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

• Label: 117.3.p
• Level: $117$
• Weight: $3$
• Character orbit: 117.p
• Rep. character: $\chi_{117}(35,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $20$
• Newform subspaces: $1$
• Sturm bound: $42$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	117.p (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(42\)
Trace bound:			\(0\)

Dimensions

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

	Total	New	Old
Modular forms	64	20	44
Cusp forms	48	20	28
Eisenstein series	16	0	16

Trace form

20 * q + 24 * q^4 - 6 * q^7 + 12 * q^10 - 2 * q^13 - 104 * q^16 - 92 * q^19 + 44 * q^22 - 116 * q^25 + 76 * q^28 - 156 * q^31 + 80 * q^34 + 148 * q^37 + 328 * q^40 + 186 * q^43 + 164 * q^46 + 8 * q^49 + 392 * q^52 - 208 * q^55 - 236 * q^58 + 210 * q^61 - 1568 * q^64 - 158 * q^67 - 1048 * q^70 + 156 * q^73 + 764 * q^76 - 132 * q^79 + 648 * q^82 + 44 * q^85 + 372 * q^88 + 834 * q^91 - 220 * q^94 - 14 * q^97

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.p.a	$117$	$3$	117.p	39.i	$6$	$20$	$10$	$3.188$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓	✓	✓	117.3.p.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)	$2^{2}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{5}q^{2}+(2\beta _{3}-\beta _{4})q^{4}+(\beta _{8}-\beta _{15}+\cdots)q^{5}+\cdots\)

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

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