Embedded newform 117.2.r.b.49.8

• Label: 117.2.r.b.49.8
• Level: $117$
• Weight: $2$
• Character: 117.49
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $22$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.8
Character	\(\chi\)	\(=\)	117.49
Dual form			117.2.r.b.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.677814 + 0.391336i) q^{2} +(1.22535 - 1.22414i) q^{3} +(-0.693712 - 1.20154i) q^{4} +(-0.0536139 - 0.0309540i) q^{5} +(1.30961 - 0.350219i) q^{6} +3.75567i q^{7} -2.65124i q^{8} +(0.00294547 - 3.00000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 2 q^{3} + 10 q^{4} + 3 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(92\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.677814	+	0.391336i	0.479287	+	0.276716i	0.720119	−	0.693850i	\(-0.244087\pi\)
							−0.240832	+	0.970567i	\(0.577420\pi\)
\(3\)	1.22535	−	1.22414i	0.707454	−	0.706760i
							\(5\)	−0.0536139	−	0.0309540i	−0.0239769	−	0.0138431i	0.487964	−	0.872864i	\(-0.337740\pi\)
−0.511941	+	0.859021i	\(0.671073\pi\)
\(7\)			3.75567i			1.41951i	0.704449	+	0.709754i	\(0.251194\pi\)
							−0.704449	+	0.709754i	\(0.748806\pi\)
\(11\)	−1.11071	−	0.641267i	−0.334891	−	0.193349i	0.323120	−	0.946358i	\(-0.395268\pi\)
							−0.658010	+	0.753009i	\(0.728602\pi\)
\(13\)	1.65180	+	3.20493i	0.458127	+	0.888887i
							\(17\)	−2.74383	+	4.75245i	−0.665476	+	1.15264i	0.313681	+	0.949529i	\(0.398438\pi\)
−0.979156	+	0.203109i	\(0.934895\pi\)
\(19\)	2.72231	+	1.57173i	0.624540	+	0.360579i	0.778635	−	0.627478i	\(-0.215913\pi\)
							−0.154094	+	0.988056i	\(0.549246\pi\)
\(23\)	−5.65979			−1.18015			−0.590074	−	0.807349i	\(-0.700901\pi\)
							−0.590074	+	0.807349i	\(0.700901\pi\)
\(29\)	3.56722	−	6.17860i	0.662416	−	1.14734i	−0.317563	−	0.948237i	\(-0.602865\pi\)
							0.979979	−	0.199101i	\(-0.0638021\pi\)
\(31\)	3.62246	+	2.09143i	0.650613	+	0.375632i	0.788691	−	0.614790i	\(-0.210759\pi\)
							−0.138078	+	0.990421i	\(0.544092\pi\)
\(37\)	−4.03923	+	2.33205i	−0.664045	+	0.383386i	−0.793816	−	0.608158i	\(-0.791909\pi\)
							0.129772	+	0.991544i	\(0.458576\pi\)
\(41\)		−	9.88981i		−	1.54453i	−0.635302	−	0.772264i	\(-0.719124\pi\)
							0.635302	−	0.772264i	\(-0.280876\pi\)
\(43\)	4.22761			0.644704			0.322352	−	0.946620i	\(-0.395526\pi\)
							0.322352	+	0.946620i	\(0.395526\pi\)
\(47\)	2.73495	−	1.57902i	0.398933	−	0.230324i	−0.287090	−	0.957904i	\(-0.592688\pi\)
							0.686024	+	0.727579i	\(0.259355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.r.b.49.8	yes	22
3.2	odd	2		351.2.r.b.10.4		22
9.2	odd	6		351.2.l.b.127.8		22
9.7	even	3		117.2.l.b.88.4	yes	22
13.4	even	6		117.2.l.b.4.8	✓	22
39.17	odd	6		351.2.l.b.199.4		22
117.43	even	6	inner	117.2.r.b.43.8	yes	22
117.56	odd	6		351.2.r.b.316.4		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.l.b.4.8	✓	22	13.4	even	6
117.2.l.b.88.4	yes	22	9.7	even	3
117.2.r.b.43.8	yes	22	117.43	even	6	inner
117.2.r.b.49.8	yes	22	1.1	even	1	trivial
351.2.l.b.127.8		22	9.2	odd	6
351.2.l.b.199.4		22	39.17	odd	6
351.2.r.b.10.4		22	3.2	odd	2
351.2.r.b.316.4		22	117.56	odd	6