Space of modular forms of level 392, weight 1, and Character 392.k

• Label: 392.1.k
• Level: $392$
• Weight: $1$
• Character orbit: 392.k
• Rep. character: $\chi_{392}(67,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $56$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	392.k (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(56\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(392, [\chi])\).

	Total	New	Old
Modular forms	22	14	8
Cusp forms	6	6	0
Eisenstein series	16	8	8

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	6	0	0	0

Trace form

\( 6 q + q^{2} - 3 q^{4} - 2 q^{8} - q^{9} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(392, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
392.1.k.a	$392$	$1$	392.k	56.k	$6$	$2$	$1$	$0.196$	\(\Q(\sqrt{-3}) \)	$D_{2}$	\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{14}) \)		$8$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\zeta_{6}q^{2}+\zeta_{6}^{2}q^{4}+q^{8}+\zeta_{6}q^{9}-\zeta_{6}^{2}q^{11}+\cdots\)
392.1.k.b	$392$	$1$	392.k	56.k	$6$	$4$	$2$	$0.196$	\(\Q(\sqrt{2}, \sqrt{-3})\)	$D_{4}$	\(\Q(\sqrt{-2}) \)	None		$8$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\beta _{2}q^{2}-\beta _{1}q^{3}+(-1-\beta _{2})q^{4}+\beta _{3}q^{6}+\cdots\)