Space of modular forms of level 117, weight 2, and Character 117.e

• Label: 117.2.e
• Level: $117$
• Weight: $2$
• Character orbit: 117.e
• Rep. character: $\chi_{117}(40,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $24$
• Newform subspaces: $3$
• Sturm bound: $28$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(28\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(117, [\chi])\).

	Total	New	Old
Modular forms	32	24	8
Cusp forms	24	24	0
Eisenstein series	8	0	8

Trace form

24 * q - 2 * q^2 - 4 * q^3 - 12 * q^4 - 2 * q^5 + 6 * q^6 + 12 * q^8 - 12 * q^9 - 6 * q^11 - 6 * q^12 + 14 * q^14 - 10 * q^15 - 12 * q^16 - 24 * q^17 + 22 * q^18 - 12 * q^19 + 10 * q^20 - 10 * q^21 + 6 * q^22 + 2 * q^23 - 12 * q^24 - 6 * q^25 + 12 * q^26 + 2 * q^27 + 12 * q^29 - 26 * q^30 + 6 * q^31 - 14 * q^32 + 38 * q^33 - 6 * q^34 + 12 * q^35 + 26 * q^36 - 12 * q^37 - 16 * q^38 - 12 * q^40 - 10 * q^41 + 54 * q^42 + 6 * q^43 - 16 * q^44 - 20 * q^45 - 12 * q^47 + 26 * q^48 - 38 * q^50 + 4 * q^51 - 56 * q^53 + 6 * q^54 + 12 * q^55 + 40 * q^56 - 12 * q^58 + 2 * q^59 - 12 * q^60 + 12 * q^61 + 16 * q^62 - 8 * q^63 + 48 * q^64 - 8 * q^65 - 18 * q^66 + 12 * q^67 + 48 * q^68 - 22 * q^69 + 56 * q^71 - 66 * q^72 - 36 * q^73 - 34 * q^74 + 32 * q^75 + 6 * q^76 + 10 * q^77 - 10 * q^78 - 12 * q^79 + 80 * q^80 - 36 * q^81 - 12 * q^82 + 2 * q^84 + 6 * q^86 + 10 * q^87 + 6 * q^88 + 56 * q^89 + 24 * q^90 + 56 * q^92 + 72 * q^93 + 44 * q^95 - 148 * q^96 + 24 * q^97 - 124 * q^98 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(117, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
117.2.e.a	$117$	$2$	117.e	9.c	$3$	$2$	$1$	$0.934$	\(\Q(\sqrt{-3}) \)	None		✓			117.2.e.a	$2$	$1$	\(-2\)	\(-3\)	\(-4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
117.2.e.b	$117$	$2$	117.e	9.c	$3$	$10$	$5$	$0.934$	10.0.\(\cdots\).1	None		✓			117.2.e.b	$2$	$0$	\(-2\)	\(1\)	\(-1\)	\(2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{9}q^{2}+(\beta _{1}-\beta _{5}+\beta _{9})q^{3}+(\beta _{1}-\beta _{6}+\cdots)q^{4}+\cdots\)
117.2.e.c	$117$	$2$	117.e	9.c	$3$	$12$	$6$	$0.934$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓	✓		117.2.e.c	$2$	$0$	\(2\)	\(-2\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{3}-\beta _{7}-\beta _{8}+\beta _{10}-\beta _{11})q^{2}+\cdots\)