Space of modular forms of level 336, weight 6, and Character 336.bl

• Label: 336.6.bl
• Level: $336$
• Weight: $6$
• Character orbit: 336.bl
• Rep. character: $\chi_{336}(31,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $80$
• Newform subspaces: $8$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.bl (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(336, [\chi])\).

	Total	New	Old
Modular forms	664	80	584
Cusp forms	616	80	536
Eisenstein series	48	0	48

Trace form

\( 80 q - 3240 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(336, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
336.6.bl.a	$336$	$6$	336.bl	28.f	$6$	$2$	$1$	$53.889$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-9\)	\(-126\)	\(49\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-9+9\zeta_{6})q^{3}+(-84+42\zeta_{6})q^{5}+\cdots\)
336.6.bl.b	$336$	$6$	336.bl	28.f	$6$	$2$	$1$	$53.889$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(9\)	\(-126\)	\(-49\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(9-9\zeta_{6})q^{3}+(-84+42\zeta_{6})q^{5}+\cdots\)
336.6.bl.c	$336$	$6$	336.bl	28.f	$6$	$10$	$5$	$53.889$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(0\)	\(-45\)	\(93\)	\(169\)	$2^{10}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q-9\beta _{1}q^{3}+(6+6\beta _{1}+\beta _{2}+\beta _{5})q^{5}+\cdots\)
336.6.bl.d	$336$	$6$	336.bl	28.f	$6$	$10$	$5$	$53.889$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(0\)	\(45\)	\(93\)	\(-169\)	$2^{10}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q+9\beta _{1}q^{3}+(6+6\beta _{1}+\beta _{2}+\beta _{5})q^{5}+\cdots\)
336.6.bl.e	$336$	$6$	336.bl	28.f	$6$	$14$	$7$	$53.889$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(-63\)	\(-33\)	\(13\)	$2^{22}\cdot 3^{8}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q+9\beta _{1}q^{3}+(-1+2\beta _{1}-\beta _{2})q^{5}+(10+\cdots)q^{7}+\cdots\)
336.6.bl.f	$336$	$6$	336.bl	28.f	$6$	$14$	$7$	$53.889$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(-63\)	\(66\)	\(-113\)	$2^{24}\cdot 3^{6}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q+9\beta _{1}q^{3}+(3-3\beta _{1}+\beta _{6})q^{5}+(-5+\cdots)q^{7}+\cdots\)
336.6.bl.g	$336$	$6$	336.bl	28.f	$6$	$14$	$7$	$53.889$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(63\)	\(-33\)	\(-13\)	$2^{22}\cdot 3^{8}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q-9\beta _{1}q^{3}+(-1+2\beta _{1}-\beta _{2})q^{5}+(-10+\cdots)q^{7}+\cdots\)
336.6.bl.h	$336$	$6$	336.bl	28.f	$6$	$14$	$7$	$53.889$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(63\)	\(66\)	\(113\)	$2^{24}\cdot 3^{6}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q-9\beta _{1}q^{3}+(3-3\beta _{1}+\beta _{6})q^{5}+(5-8\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

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