Space of modular forms of level 117, weight 1, and Character 117.j

• Label: 117.1.j
• Level: $117$
• Weight: $1$
• Character orbit: 117.j
• Rep. character: $\chi_{117}(73,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $2$
• Newform subspaces: $1$
• Sturm bound: $14$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(14\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(117, [\chi])\).

	Total	New	Old
Modular forms	10	4	6
Cusp forms	2	2	0
Eisenstein series	8	2	6

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

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

Trace form

\( 2 q - 2 q^{7} - 2 q^{16} - 2 q^{19} + 2 q^{28} + 2 q^{31} + 2 q^{37} + 2 q^{52} - 2 q^{67} - 2 q^{73} - 2 q^{76} - 2 q^{91} + 2 q^{97}+O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(117, [\chi])\) 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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
117.1.j.a	$117$	$1$	117.j	13.d	$4$	$2$	$1$	$0.058$	\(\Q(\sqrt{-1}) \)	$D_{4}$	\(\Q(\sqrt{-3}) \)	None		✓	✓	✓	117.1.j.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$1$		\(q+i q^{4}+(-i-1)q^{7}-i q^{13}-q^{16}+\cdots\)