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

• Label: 336.4.b
• Level: $336$
• Weight: $4$
• Character orbit: 336.b
• Rep. character: $\chi_{336}(223,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $6$
• Sturm bound: $256$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.b (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$28$
Character field:			$\Q$
Newform subspaces:			$6$
Sturm bound:			$256$
Trace bound:			$7$
Distinguishing $T_p$:			$5$, $19$

Dimensions

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

	Total	New	Old
Modular forms	204	24	180
Cusp forms	180	24	156
Eisenstein series	24	0	24

Trace form

$24 q + 216 q^{9} + 60 q^{21} - 1104 q^{25} + 504 q^{37} + 1104 q^{49} + 2352 q^{53} - 168 q^{57} - 1680 q^{65} - 2304 q^{77} + 1944 q^{81} + 408 q^{85} - 3216 q^{93}+O(q^{100})$

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.b.a	$336$	$4$	336.b	28.d	$2$	$2$	$2$	$19.825$	$\Q(\sqrt{-6})$	None		✓			336.4.b.a	$2$	$0$	$0$	$-6$	$0$	$-34$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q-3q^{3}+\beta q^{5}+(-17-3\beta )q^{7}+9q^{9}+\cdots$
336.4.b.b	$336$	$4$	336.b	28.d	$2$	$2$	$2$	$19.825$	$\Q(\sqrt{-3})$	None		✓			336.4.b.b	$2$	$0$	$0$	$-6$	$0$	$28$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q-3 q^{3}-8\beta q^{5}+(-7\beta+14)q^{7}+\cdots$
336.4.b.c	$336$	$4$	336.b	28.d	$2$	$2$	$2$	$19.825$	$\Q(\sqrt{-3})$	None		✓			336.4.b.b	$2$	$0$	$0$	$6$	$0$	$-28$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+3 q^{3}-8\beta q^{5}+(7\beta-14)q^{7}+9 q^{9}+\cdots$
336.4.b.d	$336$	$4$	336.b	28.d	$2$	$2$	$2$	$19.825$	$\Q(\sqrt{-6})$	None		✓			336.4.b.a	$2$	$0$	$0$	$6$	$0$	$34$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+3q^{3}+\beta q^{5}+(17+3\beta )q^{7}+9q^{9}+\cdots$
336.4.b.e	$336$	$4$	336.b	28.d	$2$	$8$	$8$	$19.825$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None		✓			336.4.b.e	$2$	$0$	$0$	$-24$	$0$	$-4$	$2^{15}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q-3q^{3}+\beta _{5}q^{5}-\beta _{2}q^{7}+9q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots$
336.4.b.f	$336$	$4$	336.b	28.d	$2$	$8$	$8$	$19.825$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None		✓			336.4.b.e	$2$	$0$	$0$	$24$	$0$	$4$	$2^{15}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+3q^{3}+\beta _{5}q^{5}+\beta _{2}q^{7}+9q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots$

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}$