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

• Label: 336.4.h
• Level: $336$
• Weight: $4$
• Character orbit: 336.h
• Rep. character: $\chi_{336}(239,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $2$
• Sturm bound: $256$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.h (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$12$
Character field:			$\Q$
Newform subspaces:			$2$
Sturm bound:			$256$
Trace bound:			$1$
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	36	168
Cusp forms	180	36	144
Eisenstein series	24	0	24

Trace form

$36 q - 60 q^{9} - 900 q^{25} - 1152 q^{33} - 1584 q^{37} + 984 q^{45} - 1764 q^{49} - 3456 q^{57} - 1872 q^{61} + 744 q^{69} + 4968 q^{73} - 1836 q^{81} - 7272 q^{85} + 2472 q^{93} + 216 q^{97}+O(q^{100})$

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.h.a	$336$	$4$	336.h	12.b	$2$	$12$	$12$	$19.825$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		✓			336.4.h.a	$4$	$0$	$0$	$0$	$0$	$0$	$2^{7}\cdot 3^{2}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{5}q^{3}+\beta _{8}q^{5}-\beta _{1}q^{7}+(6-\beta _{2}+\cdots)q^{9}+\cdots$
336.4.h.b	$336$	$4$	336.h	12.b	$2$	$24$	$24$	$19.825$		None		✓	✓		336.4.h.b	$4$	$0$	$0$	$0$	$0$	$0$		$\mathrm{SU}(2)[C_{2}]$

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

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