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

• Label: 21.6.e
• Level: $21$
• Weight: $6$
• Character orbit: 21.e
• Rep. character: $\chi_{21}(4,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $14$
• Newform subspaces: $3$
• Sturm bound: $16$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	21.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
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	14	16
Cusp forms	22	14	8
Eisenstein series	8	0	8

Trace form

14 * q + 2 * q^2 - 9 * q^3 - 106 * q^4 + 22 * q^5 + 144 * q^6 + 167 * q^7 - 264 * q^8 - 567 * q^9 - 1182 * q^10 + 466 * q^11 - 288 * q^12 + 2158 * q^13 + 4558 * q^14 + 396 * q^15 - 3034 * q^16 - 1848 * q^17 + 162 * q^18 - 2657 * q^19 + 1448 * q^20 - 4068 * q^21 + 4836 * q^22 - 5232 * q^23 - 3402 * q^24 - 205 * q^25 + 17030 * q^26 + 1458 * q^27 - 9358 * q^28 + 1480 * q^29 + 5544 * q^30 - 725 * q^31 - 11384 * q^32 - 11430 * q^33 - 1176 * q^34 - 19234 * q^35 + 17172 * q^36 + 15187 * q^37 + 47750 * q^38 + 1395 * q^39 - 3318 * q^40 - 50812 * q^41 - 16344 * q^42 + 37066 * q^43 - 26632 * q^44 + 1782 * q^45 - 27588 * q^46 + 17118 * q^47 + 63216 * q^48 - 36763 * q^49 - 61292 * q^50 + 11664 * q^51 + 5428 * q^52 + 47652 * q^53 - 5832 * q^54 + 187944 * q^55 + 80784 * q^56 - 29466 * q^57 - 45198 * q^58 - 67764 * q^59 - 78426 * q^60 - 51854 * q^61 - 183420 * q^62 + 14013 * q^63 - 228412 * q^64 - 40370 * q^65 - 2988 * q^66 + 7309 * q^67 + 66612 * q^68 + 154224 * q^69 + 149490 * q^70 + 306180 * q^71 + 10692 * q^72 - 65927 * q^73 + 240570 * q^74 - 48807 * q^75 + 222184 * q^76 - 204544 * q^77 - 238428 * q^78 + 48997 * q^79 + 97676 * q^80 - 45927 * q^81 - 5268 * q^82 - 430740 * q^83 + 47160 * q^84 - 132912 * q^85 - 137330 * q^86 - 3060 * q^87 - 332442 * q^88 + 163120 * q^89 + 191484 * q^90 - 136925 * q^91 + 833112 * q^92 + 121905 * q^93 + 5988 * q^94 + 284842 * q^95 - 176886 * q^96 - 86924 * q^97 - 40636 * q^98 - 75492 * 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.e.a	$21$	$6$	21.e	7.c	$3$	$2$	$1$	$3.368$	\(\Q(\sqrt{-3}) \)	None		✓			21.6.e.a	$2$	$0$	\(2\)	\(9\)	\(-11\)	\(259\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}+9\zeta_{6}q^{3}+28\zeta_{6}q^{4}+\cdots\)
21.6.e.b	$21$	$6$	21.e	7.c	$3$	$4$	$2$	$3.368$	\(\Q(\sqrt{-3}, \sqrt{-83})\)	None		✓			21.6.e.b	$2$	$0$	\(3\)	\(18\)	\(33\)	\(-350\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}-\beta _{3})q^{2}-9\beta _{1}q^{3}+(31\beta _{1}+\cdots)q^{4}+\cdots\)
21.6.e.c	$21$	$6$	21.e	7.c	$3$	$8$	$4$	$3.368$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓	✓		21.6.e.c	$2$	$0$	\(-3\)	\(-36\)	\(0\)	\(258\)	$3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{2}+9\beta _{2}q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(21, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(21, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)