Space of modular forms of level 135, weight 4, and trivial character

• Label: 135.4.a
• Level: $135$
• Weight: $4$
• Character orbit: 135.a
• Rep. character: $\chi_{135}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $8$
• Sturm bound: $72$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(72\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(135))\).

	Total	New	Old
Modular forms	60	16	44
Cusp forms	48	16	32
Eisenstein series	12	0	12

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
\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(6\)
Plus space		\(+\)	\(10\)
Minus space		\(-\)	\(6\)

Trace form

16 * q + 58 * q^4 + 68 * q^7 + 30 * q^10 - 100 * q^13 + 754 * q^16 - 64 * q^19 - 348 * q^22 + 400 * q^25 + 236 * q^28 + 296 * q^31 + 570 * q^34 - 412 * q^37 + 480 * q^40 - 736 * q^43 - 3090 * q^46 - 540 * q^49 - 772 * q^52 + 60 * q^55 - 3744 * q^58 + 668 * q^61 + 4600 * q^64 + 4244 * q^67 - 540 * q^70 + 788 * q^73 - 3814 * q^76 + 716 * q^79 - 2556 * q^82 + 120 * q^85 - 10212 * q^88 - 2084 * q^91 + 4116 * q^94 + 3212 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(135))\) 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
135.4.a.a	$135$	$4$	135.a	1.a	$1$	$1$	$1$	$7.965$	\(\Q\)	None	✓	✓			135.4.a.a	$1$	$1$	\(-2\)	\(0\)	\(5\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-4q^{4}+5q^{5}+24q^{8}-10q^{10}+\cdots\)
135.4.a.b	$135$	$4$	135.a	1.a	$1$	$1$	$1$	$7.965$	\(\Q\)	None	✓	✓			135.4.a.b	$1$	$0$	\(-1\)	\(0\)	\(-5\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}-5q^{5}-6q^{7}+15q^{8}+\cdots\)
135.4.a.c	$135$	$4$	135.a	1.a	$1$	$1$	$1$	$7.965$	\(\Q\)	None	✓	✓			135.4.a.b	$1$	$1$	\(1\)	\(0\)	\(5\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}+5q^{5}-6q^{7}-15q^{8}+\cdots\)
135.4.a.d	$135$	$4$	135.a	1.a	$1$	$1$	$1$	$7.965$	\(\Q\)	None	✓	✓			135.4.a.a	$1$	$1$	\(2\)	\(0\)	\(-5\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-4q^{4}-5q^{5}-24q^{8}-10q^{10}+\cdots\)
135.4.a.e	$135$	$4$	135.a	1.a	$1$	$3$	$3$	$7.965$	3.3.1772.1	None	✓	✓	✓		135.4.a.e	$1$	$1$	\(-5\)	\(0\)	\(-15\)	\(-4\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(7-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
135.4.a.f	$135$	$4$	135.a	1.a	$1$	$3$	$3$	$7.965$	3.3.5637.1	None	✓	✓			135.4.a.f	$1$	$0$	\(-1\)	\(0\)	\(15\)	\(44\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)
135.4.a.g	$135$	$4$	135.a	1.a	$1$	$3$	$3$	$7.965$	3.3.5637.1	None	✓	✓	✓		135.4.a.f	$1$	$0$	\(1\)	\(0\)	\(-15\)	\(44\)		$+$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(7+\beta _{1}+\beta _{2})q^{4}-5q^{5}+\cdots\)
135.4.a.h	$135$	$4$	135.a	1.a	$1$	$3$	$3$	$7.965$	3.3.1772.1	None	✓	✓			135.4.a.e	$1$	$0$	\(5\)	\(0\)	\(15\)	\(-4\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(7-3\beta _{1}+\beta _{2})q^{4}+5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(135))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(135)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)