Space of modular forms of level 117, weight 3, and Character 117.j

• Label: 117.3.j
• Level: $117$
• Weight: $3$
• Character orbit: 117.j
• Rep. character: $\chi_{117}(73,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $20$
• Newform subspaces: $3$
• Sturm bound: $42$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(42\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(117, [\chi])\).

	Total	New	Old
Modular forms	64	24	40
Cusp forms	48	20	28
Eisenstein series	16	4	12

Trace form

20 * q + 12 * q^5 + 4 * q^7 - 12 * q^8 + 24 * q^11 + 12 * q^14 + 28 * q^16 + 16 * q^19 - 60 * q^20 - 24 * q^22 + 120 * q^26 - 52 * q^28 - 96 * q^29 - 64 * q^31 - 168 * q^32 + 36 * q^34 - 36 * q^35 - 112 * q^37 - 108 * q^40 + 84 * q^41 + 264 * q^44 + 156 * q^46 - 96 * q^47 + 108 * q^50 - 16 * q^52 + 312 * q^53 + 144 * q^55 + 60 * q^58 - 372 * q^59 - 168 * q^61 + 36 * q^65 + 52 * q^67 - 612 * q^68 - 120 * q^70 - 288 * q^71 + 88 * q^73 - 36 * q^74 + 136 * q^76 - 528 * q^79 + 420 * q^80 + 264 * q^83 + 456 * q^85 + 24 * q^86 + 252 * q^89 - 188 * q^91 + 1176 * q^92 + 444 * q^94 + 548 * q^97 - 948 * q^98

Decomposition of \(S_{3}^{\mathrm{new}}(117, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
117.3.j.a	$117$	$3$	117.j	13.d	$4$	$4$	$2$	$3.188$	\(\Q(i, \sqrt{10})\)	None					13.3.d.a	$2$	$0$	\(4\)	\(0\)	\(-8\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}+3\beta _{2}+2\beta _{3})q^{4}+\cdots\)
117.3.j.b	$117$	$3$	117.j	13.d	$4$	$8$	$4$	$3.188$	8.0.\(\cdots\).10	None					39.3.g.a	$2$	$0$	\(-4\)	\(0\)	\(20\)	\(8\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{2}+(-1-\beta _{1}-\beta _{4}-\beta _{7})q^{4}+\cdots\)
117.3.j.c	$117$	$3$	117.j	13.d	$4$	$8$	$4$	$3.188$	8.0.897122304.10	None		✓			117.3.j.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+(\beta _{3}+\beta _{7})q^{4}-\beta _{1}q^{5}+(1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(117, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(117, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)