Embedded newform 161.2.m.a.100.13

• Label: 161.2.m.a.100.13
• Level: $161$
• Weight: $2$
• Character: 161.100
• 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.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.28559147254\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(14\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			100.13
Character	\(\chi\)	\(=\)	161.100
Dual form			161.2.m.a.95.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.59143 + 1.25151i) q^{2} +(0.924163 - 1.29781i) q^{3} +(0.494839 + 2.03976i) q^{4} +(-1.87893 + 0.362134i) q^{5} +(3.09496 - 0.908761i) q^{6} +(2.37547 - 1.16496i) q^{7} +(-0.0831964 + 0.182175i) q^{8} +(0.150982 + 0.436235i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 11 q^{2} - 9 q^{3} + q^{4} - 11 q^{5} - 52 q^{6} - 20 q^{7} - 32 q^{8} + 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(24\)	\(120\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.59143	+	1.25151i	1.12531	+	0.884952i	0.994320	−	0.106431i	\(-0.0339425\pi\)
							0.130989	+	0.991384i	\(0.458185\pi\)
\(3\)	0.924163	−	1.29781i	0.533566	−	0.749288i	−0.456526	−	0.889710i	\(-0.650906\pi\)
							0.990092	+	0.140422i	\(0.0448459\pi\)
\(5\)	−1.87893	+	0.362134i	−0.840283	+	0.161951i	−0.591194	−	0.806529i	\(-0.701343\pi\)
							−0.249089	+	0.968481i	\(0.580131\pi\)
\(7\)	2.37547	−	1.16496i	0.897844	−	0.440313i
							\(11\)	−4.40736	+	3.46599i	−1.32887	+	1.04504i	−0.334139	+	0.942524i	\(0.608445\pi\)
−0.994732	+	0.102511i	\(0.967312\pi\)
\(13\)	−2.49215	+	1.60161i	−0.691198	+	0.444206i	−0.838512	−	0.544884i	\(-0.816574\pi\)
							0.147313	+	0.989090i	\(0.452937\pi\)
\(17\)	−4.95695	−	4.72645i	−1.20224	−	1.14633i	−0.985424	−	0.170118i	\(-0.945585\pi\)
							−0.216814	−	0.976213i	\(-0.569567\pi\)
\(19\)	2.93332	−	2.79692i	0.672950	−	0.641657i	−0.274490	−	0.961590i	\(-0.588509\pi\)
							0.947440	+	0.319933i	\(0.103660\pi\)
\(23\)	−4.56180	+	1.47987i	−0.951200	+	0.308574i
							\(29\)	6.07219	−	1.78296i	1.12758	−	0.331087i	0.335822	−	0.941925i	\(-0.390986\pi\)
0.791755	+	0.610839i	\(0.209168\pi\)
\(31\)	−0.110593	+	0.0105604i	−0.0198632	+	0.00189670i	−0.104983	−	0.994474i	\(-0.533479\pi\)
							0.0851197	+	0.996371i	\(0.472873\pi\)
\(37\)	0.464468	+	1.34199i	0.0763581	+	0.220622i	0.976719	−	0.214521i	\(-0.0688190\pi\)
							−0.900361	+	0.435143i	\(0.856698\pi\)
\(41\)	−0.915016	−	1.05598i	−0.142901	−	0.164917i	0.679787	−	0.733410i	\(-0.262072\pi\)
							−0.822688	+	0.568493i	\(0.807527\pi\)
\(43\)	3.80331	+	8.32808i	0.579999	+	1.27002i	0.941300	+	0.337571i	\(0.109605\pi\)
							−0.361301	+	0.932449i	\(0.617667\pi\)
\(47\)	0.0620795	−	0.107525i	0.00905523	−	0.0156841i	−0.861462	−	0.507822i	\(-0.830451\pi\)
							0.870518	+	0.492137i	\(0.163784\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.m.a.100.13	yes	280
7.4	even	3	inner	161.2.m.a.123.2	yes	280
23.3	even	11	inner	161.2.m.a.72.2	✓	280
161.95	even	33	inner	161.2.m.a.95.13	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.m.a.72.2	✓	280	23.3	even	11	inner
161.2.m.a.95.13	yes	280	161.95	even	33	inner
161.2.m.a.100.13	yes	280	1.1	even	1	trivial
161.2.m.a.123.2	yes	280	7.4	even	3	inner