Space of modular forms of level 108, weight 4, and trivial character

• Label: 108.4.a
• Level: $108$
• Weight: $4$
• Character orbit: 108.a
• Rep. character: $\chi_{108}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $72$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(72\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(108))\).

	Total	New	Old
Modular forms	63	4	59
Cusp forms	45	4	41
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
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(2\)

Trace form

\( 4 q - 22 q^{7} + 14 q^{13} + 140 q^{19} - 338 q^{25} + 98 q^{31} + 662 q^{37} - 1108 q^{43} + 288 q^{49} + 1134 q^{55} - 742 q^{61} - 292 q^{67} - 868 q^{73} - 4 q^{79} + 1296 q^{85} + 2272 q^{91} - 112 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(108))\) 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
108.4.a.a	$108$	$4$	108.a	1.a	$1$	$1$	$1$	$6.372$	\(\Q\)	None	✓	✓			108.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-9\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{5}-q^{7}-63q^{11}-28q^{13}-72q^{17}+\cdots\)
108.4.a.b	$108$	$4$	108.a	1.a	$1$	$1$	$1$	$6.372$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			108.4.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-37\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-37q^{7}-19q^{13}-163q^{19}-5^{3}q^{25}+\cdots\)
108.4.a.c	$108$	$4$	108.a	1.a	$1$	$1$	$1$	$6.372$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			108.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(17\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+17q^{7}+89q^{13}+107q^{19}-5^{3}q^{25}+\cdots\)
108.4.a.d	$108$	$4$	108.a	1.a	$1$	$1$	$1$	$6.372$	\(\Q\)	None	✓	✓			108.4.a.a	$1$	$0$	\(0\)	\(0\)	\(9\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{5}-q^{7}+63q^{11}-28q^{13}+72q^{17}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(108))\) into lower level spaces

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