Embedded newform 322.2.o.b.5.4

• Label: 322.2.o.b.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	322.5
Dual form			322.2.o.b.129.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.723734 - 0.690079i) q^{2} +(0.141475 - 0.734043i) q^{3} +(0.0475819 - 0.998867i) q^{4} +(1.50572 + 1.18411i) q^{5} +(-0.404157 - 0.628881i) q^{6} +(2.62746 + 0.310594i) q^{7} +(-0.654861 - 0.755750i) q^{8} +(2.26630 + 0.907289i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

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
							\(3\)	0.141475	−	0.734043i	0.0816808	−	0.423800i	−0.917952	−	0.396692i	\(-0.870158\pi\)
0.999633	−	0.0271080i	\(-0.00862981\pi\)
\(5\)	1.50572	+	1.18411i	0.673377	+	0.529549i	0.895271	−	0.445521i	\(-0.146982\pi\)
							−0.221895	+	0.975071i	\(0.571224\pi\)
\(7\)	2.62746	+	0.310594i	0.993085	+	0.117393i
							\(11\)	−1.85081	+	1.94108i	−0.558041	+	0.585256i	−0.940957	−	0.338527i	\(-0.890071\pi\)
0.382916	+	0.923783i	\(0.374920\pi\)
\(13\)	−2.50152	−	1.14240i	−0.693796	−	0.316846i	0.0371226	−	0.999311i	\(-0.488181\pi\)
							−0.730919	+	0.682465i	\(0.760908\pi\)
\(17\)	−3.89558	−	2.00831i	−0.944816	−	0.487086i	−0.0842607	−	0.996444i	\(-0.526853\pi\)
							−0.860555	+	0.509357i	\(0.829883\pi\)
\(19\)	1.06585	−	0.549483i	0.244522	−	0.126060i	−0.331610	−	0.943417i	\(-0.607592\pi\)
							0.576132	+	0.817357i	\(0.304561\pi\)
\(23\)	3.04892	−	3.70190i	0.635743	−	0.771900i
							\(29\)	−6.53982	+	4.20289i	−1.21441	+	0.780456i	−0.981392	−	0.192015i	\(-0.938498\pi\)
−0.233022	+	0.972472i	\(0.574861\pi\)
\(31\)	1.70364	+	0.589636i	0.305983	+	0.105902i	0.475745	−	0.879583i	\(-0.342179\pi\)
							−0.169762	+	0.985485i	\(0.554300\pi\)
\(37\)	−3.03679	+	7.58554i	−0.499245	+	1.24705i	0.438186	+	0.898884i	\(0.355621\pi\)
							−0.937431	+	0.348170i	\(0.886803\pi\)
\(41\)	−7.91365	−	1.13781i	−1.23590	−	0.177696i	−0.506779	−	0.862076i	\(-0.669164\pi\)
							−0.729126	+	0.684380i	\(0.760073\pi\)
\(43\)	−0.349061	−	0.302463i	−0.0532313	−	0.0461252i	0.627839	−	0.778343i	\(-0.283940\pi\)
							−0.681070	+	0.732218i	\(0.738485\pi\)
\(47\)	−7.31431	−	4.22292i	−1.06690	−	0.615976i	−0.139568	−	0.990212i	\(-0.544571\pi\)
							−0.927333	+	0.374237i	\(0.877905\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.5.4	✓	160
7.3	odd	6	inner	322.2.o.b.143.4	yes	160
23.14	odd	22	inner	322.2.o.b.313.4	yes	160
161.129	even	66	inner	322.2.o.b.129.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.5.4	✓	160	1.1	even	1	trivial
322.2.o.b.129.4	yes	160	161.129	even	66	inner
322.2.o.b.143.4	yes	160	7.3	odd	6	inner
322.2.o.b.313.4	yes	160	23.14	odd	22	inner