Embedded newform 177.2.f.a.65.6

• Label: 177.2.f.a.65.6
• Level: $177$
• Weight: $2$
• Character: 177.65
• 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			65.6
Character	\(\chi\)	\(=\)	177.65
Dual form			177.2.f.a.128.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.472909 - 1.18691i) q^{2} +(-1.42996 + 0.977348i) q^{3} +(0.266872 - 0.252795i) q^{4} +(-0.187027 + 0.158862i) q^{5} +(1.83627 + 1.23504i) q^{6} +(2.43110 - 1.46275i) q^{7} +(-2.74539 - 1.27015i) q^{8} +(1.08958 - 2.79514i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{51}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.472909	−	1.18691i	−0.334397	−	0.839274i	−0.995994	−	0.0894147i	\(-0.971500\pi\)
							0.661597	−	0.749860i	\(-0.269879\pi\)
\(3\)	−1.42996	+	0.977348i	−0.825589	+	0.564272i
							\(5\)	−0.187027	+	0.158862i	−0.0836409	+	0.0710452i	−0.688231	−	0.725491i	\(-0.741613\pi\)
0.604591	+	0.796536i	\(0.293337\pi\)
\(7\)	2.43110	−	1.46275i	0.918870	−	0.552866i	0.0242561	−	0.999706i	\(-0.492278\pi\)
							0.894614	+	0.446840i	\(0.147451\pi\)
\(11\)	−0.109776	−	2.02469i	−0.0330986	−	0.610468i	−0.967286	−	0.253687i	\(-0.918357\pi\)
							0.934188	−	0.356782i	\(-0.116126\pi\)
\(13\)	0.464390	−	2.10975i	0.128799	−	0.585138i	−0.867368	−	0.497667i	\(-0.834190\pi\)
							0.996167	−	0.0874715i	\(-0.0278787\pi\)
\(17\)	1.22526	−	2.03640i	0.297169	−	0.493899i	−0.671916	−	0.740628i	\(-0.734528\pi\)
							0.969084	+	0.246729i	\(0.0793558\pi\)
\(19\)	−1.36242	−	4.90699i	−0.312560	−	1.12574i	−0.938332	−	0.345735i	\(-0.887630\pi\)
							0.625772	−	0.780006i	\(-0.284784\pi\)
\(23\)	−1.16581	+	1.71944i	−0.243088	+	0.358528i	−0.929587	−	0.368603i	\(-0.879836\pi\)
							0.686499	+	0.727131i	\(0.259147\pi\)
\(29\)	6.90693	+	2.75197i	1.28258	+	0.511028i	0.909088	−	0.416604i	\(-0.136780\pi\)
							0.373496	+	0.927632i	\(0.378159\pi\)
\(31\)	1.87329	+	0.520118i	0.336454	+	0.0934159i	0.431646	−	0.902043i	\(-0.357933\pi\)
							−0.0951926	+	0.995459i	\(0.530347\pi\)
\(37\)	2.61159	+	5.64487i	0.429343	+	0.928010i	0.994667	+	0.103141i	\(0.0328893\pi\)
							−0.565323	+	0.824869i	\(0.691249\pi\)
\(41\)	9.17878	−	6.22337i	1.43348	−	0.971927i	0.436082	−	0.899907i	\(-0.356366\pi\)
							0.997403	−	0.0720196i	\(-0.0229444\pi\)
\(43\)	−9.20994	−	0.499348i	−1.40450	−	0.0761499i	−0.663763	−	0.747943i	\(-0.731042\pi\)
							−0.740739	+	0.671793i	\(0.765525\pi\)
\(47\)	−1.56059	+	1.83726i	−0.227635	+	0.267992i	−0.864108	−	0.503306i	\(-0.832117\pi\)
							0.636473	+	0.771299i	\(0.280393\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.65.6	✓	504
3.2	odd	2	inner	177.2.f.a.65.13	yes	504
59.10	odd	58	inner	177.2.f.a.128.13	yes	504
177.128	even	58	inner	177.2.f.a.128.6	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.65.6	✓	504	1.1	even	1	trivial
177.2.f.a.65.13	yes	504	3.2	odd	2	inner
177.2.f.a.128.6	yes	504	177.128	even	58	inner
177.2.f.a.128.13	yes	504	59.10	odd	58	inner