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

• Label: 336.4.bl
• Level: $336$
• Weight: $4$
• Character orbit: 336.bl
• Rep. character: $\chi_{336}(31,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $48$
• Newform subspaces: $10$
• Sturm bound: $256$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.bl (of order $6$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$28$
Character field:			$\Q(\zeta_{6})$
Newform subspaces:			$10$
Sturm bound:			$256$
Trace bound:			$5$
Distinguishing $T_p$:			$5$, $11$

Dimensions

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

	Total	New	Old
Modular forms	408	48	360
Cusp forms	360	48	312
Eisenstein series	48	0	48

Trace form

$48 q - 216 q^{9} + 120 q^{21} + 348 q^{25} + 108 q^{33} + 252 q^{37} - 168 q^{49} + 1176 q^{53} + 168 q^{57} + 1800 q^{61} - 840 q^{65} + 324 q^{73} + 2304 q^{77} - 1944 q^{81} - 6240 q^{85} + 444 q^{93}+O(q^{100})$

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
336.4.bl.a	$336$	$4$	336.bl	28.f	$6$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None		✓			336.4.bl.a	$2$	$0$	$0$	$-3$	$-6$	$35$	$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-3+3\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+\cdots$
336.4.bl.b	$336$	$4$	336.bl	28.f	$6$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None		✓			336.4.bl.b	$2$	$0$	$0$	$-3$	$-3$	$-7$	$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-3+3\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}+(7+\cdots)q^{7}+\cdots$
336.4.bl.c	$336$	$4$	336.bl	28.f	$6$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None		✓			336.4.bl.a	$2$	$0$	$0$	$3$	$-6$	$-35$	$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(3-3\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+(-14+\cdots)q^{7}+\cdots$
336.4.bl.d	$336$	$4$	336.bl	28.f	$6$	$2$	$1$	$19.825$	$\Q(\sqrt{-3})$	None		✓			336.4.bl.b	$2$	$1$	$0$	$3$	$-3$	$7$	$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(3-3\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}+(-7+\cdots)q^{7}+\cdots$
336.4.bl.e	$336$	$4$	336.bl	28.f	$6$	$6$	$3$	$19.825$	6.0.419349987.3	None		✓			336.4.bl.e	$2$	$0$	$0$	$-9$	$-21$	$-19$	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	$q-3\beta _{1}q^{3}+(-2-2\beta _{1}+2\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots$
336.4.bl.f	$336$	$4$	336.bl	28.f	$6$	$6$	$3$	$19.825$	6.0.633537072.3	None		✓			336.4.bl.f	$2$	$0$	$0$	$-9$	$12$	$-7$	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	$q+3\beta _{2}q^{3}+(1-\beta _{2}+\beta _{3}+\beta _{4})q^{5}+\cdots$
336.4.bl.g	$336$	$4$	336.bl	28.f	$6$	$6$	$3$	$19.825$	6.0.419349987.3	None		✓			336.4.bl.e	$2$	$0$	$0$	$9$	$-21$	$19$	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	$q+3\beta _{1}q^{3}+(-2-2\beta _{1}+2\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots$
336.4.bl.h	$336$	$4$	336.bl	28.f	$6$	$6$	$3$	$19.825$	6.0.633537072.3	None		✓			336.4.bl.f	$2$	$0$	$0$	$9$	$12$	$7$	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{6}]$	$q-3\beta _{2}q^{3}+(1-\beta _{2}+\beta _{3}+\beta _{4})q^{5}+\cdots$
336.4.bl.i	$336$	$4$	336.bl	28.f	$6$	$8$	$4$	$19.825$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		✓			336.4.bl.i	$2$	$0$	$0$	$-12$	$18$	$-12$	$2^{6}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{6}]$	$q-3\beta _{1}q^{3}+(1+2\beta _{1}+\beta _{4}+\beta _{6})q^{5}+\cdots$
336.4.bl.j	$336$	$4$	336.bl	28.f	$6$	$8$	$4$	$19.825$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		✓			336.4.bl.i	$2$	$0$	$0$	$12$	$18$	$12$	$2^{6}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{6}]$	$q+3\beta _{1}q^{3}+(1+2\beta _{1}+\beta _{4}+\beta _{6})q^{5}+\cdots$

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

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