Space of modular forms of level 1170, weight 2, and Character 1170.s

• Label: 1170.2.s
• Level: $1170$
• Weight: $2$
• Character orbit: 1170.s
• Rep. character: $\chi_{1170}(161,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $48$
• Newform subspaces: $7$
• Sturm bound: $504$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.s (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(504\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1170, [\chi])\).

	Total	New	Old
Modular forms	536	48	488
Cusp forms	472	48	424
Eisenstein series	64	0	64

Trace form

\( 48 q - 16 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1170, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1170.2.s.a	$1170$	$2$	1170.s	39.f	$4$	$4$	$2$	$9.342$	\(\Q(\zeta_{8})\)	None		$4$	$1$	\(0\)	\(0\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+\zeta_{8}^{3}q^{5}+(-3+\cdots)q^{7}+\cdots\)
1170.2.s.b	$1170$	$2$	1170.s	39.f	$4$	$4$	$2$	$9.342$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+\zeta_{8}^{3}q^{5}+(-1+\cdots)q^{7}+\cdots\)
1170.2.s.c	$1170$	$2$	1170.s	39.f	$4$	$4$	$2$	$9.342$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+\zeta_{8}q^{5}+(3+3\zeta_{8}^{2}+\cdots)q^{7}+\cdots\)
1170.2.s.d	$1170$	$2$	1170.s	39.f	$4$	$4$	$2$	$9.342$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}-\zeta_{8}q^{5}+(3+3\zeta_{8}^{2}+\cdots)q^{7}+\cdots\)
1170.2.s.e	$1170$	$2$	1170.s	39.f	$4$	$8$	$4$	$9.342$	8.0.1871773696.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}+\beta _{1}q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
1170.2.s.f	$1170$	$2$	1170.s	39.f	$4$	$12$	$6$	$9.342$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{2}-\beta _{4}q^{4}+\beta _{2}q^{5}+(-1+\beta _{4}+\cdots)q^{7}+\cdots\)
1170.2.s.g	$1170$	$2$	1170.s	39.f	$4$	$12$	$6$	$9.342$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{5}q^{2}+\beta _{2}q^{4}+\beta _{5}q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1170, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1170, [\chi]) \cong \)