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

• Label: 1170.2.e
• Level: $1170$
• Weight: $2$
• Character orbit: 1170.e
• Rep. character: $\chi_{1170}(469,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $8$
• Sturm bound: $504$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(504\)
Trace bound:			\(11\)
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	268	30	238
Cusp forms	236	30	206
Eisenstein series	32	0	32

Trace form

\( 30 q - 30 q^{4} - 8 q^{5} + O(q^{10}) \)

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.e.a	$1170$	$2$	1170.e	5.b	$2$	$2$	$2$	$9.342$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+(-2-i)q^{5}+2iq^{7}+\cdots\)
1170.2.e.b	$1170$	$2$	1170.e	5.b	$2$	$2$	$2$	$9.342$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+(-2+i)q^{5}+4iq^{7}+\cdots\)
1170.2.e.c	$1170$	$2$	1170.e	5.b	$2$	$2$	$2$	$9.342$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+(-2-i)q^{5}+4iq^{7}+\cdots\)
1170.2.e.d	$1170$	$2$	1170.e	5.b	$2$	$2$	$2$	$9.342$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{2}-q^{4}+(2+i)q^{5}+iq^{8}+(1+\cdots)q^{10}+\cdots\)
1170.2.e.e	$1170$	$2$	1170.e	5.b	$2$	$4$	$4$	$9.342$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-q^{4}-\beta _{2}q^{5}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1170.2.e.f	$1170$	$2$	1170.e	5.b	$2$	$6$	$6$	$9.342$	6.0.3534400.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}-q^{4}+(-\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
1170.2.e.g	$1170$	$2$	1170.e	5.b	$2$	$6$	$6$	$9.342$	6.0.350464.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{2}-q^{4}+(-\beta _{2}+\beta _{5})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1170.2.e.h	$1170$	$2$	1170.e	5.b	$2$	$6$	$6$	$9.342$	6.0.350464.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}-q^{4}+(\beta _{2}-\beta _{5})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1170, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 3}\)