Embedded newform 177.2.f.a.2.17

• Label: 177.2.f.a.2.17
• Level: $177$
• Weight: $2$
• Character: 177.2
• 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			2.17
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.132697 - 2.44745i) q^{2} +(-0.571873 + 1.63492i) q^{3} +(-3.98412 - 0.433299i) q^{4} +(0.468889 - 1.39161i) q^{5} +(3.92549 + 1.61658i) q^{6} +(-2.78535 - 4.10808i) q^{7} +(-0.796087 + 4.85592i) q^{8} +(-2.34592 - 1.86993i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.132697	−	2.44745i	0.0938308	−	1.73061i	−0.451400	−	0.892321i	\(-0.649075\pi\)
							0.545231	−	0.838286i	\(-0.316442\pi\)
\(3\)	−0.571873	+	1.63492i	−0.330171	+	0.943921i
							\(5\)	0.468889	−	1.39161i	0.209694	−	0.622349i	−0.790260	−	0.612771i	\(-0.790055\pi\)
0.999954	−	0.00957756i	\(-0.00304868\pi\)
\(7\)	−2.78535	−	4.10808i	−1.05276	−	1.55271i	−0.813336	−	0.581795i	\(-0.802351\pi\)
							−0.239427	−	0.970914i	\(-0.576960\pi\)
\(11\)	1.28003	−	0.281755i	0.385943	−	0.0849524i	−0.0177609	−	0.999842i	\(-0.505654\pi\)
							0.403704	+	0.914890i	\(0.367723\pi\)
\(13\)	−1.67356	+	1.42154i	−0.464163	+	0.394264i	−0.848724	−	0.528836i	\(-0.822629\pi\)
							0.384561	+	0.923100i	\(0.374353\pi\)
\(17\)	2.56533	+	1.73934i	0.622184	+	0.421851i	0.831132	−	0.556076i	\(-0.187694\pi\)
							−0.208948	+	0.977927i	\(0.567004\pi\)
\(19\)	1.28082	−	2.41589i	0.293841	−	0.554242i	−0.692006	−	0.721892i	\(-0.743273\pi\)
							0.985847	+	0.167650i	\(0.0536177\pi\)
\(23\)	6.28524	−	5.95370i	1.31056	−	1.24143i	0.359237	−	0.933246i	\(-0.383037\pi\)
							0.951326	−	0.308186i	\(-0.0997218\pi\)
\(29\)	2.27580	−	0.123390i	0.422606	−	0.0229130i	0.158390	−	0.987377i	\(-0.449370\pi\)
							0.264216	+	0.964464i	\(0.414887\pi\)
\(31\)	−8.24037	+	4.36877i	−1.48001	+	0.784654i	−0.995910	−	0.0903562i	\(-0.971199\pi\)
							−0.484105	+	0.875010i	\(0.660855\pi\)
\(37\)	3.07229	−	0.503676i	0.505081	−	0.0828039i	0.0961460	−	0.995367i	\(-0.469348\pi\)
							0.408935	+	0.912563i	\(0.365900\pi\)
\(41\)	1.32775	−	1.40168i	0.207359	−	0.218906i	−0.614018	−	0.789292i	\(-0.710448\pi\)
							0.821377	+	0.570386i	\(0.193206\pi\)
\(43\)	1.41918	−	6.44742i	0.216424	−	0.983222i	−0.735652	−	0.677360i	\(-0.763124\pi\)
							0.952076	−	0.305863i	\(-0.0989448\pi\)
\(47\)	0.0951974	−	0.0320757i	0.0138860	−	0.00467873i	−0.312350	−	0.949967i	\(-0.601116\pi\)
							0.326236	+	0.945288i	\(0.394220\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.2.17	yes	504
3.2	odd	2	inner	177.2.f.a.2.2	✓	504
59.30	odd	58	inner	177.2.f.a.89.2	yes	504
177.89	even	58	inner	177.2.f.a.89.17	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.2.2	✓	504	3.2	odd	2	inner
177.2.f.a.2.17	yes	504	1.1	even	1	trivial
177.2.f.a.89.2	yes	504	59.30	odd	58	inner
177.2.f.a.89.17	yes	504	177.89	even	58	inner