Space of modular forms of level 315, weight 3, and Character 315.bg

• Label: 315.3.bg
• Level: $315$
• Weight: $3$
• Character orbit: 315.bg
• Rep. character: $\chi_{315}(34,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $184$
• Newform subspaces: $3$
• Sturm bound: $144$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	315.bg (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 315 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(144\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(315, [\chi])\).

	Total	New	Old
Modular forms	200	200	0
Cusp forms	184	184	0
Eisenstein series	16	16	0

Trace form

\( 184 q + 172 q^{4} + 12 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(315, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
315.3.bg.a	$315$	$3$	315.bg	315.ag	$6$	$4$	$2$	$8.583$	\(\Q(\sqrt{-3}, \sqrt{-35})\)	\(\Q(\sqrt{-35}) \)		$4$	$0$	\(0\)	\(-2\)	\(-10\)	\(14\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-1+\beta _{1}-\beta _{3})q^{3}-4\beta _{2}q^{4}-5\beta _{2}q^{5}+\cdots\)
315.3.bg.b	$315$	$3$	315.bg	315.ag	$6$	$4$	$2$	$8.583$	\(\Q(\sqrt{-3}, \sqrt{-35})\)	\(\Q(\sqrt{-35}) \)		$4$	$0$	\(0\)	\(2\)	\(10\)	\(-14\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(\beta _{1}-\beta _{3})q^{3}-4\beta _{2}q^{4}+5\beta _{2}q^{5}+\cdots\)
315.3.bg.c	$315$	$3$	315.bg	315.ag	$6$	$176$	$88$	$8.583$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$