Embedded newform 177.3.g.a.124.17

• Label: 177.3.g.a.124.17
• 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			124.17
Character	\(\chi\)	\(=\)	177.124
Dual form			177.3.g.a.10.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.78888 - 1.11119i) q^{2} +(1.72190 - 0.187268i) q^{3} +(3.63913 - 3.44717i) q^{4} +(-3.64648 - 4.29297i) q^{5} +(4.59408 - 2.43563i) q^{6} +(3.88004 - 2.33454i) q^{7} +(1.27644 - 2.75897i) q^{8} +(2.92986 - 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.78888	−	1.11119i	1.39444	−	0.555596i	0.452617	−	0.891705i	\(-0.350490\pi\)
							0.941823	+	0.336109i	\(0.109111\pi\)
\(3\)	1.72190	−	0.187268i	0.573966	−	0.0624225i
							\(5\)	−3.64648	−	4.29297i	−0.729297	−	0.858594i	0.265110	−	0.964218i	\(-0.414592\pi\)
−0.994406	+	0.105624i	\(0.966316\pi\)
\(7\)	3.88004	−	2.33454i	0.554291	−	0.333506i	−0.210696	−	0.977552i	\(-0.567573\pi\)
							0.764987	+	0.644046i	\(0.222745\pi\)
\(11\)	6.59997	−	0.357840i	0.599998	−	0.0325309i	0.248361	−	0.968668i	\(-0.420108\pi\)
							0.351637	+	0.936137i	\(0.385625\pi\)
\(13\)	−1.02564	+	4.65955i	−0.0788958	+	0.358427i	−0.999484	−	0.0321065i	\(-0.989778\pi\)
							0.920589	+	0.390534i	\(0.127709\pi\)
\(17\)	−12.2257	−	7.35597i	−0.719160	−	0.432704i	0.108381	−	0.994109i	\(-0.465433\pi\)
							−0.827541	+	0.561405i	\(0.810261\pi\)
\(19\)	5.79495	+	20.8715i	0.304997	+	1.09850i	0.944176	+	0.329443i	\(0.106861\pi\)
							−0.639178	+	0.769059i	\(0.720725\pi\)
\(23\)	7.28314	+	4.93809i	0.316658	+	0.214700i	0.709153	−	0.705055i	\(-0.249078\pi\)
							−0.392495	+	0.919754i	\(0.628388\pi\)
\(29\)	−19.2637	+	48.3483i	−0.664267	+	1.66718i	0.0778934	+	0.996962i	\(0.475181\pi\)
							−0.742160	+	0.670222i	\(0.766199\pi\)
\(31\)	6.84490	+	1.90048i	0.220803	+	0.0613057i	0.376168	−	0.926551i	\(-0.377242\pi\)
							−0.155365	+	0.987857i	\(0.549655\pi\)
\(37\)	−1.82822	−	3.95163i	−0.0494114	−	0.106801i	0.881297	−	0.472562i	\(-0.156671\pi\)
							−0.930709	+	0.365761i	\(0.880809\pi\)
\(41\)	−44.2995	−	65.3369i	−1.08048	−	1.59358i	−0.764274	−	0.644892i	\(-0.776902\pi\)
							−0.316203	−	0.948692i	\(-0.602408\pi\)
\(43\)	23.0978	+	1.25233i	0.537159	+	0.0291239i	0.320723	−	0.947173i	\(-0.396074\pi\)
							0.216436	+	0.976297i	\(0.430557\pi\)
\(47\)	−17.6521	−	14.9938i	−0.375577	−	0.319018i	0.439640	−	0.898174i	\(-0.355106\pi\)
							−0.815216	+	0.579156i	\(0.803382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.124.17	yes	560
59.10	odd	58	inner	177.3.g.a.10.17	✓	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.17	✓	560	59.10	odd	58	inner
177.3.g.a.124.17	yes	560	1.1	even	1	trivial