Embedded newform 322.2.o.b.17.6

• Label: 322.2.o.b.17.6
• 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			17.6
Character	\(\chi\)	\(=\)	322.17
Dual form			322.2.o.b.19.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.995472 + 0.0950560i) q^{2} +(0.519459 + 0.544793i) q^{3} +(0.981929 - 0.189251i) q^{4} +(2.76848 + 1.42725i) q^{5} +(-0.568893 - 0.492948i) q^{6} +(2.09567 + 1.61498i) q^{7} +(-0.959493 + 0.281733i) q^{8} +(0.115784 - 2.43061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

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
							\(3\)	0.519459	+	0.544793i	0.299910	+	0.314536i	0.856395	−	0.516321i	\(-0.172699\pi\)
−0.556485	+	0.830857i	\(0.687850\pi\)
\(5\)	2.76848	+	1.42725i	1.23810	+	0.638286i	0.947866	−	0.318668i	\(-0.103236\pi\)
							0.290237	+	0.956955i	\(0.406266\pi\)
\(7\)	2.09567	+	1.61498i	0.792090	+	0.610405i
							\(11\)	−0.113387	+	1.18744i	−0.0341873	+	0.358026i	0.962150	+	0.272519i	\(0.0878569\pi\)
−0.996338	+	0.0855065i	\(0.972749\pi\)
\(13\)	−2.10990	+	0.303358i	−0.585182	+	0.0841365i	−0.428546	−	0.903520i	\(-0.640974\pi\)
							−0.156636	+	0.987656i	\(0.550065\pi\)
\(17\)	−0.935929	−	2.70419i	−0.226996	−	0.655862i	−0.999776	−	0.0211874i	\(-0.993255\pi\)
							0.772779	−	0.634675i	\(-0.218866\pi\)
\(19\)	0.200738	−	0.579996i	0.0460525	−	0.133060i	−0.919630	−	0.392786i	\(-0.871511\pi\)
							0.965682	+	0.259726i	\(0.0836324\pi\)
\(23\)	−3.68249	+	3.07234i	−0.767851	+	0.640628i
							\(29\)	−0.637413	+	0.735614i	−0.118365	+	0.136600i	−0.811839	−	0.583881i	\(-0.801533\pi\)
0.693475	+	0.720481i	\(0.256079\pi\)
\(31\)	10.7055	+	2.59713i	1.92277	+	0.466459i	0.995827	+	0.0912583i	\(0.0290889\pi\)
							0.926944	+	0.375201i	\(0.122426\pi\)
\(37\)	−5.60237	−	0.266874i	−0.921023	−	0.0438738i	−0.418288	−	0.908314i	\(-0.637370\pi\)
							−0.502735	+	0.864441i	\(0.667673\pi\)
\(41\)	0.262463	+	0.408401i	0.0409899	+	0.0637815i	0.861144	−	0.508361i	\(-0.169749\pi\)
							−0.820154	+	0.572143i	\(0.806112\pi\)
\(43\)	2.03568	−	6.93290i	0.310439	−	1.05726i	−0.645517	−	0.763746i	\(-0.723358\pi\)
							0.955956	−	0.293511i	\(-0.0948238\pi\)
\(47\)	−7.68440	+	4.43659i	−1.12088	+	0.647143i	−0.941627	−	0.336659i	\(-0.890703\pi\)
							−0.179258	+	0.983802i	\(0.557370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.17.6	✓	160
7.5	odd	6	inner	322.2.o.b.201.6	yes	160
23.19	odd	22	inner	322.2.o.b.157.6	yes	160
161.19	even	66	inner	322.2.o.b.19.6	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.17.6	✓	160	1.1	even	1	trivial
322.2.o.b.19.6	yes	160	161.19	even	66	inner
322.2.o.b.157.6	yes	160	23.19	odd	22	inner
322.2.o.b.201.6	yes	160	7.5	odd	6	inner