Embedded newform 322.2.o.b.17.8

• Label: 322.2.o.b.17.8
• Level: $322$
• Weight: $2$
• Character: 322.17
• 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			17.8
Character	\(\chi\)	\(=\)	322.17
Dual form			322.2.o.b.19.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.995472 + 0.0950560i) q^{2} +(2.19647 + 2.30360i) q^{3} +(0.981929 - 0.189251i) q^{4} +(1.78887 + 0.922229i) q^{5} +(-2.40550 - 2.08438i) q^{6} +(-2.58627 + 0.557860i) q^{7} +(-0.959493 + 0.281733i) q^{8} +(-0.339309 + 7.12298i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{22}\right)\)

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
							\(3\)	2.19647	+	2.30360i	1.26814	+	1.32998i	0.919686	+	0.392654i	\(0.128443\pi\)
0.348449	+	0.937328i	\(0.386708\pi\)
\(5\)	1.78887	+	0.922229i	0.800009	+	0.412433i	0.809206	−	0.587524i	\(-0.199897\pi\)
							−0.00919750	+	0.999958i	\(0.502928\pi\)
\(7\)	−2.58627	+	0.557860i	−0.977518	+	0.210851i
							\(11\)	0.0475777	−	0.498256i	0.0143452	−	0.150230i	−0.985461	−	0.169900i	\(-0.945656\pi\)
0.999807	+	0.0196701i	\(0.00626160\pi\)
\(13\)	0.742101	−	0.106698i	0.205822	−	0.0295927i	−0.0386327	−	0.999253i	\(-0.512300\pi\)
							0.244454	+	0.969661i	\(0.421391\pi\)
\(17\)	−1.39743	−	4.03762i	−0.338928	−	0.979267i	−0.977725	−	0.209892i	\(-0.932689\pi\)
							0.638797	−	0.769375i	\(-0.279432\pi\)
\(19\)	0.609832	−	1.76200i	0.139905	−	0.404229i	−0.853181	−	0.521614i	\(-0.825330\pi\)
							0.993086	+	0.117385i	\(0.0374512\pi\)
\(23\)	3.54841	−	3.22626i	0.739895	−	0.672723i
							\(29\)	−3.01521	+	3.47974i	−0.559911	+	0.646172i	−0.963163	−	0.268917i	\(-0.913334\pi\)
0.403252	+	0.915089i	\(0.367880\pi\)
\(31\)	5.97117	+	1.44859i	1.07245	+	0.260174i	0.732847	−	0.680394i	\(-0.238191\pi\)
							0.339606	+	0.940568i	\(0.389706\pi\)
\(37\)	10.5623	+	0.503146i	1.73644	+	0.0827167i	0.891745	−	0.452537i	\(-0.149481\pi\)
							0.844691	+	0.535254i	\(0.179784\pi\)
\(41\)	−3.02517	−	4.70725i	−0.472452	−	0.735150i	0.520475	−	0.853877i	\(-0.325755\pi\)
							−0.992927	+	0.118727i	\(0.962119\pi\)
\(43\)	−0.955617	+	3.25453i	−0.145730	+	0.496312i	−0.999712	−	0.0239911i	\(-0.992363\pi\)
							0.853982	+	0.520303i	\(0.174181\pi\)
\(47\)	−9.32337	+	5.38285i	−1.35995	+	0.785169i	−0.989617	−	0.143729i	\(-0.954091\pi\)
							−0.370336	+	0.928898i	\(0.620757\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.17.8	✓	160
7.5	odd	6	inner	322.2.o.b.201.8	yes	160
23.19	odd	22	inner	322.2.o.b.157.8	yes	160
161.19	even	66	inner	322.2.o.b.19.8	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.17.8	✓	160	1.1	even	1	trivial
322.2.o.b.19.8	yes	160	161.19	even	66	inner
322.2.o.b.157.8	yes	160	23.19	odd	22	inner
322.2.o.b.201.8	yes	160	7.5	odd	6	inner