Embedded newform 322.2.o.b.5.7

• Label: 322.2.o.b.5.7
• 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			5.7
Character	\(\chi\)	\(=\)	322.5
Dual form			322.2.o.b.129.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.723734 - 0.690079i) q^{2} +(0.410973 - 2.13233i) q^{3} +(0.0475819 - 0.998867i) q^{4} +(3.20038 + 2.51681i) q^{5} +(-1.17404 - 1.82684i) q^{6} +(-2.40619 - 1.10010i) q^{7} +(-0.654861 - 0.755750i) q^{8} +(-1.59282 - 0.637669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

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
							\(3\)	0.410973	−	2.13233i	0.237275	−	1.23110i	−0.646400	−	0.762999i	\(-0.723726\pi\)
0.883675	−	0.468101i	\(-0.155062\pi\)
\(5\)	3.20038	+	2.51681i	1.43125	+	1.12555i	0.971370	+	0.237572i	\(0.0763516\pi\)
							0.459885	+	0.887979i	\(0.347891\pi\)
\(7\)	−2.40619	−	1.10010i	−0.909456	−	0.415800i
							\(11\)	2.11169	−	2.21468i	0.636698	−	0.667750i	−0.323771	−	0.946135i	\(-0.604951\pi\)
0.960469	+	0.278385i	\(0.0897993\pi\)
\(13\)	3.09022	+	1.41126i	0.857073	+	0.391412i	0.794963	−	0.606657i	\(-0.207490\pi\)
							0.0621092	+	0.998069i	\(0.480217\pi\)
\(17\)	−4.04342	−	2.08453i	−0.980672	−	0.505572i	−0.108144	−	0.994135i	\(-0.534491\pi\)
							−0.872528	+	0.488564i	\(0.837521\pi\)
\(19\)	−6.02732	+	3.10730i	−1.38276	+	0.712864i	−0.979353	−	0.202156i	\(-0.935205\pi\)
							−0.403409	+	0.915020i	\(0.632175\pi\)
\(23\)	−4.15906	+	2.38794i	−0.867223	+	0.497920i
							\(29\)	−4.40902	+	2.83351i	−0.818735	+	0.526169i	−0.881680	−	0.471848i	\(-0.843587\pi\)
0.0629451	+	0.998017i	\(0.479951\pi\)
\(31\)	6.66654	+	2.30731i	1.19735	+	0.414405i	0.851735	−	0.523973i	\(-0.175551\pi\)
							0.345611	+	0.938378i	\(0.387672\pi\)
\(37\)	0.532554	−	1.33025i	0.0875513	−	0.218693i	−0.878130	−	0.478422i	\(-0.841209\pi\)
							0.965682	+	0.259729i	\(0.0836333\pi\)
\(41\)	1.94830	+	0.280124i	0.304274	+	0.0437479i	0.292761	−	0.956186i	\(-0.405426\pi\)
							0.0115131	+	0.999934i	\(0.496335\pi\)
\(43\)	−2.80847	−	2.43356i	−0.428288	−	0.371114i	0.413878	−	0.910333i	\(-0.364174\pi\)
							−0.842166	+	0.539219i	\(0.818720\pi\)
\(47\)	6.02282	+	3.47727i	0.878518	+	0.507213i	0.870169	−	0.492753i	\(-0.164009\pi\)
							0.00834855	+	0.999965i	\(0.497343\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.5.7	✓	160
7.3	odd	6	inner	322.2.o.b.143.7	yes	160
23.14	odd	22	inner	322.2.o.b.313.7	yes	160
161.129	even	66	inner	322.2.o.b.129.7	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.5.7	✓	160	1.1	even	1	trivial
322.2.o.b.129.7	yes	160	161.129	even	66	inner
322.2.o.b.143.7	yes	160	7.3	odd	6	inner
322.2.o.b.313.7	yes	160	23.14	odd	22	inner