Space of modular forms of level 684, weight 3, and Character 684.h

• Label: 684.3.h
• Level: $684$
• Weight: $3$
• Character orbit: 684.h
• Rep. character: $\chi_{684}(37,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $5$
• Sturm bound: $360$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(360\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(684, [\chi])\).

	Total	New	Old
Modular forms	252	16	236
Cusp forms	228	16	212
Eisenstein series	24	0	24

Trace form

\( 16 q + q^{5} - 13 q^{7} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(684, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
684.3.h.a	$684$	$3$	684.h	19.b	$2$	$2$	$2$	$18.638$	\(\Q(\sqrt{57}) \)	\(\Q(\sqrt{-19}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-9\)	\(5\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-4-\beta )q^{5}+(4-3\beta )q^{7}+(4-5\beta )q^{11}+\cdots\)
684.3.h.b	$684$	$3$	684.h	19.b	$2$	$2$	$2$	$18.638$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{7}+\zeta_{6}q^{13}+(-13-\zeta_{6})q^{19}+\cdots\)
684.3.h.c	$684$	$3$	684.h	19.b	$2$	$2$	$2$	$18.638$	\(\Q(\sqrt{-29}) \)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+4q^{5}-q^{7}-14q^{11}-\beta q^{13}-23q^{17}+\cdots\)
684.3.h.d	$684$	$3$	684.h	19.b	$2$	$4$	$4$	$18.638$	\(\Q(\sqrt{3}, \sqrt{19})\)	\(\Q(\sqrt{-19}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$2^{3}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{5}+(-3-\beta _{3})q^{7}-\beta _{2}q^{11}+\cdots\)
684.3.h.e	$684$	$3$	684.h	19.b	$2$	$6$	$6$	$18.638$	6.0.219615408.1	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)	$2^{8}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{5}-\beta _{2}q^{7}+(4+\beta _{2})q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(684, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(114, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 2}\)