Space of modular forms of level 36, weight 22, and trivial character

• Label: 36.22.a
• Level: $36$
• Weight: $22$
• Character orbit: 36.a
• Rep. character: $\chi_{36}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $4$
• Sturm bound: $132$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	36.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(132\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(36))\).

	Total	New	Old
Modular forms	132	9	123
Cusp forms	120	9	111
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.

\(2\)	\(3\)	Fricke	Dim
\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	$+$	\(5\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(4\)

Trace form

\( 9 q - 31249134 q^{5} - 43147656 q^{7} + O(q^{10}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(36))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
36.22.a.a	$36$	$22$	36.a	1.a	$1$	$1$	$1$	$100.612$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(11268090\)	\(281914136\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+11268090q^{5}+281914136q^{7}+\cdots\)
36.22.a.b	$36$	$22$	36.a	1.a	$1$	$2$	$2$	$100.612$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-28827900\)	\(509669728\)		$-$	$-$	$+$	$2^{7}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-14413950-5\beta )q^{5}+(254834864+\cdots)q^{7}+\cdots\)
36.22.a.c	$36$	$22$	36.a	1.a	$1$	$2$	$2$	$100.612$	\(\Q(\sqrt{2161}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-13689324\)	\(-260508080\)		$-$	$-$	$+$	$2^{8}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-6844662-68\beta )q^{5}+(-130254040+\cdots)q^{7}+\cdots\)
36.22.a.d	$36$	$22$	36.a	1.a	$1$	$4$	$4$	$100.612$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-574223440\)		$-$	$+$	$-$	$2^{26}\cdot 3^{12}\cdot 5\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(-143555860-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(36))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(36)) \cong \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 9}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)