Space of modular forms of level 108, weight 2, and trivial character

• Label: 108.2.a
• Level: $108$
• Weight: $2$
• Character orbit: 108.a
• Rep. character: $\chi_{108}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $36$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	108.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(36\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(108))\).

	Total	New	Old
Modular forms	27	1	26
Cusp forms	10	1	9
Eisenstein series	17	0	17

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
\(+\)	\(+\)	\(+\)		\(6\)	\(0\)	\(6\)		\(1\)	\(0\)	\(1\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(9\)	\(0\)	\(9\)		\(3\)	\(0\)	\(3\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(6\)	\(1\)	\(5\)		\(3\)	\(1\)	\(2\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(3\)	\(0\)	\(3\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(12\)	\(0\)	\(12\)		\(4\)	\(0\)	\(4\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(15\)	\(1\)	\(14\)		\(6\)	\(1\)	\(5\)		\(9\)	\(0\)	\(9\)

Trace form

\( q + 5 q^{7} - 7 q^{13} - q^{19} - 5 q^{25} - 4 q^{31} - q^{37} + 8 q^{43} + 18 q^{49} - 13 q^{61} + 11 q^{67} + 17 q^{73} - 13 q^{79} - 35 q^{91} + 5 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(108))\) 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
108.2.a.a	$108$	$2$	108.a	1.a	$1$	$1$	$1$	$0.862$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓	✓	✓	108.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+5q^{7}-7q^{13}-q^{19}-5q^{25}-4q^{31}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(108))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(108)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)