Space of modular forms of level 1365, weight 1, and Character 1365.bg

• Label: 1365.1.bg
• Level: $1365$
• Weight: $1$
• Character orbit: 1365.bg
• Rep. character: $\chi_{1365}(629,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $8$
• Newform subspaces: $2$
• Sturm bound: $224$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1365 = 3 \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1365.bg (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 1365 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(224\)
Trace bound:			\(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1365, [\chi])\).

	Total	New	Old
Modular forms	24	24	0
Cusp forms	8	8	0
Eisenstein series	16	16	0

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	8	0	0	0

Trace form

\( 8 q + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1365, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1365.1.bg.a	$1365$	$1$	1365.bg	1365.ag	$4$	$4$	$2$	$0.681$	\(\Q(\zeta_{8})\)	$D_{4}$	\(\Q(\sqrt{-35}) \)	None		✓			1365.1.bg.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\zeta_{8}^{3}q^{3}-\zeta_{8}^{2}q^{4}+\zeta_{8}^{3}q^{5}-\zeta_{8}^{3}q^{7}+\cdots\)
1365.1.bg.b	$1365$	$1$	1365.bg	1365.ag	$4$	$4$	$2$	$0.681$	\(\Q(\zeta_{8})\)	$D_{4}$	\(\Q(\sqrt{-35}) \)	None		✓			1365.1.bg.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\zeta_{8}q^{3}-\zeta_{8}^{2}q^{4}+\zeta_{8}^{3}q^{5}+\zeta_{8}^{3}q^{7}+\cdots\)