Space of modular forms of level 693, weight 2, and Character 693.j

• Label: 693.2.j
• Level: $693$
• Weight: $2$
• Character orbit: 693.j
• Rep. character: $\chi_{693}(232,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $120$
• Newform subspaces: $9$
• Sturm bound: $192$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.j (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(192\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(693, [\chi])\).

	Total	New	Old
Modular forms	200	120	80
Cusp forms	184	120	64
Eisenstein series	16	0	16

Trace form

\( 120 q + 2 q^{3} - 60 q^{4} - 2 q^{5} + 12 q^{6} + 24 q^{8} + 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(693, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
693.2.j.a	$693$	$2$	693.j	9.c	$3$	$2$	$1$	$5.534$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(3\)	\(2\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
693.2.j.b	$693$	$2$	693.j	9.c	$3$	$2$	$1$	$5.534$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(1\)	\(-3\)	\(3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
693.2.j.c	$693$	$2$	693.j	9.c	$3$	$2$	$1$	$5.534$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(2\)	\(3\)	\(1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
693.2.j.d	$693$	$2$	693.j	9.c	$3$	$6$	$3$	$5.534$	6.0.4406832.1	None		$2$	$0$	\(1\)	\(9\)	\(-1\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{3}+\beta _{5})q^{2}+(2-\beta _{4})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
693.2.j.e	$693$	$2$	693.j	9.c	$3$	$12$	$6$	$5.534$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(-1\)	\(-1\)	\(12\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{4}+\beta _{8})q^{2}+(\beta _{2}-\beta _{8})q^{3}+(-2+\cdots)q^{4}+\cdots\)
693.2.j.f	$693$	$2$	693.j	9.c	$3$	$18$	$9$	$5.534$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		$2$	$0$	\(-1\)	\(1\)	\(-6\)	\(-9\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+\beta _{13}q^{3}+(-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
693.2.j.g	$693$	$2$	693.j	9.c	$3$	$20$	$10$	$5.534$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(2\)	\(-8\)	\(-10\)	\(-10\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{7}q^{2}+\beta _{4}q^{3}+(\beta _{7}+\beta _{12}-\beta _{13}+\cdots)q^{4}+\cdots\)
693.2.j.h	$693$	$2$	693.j	9.c	$3$	$28$	$14$	$5.534$		None		$2$	$0$	\(-1\)	\(-2\)	\(-4\)	\(14\)		$\mathrm{SU}(2)[C_{3}]$
693.2.j.i	$693$	$2$	693.j	9.c	$3$	$30$	$15$	$5.534$		None		$2$	$0$	\(-2\)	\(0\)	\(1\)	\(15\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{2}^{\mathrm{old}}(693, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(693, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 2}\)