Embedded newform 177.2.f.a.23.1

• Label: 177.2.f.a.23.1
• 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			23.1
Character	\(\chi\)	\(=\)	177.23
Dual form			177.2.f.a.77.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88771 + 1.78814i) q^{2} +(-1.60218 - 0.658044i) q^{3} +(0.257749 - 4.75391i) q^{4} +(-3.64954 - 0.598312i) q^{5} +(4.20113 - 1.62272i) q^{6} +(-0.920260 + 1.73579i) q^{7} +(4.64745 + 5.47140i) q^{8} +(2.13396 + 2.10861i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88771	+	1.78814i	−1.33481	+	1.26440i	−0.396757	+	0.917924i	\(0.629864\pi\)
							−0.938058	+	0.346479i	\(0.887377\pi\)
\(3\)	−1.60218	−	0.658044i	−0.925019	−	0.379922i
							\(5\)	−3.64954	−	0.598312i	−1.63213	−	0.267573i	−0.725098	−	0.688645i	\(-0.758206\pi\)
−0.907027	+	0.421072i	\(0.861654\pi\)
\(7\)	−0.920260	+	1.73579i	−0.347826	+	0.656069i	−0.994271	−	0.106892i	\(-0.965910\pi\)
							0.646445	+	0.762961i	\(0.276255\pi\)
\(11\)	5.58591	−	0.607504i	1.68421	−	0.183169i	0.784645	−	0.619946i	\(-0.212845\pi\)
							0.899570	+	0.436777i	\(0.143880\pi\)
\(13\)	0.293437	−	0.634254i	0.0813848	−	0.175910i	−0.862611	−	0.505868i	\(-0.831172\pi\)
							0.943996	+	0.329958i	\(0.107034\pi\)
\(17\)	0.992908	−	0.526407i	0.240816	−	0.127672i	−0.343630	−	0.939105i	\(-0.611657\pi\)
							0.584446	+	0.811433i	\(0.301312\pi\)
\(19\)	0.211729	−	0.127393i	0.0485738	−	0.0292259i	−0.491060	−	0.871126i	\(-0.663390\pi\)
							0.539634	+	0.841900i	\(0.318563\pi\)
\(23\)	−1.25446	−	3.14845i	−0.261573	−	0.656498i	0.738265	−	0.674511i	\(-0.235645\pi\)
							−0.999838	+	0.0180130i	\(0.994266\pi\)
\(29\)	3.31532	−	3.49994i	0.615640	−	0.649923i	−0.341387	−	0.939923i	\(-0.610897\pi\)
							0.957027	+	0.290000i	\(0.0936552\pi\)
\(31\)	−0.316746	+	0.526437i	−0.0568893	+	0.0945508i	−0.883997	−	0.467493i	\(-0.845157\pi\)
							0.827107	+	0.562044i	\(0.189985\pi\)
\(37\)	3.47371	+	2.95060i	0.571074	+	0.485075i	0.885774	−	0.464116i	\(-0.153628\pi\)
							−0.314700	+	0.949191i	\(0.601904\pi\)
\(41\)	6.17403	+	2.45996i	0.964221	+	0.384181i	0.798502	−	0.601992i	\(-0.205626\pi\)
							0.165719	+	0.986173i	\(0.447006\pi\)
\(43\)	0.785821	−	7.22551i	0.119837	−	1.10188i	−0.767593	−	0.640937i	\(-0.778546\pi\)
							0.887430	−	0.460942i	\(-0.152489\pi\)
\(47\)	0.805550	+	4.91364i	0.117502	+	0.716728i	0.978208	+	0.207627i	\(0.0665739\pi\)
							−0.860707	+	0.509101i	\(0.829978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.23.1	✓	504
3.2	odd	2	inner	177.2.f.a.23.18	yes	504
59.18	odd	58	inner	177.2.f.a.77.18	yes	504
177.77	even	58	inner	177.2.f.a.77.1	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.23.1	✓	504	1.1	even	1	trivial
177.2.f.a.23.18	yes	504	3.2	odd	2	inner
177.2.f.a.77.1	yes	504	177.77	even	58	inner
177.2.f.a.77.18	yes	504	59.18	odd	58	inner