Space of modular forms of level 9, weight 22, and trivial character

• 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: $2$

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.

\(3\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(11\)	\(3\)	\(8\)		\(9\)	\(3\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)		\(12\)	\(6\)	\(6\)		\(10\)	\(5\)	\(5\)		\(2\)	\(1\)	\(1\)

Trace form

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

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3
9.22.a.a	$9$	$22$	9.a	1.a	$1$	$1$	$1$	$25.153$	\(\Q\)	None	✓				3.22.a.b	$1$	$0$	\(-1728\)	\(0\)	\(41512770\)	\(538429808\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-12^{3}q^{2}+888832q^{4}+41512770q^{5}+\cdots\)
9.22.a.b	$9$	$22$	9.a	1.a	$1$	$1$	$1$	$25.153$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			9.22.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(1123983020\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{21}q^{4}+1123983020q^{7}-370076825230q^{13}+\cdots\)
9.22.a.c	$9$	$22$	9.a	1.a	$1$	$1$	$1$	$25.153$	\(\Q\)	None	✓				1.22.a.a	$1$	$0$	\(288\)	\(0\)	\(-21640950\)	\(-768078808\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+288q^{2}-2014208q^{4}-21640950q^{5}+\cdots\)
9.22.a.d	$9$	$22$	9.a	1.a	$1$	$1$	$1$	$25.153$	\(\Q\)	None	✓				3.22.a.a	$1$	$0$	\(2844\)	\(0\)	\(-3109950\)	\(363303920\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2844q^{2}+5991184q^{4}-3109950q^{5}+\cdots\)
9.22.a.e	$9$	$22$	9.a	1.a	$1$	$2$	$2$	$25.153$	\(\Q(\sqrt{649}) \)	None	✓		✓		3.22.a.c	$1$	$0$	\(-666\)	\(0\)	\(-996876\)	\(679896112\)		$-$	$-$	$2\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-333-\beta )q^{2}+(589618+666\beta )q^{4}+\cdots\)
9.22.a.f	$9$	$22$	9.a	1.a	$1$	$2$	$2$	$25.153$	\(\Q(\sqrt{3085}) \)	None	✓	✓	✓		9.22.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-2038061480\)		$+$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+1901008q^{4}+40\beta q^{5}-1019030740q^{7}+\cdots\)

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

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