Embedded newform 322.2.o.a.227.3

• Label: 322.2.o.a.227.3
• Level: $322$
• Weight: $2$
• Character: 322.227
• 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			227.3
Character	\(\chi\)	\(=\)	322.227
Dual form			322.2.o.a.61.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.786053 - 0.618159i) q^{2} +(-1.41633 + 1.00856i) q^{3} +(0.235759 - 0.971812i) q^{4} +(2.69009 + 0.518472i) q^{5} +(-0.489857 + 1.66830i) q^{6} +(-0.151524 + 2.64141i) q^{7} +(-0.415415 - 0.909632i) q^{8} +(0.00758268 - 0.0219087i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.786053	−	0.618159i	0.555823	−	0.437104i
							\(3\)	−1.41633	+	1.00856i	−0.817717	+	0.582294i	−0.910555	−	0.413387i	\(-0.864346\pi\)
0.0928379	+	0.995681i	\(0.470406\pi\)
\(5\)	2.69009	+	0.518472i	1.20304	+	0.231868i	0.751106	−	0.660182i	\(-0.229521\pi\)
							0.451937	+	0.892050i	\(0.350733\pi\)
\(7\)	−0.151524	+	2.64141i	−0.0572705	+	0.998359i
							\(11\)	−1.48124	+	1.88354i	−0.446609	+	0.567910i	−0.956277	−	0.292461i	\(-0.905526\pi\)
0.509668	+	0.860371i	\(0.329768\pi\)
\(13\)	1.06740	−	1.66091i	0.296044	−	0.460654i	−0.661087	−	0.750310i	\(-0.729904\pi\)
							0.957131	+	0.289656i	\(0.0935408\pi\)
\(17\)	2.58886	−	2.46847i	0.627890	−	0.598692i	−0.307828	−	0.951442i	\(-0.599602\pi\)
							0.935717	+	0.352750i	\(0.114754\pi\)
\(19\)	4.86430	+	4.63810i	1.11595	+	1.06405i	0.997469	+	0.0710959i	\(0.0226496\pi\)
							0.118477	+	0.992957i	\(0.462199\pi\)
\(23\)	3.07733	+	3.67832i	0.641668	+	0.766983i
							\(29\)	−0.522690	−	0.153476i	−0.0970611	−	0.0284997i	0.232841	−	0.972515i	\(-0.425198\pi\)
−0.329902	+	0.944015i	\(0.607016\pi\)
\(31\)	0.427413	−	4.47607i	0.0767656	−	0.803926i	−0.872326	−	0.488925i	\(-0.837389\pi\)
							0.949091	−	0.315001i	\(-0.102005\pi\)
\(37\)	0.528231	+	0.182822i	0.0868406	+	0.0300558i	0.370143	−	0.928975i	\(-0.379309\pi\)
							−0.283302	+	0.959031i	\(0.591430\pi\)
\(41\)	−4.84797	−	4.20079i	−0.757126	−	0.656054i	0.188218	−	0.982127i	\(-0.439729\pi\)
							−0.945344	+	0.326074i	\(0.894274\pi\)
\(43\)	−4.13034	−	1.88627i	−0.629872	−	0.287653i	0.0747867	−	0.997200i	\(-0.476172\pi\)
							−0.704658	+	0.709547i	\(0.748900\pi\)
\(47\)	−0.308402	+	0.178056i	−0.0449851	+	0.0259722i	−0.522324	−	0.852747i	\(-0.674935\pi\)
							0.477339	+	0.878719i	\(0.341601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.227.3	yes	160
7.5	odd	6	inner	322.2.o.a.89.6	yes	160
23.15	odd	22	inner	322.2.o.a.199.6	yes	160
161.61	even	66	inner	322.2.o.a.61.3	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.61.3	✓	160	161.61	even	66	inner
322.2.o.a.89.6	yes	160	7.5	odd	6	inner
322.2.o.a.199.6	yes	160	23.15	odd	22	inner
322.2.o.a.227.3	yes	160	1.1	even	1	trivial