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

• Label: 315.3.e
• Level: $315$
• Weight: $3$
• Character orbit: 315.e
• Rep. character: $\chi_{315}(244,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $5$
• Sturm bound: $144$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	315.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 35 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(144\)
Trace bound:			\(11\)
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	104	42	62
Cusp forms	88	38	50
Eisenstein series	16	4	12

Trace form

\( 38 q - 76 q^{4} + O(q^{10}) \)

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.e.a	$315$	$3$	315.e	35.c	$2$	$1$	$1$	$8.583$	\(\Q\)	\(\Q(\sqrt{-35}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-5\)	\(-7\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+4q^{4}-5q^{5}-7q^{7}+13q^{11}+19q^{13}+\cdots\)
315.3.e.b	$315$	$3$	315.e	35.c	$2$	$1$	$1$	$8.583$	\(\Q\)	\(\Q(\sqrt{-35}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(5\)	\(7\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+4q^{4}+5q^{5}+7q^{7}+13q^{11}-19q^{13}+\cdots\)
315.3.e.c	$315$	$3$	315.e	35.c	$2$	$4$	$4$	$8.583$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-5q^{4}+\beta _{2}q^{5}+(-\beta _{1}-2\beta _{3})q^{7}+\cdots\)
315.3.e.d	$315$	$3$	315.e	35.c	$2$	$16$	$16$	$8.583$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{7}q^{2}+(-2+\beta _{1})q^{4}+\beta _{2}q^{5}-\beta _{4}q^{7}+\cdots\)
315.3.e.e	$315$	$3$	315.e	35.c	$2$	$16$	$16$	$8.583$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-2+\beta _{4})q^{4}+(-\beta _{8}+\beta _{10}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

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