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

• Label: 684.5.y
• Level: $684$
• Weight: $5$
• Character orbit: 684.y
• Rep. character: $\chi_{684}(145,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $68$
• Newform subspaces: $6$
• Sturm bound: $600$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(600\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	984	68	916
Cusp forms	936	68	868
Eisenstein series	48	0	48

Trace form

\( 68 q + 9 q^{5} + 60 q^{7} + O(q^{10}) \)

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.y.a	$684$	$5$	684.y	19.d	$6$	$2$	$1$	$70.705$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-142\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q-71q^{7}+(-322+161\zeta_{6})q^{13}+(-185+\cdots)q^{19}+\cdots\)
684.5.y.b	$684$	$5$	684.y	19.d	$6$	$2$	$1$	$70.705$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-46\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q-23q^{7}+(30-15\zeta_{6})q^{13}+(231-416\zeta_{6})q^{19}+\cdots\)
684.5.y.c	$684$	$5$	684.y	19.d	$6$	$12$	$6$	$70.705$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-9\)	\(-52\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{3}+\beta _{5}+\beta _{8})q^{5}+(-4+\beta _{1}+\cdots)q^{7}+\cdots\)
684.5.y.d	$684$	$5$	684.y	19.d	$6$	$14$	$7$	$70.705$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-12\)	\(-110\)	$2^{6}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+2\beta _{3})q^{5}+(-8-\beta _{4})q^{7}+(6+\cdots)q^{11}+\cdots\)
684.5.y.e	$684$	$5$	684.y	19.d	$6$	$14$	$7$	$70.705$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(30\)	\(106\)	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}+\beta _{2}-4\beta _{3})q^{5}+(8+\beta _{4})q^{7}+\cdots\)
684.5.y.f	$684$	$5$	684.y	19.d	$6$	$24$	$12$	$70.705$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(304\)		$\mathrm{SU}(2)[C_{6}]$

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

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