Space of modular forms of level 336, weight 4, and Character 336.k

• Label: 336.4.k
• Level: $336$
• Weight: $4$
• Character orbit: 336.k
• Rep. character: $\chi_{336}(209,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $5$
• Sturm bound: $256$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.k (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(256\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(336, [\chi])\).

	Total	New	Old
Modular forms	204	50	154
Cusp forms	180	46	134
Eisenstein series	24	4	20

Trace form

\( 46 q + 20 q^{7} - 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(336, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
336.4.k.a	$336$	$4$	336.k	21.c	$2$	$2$	$2$	$19.825$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$2\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{6}q^{3}+(10-3\zeta_{6})q^{7}-3^{3}q^{9}+12\zeta_{6}q^{13}+\cdots\)
336.4.k.b	$336$	$4$	336.k	21.c	$2$	$4$	$4$	$19.825$	\(\Q(\sqrt{-6}, \sqrt{-17})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{3})q^{3}+(-\beta _{1}-2\beta _{3})q^{5}+\cdots\)
336.4.k.c	$336$	$4$	336.k	21.c	$2$	$8$	$8$	$19.825$	8.0.\(\cdots\).13	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{5}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{3}+(\beta _{2}-\beta _{6}-\beta _{7})q^{5}+(-1+\cdots)q^{7}+\cdots\)
336.4.k.d	$336$	$4$	336.k	21.c	$2$	$8$	$8$	$19.825$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(20\)	$2^{7}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-\beta _{5}q^{5}+(3+\beta _{7})q^{7}+(-2+\cdots)q^{9}+\cdots\)
336.4.k.e	$336$	$4$	336.k	21.c	$2$	$24$	$24$	$19.825$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{4}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)