Space of modular forms of level 684, weight 2, and Character 684.bg

• Label: 684.2.bg
• Level: $684$
• Weight: $2$
• Character orbit: 684.bg
• Rep. character: $\chi_{684}(191,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $216$
• Newform subspaces: $3$
• Sturm bound: $240$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.bg (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 36 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(240\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(684, [\chi])\).

	Total	New	Old
Modular forms	248	216	32
Cusp forms	232	216	16
Eisenstein series	16	0	16

Trace form

216 * q + 6 * q^6 - 22 * q^12 - 30 * q^14 - 16 * q^18 - 8 * q^21 - 28 * q^24 + 108 * q^25 - 24 * q^29 + 36 * q^30 + 60 * q^32 + 22 * q^36 + 20 * q^42 - 40 * q^45 - 14 * q^48 + 108 * q^49 - 78 * q^50 - 18 * q^52 - 64 * q^54 - 84 * q^56 - 18 * q^58 - 14 * q^60 - 36 * q^64 + 42 * q^66 + 48 * q^68 + 64 * q^69 + 36 * q^72 + 48 * q^77 + 32 * q^78 + 40 * q^81 - 36 * q^82 - 94 * q^84 + 60 * q^86 + 24 * q^88 - 58 * q^90 - 114 * q^92 - 80 * q^93 + 42 * q^94 + 76 * q^96

Decomposition of \(S_{2}^{\mathrm{new}}(684, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
684.2.bg.a	$684$	$2$	684.bg	36.h	$6$	$4$	$2$	$5.462$	\(\Q(\zeta_{12})\)	None		✓			684.2.bg.a	$2$	$1$	\(-2\)	\(0\)	\(-6\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(1-2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
684.2.bg.b	$684$	$2$	684.bg	36.h	$6$	$4$	$2$	$5.462$	\(\Q(\zeta_{12})\)	None		✓			684.2.bg.a	$2$	$0$	\(2\)	\(0\)	\(-6\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-1+2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
684.2.bg.c	$684$	$2$	684.bg	36.h	$6$	$208$	$104$	$5.462$		None		✓	✓		684.2.bg.c	$4$	$0$	\(0\)	\(0\)	\(12\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(684, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)