Embedded newform 177.3.g.a.10.18

• Label: 177.3.g.a.10.18
• 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			10.18
Character	\(\chi\)	\(=\)	177.10
Dual form			177.3.g.a.124.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.16149 + 1.25965i) q^{2} +(-1.72190 - 0.187268i) q^{3} +(5.50432 + 5.21397i) q^{4} +(3.15169 - 3.71045i) q^{5} +(-5.20787 - 2.76104i) q^{6} +(6.22205 + 3.74368i) q^{7} +(5.11821 + 11.0628i) q^{8} +(2.92986 + 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.16149	+	1.25965i	1.58075	+	0.629827i	0.984877	−	0.173255i	\(-0.0554284\pi\)
							0.595869	+	0.803082i	\(0.296808\pi\)
\(3\)	−1.72190	−	0.187268i	−0.573966	−	0.0624225i
							\(5\)	3.15169	−	3.71045i	0.630337	−	0.742090i	−0.350576	−	0.936534i	\(-0.614014\pi\)
0.980914	+	0.194444i	\(0.0622902\pi\)
\(7\)	6.22205	+	3.74368i	0.888864	+	0.534812i	0.885215	−	0.465181i	\(-0.154011\pi\)
							0.00364895	+	0.999993i	\(0.498839\pi\)
\(11\)	−9.16309	−	0.496808i	−0.833008	−	0.0451644i	−0.367300	−	0.930103i	\(-0.619718\pi\)
							−0.465708	+	0.884938i	\(0.654200\pi\)
\(13\)	1.92085	+	8.72651i	0.147758	+	0.671270i	0.991014	+	0.133758i	\(0.0427046\pi\)
							−0.843256	+	0.537512i	\(0.819364\pi\)
\(17\)	−4.17022	+	2.50914i	−0.245307	+	0.147596i	−0.632898	−	0.774235i	\(-0.718135\pi\)
							0.387591	+	0.921831i	\(0.373307\pi\)
\(19\)	4.75485	−	17.1254i	0.250255	−	0.901337i	−0.726533	−	0.687131i	\(-0.758870\pi\)
							0.976788	−	0.214206i	\(-0.0687164\pi\)
\(23\)	−18.7318	+	12.7005i	−0.814425	+	0.552194i	−0.895783	−	0.444491i	\(-0.853385\pi\)
							0.0813580	+	0.996685i	\(0.474074\pi\)
\(29\)	−15.3293	−	38.4735i	−0.528595	−	1.32667i	−0.914668	−	0.404205i	\(-0.867548\pi\)
							0.386073	−	0.922468i	\(-0.373831\pi\)
\(31\)	−24.0224	+	6.66979i	−0.774916	+	0.215155i	−0.632404	−	0.774639i	\(-0.717932\pi\)
							−0.142512	+	0.989793i	\(0.545518\pi\)
\(37\)	17.2974	−	37.3877i	0.467497	−	1.01048i	−0.520193	−	0.854049i	\(-0.674140\pi\)
							0.987689	−	0.156429i	\(-0.0499981\pi\)
\(41\)	3.25961	−	4.80757i	0.0795028	−	0.117258i	−0.785859	−	0.618406i	\(-0.787779\pi\)
							0.865361	+	0.501149i	\(0.167089\pi\)
\(43\)	−66.1289	+	3.58540i	−1.53788	+	0.0833815i	−0.803479	−	0.595333i	\(-0.797020\pi\)
							−0.734402	+	0.678714i	\(0.762537\pi\)
\(47\)	20.3111	−	17.2524i	0.432152	−	0.367073i	−0.404749	−	0.914428i	\(-0.632641\pi\)
							0.836901	+	0.547355i	\(0.184365\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.10.18	✓	560
59.6	odd	58	inner	177.3.g.a.124.18	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.18	✓	560	1.1	even	1	trivial
177.3.g.a.124.18	yes	560	59.6	odd	58	inner