Space of modular forms of level 9, weight 10, and trivial character

• Label: 9.10.a
• Level: $9$
• Weight: $10$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $10$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(10\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	11	4	7
Cusp forms	7	3	4
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
\(+\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)		\(6\)	\(3\)	\(3\)		\(4\)	\(2\)	\(2\)		\(2\)	\(1\)	\(1\)

Trace form

3 * q + 18 * q^2 + 84 * q^4 + 2844 * q^5 - 7932 * q^7 + 22392 * q^8 + 19764 * q^10 - 22608 * q^11 - 1938 * q^13 - 325584 * q^14 + 82704 * q^16 + 171180 * q^17 - 239592 * q^19 + 742536 * q^20 + 327240 * q^22 + 2732976 * q^23 - 1791879 * q^25 - 6016644 * q^26 + 1212576 * q^28 + 1950012 * q^29 + 8392740 * q^31 - 10875168 * q^32 - 8921988 * q^34 + 8079120 * q^35 - 13509822 * q^37 + 18758664 * q^38 + 32144688 * q^40 + 11576988 * q^41 - 44083272 * q^43 + 2815632 * q^44 + 3158352 * q^46 - 51281136 * q^47 + 140586363 * q^49 - 15134994 * q^50 - 185266920 * q^52 - 94588884 * q^53 - 34271424 * q^55 + 71144640 * q^56 + 323595972 * q^58 + 106386624 * q^59 - 265029078 * q^61 + 261185184 * q^62 - 212374464 * q^64 - 151342488 * q^65 + 486602160 * q^67 - 137315304 * q^68 - 463307040 * q^70 + 80513136 * q^71 + 25718154 * q^73 - 437254164 * q^74 + 937970448 * q^76 - 186280416 * q^77 - 826612284 * q^79 - 263981664 * q^80 + 26522316 * q^82 + 1020687984 * q^83 + 285268392 * q^85 + 259740792 * q^86 - 280716192 * q^88 - 90873684 * q^89 - 525354648 * q^91 + 428535072 * q^92 - 605638944 * q^94 + 999662976 * q^95 + 1628401650 * q^97 - 1503597438 * q^98

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.10.a.a	$9$	$10$	9.a	1.a	$1$	$1$	$1$	$4.635$	\(\Q\)	None	✓				3.10.a.b	$1$	$0$	\(-18\)	\(0\)	\(1530\)	\(9128\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-18q^{2}-188q^{4}+1530q^{5}+9128q^{7}+\cdots\)
9.10.a.b	$9$	$10$	9.a	1.a	$1$	$1$	$1$	$4.635$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓	✓	✓	9.10.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-12580\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{9}q^{4}-12580q^{7}+118370q^{13}+\cdots\)
9.10.a.c	$9$	$10$	9.a	1.a	$1$	$1$	$1$	$4.635$	\(\Q\)	None	✓				3.10.a.a	$1$	$0$	\(36\)	\(0\)	\(1314\)	\(-4480\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6^{2}q^{2}+28^{2}q^{4}+1314q^{5}-4480q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(9)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)