Embedded newform 322.2.o.a.61.8

• Label: 322.2.o.a.61.8
• Level: $322$
• Weight: $2$
• Character: 322.61
• 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			61.8
Character	\(\chi\)	\(=\)	322.61
Dual form			322.2.o.a.227.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.786053 + 0.618159i) q^{2} +(2.29508 + 1.63432i) q^{3} +(0.235759 + 0.971812i) q^{4} +(-1.84478 + 0.355552i) q^{5} +(0.793784 + 2.70338i) q^{6} +(2.64261 - 0.128809i) q^{7} +(-0.415415 + 0.909632i) q^{8} +(1.61518 + 4.66675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{22}\right)\)

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
							\(3\)	2.29508	+	1.63432i	1.32506	+	0.943573i	0.999998	−	0.00211937i	\(-0.000674618\pi\)
0.325064	+	0.945692i	\(0.394614\pi\)
\(5\)	−1.84478	+	0.355552i	−0.825011	+	0.159008i	−0.584247	−	0.811576i	\(-0.698610\pi\)
							−0.240764	+	0.970584i	\(0.577398\pi\)
\(7\)	2.64261	−	0.128809i	0.998814	−	0.0486852i
							\(11\)	−3.67197	−	4.66929i	−1.10714	−	1.40784i	−0.903389	−	0.428822i	\(-0.858929\pi\)
−0.203752	−	0.979022i	\(-0.565314\pi\)
\(13\)	−0.399477	−	0.621599i	−0.110795	−	0.172400i	0.781428	−	0.623995i	\(-0.214492\pi\)
							−0.892223	+	0.451595i	\(0.850855\pi\)
\(17\)	−0.147918	−	0.141040i	−0.0358755	−	0.0342072i	0.671929	−	0.740615i	\(-0.265466\pi\)
							−0.707805	+	0.706408i	\(0.750314\pi\)
\(19\)	3.72270	−	3.54959i	0.854047	−	0.814332i	−0.130135	−	0.991496i	\(-0.541541\pi\)
							0.984181	+	0.177164i	\(0.0566924\pi\)
\(23\)	−1.17894	+	4.64867i	−0.245825	+	0.969314i
							\(29\)	0.470884	−	0.138264i	0.0874409	−	0.0256750i	−0.237719	−	0.971334i	\(-0.576400\pi\)
0.325160	+	0.945659i	\(0.394582\pi\)
\(31\)	−0.286716	−	3.00262i	−0.0514957	−	0.539287i	−0.984005	−	0.178141i	\(-0.942992\pi\)
							0.932509	−	0.361146i	\(-0.117614\pi\)
\(37\)	9.37724	−	3.24549i	1.54161	−	0.533556i	0.581588	−	0.813484i	\(-0.302432\pi\)
							0.960021	+	0.279928i	\(0.0903105\pi\)
\(41\)	−1.43752	+	1.24562i	−0.224503	+	0.194533i	−0.759848	−	0.650100i	\(-0.774727\pi\)
							0.535346	+	0.844633i	\(0.320181\pi\)
\(43\)	−8.36489	+	3.82012i	−1.27563	+	0.582562i	−0.934001	−	0.357271i	\(-0.883707\pi\)
							−0.341633	+	0.939834i	\(0.610980\pi\)
\(47\)	0.414175	+	0.239124i	0.0604137	+	0.0348799i	0.529903	−	0.848059i	\(-0.322228\pi\)
							−0.469489	+	0.882938i	\(0.655562\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.61.8	✓	160
7.3	odd	6	inner	322.2.o.a.199.1	yes	160
23.20	odd	22	inner	322.2.o.a.89.1	yes	160
161.66	even	66	inner	322.2.o.a.227.8	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.61.8	✓	160	1.1	even	1	trivial
322.2.o.a.89.1	yes	160	23.20	odd	22	inner
322.2.o.a.199.1	yes	160	7.3	odd	6	inner
322.2.o.a.227.8	yes	160	161.66	even	66	inner