Space of modular forms of level 9, weight 20, and trivial character

• Label: 9.20.a
• Level: $9$
• Weight: $20$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $4$
• Sturm bound: $20$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(20\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	21	9	12
Cusp forms	17	8	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
\(+\)		\(11\)	\(4\)	\(7\)		\(9\)	\(4\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)		\(10\)	\(5\)	\(5\)		\(8\)	\(4\)	\(4\)		\(2\)	\(1\)	\(1\)

Trace form

8 * q - 54 * q^2 + 1947188 * q^4 - 7155000 * q^5 + 67656016 * q^7 - 437224824 * q^8 - 76895100 * q^10 + 9413141328 * q^11 + 15106490176 * q^13 - 147516385248 * q^14 + 254566114448 * q^16 + 224630946360 * q^17 + 2975892236080 * q^19 - 10116437575320 * q^20 + 20053471753800 * q^22 - 21305347490592 * q^23 + 71318486016680 * q^25 - 119499657822468 * q^26 + 221015913716608 * q^28 - 294303607912152 * q^29 + 365861519539888 * q^31 - 255963102683616 * q^32 - 228865120334100 * q^34 + 723751772538240 * q^35 - 2183367247210064 * q^37 + 2552414826507912 * q^38 - 3167414862266160 * q^40 + 3273867011176344 * q^41 - 7177793550922736 * q^43 + 16625805438169296 * q^44 - 22059956526059664 * q^46 + 573025291377408 * q^47 + 38962117834925064 * q^49 - 43223470119877050 * q^50 + 65615512079323000 * q^52 - 39434842655774712 * q^53 - 34313358598057440 * q^55 - 89766470317645440 * q^56 + 249846895535981076 * q^58 - 95938361858559024 * q^59 + 117580349704018672 * q^61 + 18747861357190608 * q^62 - 685105604933934016 * q^64 + 146661317593193520 * q^65 + 75708328470082960 * q^67 + 1131204480221373048 * q^68 - 1080530824050498240 * q^70 + 206260824851668512 * q^71 - 618209070325073408 * q^73 + 1228047324212502252 * q^74 - 860658637369876016 * q^76 - 1265844729468742272 * q^77 + 2478544244620623856 * q^79 - 971566484464257120 * q^80 + 3442829699001633372 * q^82 - 4286244831162491088 * q^83 + 4148944292037239280 * q^85 - 2155307319770659272 * q^86 - 210929922899904096 * q^88 - 11668039387458820776 * q^89 + 19031843391710886176 * q^91 + 4903758958447250784 * q^92 - 28830008702069947584 * q^94 - 181978724895913440 * q^95 - 76384917331328960 * q^97 + 49649812796286118746 * q^98

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.20.a.a	$9$	$20$	9.a	1.a	$1$	$1$	$1$	$20.594$	\(\Q\)	None	✓				1.20.a.a	$1$	$1$	\(-456\)	\(0\)	\(2377410\)	\(-16917544\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-456q^{2}-316352q^{4}+2377410q^{5}+\cdots\)
9.20.a.b	$9$	$20$	9.a	1.a	$1$	$1$	$1$	$20.594$	\(\Q\)	None	✓				3.20.a.a	$1$	$1$	\(1104\)	\(0\)	\(-3516270\)	\(-195590584\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+1104q^{2}+694528q^{4}-3516270q^{5}+\cdots\)
9.20.a.c	$9$	$20$	9.a	1.a	$1$	$2$	$2$	$20.594$	\(\Q(\sqrt{87481}) \)	None	✓		✓		3.20.a.b	$1$	$1$	\(-702\)	\(0\)	\(-6016140\)	\(113892064\)		$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-351-\beta )q^{2}+(386242+702\beta )q^{4}+\cdots\)
9.20.a.d	$9$	$20$	9.a	1.a	$1$	$4$	$4$	$20.594$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	9.20.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(166272080\)		$+$	$+$	$2^{9}\cdot 3^{10}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(199132+\beta _{3})q^{4}+(-1304\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_0(9)) \simeq \) \(S_{20}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)