Embedded newform 177.2.f.a.23.3

• Label: 177.2.f.a.23.3
• Level: $177$
• Weight: $2$
• Character: 177.23
• 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			23.3
Character	\(\chi\)	\(=\)	177.23
Dual form			177.2.f.a.77.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.66142 + 1.57378i) q^{2} +(-1.58760 + 0.692478i) q^{3} +(0.175253 - 3.23234i) q^{4} +(3.57617 + 0.586283i) q^{5} +(1.54786 - 3.64903i) q^{6} +(2.12369 - 4.00571i) q^{7} +(1.83278 + 2.15772i) q^{8} +(2.04095 - 2.19876i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{15}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.66142	+	1.57378i	−1.17480	+	1.11283i	−0.183255	+	0.983065i	\(0.558664\pi\)
							−0.991545	+	0.129765i	\(0.958578\pi\)
\(3\)	−1.58760	+	0.692478i	−0.916601	+	0.399802i
							\(5\)	3.57617	+	0.586283i	1.59931	+	0.262194i	0.894455	−	0.447157i	\(-0.147564\pi\)
0.704855	+	0.709351i	\(0.251012\pi\)
\(7\)	2.12369	−	4.00571i	0.802680	−	1.51401i	−0.0542188	−	0.998529i	\(-0.517267\pi\)
							0.856898	−	0.515485i	\(-0.172388\pi\)
\(11\)	0.784368	−	0.0853052i	0.236496	−	0.0257205i	0.0108965	−	0.999941i	\(-0.496531\pi\)
							0.225599	+	0.974220i	\(0.427566\pi\)
\(13\)	−1.08754	+	2.35068i	−0.301630	+	0.651963i	−0.997827	−	0.0658882i	\(-0.979012\pi\)
							0.696197	+	0.717851i	\(0.254874\pi\)
\(17\)	−1.62753	+	0.862863i	−0.394734	+	0.209275i	−0.653939	−	0.756548i	\(-0.726885\pi\)
							0.259204	+	0.965823i	\(0.416540\pi\)
\(19\)	3.88386	−	2.33684i	0.891019	−	0.536108i	0.00512011	−	0.999987i	\(-0.498370\pi\)
							0.885899	+	0.463879i	\(0.153543\pi\)
\(23\)	−1.49250	−	3.74589i	−0.311207	−	0.781071i	−0.998503	−	0.0546938i	\(-0.982582\pi\)
							0.687296	−	0.726378i	\(-0.258798\pi\)
\(29\)	−2.81604	+	2.97286i	−0.522925	+	0.552046i	−0.932939	−	0.360034i	\(-0.882765\pi\)
							0.410014	+	0.912079i	\(0.365524\pi\)
\(31\)	0.742665	−	1.23432i	0.133387	−	0.221690i	−0.782964	−	0.622067i	\(-0.786293\pi\)
							0.916351	+	0.400377i	\(0.131121\pi\)
\(37\)	−1.68061	−	1.42752i	−0.276290	−	0.234683i	0.498518	−	0.866879i	\(-0.333878\pi\)
							−0.774808	+	0.632196i	\(0.782154\pi\)
\(41\)	9.58894	+	3.82058i	1.49754	+	0.596675i	0.967853	−	0.251518i	\(-0.0809297\pi\)
							0.529688	+	0.848193i	\(0.322309\pi\)
\(43\)	−0.958084	+	8.80944i	−0.146106	+	1.34343i	0.659825	+	0.751419i	\(0.270630\pi\)
							−0.805932	+	0.592008i	\(0.798335\pi\)
\(47\)	−0.581228	−	3.54534i	−0.0847809	−	0.517140i	−0.994739	−	0.102441i	\(-0.967335\pi\)
							0.909958	−	0.414700i	\(-0.136113\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.23.3	✓	504
3.2	odd	2	inner	177.2.f.a.23.16	yes	504
59.18	odd	58	inner	177.2.f.a.77.16	yes	504
177.77	even	58	inner	177.2.f.a.77.3	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.23.3	✓	504	1.1	even	1	trivial
177.2.f.a.23.16	yes	504	3.2	odd	2	inner
177.2.f.a.77.3	yes	504	177.77	even	58	inner
177.2.f.a.77.16	yes	504	59.18	odd	58	inner