Embedded newform 322.2.o.b.171.5

• Label: 322.2.o.b.171.5
• Level: $322$
• Weight: $2$
• Character: 322.171
• 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			171.5
Character	\(\chi\)	\(=\)	322.171
Dual form			322.2.o.b.145.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.327068 + 0.945001i) q^{2} +(0.00418074 + 0.00810951i) q^{3} +(-0.786053 - 0.618159i) q^{4} +(-2.13096 + 0.203482i) q^{5} +(-0.00903088 + 0.00129844i) q^{6} +(0.866871 - 2.49971i) q^{7} +(0.841254 - 0.540641i) q^{8} +(1.74012 - 2.44366i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.327068	+	0.945001i	−0.231272	+	0.668216i
							\(3\)	0.00418074	+	0.00810951i	0.00241375	+	0.00468203i	0.890039	−	0.455884i	\(-0.150677\pi\)
−0.887625	+	0.460566i	\(0.847646\pi\)
\(5\)	−2.13096	+	0.203482i	−0.952993	+	0.0909998i	−0.559956	−	0.828523i	\(-0.689182\pi\)
							−0.393038	+	0.919522i	\(0.628576\pi\)
\(7\)	0.866871	−	2.49971i	0.327646	−	0.944800i
							\(11\)	−5.57368	+	1.92907i	−1.68053	+	0.581637i	−0.989925	−	0.141596i	\(-0.954777\pi\)
−0.690605	+	0.723233i	\(0.742655\pi\)
\(13\)	−1.50935	−	5.14037i	−0.418618	−	1.42568i	−0.851558	−	0.524260i	\(-0.824342\pi\)
							0.432940	−	0.901423i	\(-0.357476\pi\)
\(17\)	1.20488	+	0.482361i	0.292226	+	0.116990i	0.513144	−	0.858303i	\(-0.328481\pi\)
							−0.220917	+	0.975293i	\(0.570905\pi\)
\(19\)	7.34455	−	2.94031i	1.68495	−	0.674554i	0.686345	−	0.727276i	\(-0.259214\pi\)
							0.998609	+	0.0527216i	\(0.0167896\pi\)
\(23\)	−2.01632	−	4.35138i	−0.420431	−	0.907325i
							\(29\)	−0.187547	−	1.30442i	−0.0348266	−	0.242224i	0.964971	−	0.262358i	\(-0.0845000\pi\)
−0.999797	+	0.0201333i	\(0.993591\pi\)
\(31\)	−0.414757	+	0.0197573i	−0.0744925	+	0.00354851i	−0.0847951	−	0.996398i	\(-0.527024\pi\)
							0.0103026	+	0.999947i	\(0.496721\pi\)
\(37\)	1.05108	+	0.748471i	0.172797	+	0.123048i	0.663183	−	0.748457i	\(-0.269205\pi\)
							−0.490387	+	0.871505i	\(0.663144\pi\)
\(41\)	−0.568187	−	0.259483i	−0.0887360	−	0.0405244i	0.370553	−	0.928811i	\(-0.379168\pi\)
							−0.459289	+	0.888287i	\(0.651896\pi\)
\(43\)	−4.65588	+	7.24469i	−0.710015	+	1.10480i	0.279468	+	0.960155i	\(0.409842\pi\)
							−0.989483	+	0.144650i	\(0.953794\pi\)
\(47\)	5.96515	−	3.44398i	0.870107	−	0.502356i	0.00272307	−	0.999996i	\(-0.499133\pi\)
							0.867384	+	0.497640i	\(0.165800\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.171.5	yes	160
7.5	odd	6	inner	322.2.o.b.33.5	✓	160
23.7	odd	22	inner	322.2.o.b.283.5	yes	160
161.145	even	66	inner	322.2.o.b.145.5	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.33.5	✓	160	7.5	odd	6	inner
322.2.o.b.145.5	yes	160	161.145	even	66	inner
322.2.o.b.171.5	yes	160	1.1	even	1	trivial
322.2.o.b.283.5	yes	160	23.7	odd	22	inner