Space of modular forms of level 12, weight 22, and trivial character

• Label: 12.22.a
• Level: $12$
• Weight: $22$
• Character orbit: 12.a
• Rep. character: $\chi_{12}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $44$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	12.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(44\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(12))\).

	Total	New	Old
Modular forms	45	3	42
Cusp forms	39	3	36
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
\(+\)	\(+\)	\(+\)		\(11\)	\(0\)	\(11\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(12\)	\(0\)	\(12\)		\(10\)	\(0\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(11\)	\(2\)	\(9\)		\(10\)	\(2\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(11\)	\(1\)	\(10\)		\(10\)	\(1\)	\(9\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(22\)	\(1\)	\(21\)		\(19\)	\(1\)	\(18\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(23\)	\(2\)	\(21\)		\(20\)	\(2\)	\(18\)		\(3\)	\(0\)	\(3\)

Trace form

3 * q - 59049 * q^3 + 17559810 * q^5 + 791583864 * q^7 + 10460353203 * q^9 - 112222937772 * q^11 - 1258424621670 * q^13 - 2367628113510 * q^15 + 2953244684214 * q^17 + 25208162419020 * q^19 - 13448739952008 * q^21 + 157775918327784 * q^23 + 435012492283725 * q^25 - 205891132094649 * q^27 - 4359534278069718 * q^29 + 3420045112432992 * q^31 + 2354801655303396 * q^33 + 58943170624401360 * q^35 + 16633045822244802 * q^37 + 21271071576651570 * q^39 - 58021202638110690 * q^41 + 91890557649844404 * q^43 + 61227271592523810 * q^45 + 639021428445851856 * q^47 + 801340454562513867 * q^49 + 76201133615357022 * q^51 - 605389055944970862 * q^53 + 223229472927915960 * q^55 - 2032878440091834180 * q^57 + 111533213628510372 * q^59 - 13790848405425405654 * q^61 + 2760082269078505464 * q^63 - 15411720181369637700 * q^65 + 5308269714319267356 * q^67 + 1914051812556412872 * q^69 + 31114565784489756600 * q^71 - 2692016962386256578 * q^73 - 67005681755538219975 * q^75 + 8172031342996112544 * q^77 - 115367151634837499856 * q^79 + 36472996377170786403 * q^81 + 224057353676040396684 * q^83 + 498730577924263463460 * q^85 - 7791544403279033886 * q^87 + 751807239847095025710 * q^89 - 697445725818940231920 * q^91 - 574631299589364832896 * q^93 + 519788551712918229000 * q^95 - 1581877116210731515482 * q^97 - 391297188857803294572 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(12))\) 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
12.22.a.a	$12$	$22$	12.a	1.a	$1$	$1$	$1$	$33.537$	\(\Q\)	None	✓	✓	✓	✓	12.22.a.a	$1$	$1$	\(0\)	\(59049\)	\(-11268090\)	\(281914136\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3^{10}q^{3}-11268090q^{5}+281914136q^{7}+\cdots\)
12.22.a.b	$12$	$22$	12.a	1.a	$1$	$2$	$2$	$33.537$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓	✓	12.22.a.b	$1$	$0$	\(0\)	\(-118098\)	\(28827900\)	\(509669728\)		$-$	$+$	$-$	$2^{7}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-3^{10}q^{3}+(14413950-5\beta )q^{5}+(254834864+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(12))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(12)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)