Space of modular forms of level 315, weight 2, and Character 315.p

• Label: 315.2.p
• Level: $315$
• Weight: $2$
• Character orbit: 315.p
• Rep. character: $\chi_{315}(118,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $36$
• Newform subspaces: $5$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.p (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 35 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

	Total	New	Old
Modular forms	112	44	68
Cusp forms	80	36	44
Eisenstein series	32	8	24

Trace form

\( 36 q + 4 q^{2} - 4 q^{7} - 16 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
315.2.p.a	$315$	$2$	315.p	35.f	$4$	$4$	$2$	$2.515$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(-2+\beta _{2})q^{5}+\cdots\)
315.2.p.b	$315$	$2$	315.p	35.f	$4$	$4$	$2$	$2.515$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(8\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(2-\beta _{2})q^{5}+(1+\cdots)q^{7}+\cdots\)
315.2.p.c	$315$	$2$	315.p	35.f	$4$	$4$	$2$	$2.515$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(4\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{2})q^{2}+\beta _{1}q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots\)
315.2.p.d	$315$	$2$	315.p	35.f	$4$	$8$	$4$	$2.515$	8.0.40960000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}-\beta _{3}q^{4}+\beta _{5}q^{5}+(-1-\beta _{3}+\cdots)q^{7}+\cdots\)
315.2.p.e	$315$	$2$	315.p	35.f	$4$	$16$	$8$	$2.515$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{2}+(-\beta _{6}-\beta _{7}+\beta _{13})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

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