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.

\(3\)	Dim
\(+\)	\(3\)
\(-\)	\(5\)

Trace form

\( 8 q + 738 q^{2} + 7749908 q^{4} + 15764994 q^{5} - 100527428 q^{7} + 9551476440 q^{8} + O(q^{10}) \)

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

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