Embedded newform 322.2.o.b.103.2

• Label: 322.2.o.b.103.2
• 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			103.2
Character	\(\chi\)	\(=\)	322.103
Dual form			322.2.o.b.297.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0475819 + 0.998867i) q^{2} +(-0.688397 - 1.71953i) q^{3} +(-0.995472 + 0.0950560i) q^{4} +(-0.474539 + 1.95608i) q^{5} +(1.68483 - 0.769435i) q^{6} +(-2.42652 - 1.05452i) q^{7} +(-0.142315 - 0.989821i) q^{8} +(-0.311697 + 0.297202i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 8 q^{2} - 6 q^{3} + 8 q^{4} + 11 q^{7} - 16 q^{8} + 12 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.688397	−	1.71953i	−0.397446	−	0.992772i	−0.983241	−	0.182312i	\(-0.941642\pi\)
0.585795	−	0.810459i	\(-0.300782\pi\)
\(5\)	−0.474539	+	1.95608i	−0.212220	+	0.874784i	0.761707	+	0.647922i	\(0.224362\pi\)
							−0.973927	+	0.226862i	\(0.927153\pi\)
\(7\)	−2.42652	−	1.05452i	−0.917138	−	0.398570i
							\(11\)	−4.07832	−	0.194274i	−1.22966	−	0.0585759i	−0.577330	−	0.816511i	\(-0.695905\pi\)
−0.652330	+	0.757935i	\(0.726208\pi\)
\(13\)	−1.92663	−	1.66943i	−0.534350	−	0.463017i	0.345398	−	0.938456i	\(-0.387744\pi\)
							−0.879749	+	0.475439i	\(0.842289\pi\)
\(17\)	−2.34406	+	3.29177i	−0.568518	+	0.798372i	−0.994319	−	0.106445i	\(-0.966053\pi\)
							0.425800	+	0.904817i	\(0.359993\pi\)
\(19\)	−2.09838	−	2.94677i	−0.481402	−	0.676035i	0.500432	−	0.865776i	\(-0.333175\pi\)
							−0.981834	+	0.189741i	\(0.939235\pi\)
\(23\)	−2.84473	+	3.86102i	−0.593168	+	0.805079i
							\(29\)	−2.89387	−	6.33669i	−0.537378	−	1.17669i	−0.962431	−	0.271525i	\(-0.912472\pi\)
0.425054	−	0.905168i	\(-0.360255\pi\)
\(31\)	−4.56713	−	5.80757i	−0.820280	−	1.04307i	−0.998256	−	0.0590359i	\(-0.981197\pi\)
							0.177976	−	0.984035i	\(-0.443045\pi\)
\(37\)	1.48584	+	1.55830i	0.244270	+	0.256184i	0.834444	−	0.551093i	\(-0.185789\pi\)
							−0.590173	+	0.807277i	\(0.700941\pi\)
\(41\)	1.78930	+	6.09380i	0.279442	+	0.951692i	0.972906	+	0.231201i	\(0.0742654\pi\)
							−0.693464	+	0.720491i	\(0.743916\pi\)
\(43\)	9.92091	+	1.42641i	1.51292	+	0.217526i	0.848245	−	0.529604i	\(-0.177659\pi\)
							0.664679	+	0.747129i	\(0.268568\pi\)
\(47\)	0.657016	+	0.379328i	0.0958356	+	0.0553307i	0.547152	−	0.837033i	\(-0.315712\pi\)
							−0.451316	+	0.892364i	\(0.649045\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.o.b.103.2	✓	160
7.3	odd	6	inner	322.2.o.b.241.7	yes	160
23.21	odd	22	inner	322.2.o.b.159.7	yes	160
161.136	even	66	inner	322.2.o.b.297.2	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.o.b.103.2	✓	160	1.1	even	1	trivial
322.2.o.b.159.7	yes	160	23.21	odd	22	inner
322.2.o.b.241.7	yes	160	7.3	odd	6	inner
322.2.o.b.297.2	yes	160	161.136	even	66	inner