Space of modular forms of level 684, weight 1, and Character 684.h

• Label: 684.1.h
• Level: $684$
• Weight: $1$
• Character orbit: 684.h
• Rep. character: $\chi_{684}(37,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $120$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(120\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(684, [\chi])\).

	Total	New	Old
Modular forms	21	3	18
Cusp forms	9	3	6
Eisenstein series	12	0	12

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

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	3	0	0	0

Trace form

\( 3 q + q^{5} + q^{7} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(684, [\chi])\) 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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
684.1.h.a	$684$	$1$	684.h	19.b	$2$	$1$	$1$	$0.341$	\(\Q\)	$D_{3}$	\(\Q(\sqrt{-19}) \)	None	✓				76.1.c.a	$2$	$0$	\(0\)	\(0\)	\(1\)	\(-1\)	$1$		\(q+q^{5}-q^{7}+q^{11}+q^{17}+q^{19}-2q^{23}+\cdots\)
684.1.h.b	$684$	$1$	684.h	19.b	$2$	$2$	$2$	$0.341$	\(\Q(\sqrt{3}) \)	$D_{6}$	\(\Q(\sqrt{-19}) \)	None	✓	✓	✓		684.1.h.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$1$		\(q-\beta q^{5}+q^{7}+\beta q^{11}+\beta q^{17}-q^{19}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(684, [\chi]) \simeq \) \(S_{1}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 2}\)