Space of modular forms of level 117, weight 16, and Character 117.b

• Label: 117.16.b
• Level: $117$
• Weight: $16$
• Character orbit: 117.b
• Rep. character: $\chi_{117}(64,\cdot)$
• Character field: $\Q$
• Dimension: $86$
• Newform subspaces: $6$
• Sturm bound: $224$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(224\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(117, [\chi])\).

	Total	New	Old
Modular forms	214	88	126
Cusp forms	206	86	120
Eisenstein series	8	2	6

Trace form

\( 86 q - 1376258 q^{4} + O(q^{10}) \)

Decomposition of \(S_{16}^{\mathrm{new}}(117, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
117.16.b.a	$117$	$16$	117.b	13.b	$2$	$2$	$2$	$166.951$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+2^{15}q^{4}-133951\zeta_{6}q^{7}+(-198885925+\cdots)q^{13}+\cdots\)
117.16.b.b	$117$	$16$	117.b	13.b	$2$	$4$	$4$	$166.951$	4.0.8112.1	\(\Q(\sqrt{-39}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-59\beta _{1}+8\beta _{3})q^{2}+(-2^{15}-16489\beta _{2}+\cdots)q^{4}+\cdots\)
117.16.b.c	$117$	$16$	117.b	13.b	$2$	$16$	$16$	$166.951$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{29}\cdot 3^{18}\cdot 5^{4}\cdot 13^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-14336+\beta _{2})q^{4}+(-2^{5}\beta _{1}+\cdots)q^{5}+\cdots\)
117.16.b.d	$117$	$16$	117.b	13.b	$2$	$18$	$18$	$166.951$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{50}\cdot 3^{19}\cdot 5^{4}\cdot 13^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-20037+\beta _{2})q^{4}+(-28\beta _{1}+\cdots)q^{5}+\cdots\)
117.16.b.e	$117$	$16$	117.b	13.b	$2$	$18$	$18$	$166.951$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{40}\cdot 3^{18}\cdot 5^{6}\cdot 13^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-14552+\beta _{2})q^{4}+(42\beta _{1}+\cdots)q^{5}+\cdots\)
117.16.b.f	$117$	$16$	117.b	13.b	$2$	$28$	$28$	$166.951$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{16}^{\mathrm{old}}(117, [\chi])\) into lower level spaces

\( S_{16}^{\mathrm{old}}(117, [\chi]) \cong \) \(S_{16}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)