Embedded newform 177.2.f.a.44.9

• Label: 177.2.f.a.44.9
• Level: $177$
• Weight: $2$
• Character: 177.44
• 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			44.9
Character	\(\chi\)	\(=\)	177.44
Dual form			177.2.f.a.173.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.288025 + 0.0313247i) q^{2} +(-1.57071 + 0.729971i) q^{3} +(-1.87126 + 0.411896i) q^{4} +(0.965552 + 1.27016i) q^{5} +(0.429539 - 0.259452i) q^{6} +(-1.63524 - 4.10415i) q^{7} +(1.07518 - 0.362272i) q^{8} +(1.93429 - 2.29315i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{27}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.288025	+	0.0313247i	−0.203665	+	0.0221499i	−0.209383	−	0.977834i	\(-0.567145\pi\)
							0.00571795	+	0.999984i	\(0.498180\pi\)
\(3\)	−1.57071	+	0.729971i	−0.906852	+	0.421449i
							\(5\)	0.965552	+	1.27016i	0.431808	+	0.568034i	0.959610	−	0.281334i	\(-0.0907768\pi\)
−0.527802	+	0.849367i	\(0.676984\pi\)
\(7\)	−1.63524	−	4.10415i	−0.618064	−	1.55122i	−0.820328	−	0.571893i	\(-0.806209\pi\)
							0.202264	−	0.979331i	\(-0.435170\pi\)
\(11\)	−5.62923	−	2.60436i	−1.69728	−	0.785244i	−0.997484	−	0.0708980i	\(-0.977414\pi\)
							−0.699793	−	0.714346i	\(-0.746724\pi\)
\(13\)	0.422604	−	0.0692824i	0.117209	−	0.0192155i	−0.102892	−	0.994693i	\(-0.532810\pi\)
							0.220101	+	0.975477i	\(0.429361\pi\)
\(17\)	−3.28932	−	1.31058i	−0.797777	−	0.317863i	−0.0646085	−	0.997911i	\(-0.520580\pi\)
							−0.733168	+	0.680047i	\(0.761959\pi\)
\(19\)	−3.16718	+	4.67124i	−0.726600	+	1.07166i	0.267539	+	0.963547i	\(0.413790\pi\)
							−0.994139	+	0.108108i	\(0.965521\pi\)
\(23\)	0.0588614	+	1.08563i	0.0122734	+	0.226370i	0.998235	+	0.0593952i	\(0.0189172\pi\)
							−0.985961	+	0.166975i	\(0.946600\pi\)
\(29\)	0.0717459	−	0.659693i	0.0133229	−	0.122502i	−0.985588	−	0.169165i	\(-0.945893\pi\)
							0.998911	+	0.0466627i	\(0.0148586\pi\)
\(31\)	0.299092	−	0.202790i	0.0537186	−	0.0364221i	−0.534032	−	0.845464i	\(-0.679324\pi\)
							0.587751	+	0.809042i	\(0.300014\pi\)
\(37\)	0.347793	−	1.03221i	0.0571769	−	0.169695i	−0.915259	−	0.402867i	\(-0.868014\pi\)
							0.972436	+	0.233172i	\(0.0749104\pi\)
\(41\)	−8.55770	−	0.463985i	−1.33649	−	0.0724623i	−0.628015	−	0.778201i	\(-0.716133\pi\)
							−0.708472	+	0.705739i	\(0.750615\pi\)
\(43\)	−1.03836	−	2.24438i	−0.158348	−	0.342264i	0.812121	−	0.583489i	\(-0.198313\pi\)
							−0.970469	+	0.241225i	\(0.922451\pi\)
\(47\)	1.27341	+	0.968024i	0.185747	+	0.141201i	0.693915	−	0.720057i	\(-0.255884\pi\)
							−0.508169	+	0.861258i	\(0.669677\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.44.9	✓	504
3.2	odd	2	inner	177.2.f.a.44.10	yes	504
59.55	odd	58	inner	177.2.f.a.173.10	yes	504
177.173	even	58	inner	177.2.f.a.173.9	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.44.9	✓	504	1.1	even	1	trivial
177.2.f.a.44.10	yes	504	3.2	odd	2	inner
177.2.f.a.173.9	yes	504	177.173	even	58	inner
177.2.f.a.173.10	yes	504	59.55	odd	58	inner