Space of modular forms of level 36, weight 2, and trivial character

• Label: 36.2.a
• Level: $36$
• Weight: $2$
• Character orbit: 36.a
• Rep. character: $\chi_{36}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $12$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	36.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(12\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(36))\).

	Total	New	Old
Modular forms	12	1	11
Cusp forms	1	1	0
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.

\(2\)	\(3\)	Fricke	Dim
\(-\)	\(+\)	\(-\)	\(1\)
Plus space		\(+\)	\(0\)
Minus space		\(-\)	\(1\)

Trace form

\( q - 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(36))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3
36.2.a.a	$36$	$2$	36.a	1.a	$1$	$1$	$1$	$0.287$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓	✓	✓	36.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{7}+2q^{13}+8q^{19}-5q^{25}-4q^{31}+\cdots\)