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

• Label: 336.7.f
• Level: $336$
• Weight: $7$
• Character orbit: 336.f
• Rep. character: $\chi_{336}(97,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $4$
• Sturm bound: $448$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	336.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(448\)
Trace bound:			\(7\)
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	48	348
Cusp forms	372	48	324
Eisenstein series	24	0	24

Trace form

\( 48 q - 360 q^{7} - 11664 q^{9} + O(q^{10}) \)

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.f.a	$336$	$7$	336.f	7.b	$2$	$8$	$8$	$77.298$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)	$2^{10}\cdot 3^{7}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(-2\beta _{3}+\beta _{5})q^{5}+(-3+2\beta _{1}+\cdots)q^{7}+\cdots\)
336.7.f.b	$336$	$7$	336.f	7.b	$2$	$8$	$8$	$77.298$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$2^{10}\cdot 3^{10}\cdot 7$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(2\beta _{1}+\beta _{2})q^{5}+(2+2\beta _{1}+\cdots)q^{7}+\cdots\)
336.7.f.c	$336$	$7$	336.f	7.b	$2$	$8$	$8$	$77.298$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(212\)	$2^{10}\cdot 3^{9}\cdot 7$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-2\beta _{1}-\beta _{4})q^{5}+(26+2\beta _{2}+\cdots)q^{7}+\cdots\)
336.7.f.d	$336$	$7$	336.f	7.b	$2$	$24$	$24$	$77.298$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-564\)		$\mathrm{SU}(2)[C_{2}]$

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

\( S_{7}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)