Space of modular forms of level 42, weight 9, and Character 42.c

• Label: 42.9.c
• Level: $42$
• Weight: $9$
• Character orbit: 42.c
• Rep. character: $\chi_{42}(13,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $1$
• Sturm bound: $72$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	42.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(72\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(42, [\chi])\).

	Total	New	Old
Modular forms	68	12	56
Cusp forms	60	12	48
Eisenstein series	8	0	8

Trace form

\( 12 q + 1536 q^{4} + 6420 q^{7} - 26244 q^{9} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(42, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
42.9.c.a	$42$	$9$	42.c	7.b	$2$	$12$	$12$	$17.110$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(6420\)	$2^{30}\cdot 3^{18}\cdot 7^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-\beta _{2}q^{3}+2^{7}q^{4}+(2\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)