Space of modular forms of level 9, weight 16, and trivial character

• Label: 9.16.a
• Level: $9$
• Weight: $16$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $5$
• Sturm bound: $16$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(16\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	17	7	10
Cusp forms	13	6	7
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
\(+\)		\(9\)	\(3\)	\(6\)		\(7\)	\(3\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)		\(8\)	\(4\)	\(4\)		\(6\)	\(3\)	\(3\)		\(2\)	\(1\)	\(1\)

Trace form

6 * q + 90 * q^2 + 149748 * q^4 - 111330 * q^5 - 4637508 * q^7 - 2789784 * q^8 + 72765540 * q^10 - 91787220 * q^11 + 155922528 * q^13 - 1085740992 * q^14 + 6338575248 * q^16 - 5398648326 * q^17 + 142099284 * q^19 - 13005535320 * q^20 + 19969914312 * q^22 - 2268016344 * q^23 - 914723490 * q^25 + 110758903164 * q^26 - 423973368384 * q^28 + 201269476278 * q^29 + 256723352868 * q^31 + 115812052128 * q^32 - 66406867956 * q^34 - 237548797920 * q^35 - 742162256988 * q^37 - 1258670491128 * q^38 + 5578833690000 * q^40 - 2870975534814 * q^41 + 3774514173276 * q^43 - 8906559408 * q^44 - 13698912405456 * q^46 - 186132202848 * q^47 + 964196339814 * q^49 + 18637799721750 * q^50 - 13081373394504 * q^52 + 2605172709726 * q^53 + 14532457260600 * q^55 + 24312633327360 * q^56 - 67113774788364 * q^58 - 40346133868404 * q^59 + 57964344662388 * q^61 + 11039482068336 * q^62 + 170352596954688 * q^64 - 159905667699180 * q^65 + 1735689827676 * q^67 + 18210068335800 * q^68 - 200566302007680 * q^70 - 50889630307176 * q^71 + 229110272621184 * q^73 + 220679668039788 * q^74 - 326909774547888 * q^76 + 68507717213376 * q^77 + 317743497713172 * q^79 + 568465455515040 * q^80 - 1415898324911556 * q^82 - 154016576449740 * q^83 - 88713966524220 * q^85 + 100990225245240 * q^86 + 1585183587204384 * q^88 - 572646020260446 * q^89 + 436763017451544 * q^91 - 888807497693472 * q^92 + 1410397675781760 * q^94 - 414257174849160 * q^95 - 240276113030544 * q^97 - 1374140508526422 * q^98

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.16.a.a	$9$	$16$	9.a	1.a	$1$	$1$	$1$	$12.842$	\(\Q\)	None	✓				1.16.a.a	$1$	$1$	\(-216\)	\(0\)	\(-52110\)	\(2822456\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6^{3}q^{2}+13888q^{4}-52110q^{5}+\cdots\)
9.16.a.b	$9$	$16$	9.a	1.a	$1$	$1$	$1$	$12.842$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			9.16.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(1244900\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{15}q^{4}+1244900q^{7}+397771850q^{13}+\cdots\)
9.16.a.c	$9$	$16$	9.a	1.a	$1$	$1$	$1$	$12.842$	\(\Q\)	None	✓				3.16.a.b	$1$	$1$	\(72\)	\(0\)	\(221490\)	\(-2149000\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+72q^{2}-27584q^{4}+221490q^{5}+\cdots\)
9.16.a.d	$9$	$16$	9.a	1.a	$1$	$1$	$1$	$12.842$	\(\Q\)	None	✓				3.16.a.a	$1$	$1$	\(234\)	\(0\)	\(-280710\)	\(-1373344\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+234q^{2}+21988q^{4}-280710q^{5}+\cdots\)
9.16.a.e	$9$	$16$	9.a	1.a	$1$	$2$	$2$	$12.842$	\(\Q(\sqrt{370}) \)	None	✓	✓	✓		9.16.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-5182520\)		$+$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+87112q^{4}+464\beta q^{5}-2591260q^{7}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(9)) \simeq \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)