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

• Label: 315.2.bs
• Level: $315$
• Weight: $2$
• Character orbit: 315.bs
• Rep. character: $\chi_{315}(52,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $176$
• 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.bs (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 315 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(11\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	208	208	0
Cusp forms	176	176	0
Eisenstein series	32	32	0

Trace form

176 * q - 4 * q^2 - 6 * q^3 - 6 * q^5 - 2 * q^7 - 8 * q^8 - 12 * q^10 - 18 * q^12 - 2 * q^15 - 136 * q^16 - 30 * q^17 + 22 * q^18 + 48 * q^20 - 12 * q^22 - 6 * q^23 + 2 * q^25 - 24 * q^26 - 36 * q^27 + 4 * q^28 - 14 * q^30 - 60 * q^32 - 6 * q^33 - 44 * q^35 + 40 * q^36 - 4 * q^37 - 6 * q^38 - 6 * q^40 + 12 * q^41 - 40 * q^42 - 4 * q^43 - 54 * q^45 - 20 * q^46 + 18 * q^48 - 28 * q^50 + 32 * q^51 + 42 * q^52 + 16 * q^53 + 128 * q^56 + 8 * q^57 + 22 * q^58 + 86 * q^60 - 6 * q^63 - 52 * q^65 - 12 * q^66 - 4 * q^67 + 90 * q^68 + 18 * q^70 - 64 * q^71 + 106 * q^72 - 12 * q^73 - 54 * q^75 + 48 * q^76 + 74 * q^77 - 62 * q^78 + 12 * q^80 - 64 * q^81 - 24 * q^82 - 84 * q^83 - 4 * q^85 - 40 * q^86 + 12 * q^87 + 42 * q^88 + 36 * q^90 - 16 * q^91 - 8 * q^92 + 58 * q^93 + 100 * q^95 - 168 * q^96 + 120 * q^98

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
315.2.bs.a	$315$	$2$	315.bs	315.as	$12$	$4$	$1$	$2.515$	\(\Q(\zeta_{12})\)	None		✓			315.2.bs.a	$2$	$0$	\(-2\)	\(0\)	\(2\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2})q^{2}+(\zeta_{12}-2\zeta_{12}^{3})q^{3}+\cdots\)
315.2.bs.b	$315$	$2$	315.bs	315.as	$12$	$4$	$1$	$2.515$	\(\Q(\zeta_{12})\)	None		✓			315.2.bs.b	$2$	$0$	\(-2\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1+\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+\cdots\)
315.2.bs.c	$315$	$2$	315.bs	315.as	$12$	$4$	$1$	$2.515$	\(\Q(\zeta_{12})\)	None		✓			315.2.bs.b	$2$	$0$	\(-2\)	\(6\)	\(-4\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2})q^{2}+(2-\zeta_{12}^{2})q^{3}+\cdots\)
315.2.bs.d	$315$	$2$	315.bs	315.as	$12$	$4$	$1$	$2.515$	\(\Q(\zeta_{12})\)	None		✓			315.2.bs.a	$2$	$0$	\(-2\)	\(6\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1+\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+\cdots\)
315.2.bs.e	$315$	$2$	315.bs	315.as	$12$	$160$	$40$	$2.515$		None		✓	✓		315.2.bs.e	$4$	$0$	\(4\)	\(-18\)	\(-6\)	\(0\)		$\mathrm{SU}(2)[C_{12}]$