Embedded newform 177.4.f.a.2.5

• Label: 177.4.f.a.2.5
• Level: $177$
• Weight: $4$
• Character: 177.2
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			2.5
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.273845 + 5.05077i) q^{2} +(-1.98491 - 4.80210i) q^{3} +(-17.4822 - 1.90130i) q^{4} +(-0.350795 + 1.04112i) q^{5} +(24.7978 - 8.71029i) q^{6} +(-5.00566 - 7.38279i) q^{7} +(7.84382 - 47.8452i) q^{8} +(-19.1203 + 19.0635i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.273845	+	5.05077i	−0.0968187	+	1.78572i	0.398663	+	0.917097i	\(0.369474\pi\)
							−0.495482	+	0.868618i	\(0.665008\pi\)
\(3\)	−1.98491	−	4.80210i	−0.381996	−	0.924164i
							\(5\)	−0.350795	+	1.04112i	−0.0313761	+	0.0931208i	−0.962197	−	0.272355i	\(-0.912197\pi\)
0.930821	+	0.365476i	\(0.119094\pi\)
\(7\)	−5.00566	−	7.38279i	−0.270280	−	0.398633i	0.668192	−	0.743989i	\(-0.267069\pi\)
							−0.938472	+	0.345356i	\(0.887758\pi\)
\(11\)	63.7073	−	14.0230i	1.74622	−	0.384373i	0.776796	−	0.629752i	\(-0.216843\pi\)
							0.969427	+	0.245379i	\(0.0789123\pi\)
\(13\)	8.42784	−	7.15867i	0.179805	−	0.152728i	−0.552964	−	0.833205i	\(-0.686503\pi\)
							0.732769	+	0.680477i	\(0.238228\pi\)
\(17\)	43.1724	+	29.2716i	0.615932	+	0.417613i	0.828860	−	0.559455i	\(-0.188990\pi\)
							−0.212928	+	0.977068i	\(0.568300\pi\)
\(19\)	−43.8766	+	82.7600i	−0.529788	+	0.999286i	0.463769	+	0.885956i	\(0.346497\pi\)
							−0.993557	+	0.113330i	\(0.963848\pi\)
\(23\)	25.4490	−	24.1065i	0.230716	−	0.218546i	−0.563604	−	0.826045i	\(-0.690586\pi\)
							0.794321	+	0.607499i	\(0.207827\pi\)
\(29\)	124.451	−	6.74755i	0.796897	−	0.0432065i	0.348806	−	0.937195i	\(-0.386587\pi\)
							0.448090	+	0.893988i	\(0.352104\pi\)
\(31\)	115.384	−	61.1725i	0.668500	−	0.354416i	−0.0993442	−	0.995053i	\(-0.531674\pi\)
							0.767844	+	0.640637i	\(0.221330\pi\)
\(37\)	−18.5035	+	3.03349i	−0.0822149	+	0.0134784i	−0.202749	−	0.979231i	\(-0.564988\pi\)
							0.120534	+	0.992709i	\(0.461539\pi\)
\(41\)	−246.894	+	260.642i	−0.940446	+	0.992817i	−0.999985	−	0.00540916i	\(-0.998278\pi\)
							0.0595393	+	0.998226i	\(0.481037\pi\)
\(43\)	69.8877	−	317.503i	0.247855	−	1.12602i	−0.674076	−	0.738662i	\(-0.735458\pi\)
							0.921931	−	0.387355i	\(-0.126611\pi\)
\(47\)	438.052	−	147.597i	1.35950	−	0.458069i	0.457272	−	0.889327i	\(-0.348827\pi\)
							0.902229	+	0.431258i	\(0.141930\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.2.5	✓	1624
3.2	odd	2	inner	177.4.f.a.2.54	yes	1624
59.30	odd	58	inner	177.4.f.a.89.54	yes	1624
177.89	even	58	inner	177.4.f.a.89.5	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.5	✓	1624	1.1	even	1	trivial
177.4.f.a.2.54	yes	1624	3.2	odd	2	inner
177.4.f.a.89.5	yes	1624	177.89	even	58	inner
177.4.f.a.89.54	yes	1624	59.30	odd	58	inner