Space of modular forms of level 135, weight 2, and trivial character

• Label: 135.2.a
• Level: $135$
• Weight: $2$
• Character orbit: 135.a
• Rep. character: $\chi_{135}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $36$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(36\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(135))\).

	Total	New	Old
Modular forms	24	6	18
Cusp forms	13	6	7
Eisenstein series	11	0	11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(2\)
Plus space		\(+\)	\(1\)
Minus space		\(-\)	\(5\)

Trace form

6 * q + 10 * q^4 - 2 * q^7 + 2 * q^10 + 2 * q^13 - 14 * q^16 + 2 * q^19 - 20 * q^22 + 6 * q^25 - 32 * q^28 - 8 * q^31 + 10 * q^34 + 18 * q^37 - 12 * q^40 - 4 * q^43 - 18 * q^46 + 32 * q^49 + 24 * q^52 + 8 * q^55 + 8 * q^58 + 26 * q^61 - 24 * q^64 - 50 * q^67 + 12 * q^70 + 26 * q^73 + 30 * q^76 - 38 * q^79 + 68 * q^82 + 8 * q^85 - 12 * q^88 - 10 * q^91 + 40 * q^94 + 6 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(135))\) 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	5
135.2.a.a	$135$	$2$	135.a	1.a	$1$	$1$	$1$	$1.078$	\(\Q\)	None	✓	✓	✓	✓	135.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{5}-3q^{7}+2q^{10}+\cdots\)
135.2.a.b	$135$	$2$	135.a	1.a	$1$	$1$	$1$	$1.078$	\(\Q\)	None	✓	✓			135.2.a.a	$1$	$0$	\(2\)	\(0\)	\(1\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{5}-3q^{7}+2q^{10}+\cdots\)
135.2.a.c	$135$	$2$	135.a	1.a	$1$	$2$	$2$	$1.078$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓		135.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+q^{5}+(2-2\beta )q^{7}+\cdots\)
135.2.a.d	$135$	$2$	135.a	1.a	$1$	$2$	$2$	$1.078$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	135.2.a.c	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}-q^{5}+(2-2\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(135))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(135)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)