Space of modular forms of level 117, weight 2, and trivial character

• Label: 117.2.a
• Level: $117$
• Weight: $2$
• Character orbit: 117.a
• Rep. character: $\chi_{117}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $3$
• Sturm bound: $28$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(28\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(117))\).

	Total	New	Old
Modular forms	18	5	13
Cusp forms	11	5	6
Eisenstein series	7	0	7

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

\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(5\)	\(2\)	\(3\)		\(3\)	\(2\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(6\)	\(2\)	\(4\)		\(4\)	\(2\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(4\)	\(1\)	\(3\)		\(2\)	\(1\)	\(1\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(7\)	\(1\)	\(6\)		\(4\)	\(1\)	\(3\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(11\)	\(4\)	\(7\)		\(7\)	\(4\)	\(3\)		\(4\)	\(0\)	\(4\)

Trace form

5 * q + q^2 + 3 * q^4 - 2 * q^5 + 9 * q^8 - 6 * q^10 + q^13 - 4 * q^14 - 5 * q^16 - 6 * q^17 + 4 * q^19 - 14 * q^20 - 4 * q^22 + 8 * q^23 - 5 * q^25 - 3 * q^26 - 8 * q^28 + 6 * q^29 - 11 * q^32 - 10 * q^34 + 24 * q^35 - 2 * q^37 + 8 * q^38 - 14 * q^40 - 22 * q^41 + 12 * q^43 + 8 * q^44 + 32 * q^46 + 12 * q^47 + 5 * q^49 + 7 * q^50 - q^52 - 2 * q^53 + 8 * q^55 - 20 * q^56 + 10 * q^58 - 16 * q^59 - 18 * q^61 - 20 * q^62 - 5 * q^64 - 2 * q^65 + 28 * q^67 + 30 * q^68 + 8 * q^70 - 4 * q^71 - 6 * q^73 + 14 * q^74 + 20 * q^76 + 16 * q^77 + 2 * q^80 + 6 * q^82 - 28 * q^85 + 36 * q^86 + 12 * q^88 - 22 * q^89 + 8 * q^92 - 40 * q^94 - 16 * q^95 - 14 * q^97 - 7 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(117))\) 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	13
117.2.a.a	$117$	$2$	117.a	1.a	$1$	$1$	$1$	$0.934$	\(\Q\)	None	✓		✓	✓	39.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-2q^{5}-4q^{7}+3q^{8}+2q^{10}+\cdots\)
117.2.a.b	$117$	$2$	117.a	1.a	$1$	$2$	$2$	$0.934$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	117.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+2q^{7}-\beta q^{8}-2\beta q^{11}+\cdots\)
117.2.a.c	$117$	$2$	117.a	1.a	$1$	$2$	$2$	$0.934$	\(\Q(\sqrt{2}) \)	None	✓		✓	✓	39.2.a.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-2\beta q^{5}-2\beta q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(117))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(117)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)