Space of modular forms of level 54, weight 4, and trivial character

• Label: 54.4.a
• Level: $54$
• Weight: $4$
• Character orbit: 54.a
• Rep. character: $\chi_{54}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $36$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(36\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(54))\).

	Total	New	Old
Modular forms	33	4	29
Cusp forms	21	4	17
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	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(1\)

Trace form

\( 4 q + 16 q^{4} + 44 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(54))\) 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
54.4.a.a	$54$	$4$	54.a	1.a	$1$	$1$	$1$	$3.186$	\(\Q\)	None	✓	✓	✓	✓	54.4.a.a	$1$	$1$	\(-2\)	\(0\)	\(-12\)	\(-7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-12q^{5}-7q^{7}-8q^{8}+\cdots\)
54.4.a.b	$54$	$4$	54.a	1.a	$1$	$1$	$1$	$3.186$	\(\Q\)	None	✓	✓	✓	✓	54.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(-3\)	\(29\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-3q^{5}+29q^{7}-8q^{8}+\cdots\)
54.4.a.c	$54$	$4$	54.a	1.a	$1$	$1$	$1$	$3.186$	\(\Q\)	None	✓	✓			54.4.a.b	$1$	$0$	\(2\)	\(0\)	\(3\)	\(29\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+3q^{5}+29q^{7}+8q^{8}+\cdots\)
54.4.a.d	$54$	$4$	54.a	1.a	$1$	$1$	$1$	$3.186$	\(\Q\)	None	✓	✓			54.4.a.a	$1$	$0$	\(2\)	\(0\)	\(12\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+12q^{5}-7q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(54))\) into lower level spaces

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