Space of modular forms of level 9, weight 94, and trivial character

• Label: 9.94.a
• Level: $9$
• Weight: $94$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $5$
• Sturm bound: $94$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 94 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(94\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{94}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	95	39	56
Cusp forms	91	38	53
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
\(+\)	\(15\)
\(-\)	\(23\)

Trace form

\( 38 q + 114104170891458 q^{2} + 189656938615373735537188351892 q^{4} + 327091426626964081038015997711494 q^{5} - 2780829428047563853789903681715957151668 q^{7} + 1305933451205387040844741834882524615588120 q^{8} + O(q^{10}) \)

Decomposition of \(S_{94}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
9.94.a.a	$9$	$94$	9.a	1.a	$1$	$1$	$1$	$492.953$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(36\!\cdots\!60\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{93}q^{4}+\cdots\)
9.94.a.b	$9$	$94$	9.a	1.a	$1$	$7$	$7$	$492.953$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(-43\!\cdots\!92\)	\(0\)	\(24\!\cdots\!50\)	\(-92\!\cdots\!08\)		$-$	$-$	$2^{88}\cdot 3^{47}\cdot 5^{10}\cdot 7^{6}\cdot 13^{2}\cdot 19\cdot 23\cdot 31^{2}$	$\mathrm{SU}(2)$	\(q+(-6247918101970-\beta _{1})q^{2}+\cdots\)
9.94.a.c	$9$	$94$	9.a	1.a	$1$	$8$	$8$	$492.953$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(40\!\cdots\!26\)	\(0\)	\(-38\!\cdots\!16\)	\(55\!\cdots\!48\)		$-$	$-$	$2^{90}\cdot 3^{61}\cdot 5^{9}\cdot 7^{6}\cdot 11^{2}\cdot 13\cdot 19^{2}\cdot 23^{2}\cdot 31^{3}$	$\mathrm{SU}(2)$	\(q+(5118281421091+\beta _{1})q^{2}+\cdots\)
9.94.a.d	$9$	$94$	9.a	1.a	$1$	$8$	$8$	$492.953$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(11\!\cdots\!24\)	\(0\)	\(46\!\cdots\!60\)	\(16\!\cdots\!12\)		$-$	$-$	$2^{97}\cdot 3^{61}\cdot 5^{11}\cdot 7^{7}\cdot 11^{2}\cdot 13^{2}\cdot 19\cdot 23^{2}\cdot 31^{2}$	$\mathrm{SU}(2)$	\(q+(14611668279566+\beta _{1})q^{2}+\cdots\)
9.94.a.e	$9$	$94$	9.a	1.a	$1$	$14$	$14$	$492.953$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-76\!\cdots\!80\)		$+$	$+$	$2^{181}\cdot 3^{204}\cdot 5^{27}\cdot 7^{13}\cdot 11^{2}\cdot 13^{4}\cdot 19^{2}\cdot 23^{2}\cdot 31^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6625177926194454250944644231+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{94}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{94}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{94}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{94}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)