Space of modular forms of level 21, weight 6, and trivial character

• Label: 21.6.a
• Level: $21$
• Weight: $6$
• Character orbit: 21.a
• Rep. character: $\chi_{21}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $16$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(16\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(21))\).

	Total	New	Old
Modular forms	16	4	12
Cusp forms	12	4	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	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(2\)
Plus space		\(+\)	\(1\)
Minus space		\(-\)	\(3\)

Trace form

4 * q + 10 * q^2 + 34 * q^4 + 32 * q^5 + 180 * q^6 - 98 * q^7 + 270 * q^8 + 324 * q^9 - 1092 * q^10 + 248 * q^11 + 792 * q^12 - 88 * q^13 - 1078 * q^14 - 504 * q^15 + 466 * q^16 - 3168 * q^17 + 810 * q^18 + 4400 * q^19 - 6500 * q^20 - 882 * q^21 - 1824 * q^22 - 744 * q^23 + 540 * q^24 + 14812 * q^25 + 5956 * q^26 - 1274 * q^28 - 7048 * q^29 - 792 * q^30 + 7616 * q^31 + 9590 * q^32 + 360 * q^33 - 12372 * q^34 + 6076 * q^35 + 2754 * q^36 + 1208 * q^37 - 15632 * q^38 - 1008 * q^39 - 41244 * q^40 + 17872 * q^41 - 3528 * q^42 - 27040 * q^43 + 18208 * q^44 + 2592 * q^45 - 43032 * q^46 - 11184 * q^47 + 7920 * q^48 + 9604 * q^49 + 89942 * q^50 - 11880 * q^51 + 35108 * q^52 - 42840 * q^53 + 14580 * q^54 + 41328 * q^55 + 3234 * q^56 - 24768 * q^57 + 63180 * q^58 + 6576 * q^59 - 83088 * q^60 + 7832 * q^61 + 65664 * q^62 - 7938 * q^63 + 19018 * q^64 - 193312 * q^65 + 37656 * q^66 + 5360 * q^67 + 23292 * q^68 - 24840 * q^69 + 7644 * q^70 + 4008 * q^71 + 21870 * q^72 + 101816 * q^73 + 47532 * q^74 + 115488 * q^75 - 90784 * q^76 + 31360 * q^77 - 2304 * q^78 - 58720 * q^79 - 341732 * q^80 + 26244 * q^81 - 110868 * q^82 - 6912 * q^83 - 42336 * q^84 - 149232 * q^85 + 42008 * q^86 + 182880 * q^87 + 118992 * q^88 - 16624 * q^89 - 88452 * q^90 - 39004 * q^91 + 39816 * q^92 + 112032 * q^93 + 26208 * q^94 + 438080 * q^95 + 7380 * q^96 - 279592 * q^97 + 24010 * q^98 + 20088 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\) 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
21.6.a.a	$21$	$6$	21.a	1.a	$1$	$1$	$1$	$3.368$	\(\Q\)	None	✓	✓	✓	✓	21.6.a.a	$1$	$0$	\(-6\)	\(-9\)	\(78\)	\(49\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}-9q^{3}+4q^{4}+78q^{5}+54q^{6}+\cdots\)
21.6.a.b	$21$	$6$	21.a	1.a	$1$	$1$	$1$	$3.368$	\(\Q\)	None	✓	✓	✓	✓	21.6.a.b	$1$	$1$	\(1\)	\(-9\)	\(-34\)	\(-49\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-9q^{3}-31q^{4}-34q^{5}-9q^{6}+\cdots\)
21.6.a.c	$21$	$6$	21.a	1.a	$1$	$1$	$1$	$3.368$	\(\Q\)	None	✓	✓			21.6.a.c	$1$	$0$	\(5\)	\(9\)	\(94\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+9q^{3}-7q^{4}+94q^{5}+45q^{6}+\cdots\)
21.6.a.d	$21$	$6$	21.a	1.a	$1$	$1$	$1$	$3.368$	\(\Q\)	None	✓	✓			21.6.a.d	$1$	$0$	\(10\)	\(9\)	\(-106\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+10q^{2}+9q^{3}+68q^{4}-106q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(21)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)