Space of modular forms of level 9, weight 6, and trivial character

• Label: 9.6.a
• Level: $9$
• Weight: $6$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $6$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(6\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	7	2	5
Cusp forms	3	1	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
\(+\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)		\(4\)	\(2\)	\(2\)		\(2\)	\(1\)	\(1\)		\(2\)	\(1\)	\(1\)

Trace form

q + 6 * q^2 + 4 * q^4 - 6 * q^5 - 40 * q^7 - 168 * q^8 - 36 * q^10 + 564 * q^11 + 638 * q^13 - 240 * q^14 - 1136 * q^16 - 882 * q^17 - 556 * q^19 - 24 * q^20 + 3384 * q^22 + 840 * q^23 - 3089 * q^25 + 3828 * q^26 - 160 * q^28 - 4638 * q^29 + 4400 * q^31 - 1440 * q^32 - 5292 * q^34 + 240 * q^35 - 2410 * q^37 - 3336 * q^38 + 1008 * q^40 + 6870 * q^41 + 9644 * q^43 + 2256 * q^44 + 5040 * q^46 + 18672 * q^47 - 15207 * q^49 - 18534 * q^50 + 2552 * q^52 - 33750 * q^53 - 3384 * q^55 + 6720 * q^56 - 27828 * q^58 + 18084 * q^59 + 39758 * q^61 + 26400 * q^62 + 27712 * q^64 - 3828 * q^65 - 23068 * q^67 - 3528 * q^68 + 1440 * q^70 + 4248 * q^71 - 41110 * q^73 - 14460 * q^74 - 2224 * q^76 - 22560 * q^77 + 21920 * q^79 + 6816 * q^80 + 41220 * q^82 - 82452 * q^83 + 5292 * q^85 + 57864 * q^86 - 94752 * q^88 + 94086 * q^89 - 25520 * q^91 + 3360 * q^92 + 112032 * q^94 + 3336 * q^95 + 49442 * q^97 - 91242 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.6.a.a	$9$	$6$	9.a	1.a	$1$	$1$	$1$	$1.443$	\(\Q\)	None	✓		✓	✓	3.6.a.a	$1$	$0$	\(6\)	\(0\)	\(-6\)	\(-40\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{2}+4q^{4}-6q^{5}-40q^{7}-168q^{8}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(9)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)