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

• Label: 684.2.i
• Level: $684$
• Weight: $2$
• Character orbit: 684.i
• Rep. character: $\chi_{684}(229,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $36$
• Newform subspaces: $4$
• Sturm bound: $240$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(240\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

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

	Total	New	Old
Modular forms	252	36	216
Cusp forms	228	36	192
Eisenstein series	24	0	24

Trace form

36 * q - 2 * q^3 + 6 * q^5 + 10 * q^9 + 8 * q^11 + 4 * q^15 - 16 * q^17 - 18 * q^21 + 2 * q^23 - 12 * q^25 - 2 * q^27 - 12 * q^29 + 6 * q^31 + 14 * q^33 - 12 * q^35 - 12 * q^37 + 6 * q^39 - 16 * q^41 - 16 * q^45 - 10 * q^47 - 30 * q^49 + 28 * q^51 + 64 * q^53 - 36 * q^55 + 2 * q^59 - 12 * q^61 + 16 * q^63 + 18 * q^65 + 6 * q^67 - 40 * q^69 + 28 * q^71 + 24 * q^75 + 14 * q^77 + 24 * q^79 + 34 * q^81 - 32 * q^83 + 36 * q^85 + 20 * q^87 + 8 * q^89 - 24 * q^91 + 10 * q^93 + 30 * q^97 - 68 * q^99

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.i.a	$684$	$2$	684.i	9.c	$3$	$2$	$1$	$5.462$	\(\Q(\sqrt{-3}) \)	None		✓			684.2.i.a	$2$	$0$	\(0\)	\(0\)	\(-3\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-2\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(5-5\zeta_{6})q^{7}+\cdots\)
684.2.i.b	$684$	$2$	684.i	9.c	$3$	$2$	$1$	$5.462$	\(\Q(\sqrt{-3}) \)	None		✓			684.2.i.b	$2$	$0$	\(0\)	\(0\)	\(1\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-2\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+\cdots\)
684.2.i.c	$684$	$2$	684.i	9.c	$3$	$16$	$8$	$5.462$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			684.2.i.c	$2$	$0$	\(0\)	\(-1\)	\(2\)	\(3\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{3}-\beta _{13}q^{5}+(-\beta _{3}+\beta _{8}+\beta _{11}+\cdots)q^{7}+\cdots\)
684.2.i.d	$684$	$2$	684.i	9.c	$3$	$16$	$8$	$5.462$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓			684.2.i.d	$2$	$0$	\(0\)	\(-1\)	\(6\)	\(-5\)	$3^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}+\beta _{13})q^{3}+(\beta _{7}+\beta _{8})q^{5}+(-1+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(684, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 2}\)