Embedded newform 322.2.o.a.19.4

• Label: 322.2.o.a.19.4
• Level: $322$
• Weight: $2$
• Character: 322.19
• 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			19.4
Character	\(\chi\)	\(=\)	322.19
Dual form			322.2.o.a.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.995472 + 0.0950560i) q^{2} +(-0.522185 + 0.547652i) q^{3} +(0.981929 + 0.189251i) q^{4} +(-0.212151 + 0.109371i) q^{5} +(-0.571878 + 0.495536i) q^{6} +(2.45288 - 0.991667i) q^{7} +(0.959493 + 0.281733i) q^{8} +(0.115500 + 2.42465i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.995472	+	0.0950560i	0.703905	+	0.0672148i
							\(3\)	−0.522185	+	0.547652i	−0.301484	+	0.316187i	−0.856998	−	0.515320i	\(-0.827673\pi\)
0.555514	+	0.831507i	\(0.312522\pi\)
\(5\)	−0.212151	+	0.109371i	−0.0948767	+	0.0489124i	−0.505012	−	0.863112i	\(-0.668512\pi\)
							0.410135	+	0.912025i	\(0.365482\pi\)
\(7\)	2.45288	−	0.991667i	0.927100	−	0.374815i
							\(11\)	0.229568	+	2.40414i	0.0692172	+	0.724875i	0.961757	+	0.273906i	\(0.0883157\pi\)
−0.892539	+	0.450970i	\(0.851078\pi\)
\(13\)	0.109202	+	0.0157008i	0.0302871	+	0.00435463i	0.157442	−	0.987528i	\(-0.449675\pi\)
							−0.127155	+	0.991883i	\(0.540584\pi\)
\(17\)	1.99127	−	5.75339i	0.482953	−	1.39540i	−0.395893	−	0.918296i	\(-0.629565\pi\)
							0.878846	−	0.477105i	\(-0.158314\pi\)
\(19\)	0.760248	+	2.19659i	0.174413	+	0.503933i	0.997997	−	0.0632611i	\(-0.0201501\pi\)
							−0.823584	+	0.567194i	\(0.808029\pi\)
\(23\)	2.78167	−	3.90670i	0.580017	−	0.814604i
							\(29\)	5.08717	+	5.87091i	0.944664	+	1.09020i	0.995804	+	0.0915089i	\(0.0291690\pi\)
−0.0511402	+	0.998691i	\(0.516286\pi\)
\(31\)	−7.55581	+	1.83302i	−1.35706	+	0.329220i	−0.847309	−	0.531100i	\(-0.821779\pi\)
							−0.509754	+	0.860320i	\(0.670264\pi\)
\(37\)	−9.43052	+	0.449231i	−1.55037	+	0.0738531i	−0.805192	−	0.593015i	\(-0.797938\pi\)
							−0.745176	+	0.666868i	\(0.767635\pi\)
\(41\)	5.79595	−	9.01867i	0.905175	−	1.40848i	−0.00757969	−	0.999971i	\(-0.502413\pi\)
							0.912755	−	0.408508i	\(-0.133951\pi\)
\(43\)	−3.08261	−	10.4984i	−0.470094	−	1.60099i	−0.764039	−	0.645170i	\(-0.776787\pi\)
							0.293946	−	0.955822i	\(-0.405032\pi\)
\(47\)	−9.96221	−	5.75168i	−1.45314	−	0.838969i	−0.454479	−	0.890757i	\(-0.650175\pi\)
							−0.998658	+	0.0517882i	\(0.983508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.19.4	yes	160
7.3	odd	6	inner	322.2.o.a.157.4	yes	160
23.17	odd	22	inner	322.2.o.a.201.4	yes	160
161.17	even	66	inner	322.2.o.a.17.4	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.17.4	✓	160	161.17	even	66	inner
322.2.o.a.19.4	yes	160	1.1	even	1	trivial
322.2.o.a.157.4	yes	160	7.3	odd	6	inner
322.2.o.a.201.4	yes	160	23.17	odd	22	inner