Space of modular forms of level 312, weight 2, and Character 312.g

• Label: 312.2.g
• Level: $312$
• Weight: $2$
• Character orbit: 312.g
• Rep. character: $\chi_{312}(157,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $2$
• Sturm bound: $112$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(112\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(312, [\chi])\).

	Total	New	Old
Modular forms	60	24	36
Cusp forms	52	24	28
Eisenstein series	8	0	8

Trace form

\( 24 q + 4 q^{2} + 4 q^{6} + 8 q^{7} - 8 q^{8} - 24 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(312, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
312.2.g.a	$312$	$2$	312.g	8.b	$2$	$8$	$8$	$2.491$	\(\Q(\zeta_{20})\)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{20}^{7}q^{2}-\zeta_{20}^{3}q^{3}+\zeta_{20}q^{4}+(-1+\cdots)q^{5}+\cdots\)
312.2.g.b	$312$	$2$	312.g	8.b	$2$	$16$	$16$	$2.491$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(4\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{10}q^{2}+\beta _{9}q^{3}-\beta _{2}q^{4}+\beta _{14}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(312, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(312, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)