Space of modular forms of level 6, weight 8, and trivial character

• Label: 6.8.a
• Level: $6$
• Weight: $8$
• Character orbit: 6.a
• Rep. character: $\chi_{6}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $8$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 6 = 2 \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	6.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(8\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(6))\).

	Total	New	Old
Modular forms	9	1	8
Cusp forms	5	1	4
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.

\(2\)	\(3\)	Fricke	Dim
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(1\)
Minus space		\(-\)	\(0\)

Trace form

q + 8 * q^2 + 27 * q^3 + 64 * q^4 - 114 * q^5 + 216 * q^6 - 1576 * q^7 + 512 * q^8 + 729 * q^9 - 912 * q^10 + 7332 * q^11 + 1728 * q^12 - 3802 * q^13 - 12608 * q^14 - 3078 * q^15 + 4096 * q^16 - 6606 * q^17 + 5832 * q^18 + 24860 * q^19 - 7296 * q^20 - 42552 * q^21 + 58656 * q^22 + 41448 * q^23 + 13824 * q^24 - 65129 * q^25 - 30416 * q^26 + 19683 * q^27 - 100864 * q^28 - 41610 * q^29 - 24624 * q^30 + 33152 * q^31 + 32768 * q^32 + 197964 * q^33 - 52848 * q^34 + 179664 * q^35 + 46656 * q^36 - 36466 * q^37 + 198880 * q^38 - 102654 * q^39 - 58368 * q^40 - 639078 * q^41 - 340416 * q^42 - 156412 * q^43 + 469248 * q^44 - 83106 * q^45 + 331584 * q^46 - 433776 * q^47 + 110592 * q^48 + 1660233 * q^49 - 521032 * q^50 - 178362 * q^51 - 243328 * q^52 + 786078 * q^53 + 157464 * q^54 - 835848 * q^55 - 806912 * q^56 + 671220 * q^57 - 332880 * q^58 + 745140 * q^59 - 196992 * q^60 - 1660618 * q^61 + 265216 * q^62 - 1148904 * q^63 + 262144 * q^64 + 433428 * q^65 + 1583712 * q^66 - 3290836 * q^67 - 422784 * q^68 + 1119096 * q^69 + 1437312 * q^70 + 5716152 * q^71 + 373248 * q^72 + 2659898 * q^73 - 291728 * q^74 - 1758483 * q^75 + 1591040 * q^76 - 11555232 * q^77 - 821232 * q^78 + 3807440 * q^79 - 466944 * q^80 + 531441 * q^81 - 5112624 * q^82 + 2229468 * q^83 - 2723328 * q^84 + 753084 * q^85 - 1251296 * q^86 - 1123470 * q^87 + 3753984 * q^88 + 5991210 * q^89 - 664848 * q^90 + 5991952 * q^91 + 2652672 * q^92 + 895104 * q^93 - 3470208 * q^94 - 2834040 * q^95 + 884736 * q^96 - 4060126 * q^97 + 13281864 * q^98 + 5345028 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(6))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3
6.8.a.a	$6$	$8$	6.a	1.a	$1$	$1$	$1$	$1.874$	\(\Q\)	None	✓	✓	✓	✓	6.8.a.a	$1$	$0$	\(8\)	\(27\)	\(-114\)	\(-1576\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-114q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(6)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)