Embedded newform 322.2.o.b.103.3

• Label: 322.2.o.b.103.3
• Level: $322$
• Weight: $2$
• Character: 322.103
• 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			103.3
Character	\(\chi\)	\(=\)	322.103
Dual form			322.2.o.b.297.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0475819 + 0.998867i) q^{2} +(-0.0852205 - 0.212870i) q^{3} +(-0.995472 + 0.0950560i) q^{4} +(0.0734344 - 0.302701i) q^{5} +(0.208574 - 0.0952527i) q^{6} +(1.11868 - 2.39761i) q^{7} +(-0.142315 - 0.989821i) q^{8} +(2.13315 - 2.03396i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0475819	+	0.998867i	0.0336455	+	0.706306i
							\(3\)	−0.0852205	−	0.212870i	−0.0492021	−	0.122901i	0.901731	−	0.432298i	\(-0.142297\pi\)
−0.950933	+	0.309397i	\(0.899873\pi\)
\(5\)	0.0734344	−	0.302701i	0.0328408	−	0.135372i	−0.953031	−	0.302872i	\(-0.902054\pi\)
							0.985872	+	0.167500i	\(0.0535696\pi\)
\(7\)	1.11868	−	2.39761i	0.422822	−	0.906213i
							\(11\)	1.99515	+	0.0950406i	0.601560	+	0.0286558i	0.346153	−	0.938178i	\(-0.387488\pi\)
0.255406	+	0.966834i	\(0.417791\pi\)
\(13\)	−0.854871	−	0.740750i	−0.237098	−	0.205447i	0.528205	−	0.849117i	\(-0.322865\pi\)
							−0.765303	+	0.643670i	\(0.777411\pi\)
\(17\)	−0.160558	+	0.225472i	−0.0389410	+	0.0546850i	−0.833587	−	0.552388i	\(-0.813717\pi\)
							0.794646	+	0.607073i	\(0.207656\pi\)
\(19\)	4.29224	+	6.02761i	0.984708	+	1.38283i	0.923028	+	0.384733i	\(0.125706\pi\)
							0.0616801	+	0.998096i	\(0.480354\pi\)
\(23\)	−2.39500	−	4.15500i	−0.499392	−	0.866376i
							\(29\)	−1.98572	−	4.34812i	−0.368739	−	0.807426i	−0.999505	−	0.0314531i	\(-0.989987\pi\)
0.630766	−	0.775973i	\(-0.282741\pi\)
\(31\)	2.90288	+	3.69131i	0.521373	+	0.662979i	0.973423	−	0.229013i	\(-0.0735498\pi\)
							−0.452051	+	0.891992i	\(0.649307\pi\)
\(37\)	−3.03233	−	3.18022i	−0.498512	−	0.522825i	0.425618	−	0.904903i	\(-0.360057\pi\)
							−0.924130	+	0.382079i	\(0.875208\pi\)
\(41\)	−0.623957	−	2.12500i	−0.0974457	−	0.331870i	0.896313	−	0.443422i	\(-0.146236\pi\)
							−0.993759	+	0.111553i	\(0.964418\pi\)
\(43\)	−3.16569	−	0.455157i	−0.482763	−	0.0694108i	−0.103363	−	0.994644i	\(-0.532960\pi\)
							−0.379400	+	0.925233i	\(0.623869\pi\)
\(47\)	1.92378	+	1.11069i	0.280612	+	0.162011i	0.633700	−	0.773579i	\(-0.281535\pi\)
							−0.353088	+	0.935590i	\(0.614869\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.103.3	✓	160
7.3	odd	6	inner	322.2.o.b.241.6	yes	160
23.21	odd	22	inner	322.2.o.b.159.6	yes	160
161.136	even	66	inner	322.2.o.b.297.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.103.3	✓	160	1.1	even	1	trivial
322.2.o.b.159.6	yes	160	23.21	odd	22	inner
322.2.o.b.241.6	yes	160	7.3	odd	6	inner
322.2.o.b.297.3	yes	160	161.136	even	66	inner