Space of modular forms of level 9, weight 26, and trivial character

• Label: 9.26.a
• Level: $9$
• Weight: $26$
• Character orbit: 9.a
• Rep. character: $\chi_{9}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $4$
• Sturm bound: $26$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(26\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{26}(\Gamma_0(9))\).

	Total	New	Old
Modular forms	27	11	16
Cusp forms	23	10	13
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
\(+\)		\(13\)	\(4\)	\(9\)		\(11\)	\(4\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)		\(14\)	\(7\)	\(7\)		\(12\)	\(6\)	\(6\)		\(2\)	\(1\)	\(1\)

Trace form

10 * q + 4050 * q^2 + 220267348 * q^4 + 334280844 * q^5 - 40918831120 * q^7 + 288091685880 * q^8 - 11209064898636 * q^10 + 15715268450280 * q^11 - 74761346748100 * q^13 - 279806803135056 * q^14 + 5569970029723408 * q^16 - 3663997807947660 * q^17 - 21217736873658616 * q^19 + 52107134347831176 * q^20 - 70457763987482040 * q^22 + 10211937347530800 * q^23 + 1133420517164978086 * q^25 - 151760101572249732 * q^26 - 3245902148646774880 * q^28 + 2773786800594119004 * q^29 - 8033045133786377440 * q^31 + 22116218905429083360 * q^32 - 27215628268059889668 * q^34 + 39426602932449584640 * q^35 + 18268446607706969660 * q^37 + 318540211201856069640 * q^38 - 657835176129802083792 * q^40 + 781145503713308249892 * q^41 + 36975571414944336440 * q^43 + 2254010900914313367696 * q^44 - 1941282469414653512112 * q^46 + 2241742133351605489920 * q^47 - 2196918340425253819014 * q^49 + 8760281440407049589166 * q^50 - 10596633413935934040040 * q^52 + 428284381587667438860 * q^53 + 13949658121609170948816 * q^55 - 17467291510967593247040 * q^56 + 39443922956707978162500 * q^58 - 45253971693438198786072 * q^59 - 7509152988113527282708 * q^61 - 75283372091267901036000 * q^62 + 108502130134040005518400 * q^64 - 137816045072347030379928 * q^65 - 37704903622586737043800 * q^67 - 388192003918481236625640 * q^68 + 339494380072244591171040 * q^70 - 190967914817177968333872 * q^71 + 465679359436374271589780 * q^73 + 526199316467462788703724 * q^74 - 2035038633209335713564400 * q^76 + 1256909847221159799955200 * q^77 - 1431283368219208577770816 * q^79 + 3172673989155704153574816 * q^80 + 2655719126416945746477900 * q^82 + 1867990788574425981440280 * q^83 - 3527790254390856980458008 * q^85 + 2149286748163902122519160 * q^86 - 2939391296691021839417760 * q^88 + 2992722485060481035826852 * q^89 + 7744187915268002880698848 * q^91 - 2459380751814926996678880 * q^92 - 19150855182255804939678240 * q^94 - 6852384385171181748854064 * q^95 + 432380286540481372534340 * q^97 - 38189763006479864037209790 * q^98

Decomposition of \(S_{26}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.26.a.a	$9$	$26$	9.a	1.a	$1$	$1$	$1$	$35.640$	\(\Q\)	None	✓				1.26.a.a	$1$	$0$	\(48\)	\(0\)	\(741989850\)	\(39080597192\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+48q^{2}-33552128q^{4}+741989850q^{5}+\cdots\)
9.26.a.b	$9$	$26$	9.a	1.a	$1$	$2$	$2$	$35.640$	\(\Q(\sqrt{1287001}) \)	None	✓				3.26.a.a	$1$	$0$	\(324\)	\(0\)	\(-570861756\)	\(-29687385728\)		$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(162-\beta )q^{2}+(12803848-18^{2}\beta )q^{4}+\cdots\)
9.26.a.c	$9$	$26$	9.a	1.a	$1$	$3$	$3$	$35.640$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		3.26.a.b	$1$	$0$	\(3678\)	\(0\)	\(163152750\)	\(-9622572744\)		$-$	$-$	$2^{6}\cdot 3^{5}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(1226-\beta _{1})q^{2}+(30249196-1499\beta _{1}+\cdots)q^{4}+\cdots\)
9.26.a.d	$9$	$26$	9.a	1.a	$1$	$4$	$4$	$35.640$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	9.26.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-40689469840\)		$+$	$+$	$2^{15}\cdot 3^{10}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(34366048+\beta _{3})q^{4}+(-64017\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{26}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{26}^{\mathrm{old}}(\Gamma_0(9)) \simeq \) \(S_{26}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{26}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)