Space of modular forms of level 21, weight 3, and Character 21.f

• Label: 21.3.f
• Level: $21$
• Weight: $3$
• Character orbit: 21.f
• Rep. character: $\chi_{21}(10,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $6$
• Newform subspaces: $3$
• Sturm bound: $8$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	21.f (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(8\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(21, [\chi])\).

	Total	New	Old
Modular forms	14	6	8
Cusp forms	6	6	0
Eisenstein series	8	0	8

Trace form

6 * q - 2 * q^2 - 3 * q^3 - 2 * q^4 - 6 * q^5 - q^7 + 8 * q^8 + 9 * q^9 - 30 * q^10 - 14 * q^11 + 24 * q^12 + 22 * q^14 - 24 * q^15 + 22 * q^16 + 48 * q^17 + 6 * q^18 + 33 * q^19 - 36 * q^21 + 28 * q^22 - 68 * q^23 - 54 * q^24 - 9 * q^25 - 126 * q^26 - 34 * q^28 + 64 * q^29 + 48 * q^30 - 69 * q^31 + 12 * q^32 + 108 * q^33 + 114 * q^35 - 12 * q^36 + 43 * q^37 + 174 * q^38 + 9 * q^39 + 42 * q^40 - 48 * q^42 - 134 * q^43 - 108 * q^44 - 18 * q^45 + 52 * q^46 - 222 * q^47 - 27 * q^49 - 68 * q^50 + 36 * q^51 + 84 * q^52 - 32 * q^53 + 32 * q^56 - 90 * q^57 + 34 * q^58 + 84 * q^59 + 18 * q^60 + 216 * q^61 - 15 * q^63 - 28 * q^64 - 78 * q^65 - 108 * q^66 + 69 * q^67 + 36 * q^68 - 198 * q^70 + 244 * q^71 + 12 * q^72 - 129 * q^73 + 38 * q^74 + 39 * q^75 + 52 * q^77 + 204 * q^78 - 21 * q^79 + 48 * q^80 - 27 * q^81 - 144 * q^82 + 120 * q^84 - 96 * q^85 + 158 * q^86 + 54 * q^87 - 202 * q^88 - 60 * q^89 + 303 * q^91 - 168 * q^92 - 45 * q^93 - 48 * q^94 - 150 * q^95 - 234 * q^96 - 176 * q^98 - 84 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(21, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
21.3.f.a	$21$	$3$	21.f	7.d	$6$	$2$	$1$	$0.572$	\(\Q(\sqrt{-3}) \)	None		✓			21.3.f.a	$2$	$0$	\(-3\)	\(-3\)	\(9\)	\(13\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-3\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(-5+5\zeta_{6})q^{4}+\cdots\)
21.3.f.b	$21$	$3$	21.f	7.d	$6$	$2$	$1$	$0.572$	\(\Q(\sqrt{-3}) \)	None		✓			21.3.f.b	$2$	$0$	\(-1\)	\(3\)	\(-9\)	\(-7\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{6}q^{2}+(1+\zeta_{6})q^{3}+(3-3\zeta_{6})q^{4}+\cdots\)
21.3.f.c	$21$	$3$	21.f	7.d	$6$	$2$	$1$	$0.572$	\(\Q(\sqrt{-3}) \)	None		✓			21.3.f.c	$2$	$0$	\(2\)	\(-3\)	\(-6\)	\(-7\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+2\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+\cdots\)