Embedded newform 177.3.g.a.124.18

• Label: 177.3.g.a.124.18
• 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			124.18
Character	\(\chi\)	\(=\)	177.124
Dual form			177.3.g.a.10.18

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{51}{58}\right)\)	\(1\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.18	✓	560	59.10	odd	58	inner
177.3.g.a.124.18	yes	560	1.1	even	1	trivial