Embedded newform 161.2.o.a.103.12

• Label: 161.2.o.a.103.12
• Level: $161$
• Weight: $2$
• Character: 161.103
• Analytic conductor: $1.286$
• Analytic rank: $0$
• Dimension: $280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	161.o (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.28559147254\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(14\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			103.12
Character	\(\chi\)	\(=\)	161.103
Dual form			161.2.o.a.136.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.107348 + 2.25351i) q^{2} +(-0.119289 - 0.297968i) q^{3} +(-3.07582 + 0.293705i) q^{4} +(-0.411187 + 1.69494i) q^{5} +(0.658668 - 0.300804i) q^{6} +(-1.29437 + 2.30751i) q^{7} +(-0.349908 - 2.43366i) q^{8} +(2.09665 - 1.99915i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 7 q^{2} - 27 q^{3} + q^{4} - 33 q^{5} - 22 q^{7} - 56 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).

\(n\)	\(24\)	\(120\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.107348	+	2.25351i	0.0759063	+	1.59347i	0.640406	+	0.768036i	\(0.278766\pi\)
							−0.564500	+	0.825433i	\(0.690931\pi\)
\(3\)	−0.119289	−	0.297968i	−0.0688713	−	0.172032i	0.889939	−	0.456080i	\(-0.150747\pi\)
							−0.958810	+	0.284048i	\(0.908323\pi\)
\(5\)	−0.411187	+	1.69494i	−0.183888	+	0.757999i	0.802917	+	0.596090i	\(0.203280\pi\)
							−0.986806	+	0.161908i	\(0.948235\pi\)
\(7\)	−1.29437	+	2.30751i	−0.489225	+	0.872158i
							\(11\)	−5.00386	−	0.238363i	−1.50872	−	0.0718692i	−0.723171	−	0.690669i	\(-0.757316\pi\)
−0.785549	+	0.618800i	\(0.787619\pi\)
\(13\)	3.21299	+	2.78407i	0.891124	+	0.772163i	0.974508	−	0.224355i	\(-0.0720274\pi\)
							−0.0833840	+	0.996517i	\(0.526573\pi\)
\(17\)	3.35958	−	4.71787i	0.814818	−	1.14425i	−0.172423	−	0.985023i	\(-0.555160\pi\)
							0.987242	−	0.159229i	\(-0.0509009\pi\)
\(19\)	2.61821	+	3.67677i	0.600660	+	0.843508i	0.997221	−	0.0744998i	\(-0.0237360\pi\)
							−0.396561	+	0.918008i	\(0.629797\pi\)
\(23\)	4.73671	−	0.750728i	0.987672	−	0.156538i
							\(29\)	−2.28879	−	5.01174i	−0.425017	−	0.930657i	−0.994109	−	0.108384i	\(-0.965433\pi\)
0.569092	−	0.822274i	\(-0.307295\pi\)
\(31\)	5.82734	+	7.41006i	1.04662	+	1.33089i	0.941718	+	0.336404i	\(0.109211\pi\)
							0.104903	+	0.994483i	\(0.466547\pi\)
\(37\)	−0.962150	−	1.00907i	−0.158176	−	0.165891i	0.639877	−	0.768477i	\(-0.278985\pi\)
							−0.798053	+	0.602587i	\(0.794137\pi\)
\(41\)	0.869558	+	2.96144i	0.135802	+	0.462500i	0.999110	−	0.0421779i	\(-0.0134296\pi\)
							−0.863308	+	0.504677i	\(0.831611\pi\)
\(43\)	−2.52044	−	0.362385i	−0.384364	−	0.0552632i	−0.0525755	−	0.998617i	\(-0.516743\pi\)
							−0.331789	+	0.943354i	\(0.607652\pi\)
\(47\)	2.69681	+	1.55700i	0.393370	+	0.227112i	0.683619	−	0.729839i	\(-0.260405\pi\)
							−0.290249	+	0.956951i	\(0.593738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.o.a.103.12	yes	280
7.3	odd	6	inner	161.2.o.a.80.3	✓	280
23.21	odd	22	inner	161.2.o.a.159.3	yes	280
161.136	even	66	inner	161.2.o.a.136.12	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.o.a.80.3	✓	280	7.3	odd	6	inner
161.2.o.a.103.12	yes	280	1.1	even	1	trivial
161.2.o.a.136.12	yes	280	161.136	even	66	inner
161.2.o.a.159.3	yes	280	23.21	odd	22	inner