Embedded newform 322.2.o.b.157.8

• Label: 322.2.o.b.157.8
• Level: $322$
• Weight: $2$
• Character: 322.157
• 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			157.8
Character	\(\chi\)	\(=\)	322.157
Dual form			322.2.o.b.201.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.580057 - 0.814576i) q^{2} +(3.09321 + 0.750405i) q^{3} +(-0.327068 - 0.945001i) q^{4} +(0.0957636 - 2.01033i) q^{5} +(2.40550 - 2.08438i) q^{6} +(-2.32434 + 1.26390i) q^{7} +(-0.959493 - 0.281733i) q^{8} +(6.33834 + 3.26764i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.580057	−	0.814576i	0.410162	−	0.575992i
							\(3\)	3.09321	+	0.750405i	1.78587	+	0.433246i	0.985974	−	0.166898i	\(-0.0533751\pi\)
0.799892	+	0.600144i	\(0.204890\pi\)
\(5\)	0.0957636	−	2.01033i	0.0428268	−	0.899045i	−0.870588	−	0.492014i	\(-0.836261\pi\)
							0.913414	−	0.407031i	\(-0.133436\pi\)
\(7\)	−2.32434	+	1.26390i	−0.878518	+	0.477709i
							\(11\)	0.407714	−	0.290331i	0.122930	−	0.0875382i	−0.516938	−	0.856023i	\(-0.672928\pi\)
0.639868	+	0.768485i	\(0.278989\pi\)
\(13\)	−0.742101	−	0.106698i	−0.205822	−	0.0295927i	0.0386327	−	0.999253i	\(-0.487700\pi\)
							−0.244454	+	0.969661i	\(0.578609\pi\)
\(17\)	−4.19540	+	0.808597i	−1.01753	+	0.196114i	−0.670634	−	0.741789i	\(-0.733978\pi\)
							−0.346900	+	0.937902i	\(0.612766\pi\)
\(19\)	1.83085	+	0.352867i	0.420026	+	0.0809533i	0.394885	−	0.918731i	\(-0.370784\pi\)
							0.0251407	+	0.999684i	\(0.491997\pi\)
\(23\)	−4.56823	+	1.45988i	−0.952542	+	0.304406i
							\(29\)	−3.01521	−	3.47974i	−0.559911	−	0.646172i	0.403252	−	0.915089i	\(-0.367880\pi\)
−0.963163	+	0.268917i	\(0.913334\pi\)
\(31\)	4.24010	+	4.44689i	0.761544	+	0.798684i	0.984359	−	0.176176i	\(-0.0563729\pi\)
							−0.222815	+	0.974861i	\(0.571524\pi\)
\(37\)	−4.84543	+	9.39882i	−0.796584	+	1.54516i	0.0411980	+	0.999151i	\(0.486883\pi\)
							−0.837782	+	0.546006i	\(0.816148\pi\)
\(41\)	3.02517	−	4.70725i	0.472452	−	0.735150i	−0.520475	−	0.853877i	\(-0.674245\pi\)
							0.992927	+	0.118727i	\(0.0378815\pi\)
\(43\)	−0.955617	−	3.25453i	−0.145730	−	0.496312i	0.853982	−	0.520303i	\(-0.174181\pi\)
							−0.999712	+	0.0239911i	\(0.992363\pi\)
\(47\)	−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.157.8	yes	160
7.5	odd	6	inner	322.2.o.b.19.8	yes	160
23.17	odd	22	inner	322.2.o.b.17.8	✓	160
161.40	even	66	inner	322.2.o.b.201.8	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.17.8	✓	160	23.17	odd	22	inner
322.2.o.b.19.8	yes	160	7.5	odd	6	inner
322.2.o.b.157.8	yes	160	1.1	even	1	trivial
322.2.o.b.201.8	yes	160	161.40	even	66	inner