Space of modular forms of level 315, weight 4, and Character 315.j

• Label: 315.4.j
• Level: $315$
• Weight: $4$
• Character orbit: 315.j
• Rep. character: $\chi_{315}(46,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $80$
• Newform subspaces: $10$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.j (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(315, [\chi])\).

	Total	New	Old
Modular forms	304	80	224
Cusp forms	272	80	192
Eisenstein series	32	0	32

Trace form

\( 80 q + 2 q^{2} - 170 q^{4} - 10 q^{5} + 20 q^{7} + 36 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(315, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
315.4.j.a	$315$	$4$	315.j	7.c	$3$	$2$	$1$	$18.586$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(3\)	\(0\)	\(-5\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-5\zeta_{6}q^{5}+\cdots\)
315.4.j.b	$315$	$4$	315.j	7.c	$3$	$2$	$1$	$18.586$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(3\)	\(0\)	\(5\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+5\zeta_{6}q^{5}+\cdots\)
315.4.j.c	$315$	$4$	315.j	7.c	$3$	$4$	$2$	$18.586$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(-6\)	\(0\)	\(10\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+3\beta _{2}+\beta _{3})q^{2}+(-3-6\beta _{1}+\cdots)q^{4}+\cdots\)
315.4.j.d	$315$	$4$	315.j	7.c	$3$	$4$	$2$	$18.586$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(-4\)	\(0\)	\(-10\)	\(50\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+2\beta _{2}-\beta _{3})q^{2}+(2+4\beta _{1}+\cdots)q^{4}+\cdots\)
315.4.j.e	$315$	$4$	315.j	7.c	$3$	$6$	$3$	$18.586$	6.0.646154928.2	None		$2$	$0$	\(3\)	\(0\)	\(15\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{1}-\beta _{2}-\beta _{3})q^{2}+(-\beta _{1}+\beta _{4}+\cdots)q^{4}+\cdots\)
315.4.j.f	$315$	$4$	315.j	7.c	$3$	$10$	$5$	$18.586$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(-1\)	\(0\)	\(-25\)	\(56\)	$2^{2}\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2}+\beta _{3}-4\beta _{4}+\beta _{8}+\cdots)q^{4}+\cdots\)
315.4.j.g	$315$	$4$	315.j	7.c	$3$	$10$	$5$	$18.586$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(1\)	\(0\)	\(-25\)	\(-62\)	$2^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-7-\beta _{4}+7\beta _{6}+\beta _{8})q^{4}+\cdots\)
315.4.j.h	$315$	$4$	315.j	7.c	$3$	$10$	$5$	$18.586$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(3\)	\(0\)	\(25\)	\(-32\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{4})q^{2}+(-5-5\beta _{4}+\beta _{7}+\cdots)q^{4}+\cdots\)
315.4.j.i	$315$	$4$	315.j	7.c	$3$	$16$	$8$	$18.586$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(-4\)	\(0\)	\(40\)	\(22\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{6})q^{2}+(-6-\beta _{1}+\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
315.4.j.j	$315$	$4$	315.j	7.c	$3$	$16$	$8$	$18.586$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(4\)	\(0\)	\(-40\)	\(22\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{6})q^{2}+(-6-\beta _{1}+\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)