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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(20\)	\(3\)	\(17\)		\(17\)	\(3\)	\(14\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(23\)	\(3\)	\(20\)		\(20\)	\(3\)	\(17\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(22\)	\(4\)	\(18\)		\(19\)	\(4\)	\(15\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(22\)	\(2\)	\(20\)		\(19\)	\(2\)	\(17\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(42\)	\(5\)	\(37\)		\(36\)	\(5\)	\(31\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(45\)	\(7\)	\(38\)		\(39\)	\(7\)	\(32\)		\(6\)	\(0\)	\(6\)

Trace form

12 * q + 3072 * q^4 - 2028 * q^7 + 107424 * q^10 - 238602 * q^13 + 786432 * q^16 + 1246506 * q^19 - 1213344 * q^22 + 7193082 * q^25 - 519168 * q^28 + 13496190 * q^31 + 14957568 * q^34 + 16326654 * q^37 + 27500544 * q^40 - 120336060 * q^43 + 11494080 * q^46 + 343489176 * q^49 - 61082112 * q^52 - 25712910 * q^55 + 338266944 * q^58 - 480658566 * q^61 + 201326592 * q^64 + 391751610 * q^67 - 2042784 * q^70 - 964349364 * q^73 + 319105536 * q^76 - 785100738 * q^79 - 415847232 * q^82 - 1843313760 * q^85 - 310616064 * q^88 - 626166942 * q^91 + 368030016 * q^94 - 3923856408 * q^97

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

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