Space of modular forms of level 336, weight 3, and Character 336.m

• Label: 336.3.m
• Level: $336$
• Weight: $3$
• Character orbit: 336.m
• Rep. character: $\chi_{336}(127,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $3$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
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:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(336, [\chi])\).

	Total	New	Old
Modular forms	140	12	128
Cusp forms	116	12	104
Eisenstein series	24	0	24

Trace form

12 * q - 24 * q^5 - 36 * q^9 + 72 * q^13 + 24 * q^17 - 12 * q^25 - 120 * q^29 - 24 * q^37 + 120 * q^41 + 72 * q^45 - 84 * q^49 + 168 * q^53 - 120 * q^61 - 144 * q^65 - 72 * q^73 + 108 * q^81 - 96 * q^85 + 216 * q^89 + 144 * q^93 - 72 * q^97

Decomposition of \(S_{3}^{\mathrm{new}}(336, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
336.3.m.a	$336$	$3$	336.m	4.b	$2$	$4$	$4$	$9.155$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		✓			336.3.m.a	$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-5+\beta _{1})q^{5}+\beta _{3}q^{7}-3q^{9}+\cdots\)
336.3.m.b	$336$	$3$	336.m	4.b	$2$	$4$	$4$	$9.155$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		✓			336.3.m.b	$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-2q^{5}-\beta _{3}q^{7}-3q^{9}+(4\beta _{2}+\cdots)q^{11}+\cdots\)
336.3.m.c	$336$	$3$	336.m	4.b	$2$	$4$	$4$	$9.155$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		✓			336.3.m.c	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(1-\beta _{1})q^{5}-\beta _{3}q^{7}-3q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(336, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)