Space of modular forms of level 135, weight 3, and Character 135.d

• Label: 135.3.d
• Level: $135$
• Weight: $3$
• Character orbit: 135.d
• Rep. character: $\chi_{135}(134,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $7$
• Sturm bound: $54$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	135.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(54\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(135, [\chi])\).

	Total	New	Old
Modular forms	42	16	26
Cusp forms	30	16	14
Eisenstein series	12	0	12

Trace form

\( 16 q + 30 q^{4} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(135, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
135.3.d.a	$135$	$3$	135.d	15.d	$2$	$2$	$2$	$3.678$	\(\Q(\sqrt{-11}) \)	None		✓			135.3.d.a	$2$	$0$	\(-2\)	\(0\)	\(-1\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}-3q^{4}-\beta q^{5}+(-1+2\beta )q^{7}+\cdots\)
135.3.d.b	$135$	$3$	135.d	15.d	$2$	$2$	$2$	$3.678$	\(\Q(\sqrt{-1}) \)	None		✓			135.3.d.b	$2$	$0$	\(-2\)	\(0\)	\(8\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}-3 q^{4}+(\beta+4)q^{5}-2\beta q^{7}+\cdots\)
135.3.d.c	$135$	$3$	135.d	15.d	$2$	$2$	$2$	$3.678$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓	✓			135.3.d.c	$2$	$0$	\(-1\)	\(0\)	\(10\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+(-1-\beta )q^{2}+(8+\beta )q^{4}+5q^{5}+(-15+\cdots)q^{8}+\cdots\)
135.3.d.d	$135$	$3$	135.d	15.d	$2$	$2$	$2$	$3.678$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	✓	✓			135.3.d.c	$2$	$0$	\(1\)	\(0\)	\(-10\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+(1+\beta )q^{2}+(8+\beta )q^{4}-5q^{5}+(15+\cdots)q^{8}+\cdots\)
135.3.d.e	$135$	$3$	135.d	15.d	$2$	$2$	$2$	$3.678$	\(\Q(\sqrt{-1}) \)	None		✓			135.3.d.b	$2$	$0$	\(2\)	\(0\)	\(-8\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}-3 q^{4}+(\beta-4)q^{5}+2\beta q^{7}+\cdots\)
135.3.d.f	$135$	$3$	135.d	15.d	$2$	$2$	$2$	$3.678$	\(\Q(\sqrt{-11}) \)	None		✓			135.3.d.a	$2$	$0$	\(2\)	\(0\)	\(1\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}-3q^{4}+(1-\beta )q^{5}+(1-2\beta )q^{7}+\cdots\)
135.3.d.g	$135$	$3$	135.d	15.d	$2$	$4$	$4$	$3.678$	\(\Q(i, \sqrt{10})\)	None		✓	✓		135.3.d.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+6q^{4}-\beta _{2}q^{5}+\beta _{1}q^{7}-2\beta _{3}q^{8}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(135, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(135, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)