2844\) \(0\) \(-3109950\) \(363303920\) $-$ $\mathrm{SU}(2)$ \(q+2844q^{2}+5991184q^{4}-3109950q^{5}+\cdots\)
9.22.a.e 9.a 1.a $2$ $25.153$ \(\Q(\sqrt{649}) \) None \(-666\) \(0\) \(-996876\) \(679896112\) $-$ $\mathrm{SU}(2)$ \(q+(-333-\beta )q^{2}+(589618+666\beta )q^{4}+\cdots\)
9.22.a.f 9.a 1.a $2$ $25.153$ \(\Q(\sqrt{3085}) \) None \(0\) \(0\) \(0\) \(-2038061480\) $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+1901008q^{4}+40\beta q^{5}-1019030740q^{7}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)