Space of modular forms of level 3997, weight 1, and Character 3997.n

• Label: 3997.1.n
• Level: $3997$
• Weight: $1$
• Character orbit: 3997.n
• Rep. character: $\chi_{3997}(461,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $10$
• Newform subspaces: $3$
• Sturm bound: $381$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 3997 = 7 \cdot 571 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3997.n (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3997 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(381\)
Trace bound:			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(3997, [\chi])\).

	Total	New	Old
Modular forms	14	14	0
Cusp forms	10	10	0
Eisenstein series	4	4	0

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	2	4	4	0

Trace form

\( 10 q + q^{2} + 6 q^{7} + 2 q^{8} - 3 q^{9} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(3997, [\chi])\) 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
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3997.1.n.a	$3997$	$1$	3997.n	3997.n	$6$	$2$	$1$	$1.995$	\(\Q(\sqrt{-3}) \)	$D_{3}$	\(\Q(\sqrt{-7}) \)	None		$4$	$0$	\(1\)	\(0\)	\(0\)	\(2\)	$1$		\(q-\zeta_{6}^{2}q^{2}+q^{7}+q^{8}+\zeta_{6}^{2}q^{9}+\zeta_{6}q^{11}+\cdots\)
3997.1.n.b	$3997$	$1$	3997.n	3997.n	$6$	$4$	$2$	$1.995$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	$S_{4}$	None	None		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)	$1$		\(q+(-1+\beta _{2})q^{2}+\beta _{1}q^{5}+q^{7}-q^{8}+\cdots\)
3997.1.n.c	$3997$	$1$	3997.n	3997.n	$6$	$4$	$2$	$1.995$	\(\Q(\zeta_{12})\)	$A_{4}$	None	None		$4$	$0$	\(2\)	\(0\)	\(0\)	\(0\)	$1$		\(q+\zeta_{12}^{2}q^{2}-\zeta_{12}q^{3}-\zeta_{12}^{5}q^{5}-\zeta_{12}^{3}q^{6}+\cdots\)