Space of modular forms of level 6, weight 12, and trivial character

• Label: 6.12.a
• Level: $6$
• Weight: $12$
• Character orbit: 6.a
• Rep. character: $\chi_{6}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $12$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 6 = 2 \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	6.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(12\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(6))\).

	Total	New	Old
Modular forms	13	3	10
Cusp forms	9	3	6
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
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(1\)

Trace form

3 * q - 32 * q^2 + 243 * q^3 + 3072 * q^4 - 2334 * q^5 + 7776 * q^6 + 55392 * q^7 - 32768 * q^8 + 177147 * q^9 + 307008 * q^10 - 1699116 * q^11 + 248832 * q^12 + 217266 * q^13 + 335360 * q^14 - 3369438 * q^15 + 3145728 * q^16 + 8603622 * q^17 - 1889568 * q^18 + 7200204 * q^19 - 2390016 * q^20 - 21757248 * q^21 + 5811840 * q^22 - 1057800 * q^23 + 7962624 * q^24 + 37532181 * q^25 - 165855424 * q^26 + 14348907 * q^27 + 56721408 * q^28 + 70389066 * q^29 + 164275776 * q^30 - 146569368 * q^31 - 33554432 * q^32 - 214136460 * q^33 + 255268800 * q^34 + 1123978944 * q^35 + 181398528 * q^36 - 1480918566 * q^37 - 925913728 * q^38 - 611837550 * q^39 + 314376192 * q^40 + 899560158 * q^41 + 1208452608 * q^42 - 273486252 * q^43 - 1739894784 * q^44 - 137820366 * q^45 + 1348363008 * q^46 + 1552169280 * q^47 + 254803968 * q^48 + 2904631227 * q^49 - 3482708192 * q^50 - 544856058 * q^51 + 222480384 * q^52 - 1324585278 * q^53 + 459165024 * q^54 + 1121307192 * q^55 + 343408640 * q^56 - 5621075028 * q^57 - 408318912 * q^58 + 4411180500 * q^59 - 3450304512 * q^60 - 12529017966 * q^61 + 14322308864 * q^62 + 3270842208 * q^63 + 3221225472 * q^64 - 16758434292 * q^65 - 4947682176 * q^66 + 27115273932 * q^67 + 8810108928 * q^68 + 25109273592 * q^69 - 28315634688 * q^70 - 25932460536 * q^71 - 1934917632 * q^72 + 6082420686 * q^73 + 26916324416 * q^74 + 16692865317 * q^75 + 7373008896 * q^76 - 28048880640 * q^77 - 19034606016 * q^78 - 58308979368 * q^79 - 2447376384 * q^80 + 10460353203 * q^81 - 52338500928 * q^82 + 37956458316 * q^83 - 22279421952 * q^84 + 121299146244 * q^85 + 48461612672 * q^86 - 40525895862 * q^87 + 5951324160 * q^88 - 210662658 * q^89 + 18128515392 * q^90 - 49315648704 * q^91 - 1083187200 * q^92 - 7597286136 * q^93 + 127169639424 * q^94 + 13965964680 * q^95 + 8153726976 * q^96 - 49488374586 * q^97 - 150071184672 * q^98 - 100331100684 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(6))\) 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
6.12.a.a	$6$	$12$	6.a	1.a	$1$	$1$	$1$	$4.610$	\(\Q\)	None	✓	✓	✓	✓	6.12.a.a	$1$	$0$	\(-32\)	\(-243\)	\(5766\)	\(72464\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+5766q^{5}+\cdots\)
6.12.a.b	$6$	$12$	6.a	1.a	$1$	$1$	$1$	$4.610$	\(\Q\)	None	✓	✓	✓	✓	6.12.a.b	$1$	$1$	\(-32\)	\(243\)	\(-11730\)	\(-50008\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-11730q^{5}+\cdots\)
6.12.a.c	$6$	$12$	6.a	1.a	$1$	$1$	$1$	$4.610$	\(\Q\)	None	✓	✓	✓	✓	6.12.a.c	$1$	$0$	\(32\)	\(243\)	\(3630\)	\(32936\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+3630q^{5}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(6)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)