Space of modular forms of level 729, weight 2, and Character 729.c

• Label: 729.2.c
• Level: $729$
• Weight: $2$
• Character orbit: 729.c
• Rep. character: $\chi_{729}(244,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $60$
• Newform subspaces: $5$
• Sturm bound: $162$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(162\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(729, [\chi])\).

	Total	New	Old
Modular forms	198	84	114
Cusp forms	126	60	66
Eisenstein series	72	24	48

Trace form

\( 60 q - 24 q^{4} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(729, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
729.2.c.a	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(-3\)	\(0\)	\(3\)	\(-6\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{6})q^{2}+(-1+\cdots)q^{4}+\cdots\)
729.2.c.b	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	12.0.\(\cdots\).1	None		$2$	$0$	\(-3\)	\(0\)	\(-6\)	\(0\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{6}+\beta _{8}-\beta _{10})q^{2}+(-1+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
729.2.c.c	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	\(\Q(\zeta_{36})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\zeta_{36}^{7}-\zeta_{36}^{8}+\zeta_{36}^{10})q^{2}-\zeta_{36}^{11}q^{4}+\cdots\)
729.2.c.d	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(3\)	\(0\)	\(-3\)	\(-6\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}-\beta _{3}+\beta _{5}-\beta _{6}+\beta _{9})q^{2}+(-1+\cdots)q^{4}+\cdots\)
729.2.c.e	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	12.0.\(\cdots\).1	None		$2$	$0$	\(3\)	\(0\)	\(6\)	\(0\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{6}-\beta _{8}+\beta _{10})q^{2}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(729, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(729, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(243, [\chi])\)\(^{\oplus 2}\)