Space of modular forms of level 54, weight 10, and trivial character

• Label: 54.10.a
• Level: $54$
• Weight: $10$
• Character orbit: 54.a
• Rep. character: $\chi_{54}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $8$
• Sturm bound: $90$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(90\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(54))\).

	Total	New	Old
Modular forms	87	12	75
Cusp forms	75	12	63
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
\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(7\)

Trace form

\( 12 q + 3072 q^{4} - 2028 q^{7} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(54))\) 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
54.10.a.a	$54$	$10$	54.a	1.a	$1$	$1$	$1$	$27.812$	\(\Q\)	None	✓	✓			54.10.a.a	$1$	$0$	\(-16\)	\(0\)	\(-1176\)	\(-11473\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}-1176q^{5}-11473q^{7}+\cdots\)
54.10.a.b	$54$	$10$	54.a	1.a	$1$	$1$	$1$	$27.812$	\(\Q\)	None	✓	✓			54.10.a.b	$1$	$1$	\(-16\)	\(0\)	\(435\)	\(-2527\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+435q^{5}-2527q^{7}+\cdots\)
54.10.a.c	$54$	$10$	54.a	1.a	$1$	$1$	$1$	$27.812$	\(\Q\)	None	✓	✓			54.10.a.b	$1$	$1$	\(16\)	\(0\)	\(-435\)	\(-2527\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}-435q^{5}-2527q^{7}+\cdots\)
54.10.a.d	$54$	$10$	54.a	1.a	$1$	$1$	$1$	$27.812$	\(\Q\)	None	✓	✓			54.10.a.a	$1$	$1$	\(16\)	\(0\)	\(1176\)	\(-11473\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+1176q^{5}-11473q^{7}+\cdots\)
54.10.a.e	$54$	$10$	54.a	1.a	$1$	$2$	$2$	$27.812$	\(\Q(\sqrt{301}) \)	None	✓	✓	✓		54.10.a.e	$1$	$1$	\(-32\)	\(0\)	\(-1704\)	\(6538\)		$+$	$+$	$+$	$2^{3}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-852-\beta )q^{5}+\cdots\)
54.10.a.f	$54$	$10$	54.a	1.a	$1$	$2$	$2$	$27.812$	\(\Q(\sqrt{3329}) \)	None	✓	✓	✓		54.10.a.f	$1$	$0$	\(-32\)	\(0\)	\(-912\)	\(6448\)		$+$	$-$	$-$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-456-\beta )q^{5}+\cdots\)
54.10.a.g	$54$	$10$	54.a	1.a	$1$	$2$	$2$	$27.812$	\(\Q(\sqrt{3329}) \)	None	✓	✓			54.10.a.f	$1$	$0$	\(32\)	\(0\)	\(912\)	\(6448\)		$-$	$+$	$-$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(456-\beta )q^{5}+(3224+\cdots)q^{7}+\cdots\)
54.10.a.h	$54$	$10$	54.a	1.a	$1$	$2$	$2$	$27.812$	\(\Q(\sqrt{301}) \)	None	✓	✓			54.10.a.e	$1$	$0$	\(32\)	\(0\)	\(1704\)	\(6538\)		$-$	$+$	$-$	$2^{3}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(852-\beta )q^{5}+(3269+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(54))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(54)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)