Space of modular forms of level 21, weight 8, and Character 21.e

• Label: 21.8.e
• Level: $21$
• Weight: $8$
• Character orbit: 21.e
• Rep. character: $\chi_{21}(4,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $18$
• Newform subspaces: $2$
• Sturm bound: $21$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	21.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(21\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(21, [\chi])\).

	Total	New	Old
Modular forms	42	18	24
Cusp forms	34	18	16
Eisenstein series	8	0	8

Trace form

18 * q - 14 * q^2 + 27 * q^3 - 330 * q^4 + 2 * q^5 - 864 * q^6 - 705 * q^7 + 4440 * q^8 - 6561 * q^9 + 10266 * q^10 - 2038 * q^11 + 3456 * q^12 - 10134 * q^13 - 29986 * q^14 + 21276 * q^15 + 35814 * q^16 + 46200 * q^17 - 10206 * q^18 + 36147 * q^19 - 391448 * q^20 - 54540 * q^21 + 329844 * q^22 + 214320 * q^23 + 132678 * q^24 - 226659 * q^25 - 36794 * q^26 - 39366 * q^27 - 550206 * q^28 + 532376 * q^29 + 2808 * q^30 + 492987 * q^31 - 198904 * q^32 + 336366 * q^33 + 12936 * q^34 - 1209770 * q^35 + 481140 * q^36 - 192351 * q^37 - 763226 * q^38 + 68067 * q^39 + 309162 * q^40 + 726076 * q^41 + 818640 * q^42 - 1169070 * q^43 - 1403864 * q^44 + 1458 * q^45 + 878748 * q^46 - 521298 * q^47 - 339984 * q^48 + 5045499 * q^49 + 2496500 * q^50 - 629856 * q^51 - 4103724 * q^52 + 328764 * q^53 + 314928 * q^54 + 2247288 * q^55 - 1660176 * q^56 - 5517126 * q^57 - 5965974 * q^58 - 3018948 * q^59 - 533034 * q^60 + 4334166 * q^61 + 16674372 * q^62 + 1572453 * q^63 - 619548 * q^64 - 1096030 * q^65 + 1360692 * q^66 - 2244375 * q^67 - 10603740 * q^68 - 4218480 * q^69 - 5455494 * q^70 + 2424132 * q^71 - 1618380 * q^72 + 11502039 * q^73 - 2407014 * q^74 + 5783373 * q^75 - 1036920 * q^76 - 2615936 * q^77 + 9644292 * q^78 + 438693 * q^79 + 10668412 * q^80 - 4782969 * q^81 - 7833540 * q^82 - 3911364 * q^83 - 13788360 * q^84 - 23048784 * q^85 + 32388062 * q^86 + 11002932 * q^87 + 942390 * q^88 + 13577072 * q^89 - 14967828 * q^90 - 12690417 * q^91 + 5254200 * q^92 + 2274723 * q^93 - 105324 * q^94 - 23000422 * q^95 + 5227146 * q^96 - 11558244 * q^97 - 56168108 * q^98 + 2971404 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(21, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
21.8.e.a	$21$	$8$	21.e	7.c	$3$	$8$	$4$	$6.560$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			21.8.e.a	$2$	$0$	\(1\)	\(-108\)	\(-196\)	\(154\)	$2^{2}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{6}q^{2}+3^{3}\beta _{4}q^{3}+(-2\beta _{1}+5^{2}\beta _{4}+\cdots)q^{4}+\cdots\)
21.8.e.b	$21$	$8$	21.e	7.c	$3$	$10$	$5$	$6.560$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓	✓		21.8.e.b	$2$	$0$	\(-15\)	\(135\)	\(198\)	\(-859\)	$2^{2}\cdot 3^{3}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3-\beta _{1}+3\beta _{4})q^{2}+3^{3}\beta _{4}q^{3}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(21, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(21, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 2}\)