Embedded newform 161.2.o.a.10.4

• Label: 161.2.o.a.10.4
• 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			10.4
Character	\(\chi\)	\(=\)	161.10
Dual form			161.2.o.a.145.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.469553 + 1.35668i) q^{2} +(-1.33410 - 2.58780i) q^{3} +(-0.0480036 - 0.0377505i) q^{4} +(-3.27323 + 0.312556i) q^{5} +(4.13726 - 0.594848i) q^{6} +(1.13446 + 2.39019i) q^{7} +(-2.34172 + 1.50493i) q^{8} +(-3.17671 + 4.46107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 7 q^{2} - 27 q^{3} + q^{4} - 33 q^{5} - 22 q^{7} - 56 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.469553	+	1.35668i	−0.332024	+	0.959320i	0.648207	+	0.761464i	\(0.275519\pi\)
							−0.980231	+	0.197856i	\(0.936602\pi\)
\(3\)	−1.33410	−	2.58780i	−0.770246	−	1.49407i	−0.866754	−	0.498736i	\(-0.833798\pi\)
							0.0965083	−	0.995332i	\(-0.469233\pi\)
\(5\)	−3.27323	+	0.312556i	−1.46383	+	0.139779i	−0.796346	−	0.604841i	\(-0.793237\pi\)
							−0.667488	+	0.744620i	\(0.732630\pi\)
\(7\)	1.13446	+	2.39019i	0.428787	+	0.903405i
							\(11\)	−4.87710	+	1.68798i	−1.47050	+	0.508945i	−0.940718	−	0.339188i	\(-0.889848\pi\)
−0.529782	+	0.848134i	\(0.677726\pi\)
\(13\)	−0.628194	−	2.13943i	−0.174230	−	0.593372i	−0.999586	−	0.0287602i	\(-0.990844\pi\)
							0.825357	−	0.564612i	\(-0.190974\pi\)
\(17\)	3.16159	+	1.26571i	0.766797	+	0.306979i	0.721886	−	0.692012i	\(-0.243275\pi\)
							0.0449111	+	0.998991i	\(0.485700\pi\)
\(19\)	−4.14095	+	1.65779i	−0.950000	+	0.380322i	−0.794330	−	0.607487i	\(-0.792178\pi\)
							−0.155670	+	0.987809i	\(0.549754\pi\)
\(23\)	2.53420	−	4.07159i	0.528418	−	0.848984i
							\(29\)	−0.202396	−	1.40770i	−0.0375841	−	0.261403i	0.962362	−	0.271770i	\(-0.0876091\pi\)
−0.999946	+	0.0103672i	\(0.996700\pi\)
\(31\)	−4.40612	+	0.209889i	−0.791362	+	0.0376972i	−0.439371	−	0.898306i	\(-0.644799\pi\)
							−0.351992	+	0.936003i	\(0.614495\pi\)
\(37\)	3.92072	+	2.79193i	0.644562	+	0.458990i	0.855008	−	0.518615i	\(-0.173552\pi\)
							−0.210446	+	0.977605i	\(0.567492\pi\)
\(41\)	−1.60245	−	0.731813i	−0.250260	−	0.114290i	0.286336	−	0.958129i	\(-0.407563\pi\)
							−0.536596	+	0.843839i	\(0.680290\pi\)
\(43\)	−1.44547	+	2.24919i	−0.220432	+	0.342999i	−0.933803	−	0.357788i	\(-0.883531\pi\)
							0.713371	+	0.700786i	\(0.247167\pi\)
\(47\)	−3.25505	+	1.87931i	−0.474798	+	0.274125i	−0.718246	−	0.695789i	\(-0.755055\pi\)
							0.243448	+	0.969914i	\(0.421722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.o.a.10.4	✓	280
7.5	odd	6	inner	161.2.o.a.33.11	yes	280
23.7	odd	22	inner	161.2.o.a.122.11	yes	280
161.145	even	66	inner	161.2.o.a.145.4	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.o.a.10.4	✓	280	1.1	even	1	trivial
161.2.o.a.33.11	yes	280	7.5	odd	6	inner
161.2.o.a.122.11	yes	280	23.7	odd	22	inner
161.2.o.a.145.4	yes	280	161.145	even	66	inner