Embedded newform 161.2.o.a.10.1

• Label: 161.2.o.a.10.1
• Level: $161$
• Weight: $2$
• Character: 161.10
• 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			10.1
Character	\(\chi\)	\(=\)	161.10
Dual form			161.2.o.a.145.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.804744 + 2.32516i) q^{2} +(-0.737874 - 1.43128i) q^{3} +(-3.18663 - 2.50600i) q^{4} +(-0.940491 + 0.0898060i) q^{5} +(3.92174 - 0.563861i) q^{6} +(-1.53049 - 2.15815i) q^{7} +(4.25148 - 2.73226i) q^{8} +(0.236079 - 0.331527i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 7 q^{2} - 27 q^{3} + q^{4} - 33 q^{5} - 22 q^{7} - 56 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(24\)	\(120\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.804744	+	2.32516i	−0.569040	+	1.64413i	0.181562	+	0.983380i	\(0.441885\pi\)
							−0.750602	+	0.660754i	\(0.770237\pi\)
\(3\)	−0.737874	−	1.43128i	−0.426012	−	0.826347i	−0.999972	−	0.00746430i	\(-0.997624\pi\)
							0.573961	−	0.818883i	\(-0.305406\pi\)
\(5\)	−0.940491	+	0.0898060i	−0.420600	+	0.0401624i	−0.303211	−	0.952924i	\(-0.598059\pi\)
							−0.117389	+	0.993086i	\(0.537453\pi\)
\(7\)	−1.53049	−	2.15815i	−0.578472	−	0.815702i
							\(11\)	3.45878	−	1.19709i	1.04286	−	0.360938i	0.248659	−	0.968591i	\(-0.420010\pi\)
0.794202	+	0.607654i	\(0.207889\pi\)
\(13\)	−0.821934	−	2.79925i	−0.227964	−	0.776372i	−0.991443	−	0.130540i	\(-0.958329\pi\)
							0.763479	−	0.645832i	\(-0.223489\pi\)
\(17\)	−4.69007	−	1.87762i	−1.13751	−	0.455390i	−0.275007	−	0.961442i	\(-0.588680\pi\)
							−0.862503	+	0.506052i	\(0.831104\pi\)
\(19\)	−3.39406	+	1.35878i	−0.778650	+	0.311724i	−0.726727	−	0.686927i	\(-0.758959\pi\)
							−0.0519232	+	0.998651i	\(0.516535\pi\)
\(23\)	4.61547	−	1.30287i	0.962391	−	0.271667i
							\(29\)	0.332631	+	2.31350i	0.0617681	+	0.429607i	0.997117	+	0.0758800i	\(0.0241766\pi\)
−0.935349	+	0.353727i	\(0.884914\pi\)
\(31\)	3.91257	−	0.186379i	0.702719	−	0.0334746i	0.306818	−	0.951768i	\(-0.400736\pi\)
							0.395900	+	0.918293i	\(0.370433\pi\)
\(37\)	−4.86447	−	3.46397i	−0.799714	−	0.569474i	0.105505	−	0.994419i	\(-0.466354\pi\)
							−0.905219	+	0.424945i	\(0.860293\pi\)
\(41\)	−3.82293	−	1.74588i	−0.597042	−	0.272660i	0.0938867	−	0.995583i	\(-0.470071\pi\)
							−0.690929	+	0.722923i	\(0.742798\pi\)
\(43\)	4.31649	−	6.71659i	0.658259	−	1.02427i	−0.338272	−	0.941048i	\(-0.609842\pi\)
							0.996531	−	0.0832223i	\(-0.0265212\pi\)
\(47\)	3.02425	−	1.74605i	0.441133	−	0.254688i	−0.262945	−	0.964811i	\(-0.584694\pi\)
							0.704078	+	0.710123i	\(0.251360\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.o.a.10.1	✓	280
7.5	odd	6	inner	161.2.o.a.33.14	yes	280
23.7	odd	22	inner	161.2.o.a.122.14	yes	280
161.145	even	66	inner	161.2.o.a.145.1	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.o.a.10.1	✓	280	1.1	even	1	trivial
161.2.o.a.33.14	yes	280	7.5	odd	6	inner
161.2.o.a.122.14	yes	280	23.7	odd	22	inner
161.2.o.a.145.1	yes	280	161.145	even	66	inner