Embedded newform 322.2.o.a.241.1

• Label: 322.2.o.a.241.1
• Level: $322$
• Weight: $2$
• Character: 322.241
• Analytic conductor: $2.571$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.o (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			241.1
Character	\(\chi\)	\(=\)	322.241
Dual form			322.2.o.a.159.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.888835 + 0.458227i) q^{2} +(-1.97541 + 2.51194i) q^{3} +(0.580057 + 0.814576i) q^{4} +(0.841250 + 0.802131i) q^{5} +(-2.90685 + 1.32751i) q^{6} +(-1.02727 + 2.43818i) q^{7} +(0.142315 + 0.989821i) q^{8} +(-1.70031 - 7.00877i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.888835	+	0.458227i	0.628502	+	0.324015i
							\(3\)	−1.97541	+	2.51194i	−1.14050	+	1.45027i	−0.266836	+	0.963742i	\(0.585978\pi\)
−0.873666	+	0.486526i	\(0.838264\pi\)
\(5\)	0.841250	+	0.802131i	0.376219	+	0.358724i	0.854471	−	0.519499i	\(-0.173881\pi\)
							−0.478253	+	0.878222i	\(0.658730\pi\)
\(7\)	−1.02727	+	2.43818i	−0.388272	+	0.921545i
							\(11\)	−0.334219	−	0.648294i	−0.100771	−	0.195468i	0.833060	−	0.553182i	\(-0.186587\pi\)
−0.933831	+	0.357714i	\(0.883556\pi\)
\(13\)	0.469422	+	0.406756i	0.130194	+	0.112814i	0.717524	−	0.696534i	\(-0.245275\pi\)
							−0.587330	+	0.809348i	\(0.699821\pi\)
\(17\)	3.54687	+	0.338685i	0.860243	+	0.0821433i	0.515847	−	0.856681i	\(-0.327477\pi\)
							0.344397	+	0.938824i	\(0.388084\pi\)
\(19\)	−2.38855	+	0.228079i	−0.547971	+	0.0523248i	−0.365370	−	0.930862i	\(-0.619058\pi\)
							−0.182600	+	0.983187i	\(0.558451\pi\)
\(23\)	−3.90331	−	2.78643i	−0.813896	−	0.581010i
							\(29\)	2.46155	+	5.39004i	0.457098	+	1.00091i	0.988139	+	0.153559i	\(0.0490736\pi\)
−0.531041	+	0.847346i	\(0.678199\pi\)
\(31\)	−3.52568	+	8.80672i	−0.633230	+	1.58173i	0.169592	+	0.985514i	\(0.445755\pi\)
							−0.802822	+	0.596218i	\(0.796669\pi\)
\(37\)	0.488986	−	0.118627i	0.0803889	−	0.0195021i	−0.195363	−	0.980731i	\(-0.562588\pi\)
							0.275752	+	0.961229i	\(0.411073\pi\)
\(41\)	2.57890	+	8.78294i	0.402757	+	1.37166i	0.872394	+	0.488804i	\(0.162567\pi\)
							−0.469637	+	0.882860i	\(0.655615\pi\)
\(43\)	10.6665	+	1.53360i	1.62662	+	0.233872i	0.894423	−	0.447222i	\(-0.147587\pi\)
							0.732196	+	0.681094i	\(0.238496\pi\)
\(47\)	6.07931	−	3.50989i	0.886759	−	0.511970i	0.0138777	−	0.999904i	\(-0.495582\pi\)
							0.872881	+	0.487933i	\(0.162249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.241.1	yes	160
7.5	odd	6	inner	322.2.o.a.103.8	✓	160
23.21	odd	22	inner	322.2.o.a.297.8	yes	160
161.159	even	66	inner	322.2.o.a.159.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.103.8	✓	160	7.5	odd	6	inner
322.2.o.a.159.1	yes	160	161.159	even	66	inner
322.2.o.a.241.1	yes	160	1.1	even	1	trivial
322.2.o.a.297.8	yes	160	23.21	odd	22	inner