Space of modular forms of level 9, weight 8, and trivial character

• Label: 9.8.a
• Level: $9$
• Weight: $8$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $8$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(8\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	9	4	5
Cusp forms	5	3	2
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
\(+\)		\(5\)	\(2\)	\(3\)		\(3\)	\(2\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)		\(4\)	\(2\)	\(2\)		\(2\)	\(1\)	\(1\)		\(2\)	\(1\)	\(1\)

Trace form

3 * q - 6 * q^2 + 372 * q^4 - 390 * q^5 + 456 * q^7 + 1320 * q^8 - 9180 * q^10 + 948 * q^11 + 8682 * q^13 + 384 * q^14 + 19344 * q^16 - 28386 * q^17 + 57732 * q^19 + 35880 * q^20 - 236088 * q^22 + 15288 * q^23 + 102045 * q^25 + 30588 * q^26 + 126528 * q^28 - 36510 * q^29 - 273792 * q^31 - 192096 * q^32 + 1068876 * q^34 + 24960 * q^35 - 493014 * q^37 + 51720 * q^38 - 1712880 * q^40 + 629718 * q^41 + 701052 * q^43 - 87216 * q^44 + 1106352 * q^46 - 583296 * q^47 - 2331333 * q^49 - 443850 * q^50 + 3665976 * q^52 + 428058 * q^53 + 3316680 * q^55 - 84480 * q^56 - 5022540 * q^58 - 1306380 * q^59 - 1677054 * q^61 + 1660848 * q^62 - 5332416 * q^64 + 1988220 * q^65 + 7207476 * q^67 + 2611512 * q^68 - 3144960 * q^70 - 5560632 * q^71 - 2640378 * q^73 - 1611156 * q^74 + 16186704 * q^76 - 60672 * q^77 - 1514352 * q^79 - 1503840 * q^80 - 437508 * q^82 + 4376748 * q^83 - 3306420 * q^85 - 4114632 * q^86 - 22710240 * q^88 + 8528310 * q^89 + 3909072 * q^91 - 1406496 * q^92 + 24973056 * q^94 + 3361800 * q^95 - 34741794 * q^97 + 4916682 * q^98

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.8.a.a	$9$	$8$	9.a	1.a	$1$	$1$	$1$	$2.811$	\(\Q\)	None	✓		✓	✓	3.8.a.a	$1$	$1$	\(-6\)	\(0\)	\(-390\)	\(-64\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}-92q^{4}-390q^{5}-2^{6}q^{7}+\cdots\)
9.8.a.b	$9$	$8$	9.a	1.a	$1$	$2$	$2$	$2.811$	\(\Q(\sqrt{10}) \)	None	✓	✓	✓	✓	9.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(520\)		$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+232q^{4}-2^{4}\beta q^{5}+260q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(9)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)