Embedded newform 177.2.f.a.101.18

• Label: 177.2.f.a.101.18
• Level: $177$
• Weight: $2$
• Character: 177.101
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			101.18
Character	\(\chi\)	\(=\)	177.101
Dual form			177.2.f.a.170.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.57717 - 2.32615i) q^{2} +(1.22088 + 1.22859i) q^{3} +(-2.18323 - 5.47948i) q^{4} +(-0.564891 + 1.22099i) q^{5} +(4.78343 - 0.902256i) q^{6} +(-0.385009 - 1.38668i) q^{7} +(-10.7000 - 2.35524i) q^{8} +(-0.0188877 + 2.99994i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.57717	−	2.32615i	1.11522	−	1.64483i	0.464340	−	0.885657i	\(-0.346292\pi\)
							0.650885	−	0.759176i	\(-0.274398\pi\)
\(3\)	1.22088	+	1.22859i	0.704877	+	0.709329i
							\(5\)	−0.564891	+	1.22099i	−0.252627	+	0.546044i	−0.991849	−	0.127417i	\(-0.959331\pi\)
0.739222	+	0.673462i	\(0.235193\pi\)
\(7\)	−0.385009	−	1.38668i	−0.145520	−	0.524115i	−0.999989	−	0.00461361i	\(-0.998531\pi\)
							0.854469	−	0.519502i	\(-0.173882\pi\)
\(11\)	3.35245	+	3.17561i	1.01080	+	0.957481i	0.999099	−	0.0424503i	\(-0.0135164\pi\)
							0.0117021	+	0.999932i	\(0.496275\pi\)
\(13\)	0.0811226	−	0.745910i	0.0224994	−	0.206878i	−0.977492	−	0.210971i	\(-0.932337\pi\)
							0.999992	+	0.00409304i	\(0.00130286\pi\)
\(17\)	0.839088	+	0.232972i	0.203509	+	0.0565039i	0.367787	−	0.929910i	\(-0.380116\pi\)
							−0.164278	+	0.986414i	\(0.552529\pi\)
\(19\)	−1.96593	−	1.49446i	−0.451014	−	0.342852i	0.354880	−	0.934912i	\(-0.384522\pi\)
							−0.805894	+	0.592060i	\(0.798315\pi\)
\(23\)	0.762593	+	1.43840i	0.159012	+	0.299928i	0.950198	−	0.311648i	\(-0.100881\pi\)
							−0.791186	+	0.611576i	\(0.790536\pi\)
\(29\)	−4.85316	+	3.29052i	−0.901208	+	0.611034i	−0.921338	−	0.388763i	\(-0.872903\pi\)
							0.0201295	+	0.999797i	\(0.493592\pi\)
\(31\)	−4.16003	−	5.47242i	−0.747163	−	0.982876i	−0.999882	−	0.0153368i	\(-0.995118\pi\)
							0.252719	−	0.967540i	\(-0.418675\pi\)
\(37\)	−1.49935	−	6.81160i	−0.246491	−	1.11982i	−0.923426	−	0.383777i	\(-0.874623\pi\)
							0.676935	−	0.736043i	\(-0.263308\pi\)
\(41\)	−3.19707	−	1.69498i	−0.499298	−	0.264711i	0.199671	−	0.979863i	\(-0.436013\pi\)
							−0.698969	+	0.715152i	\(0.746358\pi\)
\(43\)	−0.500580	−	0.528456i	−0.0763378	−	0.0805888i	0.686708	−	0.726934i	\(-0.259055\pi\)
							−0.763045	+	0.646345i	\(0.776297\pi\)
\(47\)	4.73581	−	2.19102i	0.690789	−	0.319593i	−0.0428993	−	0.999079i	\(-0.513659\pi\)
							0.733688	+	0.679487i	\(0.237797\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.101.18	yes	504
3.2	odd	2	inner	177.2.f.a.101.1	✓	504
59.52	odd	58	inner	177.2.f.a.170.1	yes	504
177.170	even	58	inner	177.2.f.a.170.18	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.101.1	✓	504	3.2	odd	2	inner
177.2.f.a.101.18	yes	504	1.1	even	1	trivial
177.2.f.a.170.1	yes	504	59.52	odd	58	inner
177.2.f.a.170.18	yes	504	177.170	even	58	inner