Space of modular forms of level 36, weight 4, and trivial character

• Label: 36.4.a
• Level: $36$
• Weight: $4$
• Character orbit: 36.a
• Rep. character: $\chi_{36}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $24$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	36.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(24\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(36))\).

	Total	New	Old
Modular forms	24	1	23
Cusp forms	12	1	11
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(7\)	\(0\)	\(7\)		\(3\)	\(0\)	\(3\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(13\)	\(1\)	\(12\)		\(7\)	\(1\)	\(6\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(11\)	\(0\)	\(11\)		\(5\)	\(0\)	\(5\)		\(6\)	\(0\)	\(6\)

Trace form

q + 18 * q^5 + 8 * q^7 - 36 * q^11 - 10 * q^13 - 18 * q^17 - 100 * q^19 - 72 * q^23 + 199 * q^25 + 234 * q^29 - 16 * q^31 + 144 * q^35 - 226 * q^37 - 90 * q^41 + 452 * q^43 - 432 * q^47 - 279 * q^49 - 414 * q^53 - 648 * q^55 + 684 * q^59 + 422 * q^61 - 180 * q^65 + 332 * q^67 + 360 * q^71 + 26 * q^73 - 288 * q^77 + 512 * q^79 + 1188 * q^83 - 324 * q^85 + 630 * q^89 - 80 * q^91 - 1800 * q^95 - 1054 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(36))\) 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
36.4.a.a	$36$	$4$	36.a	1.a	$1$	$1$	$1$	$2.124$	\(\Q\)	None	✓		✓	✓	12.4.a.a	$1$	$0$	\(0\)	\(0\)	\(18\)	\(8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{5}+8q^{7}-6^{2}q^{11}-10q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(36))\) into lower level spaces

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