Embedded newform 322.2.o.a.171.5

• Label: 322.2.o.a.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.a.145.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.327068 - 0.945001i) q^{2} +(0.0599871 + 0.116359i) q^{3} +(-0.786053 - 0.618159i) q^{4} +(-0.756777 + 0.0722635i) q^{5} +(0.129579 - 0.0186306i) q^{6} +(2.12016 + 1.58269i) q^{7} +(-0.841254 + 0.540641i) q^{8} +(1.73023 - 2.42977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

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
							\(3\)	0.0599871	+	0.116359i	0.0346336	+	0.0671797i	0.905517	−	0.424309i	\(-0.139483\pi\)
−0.870884	+	0.491489i	\(0.836453\pi\)
\(5\)	−0.756777	+	0.0722635i	−0.338441	+	0.0323172i	−0.262893	−	0.964825i	\(-0.584677\pi\)
							−0.0755480	+	0.997142i	\(0.524071\pi\)
\(7\)	2.12016	+	1.58269i	0.801345	+	0.598202i
							\(11\)	4.23040	−	1.46416i	1.27551	−	0.441460i	0.396403	−	0.918077i	\(-0.370258\pi\)
0.879111	+	0.476617i	\(0.158137\pi\)
\(13\)	−1.10160	−	3.75170i	−0.305529	−	1.04053i	−0.958958	−	0.283547i	\(-0.908489\pi\)
							0.653430	−	0.756987i	\(-0.273329\pi\)
\(17\)	−1.51838	−	0.607870i	−0.368262	−	0.147430i	0.180148	−	0.983640i	\(-0.442342\pi\)
							−0.548410	+	0.836210i	\(0.684767\pi\)
\(19\)	1.22202	−	0.489224i	0.280351	−	0.112236i	−0.227226	−	0.973842i	\(-0.572965\pi\)
							0.507577	+	0.861606i	\(0.330541\pi\)
\(23\)	3.65030	−	3.11051i	0.761141	−	0.648587i
							\(29\)	1.26057	+	8.76743i	0.234081	+	1.62807i	0.680150	+	0.733073i	\(0.261915\pi\)
−0.446069	+	0.894999i	\(0.647176\pi\)
\(31\)	−9.12161	+	0.434516i	−1.63829	+	0.0780413i	−0.846374	−	0.532589i	\(-0.821219\pi\)
							−0.791916	+	0.610631i	\(0.790916\pi\)
\(37\)	4.53499	+	3.22935i	0.745548	+	0.530902i	0.888527	−	0.458824i	\(-0.151729\pi\)
							−0.142980	+	0.989726i	\(0.545668\pi\)
\(41\)	11.2005	+	5.11508i	1.74922	+	0.798841i	0.988771	+	0.149440i	\(0.0477472\pi\)
							0.760447	+	0.649400i	\(0.224980\pi\)
\(43\)	0.695075	−	1.08156i	0.105998	−	0.164936i	−0.784203	−	0.620505i	\(-0.786928\pi\)
							0.890201	+	0.455569i	\(0.150564\pi\)
\(47\)	−2.57447	+	1.48637i	−0.375525	+	0.216809i	−0.675869	−	0.737021i	\(-0.736232\pi\)
							0.300345	+	0.953831i	\(0.402898\pi\)

(See \(a_n\) instead)

Twists

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

    

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