Embedded newform 177.4.f.a.101.15

• Label: 177.4.f.a.101.15
• Level: $177$
• Weight: $4$
• Character: 177.101
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			101.15
Character	\(\chi\)	\(=\)	177.101
Dual form			177.4.f.a.170.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65422 + 2.43980i) q^{2} +(-2.20594 - 4.70466i) q^{3} +(-0.255051 - 0.640130i) q^{4} +(-0.218507 + 0.472294i) q^{5} +(15.1275 + 2.40051i) q^{6} +(-2.88732 - 10.3992i) q^{7} +(-21.0468 - 4.63274i) q^{8} +(-17.2676 + 20.7564i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65422	+	2.43980i	−0.584856	+	0.862599i	−0.998863	−	0.0476752i	\(-0.984819\pi\)
							0.414006	+	0.910274i	\(0.364129\pi\)
\(3\)	−2.20594	−	4.70466i	−0.424534	−	0.905412i
							\(5\)	−0.218507	+	0.472294i	−0.0195438	+	0.0422433i	−0.917102	−	0.398654i	\(-0.869478\pi\)
0.897558	+	0.440897i	\(0.145340\pi\)
\(7\)	−2.88732	−	10.3992i	−0.155901	−	0.561504i	−0.999692	−	0.0248316i	\(-0.992095\pi\)
							0.843791	−	0.536672i	\(-0.180319\pi\)
\(11\)	28.5343	+	27.0292i	0.782129	+	0.740872i	0.970468	−	0.241232i	\(-0.0775514\pi\)
							−0.188338	+	0.982104i	\(0.560310\pi\)
\(13\)	4.12006	−	37.8834i	0.0879000	−	0.808227i	−0.864346	−	0.502897i	\(-0.832268\pi\)
							0.952246	−	0.305331i	\(-0.0987670\pi\)
\(17\)	35.0448	+	9.73014i	0.499977	+	0.138818i	0.508356	−	0.861147i	\(-0.330253\pi\)
							−0.00837903	+	0.999965i	\(0.502667\pi\)
\(19\)	30.3519	+	23.0729i	0.366484	+	0.278594i	0.772167	−	0.635420i	\(-0.219173\pi\)
							−0.405683	+	0.914014i	\(0.632966\pi\)
\(23\)	40.1012	+	75.6390i	0.363552	+	0.685731i	0.996046	−	0.0888334i	\(-0.0283139\pi\)
							−0.632495	+	0.774565i	\(0.717969\pi\)
\(29\)	−117.146	+	79.4273i	−0.750123	+	0.508596i	−0.875325	−	0.483535i	\(-0.839352\pi\)
							0.125202	+	0.992131i	\(0.460042\pi\)
\(31\)	141.113	+	185.631i	0.817571	+	1.07550i	0.995804	+	0.0915076i	\(0.0291686\pi\)
							−0.178234	+	0.983988i	\(0.557038\pi\)
\(37\)	46.6429	+	211.901i	0.207245	+	0.941521i	0.959293	+	0.282413i	\(0.0911350\pi\)
							−0.752048	+	0.659108i	\(0.770934\pi\)
\(41\)	107.403	+	56.9415i	0.409111	+	0.216897i	0.660237	−	0.751058i	\(-0.270456\pi\)
							−0.251126	+	0.967954i	\(0.580801\pi\)
\(43\)	−6.28132	−	6.63111i	−0.0222766	−	0.0235171i	0.714762	−	0.699368i	\(-0.246535\pi\)
							−0.737039	+	0.675851i	\(0.763776\pi\)
\(47\)	−154.320	+	71.3962i	−0.478935	+	0.221579i	−0.644477	−	0.764624i	\(-0.722925\pi\)
							0.165542	+	0.986203i	\(0.447063\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.101.15	✓	1624
3.2	odd	2	inner	177.4.f.a.101.44	yes	1624
59.52	odd	58	inner	177.4.f.a.170.44	yes	1624
177.170	even	58	inner	177.4.f.a.170.15	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.101.15	✓	1624	1.1	even	1	trivial
177.4.f.a.101.44	yes	1624	3.2	odd	2	inner
177.4.f.a.170.15	yes	1624	177.170	even	58	inner
177.4.f.a.170.44	yes	1624	59.52	odd	58	inner