Space of modular forms of level 336, weight 7, and Character 336.m

• Label: 336.7.m
• Level: $336$
• Weight: $7$
• Character orbit: 336.m
• Rep. character: $\chi_{336}(127,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $3$
• Sturm bound: $448$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	336.m (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(448\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(336, [\chi])\).

	Total	New	Old
Modular forms	396	36	360
Cusp forms	372	36	336
Eisenstein series	24	0	24

Trace form

\( 36 q - 264 q^{5} - 8748 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(336, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
336.7.m.a	$336$	$7$	336.m	4.b	$2$	$12$	$12$	$77.298$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-376\)	\(0\)	$2^{28}\cdot 3^{2}\cdot 5^{2}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta _{1}q^{3}+(-31-\beta _{2})q^{5}+\beta _{7}q^{7}+\cdots\)
336.7.m.b	$336$	$7$	336.m	4.b	$2$	$12$	$12$	$77.298$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-88\)	\(0\)	$2^{28}\cdot 3^{10}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(-7+\beta _{1})q^{5}-\beta _{4}q^{7}-3^{5}q^{9}+\cdots\)
336.7.m.c	$336$	$7$	336.m	4.b	$2$	$12$	$12$	$77.298$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(200\)	\(0\)	$2^{28}\cdot 3^{2}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta _{6}q^{3}+(17-\beta _{2})q^{5}+\beta _{7}q^{7}-3^{5}q^{9}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)