Space of modular forms of level 9, weight 34, and trivial character

• Label: 9.34.a
• Level: $9$
• Weight: $34$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $5$
• Sturm bound: $34$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 34 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(34\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{34}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	35	14	21
Cusp forms	31	13	18
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
\(+\)	\(5\)
\(-\)	\(8\)

Trace form

\( 13 q - 56142 q^{2} + 53633173972 q^{4} + 390287748744 q^{5} - 107655133710412 q^{7} - 141750813200328 q^{8} + O(q^{10}) \)

Decomposition of \(S_{34}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.34.a.a	$9$	$34$	9.a	1.a	$1$	$1$	$1$	$62.085$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(15\!\cdots\!40\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{33}q^{4}+1589751452540q^{7}+\cdots\)
9.34.a.b	$9$	$34$	9.a	1.a	$1$	$2$	$2$	$62.085$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$0$	\(121680\)	\(0\)	\(181061536500\)	\(-67\!\cdots\!00\)		$-$	$-$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(60840-\beta )q^{2}+(7326116992-121680\beta )q^{4}+\cdots\)
9.34.a.c	$9$	$34$	9.a	1.a	$1$	$3$	$3$	$62.085$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-136620\)	\(0\)	\(260488036134\)	\(10\!\cdots\!32\)		$-$	$-$	$2^{11}\cdot 3^{6}\cdot 11$	$\mathrm{SU}(2)$	\(q+(-45540+\beta _{1})q^{2}+(4863185200+\cdots)q^{4}+\cdots\)
9.34.a.d	$9$	$34$	9.a	1.a	$1$	$3$	$3$	$62.085$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-41202\)	\(0\)	\(-51261823890\)	\(76\!\cdots\!56\)		$-$	$-$	$2^{6}\cdot 3^{6}\cdot 5\cdot 7\cdot 11$	$\mathrm{SU}(2)$	\(q+(-13734+\beta _{1})q^{2}+(-369155924+\cdots)q^{4}+\cdots\)
9.34.a.e	$9$	$34$	9.a	1.a	$1$	$4$	$4$	$62.085$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-12\!\cdots\!40\)		$+$	$+$	$2^{14}\cdot 3^{14}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(8522196688+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{34}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{34}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{34}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{34}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)