Space of modular forms of level 9, weight 18, and trivial character

• Label: 9.18.a
• Level: $9$
• Weight: $18$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $18$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(18\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	19	7	12
Cusp forms	15	6	9
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
\(+\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)		\(10\)	\(5\)	\(5\)		\(8\)	\(4\)	\(4\)		\(2\)	\(1\)	\(1\)

Trace form

6 * q - 270 * q^2 + 234708 * q^4 + 806544 * q^5 + 6458160 * q^7 - 76566600 * q^8 - 374481036 * q^10 + 923799384 * q^11 - 2296007700 * q^13 + 6391587312 * q^14 + 4085718288 * q^16 + 84518474760 * q^17 - 62355388344 * q^19 + 651425563656 * q^20 - 496342340280 * q^22 + 724535089920 * q^23 - 155397346734 * q^25 + 2916102914172 * q^26 - 74759048160 * q^28 - 1799242747776 * q^29 + 3748330542720 * q^31 - 20367629690400 * q^32 + 3286200129084 * q^34 - 31901492535120 * q^35 + 37848820569060 * q^37 - 69673042158840 * q^38 - 72386049129552 * q^40 + 49852612514376 * q^41 - 34027022273640 * q^43 + 251638092507024 * q^44 + 219822902095824 * q^46 + 505557230796720 * q^47 - 424989443622474 * q^49 + 122797506379086 * q^50 - 400769546611560 * q^52 + 355160088780480 * q^53 + 556455745858896 * q^55 - 1475076076084800 * q^56 + 449915591780100 * q^58 - 1497422030520312 * q^59 + 1522569427205844 * q^61 - 7045407610135200 * q^62 + 1702525901416512 * q^64 + 14827467465792 * q^65 - 2052350804175000 * q^67 + 3145105961728920 * q^68 - 867144826576800 * q^70 + 21202162848708192 * q^71 - 17131224427726380 * q^73 - 7647379891364628 * q^74 + 18434654480975376 * q^76 + 32541109248013920 * q^77 - 10634113244496768 * q^79 + 543351843727776 * q^80 - 4473418054179060 * q^82 + 43230229439974920 * q^83 + 18905975942078592 * q^85 - 138915358987224456 * q^86 + 15032539537999200 * q^88 + 2744379883963512 * q^89 - 2296075661688576 * q^91 - 255143197159941600 * q^92 + 32034484897176672 * q^94 + 106855668170186256 * q^95 - 14887566902661180 * q^97 + 12028777617492450 * q^98

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.18.a.a	$9$	$18$	9.a	1.a	$1$	$1$	$1$	$16.490$	\(\Q\)	None	✓				3.18.a.a	$1$	$0$	\(-204\)	\(0\)	\(163554\)	\(-20846560\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-204q^{2}-89456q^{4}+163554q^{5}+\cdots\)
9.18.a.b	$9$	$18$	9.a	1.a	$1$	$1$	$1$	$16.490$	\(\Q\)	None	✓				1.18.a.a	$1$	$0$	\(528\)	\(0\)	\(1025850\)	\(3225992\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+528q^{2}+147712q^{4}+1025850q^{5}+\cdots\)
9.18.a.c	$9$	$18$	9.a	1.a	$1$	$2$	$2$	$16.490$	\(\Q(\sqrt{14569}) \)	None	✓		✓		3.18.a.b	$1$	$0$	\(-594\)	\(0\)	\(-382860\)	\(24471568\)		$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-297-\beta )q^{2}+(88258+594\beta )q^{4}+\cdots\)
9.18.a.d	$9$	$18$	9.a	1.a	$1$	$2$	$2$	$16.490$	\(\Q(\sqrt{910}) \)	None	✓	✓	✓	✓	9.18.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-392840\)		$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-2^{5}q^{4}-280\beta q^{5}-196420q^{7}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(9)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)