Embedded newform 177.2.f.a.101.1

• Label: 177.2.f.a.101.1
• 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			101.1
Character	\(\chi\)	\(=\)	177.101
Dual form			177.2.f.a.170.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57717 + 2.32615i) q^{2} +(-1.08089 + 1.35339i) q^{3} +(-2.18323 - 5.47948i) q^{4} +(0.564891 - 1.22099i) q^{5} +(-1.44344 - 4.64884i) q^{6} +(-0.385009 - 1.38668i) q^{7} +(10.7000 + 2.35524i) q^{8} +(-0.663345 - 2.92574i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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	\(0.653708\pi\)
							−0.650885	+	0.759176i	\(0.725602\pi\)
\(3\)	−1.08089	+	1.35339i	−0.624053	+	0.781382i
							\(5\)	0.564891	−	1.22099i	0.252627	−	0.546044i	−0.739222	−	0.673462i	\(-0.764807\pi\)
0.991849	+	0.127417i	\(0.0406688\pi\)
\(7\)	−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	−0.0117021	−	0.999932i	\(-0.503725\pi\)
							−0.999099	+	0.0424503i	\(0.986484\pi\)
\(13\)	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	0.164278	−	0.986414i	\(-0.447471\pi\)
							−0.367787	+	0.929910i	\(0.619884\pi\)
\(19\)	−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	0.791186	−	0.611576i	\(-0.209464\pi\)
							−0.950198	+	0.311648i	\(0.899119\pi\)
\(29\)	4.85316	−	3.29052i	0.901208	−	0.611034i	−0.0201295	−	0.999797i	\(-0.506408\pi\)
							0.921338	+	0.388763i	\(0.127097\pi\)
\(31\)	−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	0.698969	−	0.715152i	\(-0.253642\pi\)
							−0.199671	+	0.979863i	\(0.563987\pi\)
\(43\)	−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	−0.733688	−	0.679487i	\(-0.762203\pi\)
							0.0428993	+	0.999079i	\(0.486341\pi\)

(See \(a_n\) instead)

Twists

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

    

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