Space of modular forms of level 243, weight 2, and trivial character

• Label: 243.2.a
• Level: $243$
• Weight: $2$
• Character orbit: 243.a
• Rep. character: $\chi_{243}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $6$
• Sturm bound: $54$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(54\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(243))\).

	Total	New	Old
Modular forms	36	12	24
Cusp forms	19	12	7
Eisenstein series	17	0	17

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(15\)	\(4\)	\(11\)		\(7\)	\(4\)	\(3\)		\(8\)	\(0\)	\(8\)
\(-\)		\(21\)	\(8\)	\(13\)		\(12\)	\(8\)	\(4\)		\(9\)	\(0\)	\(9\)

Trace form

12 * q + 12 * q^4 - 3 * q^7 - 3 * q^13 + 12 * q^16 - 3 * q^19 + 12 * q^25 - 12 * q^28 - 12 * q^31 - 18 * q^34 - 3 * q^37 - 18 * q^40 - 12 * q^43 - 18 * q^46 + 9 * q^49 - 12 * q^52 - 18 * q^55 - 18 * q^58 + 24 * q^61 - 6 * q^64 + 24 * q^67 - 18 * q^70 + 24 * q^73 - 30 * q^76 - 21 * q^79 + 72 * q^82 + 72 * q^85 - 36 * q^88 + 12 * q^91 - 36 * q^94 + 69 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(243))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3
243.2.a.a	$243$	$2$	243.a	1.a	$1$	$1$	$1$	$1.940$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			243.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-4q^{7}-7q^{13}+4q^{16}-q^{19}+\cdots\)
243.2.a.b	$243$	$2$	243.a	1.a	$1$	$1$	$1$	$1.940$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			243.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+5q^{7}+2q^{13}+4q^{16}+8q^{19}+\cdots\)
243.2.a.c	$243$	$2$	243.a	1.a	$1$	$2$	$2$	$1.940$	\(\Q(\sqrt{3}) \)	None	✓	✓			243.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+2\beta q^{5}-q^{7}-\beta q^{8}+\cdots\)
243.2.a.d	$243$	$2$	243.a	1.a	$1$	$2$	$2$	$1.940$	\(\Q(\sqrt{6}) \)	None	✓	✓			243.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+4q^{4}-\beta q^{5}+2q^{7}+2\beta q^{8}+\cdots\)
243.2.a.e	$243$	$2$	243.a	1.a	$1$	$3$	$3$	$1.940$	\(\Q(\zeta_{18})^+\)	None	✓	✓	✓		243.2.a.e	$1$	$1$	\(-3\)	\(0\)	\(-6\)	\(-3\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
243.2.a.f	$243$	$2$	243.a	1.a	$1$	$3$	$3$	$1.940$	\(\Q(\zeta_{18})^+\)	None	✓	✓	✓		243.2.a.e	$1$	$0$	\(3\)	\(0\)	\(6\)	\(-3\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(243))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(243)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)