Space of modular forms of level 242, weight 8, and trivial character

• Label: 242.8.a
• Level: $242$
• Weight: $8$
• Character orbit: 242.a
• Rep. character: $\chi_{242}(1,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $19$
• Sturm bound: $264$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(264\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(242))\).

	Total	New	Old
Modular forms	243	64	179
Cusp forms	219	64	155
Eisenstein series	24	0	24

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
\(+\)	\(+\)	$+$	\(16\)
\(+\)	\(-\)	$-$	\(16\)
\(-\)	\(+\)	$-$	\(14\)
\(-\)	\(-\)	$+$	\(18\)
Plus space		\(+\)	\(34\)
Minus space		\(-\)	\(30\)

Trace form

\( 64 q - 66 q^{3} + 4096 q^{4} + 70 q^{5} + 1024 q^{6} - 1120 q^{7} + 49378 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(242))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
242.8.a.a	$242$	$8$	242.a	1.a	$1$	$1$	$1$	$75.597$	\(\Q\)	None	✓	$1$	$1$	\(-8\)	\(-46\)	\(233\)	\(1238\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-46q^{3}+2^{6}q^{4}+233q^{5}+\cdots\)
242.8.a.b	$242$	$8$	242.a	1.a	$1$	$1$	$1$	$75.597$	\(\Q\)	None	✓	$1$	$1$	\(-8\)	\(-21\)	\(-551\)	\(-62\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-21q^{3}+2^{6}q^{4}-551q^{5}+\cdots\)
242.8.a.c	$242$	$8$	242.a	1.a	$1$	$1$	$1$	$75.597$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(-46\)	\(233\)	\(-1238\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-46q^{3}+2^{6}q^{4}+233q^{5}+\cdots\)
242.8.a.d	$242$	$8$	242.a	1.a	$1$	$1$	$1$	$75.597$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(-19\)	\(317\)	\(1030\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-19q^{3}+2^{6}q^{4}+317q^{5}+\cdots\)
242.8.a.e	$242$	$8$	242.a	1.a	$1$	$1$	$1$	$75.597$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(12\)	\(-210\)	\(-1016\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+12q^{3}+2^{6}q^{4}-210q^{5}+\cdots\)
242.8.a.f	$242$	$8$	242.a	1.a	$1$	$1$	$1$	$75.597$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(91\)	\(185\)	\(722\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+91q^{3}+2^{6}q^{4}+185q^{5}+\cdots\)
242.8.a.g	$242$	$8$	242.a	1.a	$1$	$2$	$2$	$75.597$	\(\Q(\sqrt{1585}) \)	None	✓	$1$	$0$	\(-16\)	\(-40\)	\(30\)	\(-432\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-8q^{2}+(-20-\beta )q^{3}+2^{6}q^{4}+(15+\cdots)q^{5}+\cdots\)
242.8.a.h	$242$	$8$	242.a	1.a	$1$	$2$	$2$	$75.597$	\(\Q(\sqrt{14881}) \)	None	✓	$1$	$1$	\(-16\)	\(-23\)	\(331\)	\(-1794\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(-11-\beta )q^{3}+2^{6}q^{4}+(165+\cdots)q^{5}+\cdots\)
242.8.a.i	$242$	$8$	242.a	1.a	$1$	$2$	$2$	$75.597$	\(\Q(\sqrt{718}) \)	None	✓	$1$	$1$	\(-16\)	\(4\)	\(-190\)	\(-2148\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-8q^{2}+(2+\beta )q^{3}+2^{6}q^{4}+(-95+\cdots)q^{5}+\cdots\)
242.8.a.j	$242$	$8$	242.a	1.a	$1$	$2$	$2$	$75.597$	\(\Q(\sqrt{1585}) \)	None	✓	$1$	$1$	\(16\)	\(-40\)	\(30\)	\(432\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-20-\beta )q^{3}+2^{6}q^{4}+(15+\cdots)q^{5}+\cdots\)
242.8.a.k	$242$	$8$	242.a	1.a	$1$	$2$	$2$	$75.597$	\(\Q(\sqrt{718}) \)	None	✓	$1$	$0$	\(16\)	\(4\)	\(-190\)	\(2148\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+8q^{2}+(2+\beta )q^{3}+2^{6}q^{4}+(-95+\cdots)q^{5}+\cdots\)
242.8.a.l	$242$	$8$	242.a	1.a	$1$	$4$	$4$	$75.597$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-32\)	\(16\)	\(404\)	\(656\)		$+$	$-$	$-$	$2^{7}\cdot 11$	$\mathrm{SU}(2)$	\(q-8q^{2}+(4+\beta _{2})q^{3}+2^{6}q^{4}+(101+\cdots)q^{5}+\cdots\)
242.8.a.m	$242$	$8$	242.a	1.a	$1$	$4$	$4$	$75.597$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(32\)	\(16\)	\(404\)	\(-656\)		$-$	$-$	$+$	$2^{7}\cdot 11$	$\mathrm{SU}(2)$	\(q+8q^{2}+(4+\beta _{2})q^{3}+2^{6}q^{4}+(101+\cdots)q^{5}+\cdots\)
242.8.a.n	$242$	$8$	242.a	1.a	$1$	$6$	$6$	$75.597$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-48\)	\(-26\)	\(-192\)	\(2490\)		$+$	$+$	$+$	$2^{5}\cdot 3^{2}\cdot 11^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}+(-4-\beta _{1}+\beta _{2})q^{3}+2^{6}q^{4}+\cdots\)
242.8.a.o	$242$	$8$	242.a	1.a	$1$	$6$	$6$	$75.597$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-48\)	\(-13\)	\(-380\)	\(928\)		$+$	$-$	$-$	$2^{2}\cdot 11^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}+(-2+\beta _{2})q^{3}+2^{6}q^{4}+(-2^{6}+\cdots)q^{5}+\cdots\)
242.8.a.p	$242$	$8$	242.a	1.a	$1$	$6$	$6$	$75.597$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(48\)	\(-26\)	\(-192\)	\(-2490\)		$-$	$+$	$-$	$2^{5}\cdot 3^{2}\cdot 11^{3}$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-4-\beta _{1}+\beta _{2})q^{3}+2^{6}q^{4}+\cdots\)
242.8.a.q	$242$	$8$	242.a	1.a	$1$	$6$	$6$	$75.597$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(48\)	\(-13\)	\(-380\)	\(-928\)		$-$	$+$	$-$	$2^{2}\cdot 11^{3}$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-2+\beta _{2})q^{3}+2^{6}q^{4}+(-2^{6}+\cdots)q^{5}+\cdots\)
242.8.a.r	$242$	$8$	242.a	1.a	$1$	$8$	$8$	$75.597$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-64\)	\(52\)	\(94\)	\(-140\)		$+$	$+$	$+$	$2^{4}\cdot 5\cdot 11^{5}$	$\mathrm{SU}(2)$	\(q-8q^{2}+(7+\beta _{1}-\beta _{2})q^{3}+2^{6}q^{4}+\cdots\)
242.8.a.s	$242$	$8$	242.a	1.a	$1$	$8$	$8$	$75.597$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(64\)	\(52\)	\(94\)	\(140\)		$-$	$-$	$+$	$2^{4}\cdot 5\cdot 11^{5}$	$\mathrm{SU}(2)$	\(q+8q^{2}+(7+\beta _{1}-\beta _{2})q^{3}+2^{6}q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(242))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(242)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)