Space of modular forms of level 242, weight 2, and trivial character

• Label: 242.2.a
• Level: $242$
• Weight: $2$
• Character orbit: 242.a
• Rep. character: $\chi_{242}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $6$
• Sturm bound: $66$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(66\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(242))\).

	Total	New	Old
Modular forms	45	10	35
Cusp forms	22	10	12
Eisenstein series	23	0	23

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

\(2\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(7\)

Trace form

10 * q - 2 * q^3 + 10 * q^4 - 2 * q^5 + 8 * q^9 - 2 * q^12 - 4 * q^15 + 10 * q^16 - 2 * q^20 - 4 * q^23 + 4 * q^25 - 2 * q^26 - 8 * q^27 - 8 * q^31 - 4 * q^34 + 8 * q^36 - 10 * q^37 - 2 * q^38 - 4 * q^42 - 10 * q^45 - 8 * q^47 - 2 * q^48 + 2 * q^49 - 6 * q^53 - 6 * q^58 - 6 * q^59 - 4 * q^60 + 10 * q^64 - 18 * q^67 - 16 * q^69 - 8 * q^70 - 12 * q^71 - 22 * q^75 - 8 * q^78 - 2 * q^80 - 14 * q^81 - 12 * q^82 - 10 * q^86 - 16 * q^89 + 64 * q^91 - 4 * q^92 + 60 * q^93 + 60 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(242))\) 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	11
242.2.a.a	$242$	$2$	242.a	1.a	$1$	$1$	$1$	$1.932$	\(\Q\)	None	✓	✓			242.2.a.a	$1$	$0$	\(-1\)	\(-2\)	\(-3\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}-3q^{5}+2q^{6}+2q^{7}+\cdots\)
242.2.a.b	$242$	$2$	242.a	1.a	$1$	$1$	$1$	$1.932$	\(\Q\)	None	✓	✓	✓	✓	242.2.a.a	$1$	$1$	\(1\)	\(-2\)	\(-3\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-3q^{5}-2q^{6}-2q^{7}+\cdots\)
242.2.a.c	$242$	$2$	242.a	1.a	$1$	$2$	$2$	$1.932$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	242.2.a.c	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+(1+\cdots)q^{6}+\cdots\)
242.2.a.d	$242$	$2$	242.a	1.a	$1$	$2$	$2$	$1.932$	\(\Q(\sqrt{5}) \)	None	✓		✓		22.2.c.a	$1$	$0$	\(-2\)	\(3\)	\(2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta )q^{3}+q^{4}+(2-2\beta )q^{5}+\cdots\)
242.2.a.e	$242$	$2$	242.a	1.a	$1$	$2$	$2$	$1.932$	\(\Q(\sqrt{3}) \)	None	✓	✓			242.2.a.c	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+(-1+\cdots)q^{6}+\cdots\)
242.2.a.f	$242$	$2$	242.a	1.a	$1$	$2$	$2$	$1.932$	\(\Q(\sqrt{5}) \)	None	✓				22.2.c.a	$1$	$0$	\(2\)	\(3\)	\(2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(2-2\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(242))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(242)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)