Embedded newform 177.2.f.a.113.13

• Label: 177.2.f.a.113.13
• Level: $177$
• Weight: $2$
• Character: 177.113
• 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			113.13
Character	\(\chi\)	\(=\)	177.113
Dual form			177.2.f.a.47.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.980077 + 0.330226i) q^{2} +(1.71351 + 0.252759i) q^{3} +(-0.740685 - 0.563054i) q^{4} +(2.53710 + 1.01087i) q^{5} +(1.59590 + 0.813569i) q^{6} +(-3.45158 + 1.59687i) q^{7} +(-1.70077 - 2.50845i) q^{8} +(2.87223 + 0.866209i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{35}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.980077	+	0.330226i	0.693019	+	0.233505i	0.643700	−	0.765278i	\(-0.277398\pi\)
							0.0493188	+	0.998783i	\(0.484295\pi\)
\(3\)	1.71351	+	0.252759i	0.989295	+	0.145930i
							\(5\)	2.53710	+	1.01087i	1.13463	+	0.452076i	0.860415	−	0.509594i	\(-0.170204\pi\)
0.274210	+	0.961670i	\(0.411584\pi\)
\(7\)	−3.45158	+	1.59687i	−1.30457	+	0.603560i	−0.944238	−	0.329262i	\(-0.893200\pi\)
							−0.360336	+	0.932823i	\(0.617338\pi\)
\(11\)	0.332179	−	1.19640i	0.100156	−	0.360729i	−0.896441	−	0.443164i	\(-0.853856\pi\)
							0.996596	+	0.0824353i	\(0.0262698\pi\)
\(13\)	−3.53258	−	1.87286i	−0.979762	−	0.519437i	−0.100070	−	0.994980i	\(-0.531907\pi\)
							−0.879692	+	0.475543i	\(0.842251\pi\)
\(17\)	0.947267	−	2.04748i	0.229746	−	0.496588i	−0.758078	−	0.652164i	\(-0.773861\pi\)
							0.987824	+	0.155576i	\(0.0497235\pi\)
\(19\)	0.558769	−	0.122994i	0.128190	−	0.0282169i	−0.150412	−	0.988623i	\(-0.548060\pi\)
							0.278602	+	0.960407i	\(0.410129\pi\)
\(23\)	−1.16141	+	7.08431i	−0.242171	+	1.47718i	0.533369	+	0.845882i	\(0.320926\pi\)
							−0.775541	+	0.631298i	\(0.782523\pi\)
\(29\)	−3.01788	−	8.95676i	−0.560407	−	1.66323i	−0.734587	−	0.678514i	\(-0.762624\pi\)
							0.174180	−	0.984714i	\(-0.444272\pi\)
\(31\)	−1.30039	+	5.90773i	−0.233557	+	1.06106i	0.703176	+	0.711016i	\(0.251765\pi\)
							−0.936733	+	0.350045i	\(0.886166\pi\)
\(37\)	−4.73021	−	3.20716i	−0.777642	−	0.527254i	0.106590	−	0.994303i	\(-0.466007\pi\)
							−0.884231	+	0.467049i	\(0.845317\pi\)
\(41\)	0.441006	−	0.0722992i	0.0688735	−	0.0112912i	−0.127247	−	0.991871i	\(-0.540614\pi\)
							0.196120	+	0.980580i	\(0.437166\pi\)
\(43\)	10.2633	−	2.84960i	1.56514	−	0.434560i	0.626492	−	0.779428i	\(-0.284490\pi\)
							0.938651	+	0.344868i	\(0.112076\pi\)
\(47\)	1.85657	+	4.65964i	0.270809	+	0.679679i	0.999989	−	0.00464268i	\(-0.00147782\pi\)
							−0.729180	+	0.684321i	\(0.760099\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.113.13	yes	504
3.2	odd	2	inner	177.2.f.a.113.6	yes	504
59.47	odd	58	inner	177.2.f.a.47.6	✓	504
177.47	even	58	inner	177.2.f.a.47.13	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.47.6	✓	504	59.47	odd	58	inner
177.2.f.a.47.13	yes	504	177.47	even	58	inner
177.2.f.a.113.6	yes	504	3.2	odd	2	inner
177.2.f.a.113.13	yes	504	1.1	even	1	trivial