Space of modular forms of level 21, weight 6, and Character 21.g

• Label: 21.6.g
• Level: $21$
• Weight: $6$
• Character orbit: 21.g
• Rep. character: $\chi_{21}(5,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $22$
• Newform subspaces: $3$
• Sturm bound: $16$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(16\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(21, [\chi])\).

	Total	New	Old
Modular forms	30	30	0
Cusp forms	22	22	0
Eisenstein series	8	8	0

Trace form

\( 22 q - 3 q^{3} + 126 q^{4} + 85 q^{7} - 81 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(21, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
21.6.g.a	$21$	$6$	21.g	21.g	$6$	$2$	$1$	$3.368$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(27\)	\(0\)	\(211\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(9+9\zeta_{6})q^{3}+(-2^{5}+2^{5}\zeta_{6})q^{4}+\cdots\)
21.6.g.b	$21$	$6$	21.g	21.g	$6$	$4$	$2$	$3.368$	\(\Q(\sqrt{-3}, \sqrt{-17})\)	None		$4$	$0$	\(0\)	\(-48\)	\(0\)	\(490\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+2\beta _{1}q^{2}+(-2^{4}+\beta _{1}+8\beta _{2}+\beta _{3})q^{3}+\cdots\)
21.6.g.c	$21$	$6$	21.g	21.g	$6$	$16$	$8$	$3.368$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(18\)	\(0\)	\(-616\)	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2}-\beta _{6})q^{3}+(12+\cdots)q^{4}+\cdots\)