Embedded newform 177.2.f.a.2.2

• Label: 177.2.f.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.132697 + 2.44745i) q^{2} +(-1.61630 - 0.622564i) q^{3} +(-3.98412 - 0.433299i) q^{4} +(-0.468889 + 1.39161i) q^{5} +(1.73817 - 3.87319i) q^{6} +(-2.78535 - 4.10808i) q^{7} +(0.796087 - 4.85592i) q^{8} +(2.22483 + 2.01250i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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	\(0.350925\pi\)
							−0.545231	+	0.838286i	\(0.683558\pi\)
\(3\)	−1.61630	−	0.622564i	−0.933169	−	0.359438i
							\(5\)	−0.468889	+	1.39161i	−0.209694	+	0.622349i	0.790260	+	0.612771i	\(0.209945\pi\)
−0.999954	+	0.00957756i	\(0.996951\pi\)
\(7\)	−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	−0.403704	−	0.914890i	\(-0.632277\pi\)
							0.0177609	+	0.999842i	\(0.494346\pi\)
\(13\)	−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	0.208948	−	0.977927i	\(-0.432996\pi\)
							−0.831132	+	0.556076i	\(0.812306\pi\)
\(19\)	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	\(0.616963\pi\)
							−0.951326	+	0.308186i	\(0.900278\pi\)
\(29\)	−2.27580	+	0.123390i	−0.422606	+	0.0229130i	−0.264216	−	0.964464i	\(-0.585113\pi\)
							−0.158390	+	0.987377i	\(0.550630\pi\)
\(31\)	−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	−0.821377	−	0.570386i	\(-0.806794\pi\)
							0.614018	+	0.789292i	\(0.289552\pi\)
\(43\)	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	−0.326236	−	0.945288i	\(-0.605780\pi\)
							0.312350	+	0.949967i	\(0.398884\pi\)

(See \(a_n\) instead)

Twists

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

    

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