Embedded newform 177.3.g.a.10.12

• Label: 177.3.g.a.10.12
• Level: $177$
• Weight: $3$
• Character: 177.10
• 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			10.12
Character	\(\chi\)	\(=\)	177.10
Dual form			177.3.g.a.124.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.782131 + 0.311630i) q^{2} +(-1.72190 - 0.187268i) q^{3} +(-2.38937 - 2.26333i) q^{4} +(-2.21295 + 2.60528i) q^{5} +(-1.28839 - 0.683062i) q^{6} +(9.58665 + 5.76810i) q^{7} +(-2.57754 - 5.57127i) q^{8} +(2.92986 + 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{7}{58}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.782131	+	0.311630i	0.391066	+	0.155815i	0.557379	−	0.830258i	\(-0.311807\pi\)
							−0.166313	+	0.986073i	\(0.553186\pi\)
\(3\)	−1.72190	−	0.187268i	−0.573966	−	0.0624225i
							\(5\)	−2.21295	+	2.60528i	−0.442590	+	0.521057i	−0.937546	−	0.347861i	\(-0.886908\pi\)
0.494956	+	0.868918i	\(0.335184\pi\)
\(7\)	9.58665	+	5.76810i	1.36952	+	0.824014i	0.994674	−	0.103072i	\(-0.0328673\pi\)
							0.374848	+	0.927086i	\(0.377695\pi\)
\(11\)	16.3553	+	0.886760i	1.48685	+	0.0806145i	0.779567	−	0.626319i	\(-0.215439\pi\)
							0.707280	+	0.706933i	\(0.249922\pi\)
\(13\)	3.90180	+	17.7261i	0.300139	+	1.36354i	0.849337	+	0.527851i	\(0.177002\pi\)
							−0.549198	+	0.835692i	\(0.685067\pi\)
\(17\)	−3.56820	+	2.14691i	−0.209894	+	0.126289i	−0.616619	−	0.787261i	\(-0.711498\pi\)
							0.406725	+	0.913551i	\(0.366671\pi\)
\(19\)	1.67569	−	6.03528i	0.0881941	−	0.317646i	−0.906273	−	0.422692i	\(-0.861085\pi\)
							0.994467	+	0.105046i	\(0.0334989\pi\)
\(23\)	19.0313	−	12.9035i	0.827447	−	0.561023i	−0.0723259	−	0.997381i	\(-0.523042\pi\)
							0.899773	+	0.436358i	\(0.143732\pi\)
\(29\)	−6.81640	−	17.1079i	−0.235048	−	0.589927i	0.763381	−	0.645948i	\(-0.223538\pi\)
							−0.998430	+	0.0560213i	\(0.982159\pi\)
\(31\)	−24.4744	+	6.79529i	−0.789498	+	0.219203i	−0.638796	−	0.769376i	\(-0.720567\pi\)
							−0.150702	+	0.988579i	\(0.548153\pi\)
\(37\)	−8.36654	+	18.0840i	−0.226123	+	0.488756i	−0.987117	−	0.160003i	\(-0.948850\pi\)
							0.760994	+	0.648759i	\(0.224712\pi\)
\(41\)	10.4448	−	15.4049i	0.254750	−	0.375728i	−0.678712	−	0.734405i	\(-0.737461\pi\)
							0.933462	+	0.358676i	\(0.116772\pi\)
\(43\)	64.8975	−	3.51864i	1.50924	−	0.0818288i	0.719167	−	0.694837i	\(-0.244524\pi\)
							0.790076	+	0.613008i	\(0.210041\pi\)
\(47\)	−52.4796	+	44.5766i	−1.11659	+	0.948438i	−0.999004	−	0.0446245i	\(-0.985791\pi\)
							−0.117584	+	0.993063i	\(0.537515\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.10.12	✓	560
59.6	odd	58	inner	177.3.g.a.124.12	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.12	✓	560	1.1	even	1	trivial
177.3.g.a.124.12	yes	560	59.6	odd	58	inner