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

• Label: 684.2.cf
• Level: $684$
• Weight: $2$
• Character orbit: 684.cf
• Rep. character: $\chi_{684}(91,\cdot)$
• Character field: $\Q(\zeta_{18})$
• Dimension: $288$
• Newform subspaces: $4$
• 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.cf (of order \(18\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 76 \)
Character field:			\(\Q(\zeta_{18})\)
Newform subspaces:			\( 4 \)
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	768	312	456
Cusp forms	672	288	384
Eisenstein series	96	24	72

Trace form

288 * q + 6 * q^2 - 12 * q^4 + 12 * q^5 + 9 * q^8 + 9 * q^10 - 24 * q^13 + 21 * q^14 - 12 * q^16 + 12 * q^17 - 18 * q^20 - 12 * q^22 - 12 * q^25 - 21 * q^26 + 12 * q^29 + 21 * q^32 + 12 * q^34 + 72 * q^38 + 6 * q^40 - 6 * q^41 + 15 * q^44 + 36 * q^46 + 102 * q^49 + 36 * q^50 + 33 * q^52 - 84 * q^58 - 48 * q^61 - 66 * q^62 - 33 * q^64 + 54 * q^65 - 36 * q^68 + 9 * q^70 + 60 * q^73 + 9 * q^74 + 114 * q^76 + 180 * q^77 + 9 * q^80 - 39 * q^82 - 132 * q^85 - 120 * q^86 + 63 * q^88 + 24 * q^89 - 36 * q^92 - 6 * q^97 - 51 * q^98

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.cf.a	$684$	$2$	684.cf	76.k	$18$	$48$	$8$	$5.462$		None					76.2.k.a	$4$	$0$	\(6\)	\(0\)	\(12\)	\(0\)		$\mathrm{SU}(2)[C_{18}]$
684.2.cf.b	$684$	$2$	684.cf	76.k	$18$	$60$	$10$	$5.462$		None					228.2.w.a	$2$	$0$	\(-3\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{18}]$
684.2.cf.c	$684$	$2$	684.cf	76.k	$18$	$60$	$10$	$5.462$		None					228.2.w.a	$2$	$0$	\(3\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{18}]$
684.2.cf.d	$684$	$2$	684.cf	76.k	$18$	$120$	$20$	$5.462$		None		✓	✓		684.2.cf.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{18}]$

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

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