Space of modular forms of level 3, weight 6, and trivial character

• Label: 3.6.a
• Level: $3$
• Weight: $6$
• Character orbit: 3.a
• Rep. character: $\chi_{3}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $2$
• Trace bound: $0$

Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(3))\).

	Total	New	Old
Modular forms	3	1	2
Cusp forms	1	1	0
Eisenstein series	2	0	2

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

\(3\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(1\)	\(0\)	\(1\)		\(0\)	\(0\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

q - 6 * q^2 + 9 * q^3 + 4 * q^4 + 6 * q^5 - 54 * q^6 - 40 * q^7 + 168 * q^8 + 81 * q^9 - 36 * q^10 - 564 * q^11 + 36 * q^12 + 638 * q^13 + 240 * q^14 + 54 * q^15 - 1136 * q^16 + 882 * q^17 - 486 * q^18 - 556 * q^19 + 24 * q^20 - 360 * q^21 + 3384 * q^22 - 840 * q^23 + 1512 * q^24 - 3089 * q^25 - 3828 * q^26 + 729 * q^27 - 160 * q^28 + 4638 * q^29 - 324 * q^30 + 4400 * q^31 + 1440 * q^32 - 5076 * q^33 - 5292 * q^34 - 240 * q^35 + 324 * q^36 - 2410 * q^37 + 3336 * q^38 + 5742 * q^39 + 1008 * q^40 - 6870 * q^41 + 2160 * q^42 + 9644 * q^43 - 2256 * q^44 + 486 * q^45 + 5040 * q^46 - 18672 * q^47 - 10224 * q^48 - 15207 * q^49 + 18534 * q^50 + 7938 * q^51 + 2552 * q^52 + 33750 * q^53 - 4374 * q^54 - 3384 * q^55 - 6720 * q^56 - 5004 * q^57 - 27828 * q^58 - 18084 * q^59 + 216 * q^60 + 39758 * q^61 - 26400 * q^62 - 3240 * q^63 + 27712 * q^64 + 3828 * q^65 + 30456 * q^66 - 23068 * q^67 + 3528 * q^68 - 7560 * q^69 + 1440 * q^70 - 4248 * q^71 + 13608 * q^72 - 41110 * q^73 + 14460 * q^74 - 27801 * q^75 - 2224 * q^76 + 22560 * q^77 - 34452 * q^78 + 21920 * q^79 - 6816 * q^80 + 6561 * q^81 + 41220 * q^82 + 82452 * q^83 - 1440 * q^84 + 5292 * q^85 - 57864 * q^86 + 41742 * q^87 - 94752 * q^88 - 94086 * q^89 - 2916 * q^90 - 25520 * q^91 - 3360 * q^92 + 39600 * q^93 + 112032 * q^94 - 3336 * q^95 + 12960 * q^96 + 49442 * q^97 + 91242 * q^98 - 45684 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\) 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
3.6.a.a	$3$	$6$	3.a	1.a	$1$	$1$	$1$	$0.481$	\(\Q\)	None	✓	✓	✓	✓	3.6.a.a	$1$	$0$	\(-6\)	\(9\)	\(6\)	\(-40\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}+9q^{3}+4q^{4}+6q^{5}-54q^{6}+\cdots\)