Space of modular forms of level 36, weight 8, and trivial character

• Label: 36.8.a
• Level: $36$
• Weight: $8$
• Character orbit: 36.a
• Rep. character: $\chi_{36}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	36.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(48\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(36))\).

	Total	New	Old
Modular forms	48	3	45
Cusp forms	36	3	33
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
\(+\)	\(+\)	\(+\)		\(13\)	\(0\)	\(13\)		\(9\)	\(0\)	\(9\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(12\)	\(0\)	\(12\)		\(8\)	\(0\)	\(8\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(11\)	\(1\)	\(10\)		\(9\)	\(1\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(12\)	\(2\)	\(10\)		\(10\)	\(2\)	\(8\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(25\)	\(2\)	\(23\)		\(19\)	\(2\)	\(17\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(23\)	\(1\)	\(22\)		\(17\)	\(1\)	\(16\)		\(6\)	\(0\)	\(6\)

Trace form

3 * q + 108 * q^5 - 228 * q^7 + 8208 * q^11 - 4314 * q^13 + 58860 * q^17 - 22008 * q^19 + 92880 * q^23 - 18591 * q^25 + 108 * q^29 + 58524 * q^31 - 614736 * q^35 + 35610 * q^37 - 418500 * q^41 + 516264 * q^43 - 1328400 * q^47 - 283797 * q^49 + 2043900 * q^53 - 606528 * q^55 + 433728 * q^59 + 365298 * q^61 + 6854760 * q^65 - 696432 * q^67 - 1643760 * q^71 - 5809350 * q^73 + 4298400 * q^77 + 9934908 * q^79 - 10620720 * q^83 - 1440504 * q^85 - 3245940 * q^89 - 10029768 * q^91 - 20131200 * q^95 + 24341298 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(36))\) 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
36.8.a.a	$36$	$8$	36.a	1.a	$1$	$1$	$1$	$11.246$	\(\Q\)	None	✓				12.8.a.b	$1$	$0$	\(0\)	\(0\)	\(-270\)	\(1112\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-270q^{5}+1112q^{7}+5724q^{11}+\cdots\)
36.8.a.b	$36$	$8$	36.a	1.a	$1$	$1$	$1$	$11.246$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓	✓	✓	36.8.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-508\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-508q^{7}-14614q^{13}-57448q^{19}+\cdots\)
36.8.a.c	$36$	$8$	36.a	1.a	$1$	$1$	$1$	$11.246$	\(\Q\)	None	✓				12.8.a.a	$1$	$0$	\(0\)	\(0\)	\(378\)	\(-832\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+378q^{5}-832q^{7}+2484q^{11}+14870q^{13}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(36))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(36)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)