Embedded newform 322.2.o.a.171.4

• Label: 322.2.o.a.171.4
• 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			171.4
Character	\(\chi\)	\(=\)	322.171
Dual form			322.2.o.a.145.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.327068 - 0.945001i) q^{2} +(-0.132722 - 0.257445i) q^{3} +(-0.786053 - 0.618159i) q^{4} +(-2.50995 + 0.239671i) q^{5} +(-0.286695 + 0.0412206i) q^{6} +(-2.51860 + 0.810332i) q^{7} +(-0.841254 + 0.540641i) q^{8} +(1.69151 - 2.37539i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 6 q^{3} + 8 q^{4} - 11 q^{7} + 16 q^{8} - 32 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.132722	−	0.257445i	−0.0766273	−	0.148636i	0.847409	−	0.530940i	\(-0.178161\pi\)
−0.924037	+	0.382304i	\(0.875131\pi\)
\(5\)	−2.50995	+	0.239671i	−1.12248	+	0.107184i	−0.639751	−	0.768583i	\(-0.720962\pi\)
							−0.482734	+	0.875767i	\(0.660356\pi\)
\(7\)	−2.51860	+	0.810332i	−0.951943	+	0.306277i
							\(11\)	−4.41247	+	1.52717i	−1.33041	+	0.460459i	−0.897515	−	0.440985i	\(-0.854629\pi\)
−0.432896	+	0.901444i	\(0.642508\pi\)
\(13\)	0.296687	+	1.01042i	0.0822861	+	0.280241i	0.990350	−	0.138592i	\(-0.0442576\pi\)
							−0.908063	+	0.418833i	\(0.862439\pi\)
\(17\)	−3.26419	−	1.30678i	−0.791683	−	0.316942i	−0.0596569	−	0.998219i	\(-0.519001\pi\)
							−0.732026	+	0.681277i	\(0.761425\pi\)
\(19\)	−4.41213	+	1.76635i	−1.01221	+	0.405229i	−0.817684	−	0.575668i	\(-0.804742\pi\)
							−0.194529	+	0.980897i	\(0.562318\pi\)
\(23\)	4.55354	−	1.50509i	0.949478	−	0.313833i
							\(29\)	−1.00049	−	6.95854i	−0.185786	−	1.29217i	−0.842774	−	0.538267i	\(-0.819079\pi\)
0.656988	−	0.753901i	\(-0.271830\pi\)
\(31\)	2.13489	−	0.101697i	0.383437	−	0.0182654i	0.145021	−	0.989429i	\(-0.453675\pi\)
							0.238416	+	0.971163i	\(0.423372\pi\)
\(37\)	0.561165	+	0.399604i	0.0922550	+	0.0656945i	0.625254	−	0.780422i	\(-0.284996\pi\)
							−0.532999	+	0.846116i	\(0.678935\pi\)
\(41\)	0.811717	+	0.370699i	0.126769	+	0.0578934i	0.477790	−	0.878474i	\(-0.341438\pi\)
							−0.351021	+	0.936368i	\(0.614165\pi\)
\(43\)	1.84943	−	2.87777i	0.282035	−	0.438856i	−0.671115	−	0.741354i	\(-0.734184\pi\)
							0.953150	+	0.302498i	\(0.0978205\pi\)
\(47\)	−2.33417	+	1.34763i	−0.340473	+	0.196572i	−0.660481	−	0.750843i	\(-0.729648\pi\)
							0.320008	+	0.947415i	\(0.396314\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.a.171.4	yes	160
7.5	odd	6	inner	322.2.o.a.33.4	✓	160
23.7	odd	22	inner	322.2.o.a.283.4	yes	160
161.145	even	66	inner	322.2.o.a.145.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.a.33.4	✓	160	7.5	odd	6	inner
322.2.o.a.145.4	yes	160	161.145	even	66	inner
322.2.o.a.171.4	yes	160	1.1	even	1	trivial
322.2.o.a.283.4	yes	160	23.7	odd	22	inner