Embedded newform 117.2.r.b.49.3

• Label: 117.2.r.b.49.3
• 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			49.3
Character	\(\chi\)	\(=\)	117.49
Dual form			117.2.r.b.43.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21740 - 0.702869i) q^{2} +(0.712485 - 1.57872i) q^{3} +(-0.0119503 - 0.0206986i) q^{4} +(2.61504 + 1.50979i) q^{5} +(-1.97702 + 1.42116i) q^{6} -3.19463i q^{7} +2.84507i q^{8} +(-1.98473 - 2.24963i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 2 q^{3} + 10 q^{4} + 3 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21740	−	0.702869i	−0.860835	−	0.497003i	0.00345668	−	0.999994i	\(-0.498900\pi\)
							−0.864292	+	0.502991i	\(0.832233\pi\)
\(3\)	0.712485	−	1.57872i	0.411353	−	0.911476i
							\(5\)	2.61504	+	1.50979i	1.16948	+	0.675199i	0.953557	−	0.301211i	\(-0.0973910\pi\)
0.215922	+	0.976411i	\(0.430724\pi\)
\(7\)		−	3.19463i		−	1.20746i	−0.797190	−	0.603728i	\(-0.793681\pi\)
							0.797190	−	0.603728i	\(-0.206319\pi\)
\(11\)	−0.687300	−	0.396813i	−0.207229	−	0.119644i	0.392794	−	0.919626i	\(-0.371509\pi\)
							−0.600023	+	0.799983i	\(0.704842\pi\)
\(13\)	−3.60510	+	0.0569665i	−0.999875	+	0.0157997i
							\(17\)	−0.0957495	+	0.165843i	−0.0232227	+	0.0402228i	−0.877403	−	0.479754i	\(-0.840726\pi\)
0.854181	+	0.519977i	\(0.174059\pi\)
\(19\)	6.28411	+	3.62813i	1.44167	+	0.832350i	0.997961	−	0.0638195i	\(-0.0203282\pi\)
							0.443711	+	0.896170i	\(0.353662\pi\)
\(23\)	2.54500			0.530670			0.265335	−	0.964156i	\(-0.414517\pi\)
							0.265335	+	0.964156i	\(0.414517\pi\)
\(29\)	−4.22442	+	7.31690i	−0.784454	+	1.35871i	0.144870	+	0.989451i	\(0.453723\pi\)
							−0.929325	+	0.369264i	\(0.879610\pi\)
\(31\)	6.22662	+	3.59494i	1.11833	+	0.645670i	0.940975	−	0.338476i	\(-0.109911\pi\)
							0.177358	+	0.984146i	\(0.443245\pi\)
\(37\)	2.25703	−	1.30310i	0.371054	−	0.214228i	−0.302865	−	0.953033i	\(-0.597943\pi\)
							0.673919	+	0.738806i	\(0.264610\pi\)
\(41\)			0.905916i			0.141480i	0.997495	+	0.0707402i	\(0.0225361\pi\)
							−0.997495	+	0.0707402i	\(0.977464\pi\)
\(43\)	−3.97309			−0.605890			−0.302945	−	0.953008i	\(-0.597970\pi\)
							−0.302945	+	0.953008i	\(0.597970\pi\)
\(47\)	6.60765	−	3.81493i	0.963825	−	0.556465i	0.0664769	−	0.997788i	\(-0.478824\pi\)
							0.897348	+	0.441323i	\(0.145491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.r.b.49.3	yes	22
3.2	odd	2		351.2.r.b.10.9		22
9.2	odd	6		351.2.l.b.127.3		22
9.7	even	3		117.2.l.b.88.9	yes	22
13.4	even	6		117.2.l.b.4.3	✓	22
39.17	odd	6		351.2.l.b.199.9		22
117.43	even	6	inner	117.2.r.b.43.3	yes	22
117.56	odd	6		351.2.r.b.316.9		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.l.b.4.3	✓	22	13.4	even	6
117.2.l.b.88.9	yes	22	9.7	even	3
117.2.r.b.43.3	yes	22	117.43	even	6	inner
117.2.r.b.49.3	yes	22	1.1	even	1	trivial
351.2.l.b.127.3		22	9.2	odd	6
351.2.l.b.199.9		22	39.17	odd	6
351.2.r.b.10.9		22	3.2	odd	2
351.2.r.b.316.9		22	117.56	odd	6