Space of modular forms of level 612, weight 1, and Character 612.bf

• Label: 612.1.bf
• Level: $612$
• Weight: $1$
• Character orbit: 612.bf
• Rep. character: $\chi_{612}(71,\cdot)$
• Character field: $\Q(\zeta_{16})$
• Dimension: $16$
• Newform subspaces: $2$
• Sturm bound: $108$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	612.bf (of order \(16\) and degree \(8\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 204 \)
Character field:			\(\Q(\zeta_{16})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(108\)
Trace bound:			\(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(612, [\chi])\).

	Total	New	Old
Modular forms	80	16	64
Cusp forms	16	16	0
Eisenstein series	64	0	64

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	16	0	0	0

Trace form

\( 16 q + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(612, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
612.1.bf.a	$612$	$1$	612.bf	204.t	$16$	$8$	$1$	$0.305$	\(\Q(\zeta_{16})\)	$D_{16}$	\(\Q(\sqrt{-1}) \)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\zeta_{16}^{3}q^{2}+\zeta_{16}^{6}q^{4}+(-\zeta_{16}-\zeta_{16}^{4}+\cdots)q^{5}+\cdots\)
612.1.bf.b	$612$	$1$	612.bf	204.t	$16$	$8$	$1$	$0.305$	\(\Q(\zeta_{16})\)	$D_{16}$	\(\Q(\sqrt{-1}) \)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+\zeta_{16}^{3}q^{2}+\zeta_{16}^{6}q^{4}+(\zeta_{16}+\zeta_{16}^{4}+\cdots)q^{5}+\cdots\)