Embedded newform 322.2.o.a.103.6

• Label: 322.2.o.a.103.6
• Level: $322$
• Weight: $2$
• Character: 322.103
• 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			103.6
Character	\(\chi\)	\(=\)	322.103
Dual form			322.2.o.a.297.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0475819 - 0.998867i) q^{2} +(0.248754 + 0.621357i) q^{3} +(-0.995472 + 0.0950560i) q^{4} +(0.422293 - 1.74071i) q^{5} +(0.608817 - 0.278037i) q^{6} +(-1.77627 - 1.96083i) q^{7} +(0.142315 + 0.989821i) q^{8} +(1.84700 - 1.76111i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0475819	−	0.998867i	−0.0336455	−	0.706306i
							\(3\)	0.248754	+	0.621357i	0.143618	+	0.358740i	0.982684	−	0.185289i	\(-0.0593222\pi\)
−0.839066	+	0.544030i	\(0.816898\pi\)
\(5\)	0.422293	−	1.74071i	0.188855	−	0.778471i	−0.796029	−	0.605258i	\(-0.793070\pi\)
							0.984884	−	0.173213i	\(-0.0554149\pi\)
\(7\)	−1.77627	−	1.96083i	−0.671368	−	0.741124i
							\(11\)	−2.70040	−	0.128636i	−0.814202	−	0.0387852i	−0.363650	−	0.931535i	\(-0.618470\pi\)
−0.450552	+	0.892750i	\(0.648773\pi\)
\(13\)	0.379440	+	0.328786i	0.105238	+	0.0911889i	0.705890	−	0.708321i	\(-0.250547\pi\)
							−0.600653	+	0.799510i	\(0.705093\pi\)
\(17\)	2.03036	−	2.85125i	0.492435	−	0.691529i	−0.491333	−	0.870972i	\(-0.663490\pi\)
							0.983768	+	0.179443i	\(0.0574295\pi\)
\(19\)	−1.64721	−	2.31318i	−0.377896	−	0.530680i	0.581201	−	0.813760i	\(-0.302583\pi\)
							−0.959096	+	0.283080i	\(0.908644\pi\)
\(23\)	−1.57498	−	4.52984i	−0.328406	−	0.944537i
							\(29\)	−1.73391	−	3.79673i	−0.321979	−	0.705036i	0.677558	−	0.735470i	\(-0.263038\pi\)
−0.999537	+	0.0304339i	\(0.990311\pi\)
\(31\)	−0.168826	−	0.214680i	−0.0303220	−	0.0385576i	0.770656	−	0.637252i	\(-0.219929\pi\)
							−0.800978	+	0.598694i	\(0.795686\pi\)
\(37\)	3.73909	+	3.92144i	0.614702	+	0.644681i	0.955338	−	0.295514i	\(-0.0954909\pi\)
							−0.340636	+	0.940195i	\(0.610642\pi\)
\(41\)	2.68844	+	9.15600i	0.419864	+	1.42993i	0.849818	+	0.527076i	\(0.176712\pi\)
							−0.429954	+	0.902851i	\(0.641470\pi\)
\(43\)	−6.01505	−	0.864833i	−0.917286	−	0.131886i	−0.332528	−	0.943093i	\(-0.607901\pi\)
							−0.584758	+	0.811208i	\(0.698811\pi\)
\(47\)	10.7225	+	6.19065i	1.56404	+	0.902999i	0.996841	+	0.0794185i	\(0.0253063\pi\)
							0.567199	+	0.823581i	\(0.308027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.103.6	✓	160
7.3	odd	6	inner	322.2.o.a.241.3	yes	160
23.21	odd	22	inner	322.2.o.a.159.3	yes	160
161.136	even	66	inner	322.2.o.a.297.6	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.103.6	✓	160	1.1	even	1	trivial
322.2.o.a.159.3	yes	160	23.21	odd	22	inner
322.2.o.a.241.3	yes	160	7.3	odd	6	inner
322.2.o.a.297.6	yes	160	161.136	even	66	inner