Embedded newform 177.3.g.a.124.14

• Label: 177.3.g.a.124.14
• Level: $177$
• Weight: $3$
• Character: 177.124
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(20\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			124.14
Character	\(\chi\)	\(=\)	177.124
Dual form			177.3.g.a.10.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.61637 - 0.644021i) q^{2} +(-1.72190 + 0.187268i) q^{3} +(-0.706089 + 0.668843i) q^{4} +(-2.42613 - 2.85626i) q^{5} +(-2.66262 + 1.41163i) q^{6} +(-4.88728 + 2.94058i) q^{7} +(-3.63289 + 7.85237i) q^{8} +(2.92986 - 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.61637	−	0.644021i	0.808186	−	0.322011i	0.0708068	−	0.997490i	\(-0.477443\pi\)
							0.737379	+	0.675479i	\(0.236063\pi\)
\(3\)	−1.72190	+	0.187268i	−0.573966	+	0.0624225i
							\(5\)	−2.42613	−	2.85626i	−0.485227	−	0.571253i	0.464004	−	0.885833i	\(-0.346412\pi\)
−0.949231	+	0.314580i	\(0.898136\pi\)
\(7\)	−4.88728	+	2.94058i	−0.698183	+	0.420083i	−0.819919	−	0.572480i	\(-0.805981\pi\)
							0.121736	+	0.992563i	\(0.461154\pi\)
\(11\)	−12.2764	+	0.665607i	−1.11604	+	0.0605097i	−0.602914	−	0.797806i	\(-0.705994\pi\)
							−0.513122	+	0.858316i	\(0.671511\pi\)
\(13\)	−0.957795	+	4.35131i	−0.0736766	+	0.334716i	−0.999012	−	0.0444325i	\(-0.985852\pi\)
							0.925336	+	0.379149i	\(0.123783\pi\)
\(17\)	−16.4052	−	9.87071i	−0.965014	−	0.580630i	−0.0564638	−	0.998405i	\(-0.517983\pi\)
							−0.908551	+	0.417775i	\(0.862810\pi\)
\(19\)	4.85655	+	17.4917i	0.255608	+	0.920617i	0.974294	+	0.225282i	\(0.0723303\pi\)
							−0.718686	+	0.695335i	\(0.755256\pi\)
\(23\)	0.619562	+	0.420073i	0.0269375	+	0.0182641i	0.574581	−	0.818447i	\(-0.305165\pi\)
							−0.547644	+	0.836711i	\(0.684475\pi\)
\(29\)	6.74579	−	16.9307i	0.232614	−	0.583816i	−0.765616	−	0.643298i	\(-0.777566\pi\)
							0.998229	+	0.0594821i	\(0.0189449\pi\)
\(31\)	−2.24533	−	0.623412i	−0.0724299	−	0.0201101i	0.231124	−	0.972924i	\(-0.425760\pi\)
							−0.303554	+	0.952814i	\(0.598173\pi\)
\(37\)	4.64360	+	10.0370i	0.125503	+	0.271270i	0.960134	−	0.279539i	\(-0.0901816\pi\)
							−0.834632	+	0.550808i	\(0.814319\pi\)
\(41\)	21.5063	+	31.7195i	0.524545	+	0.773645i	0.994008	−	0.109306i	\(-0.0348630\pi\)
							−0.469463	+	0.882952i	\(0.655553\pi\)
\(43\)	−6.98439	−	0.378682i	−0.162428	−	0.00880657i	−0.0272542	−	0.999629i	\(-0.508676\pi\)
							−0.135173	+	0.990822i	\(0.543159\pi\)
\(47\)	30.1786	+	25.6339i	0.642098	+	0.545403i	0.908197	−	0.418543i	\(-0.137459\pi\)
							−0.266099	+	0.963946i	\(0.585735\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.124.14	yes	560
59.10	odd	58	inner	177.3.g.a.10.14	✓	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.14	✓	560	59.10	odd	58	inner
177.3.g.a.124.14	yes	560	1.1	even	1	trivial