Space of modular forms of level 315, weight 5, and Character 315.e

• Label: 315.5.e
• Level: $315$
• Weight: $5$
• Character orbit: 315.e
• Rep. character: $\chi_{315}(244,\cdot)$
• Character field: $\Q$
• Dimension: $78$
• Newform subspaces: $7$
• Sturm bound: $240$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	315.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 35 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(240\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(315, [\chi])\).

	Total	New	Old
Modular forms	200	82	118
Cusp forms	184	78	106
Eisenstein series	16	4	12

Trace form

\( 78 q - 612 q^{4} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(315, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
315.5.e.a	$315$	$5$	315.e	35.c	$2$	$1$	$1$	$32.562$	\(\Q\)	\(\Q(\sqrt{-35}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-25\)	\(49\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2^{4}q^{4}-5^{2}q^{5}+7^{2}q^{7}+73q^{11}+\cdots\)
315.5.e.b	$315$	$5$	315.e	35.c	$2$	$1$	$1$	$32.562$	\(\Q\)	\(\Q(\sqrt{-35}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(25\)	\(-49\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2^{4}q^{4}+5^{2}q^{5}-7^{2}q^{7}+73q^{11}+\cdots\)
315.5.e.c	$315$	$5$	315.e	35.c	$2$	$2$	$2$	$32.562$	\(\Q(\sqrt{-6}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-10\)	\(70\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+10q^{4}+(-5-10\beta )q^{5}+(35+\cdots)q^{7}+\cdots\)
315.5.e.d	$315$	$5$	315.e	35.c	$2$	$2$	$2$	$32.562$	\(\Q(\sqrt{-6}) \)	None		$2$	$0$	\(0\)	\(0\)	\(10\)	\(-70\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+10q^{4}+(5+10\beta )q^{5}+(-35+\cdots)q^{7}+\cdots\)
315.5.e.e	$315$	$5$	315.e	35.c	$2$	$8$	$8$	$32.562$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+(-22+\beta _{2})q^{4}+(-\beta _{1}-\beta _{7})q^{5}+\cdots\)
315.5.e.f	$315$	$5$	315.e	35.c	$2$	$32$	$32$	$32.562$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
315.5.e.g	$315$	$5$	315.e	35.c	$2$	$32$	$32$	$32.562$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{5}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)