Space of modular forms of level 54, weight 8, and trivial character

• Label: 54.8.a
• Level: $54$
• Weight: $8$
• Character orbit: 54.a
• Rep. character: $\chi_{54}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $8$
• Sturm bound: $72$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(72\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(54))\).

	Total	New	Old
Modular forms	69	10	59
Cusp forms	57	10	47
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
\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(3\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(4\)

Trace form

\( 10 q + 640 q^{4} + 1286 q^{7} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(54))\) 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
54.8.a.a	$54$	$8$	54.a	1.a	$1$	$1$	$1$	$16.869$	\(\Q\)	None	✓	✓			54.8.a.a	$1$	$1$	\(-8\)	\(0\)	\(-120\)	\(377\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}-120q^{5}+377q^{7}+\cdots\)
54.8.a.b	$54$	$8$	54.a	1.a	$1$	$1$	$1$	$16.869$	\(\Q\)	None	✓	✓			54.8.a.b	$1$	$1$	\(-8\)	\(0\)	\(105\)	\(-937\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+105q^{5}-937q^{7}+\cdots\)
54.8.a.c	$54$	$8$	54.a	1.a	$1$	$1$	$1$	$16.869$	\(\Q\)	None	✓	✓			54.8.a.c	$1$	$0$	\(-8\)	\(0\)	\(312\)	\(323\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+312q^{5}+323q^{7}+\cdots\)
54.8.a.d	$54$	$8$	54.a	1.a	$1$	$1$	$1$	$16.869$	\(\Q\)	None	✓	✓			54.8.a.c	$1$	$1$	\(8\)	\(0\)	\(-312\)	\(323\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}-312q^{5}+323q^{7}+\cdots\)
54.8.a.e	$54$	$8$	54.a	1.a	$1$	$1$	$1$	$16.869$	\(\Q\)	None	✓	✓			54.8.a.b	$1$	$1$	\(8\)	\(0\)	\(-105\)	\(-937\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}-105q^{5}-937q^{7}+\cdots\)
54.8.a.f	$54$	$8$	54.a	1.a	$1$	$1$	$1$	$16.869$	\(\Q\)	None	✓	✓			54.8.a.a	$1$	$0$	\(8\)	\(0\)	\(120\)	\(377\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+120q^{5}+377q^{7}+\cdots\)
54.8.a.g	$54$	$8$	54.a	1.a	$1$	$2$	$2$	$16.869$	\(\Q(\sqrt{329}) \)	None	✓	✓	✓		54.8.a.g	$1$	$0$	\(-16\)	\(0\)	\(48\)	\(880\)		$+$	$+$	$+$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(24+\beta )q^{5}+(440+\cdots)q^{7}+\cdots\)
54.8.a.h	$54$	$8$	54.a	1.a	$1$	$2$	$2$	$16.869$	\(\Q(\sqrt{329}) \)	None	✓	✓	✓		54.8.a.g	$1$	$0$	\(16\)	\(0\)	\(-48\)	\(880\)		$-$	$-$	$+$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(-24-\beta )q^{5}+(440+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(54))\) into lower level spaces

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