Space of modular forms of level 108, weight 8, and trivial character

• Label: 108.8.a
• Level: $108$
• Weight: $8$
• Character orbit: 108.a
• Rep. character: $\chi_{108}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $5$
• Sturm bound: $144$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	108.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(144\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(108))\).

	Total	New	Old
Modular forms	135	9	126
Cusp forms	117	9	108
Eisenstein series	18	0	18

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

\( 9 q - 135 q^{7} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(108))\) 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
108.8.a.a	$108$	$8$	108.a	1.a	$1$	$1$	$1$	$33.738$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			108.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-1255\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-1255q^{7}+2009q^{13}+43091q^{19}+\cdots\)
108.8.a.b	$108$	$8$	108.a	1.a	$1$	$2$	$2$	$33.738$	\(\Q(\sqrt{1289}) \)	None	✓	✓			108.8.a.b	$1$	$0$	\(0\)	\(0\)	\(-324\)	\(-980\)		$-$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-162-\beta )q^{5}+(-490-3\beta )q^{7}+\cdots\)
108.8.a.c	$108$	$8$	108.a	1.a	$1$	$2$	$2$	$33.738$	\(\Q(\sqrt{30}) \)	None	✓	✓			108.8.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-26\)		$-$	$+$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-13q^{7}-41\beta q^{11}-691q^{13}+\cdots\)
108.8.a.d	$108$	$8$	108.a	1.a	$1$	$2$	$2$	$33.738$	\(\Q(\sqrt{195}) \)	None	✓	✓			108.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(3106\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+1553q^{7}+13\beta q^{11}-799q^{13}+\cdots\)
108.8.a.e	$108$	$8$	108.a	1.a	$1$	$2$	$2$	$33.738$	\(\Q(\sqrt{1289}) \)	None	✓	✓			108.8.a.b	$1$	$1$	\(0\)	\(0\)	\(324\)	\(-980\)		$-$	$+$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(162-\beta )q^{5}+(-490+3\beta )q^{7}+(405+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(108))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(108)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)