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 - 750 * q^10 + 1536 * q^12 + 984 * q^15 - 2122 * q^16 - 3702 * q^18 + 1095 * q^19 - 4500 * q^21 - 7076 * q^22 + 9558 * q^24 + 2695 * q^25 + 29982 * q^28 + 2082 * q^30 - 21909 * q^31 + 28674 * q^33 - 10380 * q^36 - 4915 * q^37 - 14121 * q^39 - 61590 * q^40 - 66990 * q^42 + 47126 * q^43 - 26478 * q^45 + 19000 * q^46 + 12181 * q^49 + 48528 * q^51 + 124992 * q^52 + 114534 * q^54 - 6246 * q^57 - 85022 * q^58 + 56394 * q^60 - 219216 * q^61 - 74409 * q^63 + 68188 * q^64 - 338202 * q^66 - 67027 * q^67 + 204330 * q^70 + 343920 * q^72 + 376755 * q^73 + 365025 * q^75 - 400728 * q^78 - 384343 * q^79 + 158787 * q^81 - 540288 * q^82 - 191904 * q^84 + 239016 * q^85 - 616896 * q^87 - 329866 * q^88 + 149025 * q^91 + 419337 * q^93 + 1023252 * q^94 + 1249542 * q^96 - 140304 * q^99

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
21.6.g.a	$21$	$6$	21.g	21.g	$6$	$2$	$1$	$3.368$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			21.6.g.a	$4$	$0$	\(0\)	\(27\)	\(0\)	\(211\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(9+9\zeta_{6})q^{3}+(-2^{5}+2^{5}\zeta_{6})q^{4}+\cdots\)
21.6.g.b	$21$	$6$	21.g	21.g	$6$	$4$	$2$	$3.368$	\(\Q(\sqrt{-3}, \sqrt{-17})\)	None		✓			21.6.g.b	$4$	$0$	\(0\)	\(-48\)	\(0\)	\(490\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+2\beta _{1}q^{2}+(-2^{4}+\beta _{1}+8\beta _{2}+\beta _{3})q^{3}+\cdots\)
21.6.g.c	$21$	$6$	21.g	21.g	$6$	$16$	$8$	$3.368$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		21.6.g.c	$4$	$0$	\(0\)	\(18\)	\(0\)	\(-616\)	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2}-\beta _{6})q^{3}+(12+\cdots)q^{4}+\cdots\)