Space of modular forms of level 21, weight 8, and trivial character

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

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(21\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(21))\).

	Total	New	Old
Modular forms	20	8	12
Cusp forms	16	8	8
Eisenstein series	4	0	4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(6\)	\(2\)	\(4\)		\(5\)	\(2\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(5\)	\(3\)	\(2\)		\(4\)	\(3\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(11\)	\(5\)	\(6\)		\(9\)	\(5\)	\(4\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(9\)	\(3\)	\(6\)		\(7\)	\(3\)	\(4\)		\(2\)	\(0\)	\(2\)

Trace form

8 * q + 2 * q^2 + 770 * q^4 - 776 * q^5 - 108 * q^6 + 686 * q^7 + 1854 * q^8 + 5832 * q^9 + 3852 * q^10 + 1432 * q^11 - 15336 * q^12 + 12112 * q^13 - 8918 * q^14 - 216 * q^15 + 92882 * q^16 - 40968 * q^17 + 1458 * q^18 - 97424 * q^19 + 572 * q^20 + 18522 * q^21 - 68400 * q^22 + 134808 * q^23 - 54756 * q^24 + 138632 * q^25 - 651436 * q^26 + 107702 * q^28 - 439328 * q^29 + 367848 * q^30 + 442432 * q^31 + 523654 * q^32 + 190296 * q^33 - 935460 * q^34 - 58996 * q^35 + 561330 * q^36 - 518624 * q^37 + 444368 * q^38 - 30672 * q^39 + 1448676 * q^40 + 1751048 * q^41 + 148176 * q^42 + 271216 * q^43 - 3091024 * q^44 - 565704 * q^45 + 37512 * q^46 + 1356144 * q^47 - 1884816 * q^48 + 941192 * q^49 - 2099378 * q^50 + 2017224 * q^51 - 696572 * q^52 - 1085904 * q^53 - 78732 * q^54 - 3055728 * q^55 - 3825822 * q^56 + 2204928 * q^57 - 2019924 * q^58 - 1151088 * q^59 - 4629744 * q^60 + 5183200 * q^61 + 12950208 * q^62 + 500094 * q^63 + 8451050 * q^64 + 3820624 * q^65 - 4485672 * q^66 - 6863648 * q^67 - 6501060 * q^68 - 4491288 * q^69 - 1140132 * q^70 - 3994680 * q^71 + 1351566 * q^72 - 10351520 * q^73 + 12127020 * q^74 + 3748896 * q^75 + 832192 * q^76 + 2118368 * q^77 + 1252800 * q^78 + 13872064 * q^79 + 28655804 * q^80 + 4251528 * q^81 - 21988356 * q^82 - 9046368 * q^83 + 3556224 * q^84 - 16470384 * q^85 - 4645208 * q^86 + 271296 * q^87 + 14529696 * q^88 - 353816 * q^89 + 2808108 * q^90 + 9888004 * q^91 - 15951096 * q^92 - 10010736 * q^93 + 7381056 * q^94 - 21731744 * q^95 - 16331436 * q^96 - 5445920 * q^97 + 235298 * q^98 + 1043928 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(21))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7
21.8.a.a	$21$	$8$	21.a	1.a	$1$	$1$	$1$	$6.560$	\(\Q\)	None	✓	✓	✓	✓	21.8.a.a	$1$	$1$	\(2\)	\(27\)	\(-278\)	\(-343\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3^{3}q^{3}-124q^{4}-278q^{5}+\cdots\)
21.8.a.b	$21$	$8$	21.a	1.a	$1$	$2$	$2$	$6.560$	\(\Q(\sqrt{1065}) \)	None	✓	✓	✓	✓	21.8.a.b	$1$	$1$	\(-9\)	\(-54\)	\(-360\)	\(686\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{2}-3^{3}q^{3}+(154+9\beta )q^{4}+\cdots\)
21.8.a.c	$21$	$8$	21.a	1.a	$1$	$2$	$2$	$6.560$	\(\Q(\sqrt{67}) \)	None	✓	✓	✓	✓	21.8.a.c	$1$	$0$	\(12\)	\(-54\)	\(-24\)	\(-686\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(6+\beta )q^{2}-3^{3}q^{3}+(176+12\beta )q^{4}+\cdots\)
21.8.a.d	$21$	$8$	21.a	1.a	$1$	$3$	$3$	$6.560$	3.3.2910828.1	None	✓	✓	✓	✓	21.8.a.d	$1$	$0$	\(-3\)	\(81\)	\(-114\)	\(1029\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3^{3}q^{3}+(75+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(21)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)