Space of modular forms of level 12, weight 4, and trivial character

• Label: 12.4.a
• Level: $12$
• Weight: $4$
• Character orbit: 12.a
• Rep. character: $\chi_{12}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $8$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	12.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 \(M_{4}(\Gamma_0(12))\).

	Total	New	Old
Modular forms	9	1	8
Cusp forms	3	1	2
Eisenstein series	6	0	6

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

Trace form

q + 3 * q^3 - 18 * q^5 + 8 * q^7 + 9 * q^9 + 36 * q^11 - 10 * q^13 - 54 * q^15 + 18 * q^17 - 100 * q^19 + 24 * q^21 + 72 * q^23 + 199 * q^25 + 27 * q^27 - 234 * q^29 - 16 * q^31 + 108 * q^33 - 144 * q^35 - 226 * q^37 - 30 * q^39 + 90 * q^41 + 452 * q^43 - 162 * q^45 + 432 * q^47 - 279 * q^49 + 54 * q^51 + 414 * q^53 - 648 * q^55 - 300 * q^57 - 684 * q^59 + 422 * q^61 + 72 * q^63 + 180 * q^65 + 332 * q^67 + 216 * q^69 - 360 * q^71 + 26 * q^73 + 597 * q^75 + 288 * q^77 + 512 * q^79 + 81 * q^81 - 1188 * q^83 - 324 * q^85 - 702 * q^87 - 630 * q^89 - 80 * q^91 - 48 * q^93 + 1800 * q^95 - 1054 * q^97 + 324 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(12))\) 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
12.4.a.a	$12$	$4$	12.a	1.a	$1$	$1$	$1$	$0.708$	\(\Q\)	None	✓	✓	✓	✓	12.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-18\)	\(8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-18q^{5}+8q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(12))\) into lower level spaces

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