Space of modular forms of level 36, weight 3, and Character 36.d

• Label: 36.3.d
• Level: $36$
• Weight: $3$
• Character orbit: 36.d
• Rep. character: $\chi_{36}(19,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $3$
• Sturm bound: $18$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	36.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(18\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(36, [\chi])\).

	Total	New	Old
Modular forms	16	6	10
Cusp forms	8	4	4
Eisenstein series	8	2	6

Trace form

\( 4 q + 2 q^{2} + 4 q^{4} + 4 q^{5} - 16 q^{8} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(36, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
36.3.d.a	$36$	$3$	36.d	4.b	$2$	$1$	$1$	$0.981$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(-2\)	\(0\)	\(8\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+4q^{4}+8q^{5}-8q^{8}-2^{4}q^{10}+\cdots\)
36.3.d.b	$36$	$3$	36.d	4.b	$2$	$1$	$1$	$0.981$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(2\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+4q^{4}-8q^{5}+8q^{8}-2^{4}q^{10}+\cdots\)
36.3.d.c	$36$	$3$	36.d	4.b	$2$	$2$	$2$	$0.981$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(2\)	\(0\)	\(4\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}+2q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(36, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(36, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 2}\)