Space of modular forms of level 684, weight 5, and Character 684.h

• Label: 684.5.h
• Level: $684$
• Weight: $5$
• Character orbit: 684.h
• Rep. character: $\chi_{684}(37,\cdot)$
• Character field: $\Q$
• Dimension: $34$
• Newform subspaces: $6$
• Sturm bound: $600$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(600\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(684, [\chi])\).

	Total	New	Old
Modular forms	492	34	458
Cusp forms	468	34	434
Eisenstein series	24	0	24

Trace form

\( 34 q - 9 q^{5} - 15 q^{7} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(684, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
684.5.h.a	$684$	$5$	684.h	19.b	$2$	$2$	$2$	$70.705$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(188\)	$2^{5}$	$\mathrm{U}(1)[D_{2}]$	\(q+94q^{7}+11\zeta_{6}q^{13}+(23+13\zeta_{6})q^{19}+\cdots\)
684.5.h.b	$684$	$5$	684.h	19.b	$2$	$2$	$2$	$70.705$	\(\Q(\sqrt{57}) \)	\(\Q(\sqrt{-19}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(31\)	\(73\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+(17+3\beta )q^{5}+(39+5\beta )q^{7}+(-119+\cdots)q^{11}+\cdots\)
684.5.h.c	$684$	$5$	684.h	19.b	$2$	$4$	$4$	$70.705$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-22\)	\(24\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-6+\beta _{3})q^{5}+(7-2\beta _{3})q^{7}+(48+\cdots)q^{11}+\cdots\)
684.5.h.d	$684$	$5$	684.h	19.b	$2$	$4$	$4$	$70.705$	\(\Q(\sqrt{3}, \sqrt{19})\)	\(\Q(\sqrt{-19}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-146\)	$2^{4}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-2\beta _{1}-\beta _{2})q^{5}+(-34+5\beta _{3})q^{7}+\cdots\)
684.5.h.e	$684$	$5$	684.h	19.b	$2$	$8$	$8$	$70.705$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-68\)	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{5}+(-9+\beta _{1})q^{7}+(-2\beta _{4}+\beta _{5}+\cdots)q^{11}+\cdots\)
684.5.h.f	$684$	$5$	684.h	19.b	$2$	$14$	$14$	$70.705$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-18\)	\(-86\)	$2^{20}\cdot 3^{13}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{5}+(-6+\beta _{2})q^{7}+(-18+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(684, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(114, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 2}\)