Space of modular forms of level 9, weight 12, and trivial character

• Label: 9.12.a
• Level: $9$
• Weight: $12$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $3$
• Sturm bound: $12$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(12\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	13	5	8
Cusp forms	9	4	5
Eisenstein series	4	1	3

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
\(+\)		\(7\)	\(2\)	\(5\)		\(5\)	\(2\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)		\(6\)	\(3\)	\(3\)		\(4\)	\(2\)	\(2\)		\(2\)	\(1\)	\(1\)

Trace form

4 * q - 54 * q^2 + 3508 * q^4 + 540 * q^5 + 71696 * q^7 - 239544 * q^8 + 594180 * q^10 - 1172448 * q^11 + 1713776 * q^13 + 1763424 * q^14 - 5059952 * q^16 + 3821580 * q^17 - 29453392 * q^19 + 28783080 * q^20 + 20792520 * q^22 - 33955632 * q^23 + 109740340 * q^25 - 73630404 * q^26 - 32545792 * q^28 - 139157892 * q^29 + 108538448 * q^31 + 215594784 * q^32 - 494588052 * q^34 - 68197680 * q^35 + 463378616 * q^37 + 1777763592 * q^38 - 2203896240 * q^40 - 1206953892 * q^41 + 2206321904 * q^43 - 1787357232 * q^44 + 1790729712 * q^46 - 1131607152 * q^47 - 107107836 * q^49 + 947334150 * q^50 + 4662811640 * q^52 - 2196361332 * q^53 - 4455675360 * q^55 + 5719109760 * q^56 - 16399512684 * q^58 + 4633896816 * q^59 + 5721657464 * q^61 + 2704881168 * q^62 + 4989370432 * q^64 + 6905043720 * q^65 - 28417068400 * q^67 - 22613987592 * q^68 + 79161094080 * q^70 + 5039600976 * q^71 + 16675082432 * q^73 - 56222904276 * q^74 - 104166445360 * q^76 + 26657870688 * q^77 + 63074361008 * q^79 + 15795280800 * q^80 + 51819654492 * q^82 + 20566645632 * q^83 - 251722500120 * q^85 + 74308863672 * q^86 + 169487151264 * q^88 + 50465686284 * q^89 + 77023474432 * q^91 - 34357788576 * q^92 - 330468480576 * q^94 - 156270898080 * q^95 + 299666878400 * q^97 + 53396148186 * q^98

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.12.a.a	$9$	$12$	9.a	1.a	$1$	$1$	$1$	$6.915$	\(\Q\)	None	✓				3.12.a.a	$1$	$1$	\(-78\)	\(0\)	\(5370\)	\(-27760\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-78q^{2}+4036q^{4}+5370q^{5}-27760q^{7}+\cdots\)
9.12.a.b	$9$	$12$	9.a	1.a	$1$	$1$	$1$	$6.915$	\(\Q\)	None	✓				1.12.a.a	$1$	$1$	\(24\)	\(0\)	\(-4830\)	\(-16744\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+24q^{2}-1472q^{4}-4830q^{5}-16744q^{7}+\cdots\)
9.12.a.c	$9$	$12$	9.a	1.a	$1$	$2$	$2$	$6.915$	\(\Q(\sqrt{70}) \)	None	✓	✓	✓	✓	9.12.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(116200\)		$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+472q^{4}+224\beta q^{5}+58100q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(9)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)