Embedded newform 322.2.o.b.5.5

• Label: 322.2.o.b.5.5
• Level: $322$
• Weight: $2$
• Character: 322.5
• 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			5.5
Character	\(\chi\)	\(=\)	322.5
Dual form			322.2.o.b.129.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.723734 - 0.690079i) q^{2} +(0.223563 - 1.15995i) q^{3} +(0.0475819 - 0.998867i) q^{4} +(-2.96167 - 2.32908i) q^{5} +(-0.638660 - 0.993774i) q^{6} +(-1.92017 + 1.82015i) q^{7} +(-0.654861 - 0.755750i) q^{8} +(1.48959 + 0.596343i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.723734	−	0.690079i	0.511757	−	0.487960i
							\(3\)	0.223563	−	1.15995i	0.129074	−	0.669699i	−0.858534	−	0.512757i	\(-0.828624\pi\)
0.987608	−	0.156942i	\(-0.0501637\pi\)
\(5\)	−2.96167	−	2.32908i	−1.32450	−	1.04160i	−0.995250	−	0.0973553i	\(-0.968962\pi\)
							−0.329250	−	0.944243i	\(-0.606796\pi\)
\(7\)	−1.92017	+	1.82015i	−0.725757	+	0.687951i
							\(11\)	2.52838	−	2.65169i	0.762336	−	0.799515i	−0.222144	−	0.975014i	\(-0.571305\pi\)
0.984480	+	0.175499i	\(0.0561539\pi\)
\(13\)	−3.91484	−	1.78785i	−1.08578	−	0.495859i	−0.209573	−	0.977793i	\(-0.567207\pi\)
							−0.876208	+	0.481934i	\(0.839935\pi\)
\(17\)	−1.39453	−	0.718930i	−0.338223	−	0.174366i	0.280745	−	0.959783i	\(-0.409419\pi\)
							−0.618968	+	0.785416i	\(0.712449\pi\)
\(19\)	−1.23790	+	0.638181i	−0.283993	+	0.146409i	−0.594338	−	0.804215i	\(-0.702586\pi\)
							0.310345	+	0.950624i	\(0.399556\pi\)
\(23\)	1.68458	−	4.49023i	0.351260	−	0.936278i
							\(29\)	0.301381	−	0.193686i	0.0559650	−	0.0359665i	−0.512359	−	0.858771i	\(-0.671228\pi\)
0.568324	+	0.822805i	\(0.307592\pi\)
\(31\)	7.42064	+	2.56831i	1.33279	+	0.461282i	0.898295	−	0.439393i	\(-0.144807\pi\)
							0.434492	+	0.900676i	\(0.356928\pi\)
\(37\)	2.49298	−	6.22718i	0.409844	−	1.02374i	−0.569581	−	0.821935i	\(-0.692895\pi\)
							0.979425	−	0.201806i	\(-0.0646811\pi\)
\(41\)	10.3658	+	1.49038i	1.61887	+	0.232759i	0.891347	−	0.453321i	\(-0.149761\pi\)
							0.727526	+	0.686080i	\(0.240670\pi\)
\(43\)	−7.40224	−	6.41407i	−1.12883	−	0.978137i	−0.128922	−	0.991655i	\(-0.541152\pi\)
							−0.999909	+	0.0135176i	\(0.995697\pi\)
\(47\)	−0.671377	−	0.387620i	−0.0979304	−	0.0565402i	0.450235	−	0.892910i	\(-0.351340\pi\)
							−0.548165	+	0.836370i	\(0.684674\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.5.5	✓	160
7.3	odd	6	inner	322.2.o.b.143.5	yes	160
23.14	odd	22	inner	322.2.o.b.313.5	yes	160
161.129	even	66	inner	322.2.o.b.129.5	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.5.5	✓	160	1.1	even	1	trivial
322.2.o.b.129.5	yes	160	161.129	even	66	inner
322.2.o.b.143.5	yes	160	7.3	odd	6	inner
322.2.o.b.313.5	yes	160	23.14	odd	22	inner