Embedded newform 177.3.g.a.10.3

• Label: 177.3.g.a.10.3
• 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			10.3
Character	\(\chi\)	\(=\)	177.10
Dual form			177.3.g.a.124.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.66127 - 1.06035i) q^{2} +(-1.72190 - 0.187268i) q^{3} +(3.05402 + 2.89292i) q^{4} +(-5.63025 + 6.62844i) q^{5} +(4.38386 + 2.32417i) q^{6} +(-9.66711 - 5.81651i) q^{7} +(-0.248591 - 0.537320i) q^{8} +(2.92986 + 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.66127	−	1.06035i	−1.33063	−	0.530173i	−0.406992	−	0.913432i	\(-0.633422\pi\)
							−0.923641	+	0.383259i	\(0.874802\pi\)
\(3\)	−1.72190	−	0.187268i	−0.573966	−	0.0624225i
							\(5\)	−5.63025	+	6.62844i	−1.12605	+	1.32569i	−0.186822	+	0.982394i	\(0.559819\pi\)
−0.939228	+	0.343295i	\(0.888457\pi\)
\(7\)	−9.66711	−	5.81651i	−1.38102	−	0.830930i	−0.385246	−	0.922814i	\(-0.625883\pi\)
							−0.995770	+	0.0918843i	\(0.970711\pi\)
\(11\)	3.71017	+	0.201160i	0.337288	+	0.0182872i	0.222005	−	0.975045i	\(-0.428740\pi\)
							0.115283	+	0.993333i	\(0.463223\pi\)
\(13\)	2.13917	+	9.71835i	0.164552	+	0.747565i	0.984536	+	0.175180i	\(0.0560508\pi\)
							−0.819985	+	0.572385i	\(0.806018\pi\)
\(17\)	−9.52590	+	5.73155i	−0.560347	+	0.337150i	−0.767388	−	0.641182i	\(-0.778444\pi\)
							0.207041	+	0.978332i	\(0.433617\pi\)
\(19\)	5.29744	−	19.0797i	0.278813	−	1.00419i	−0.682985	−	0.730432i	\(-0.739319\pi\)
							0.961798	−	0.273760i	\(-0.0882675\pi\)
\(23\)	14.2231	−	9.64348i	0.618394	−	0.419282i	−0.211362	−	0.977408i	\(-0.567790\pi\)
							0.829756	+	0.558126i	\(0.188479\pi\)
\(29\)	−6.46968	−	16.2377i	−0.223092	−	0.559920i	0.774242	−	0.632890i	\(-0.218131\pi\)
							−0.997334	+	0.0729703i	\(0.976752\pi\)
\(31\)	39.7409	−	11.0340i	1.28196	−	0.355935i	0.441234	−	0.897392i	\(-0.354541\pi\)
							0.840729	+	0.541457i	\(0.182127\pi\)
\(37\)	−13.7852	+	29.7962i	−0.372573	+	0.805303i	0.627080	+	0.778955i	\(0.284250\pi\)
							−0.999653	+	0.0263482i	\(0.991612\pi\)
\(41\)	−0.305876	+	0.451133i	−0.00746038	+	0.0110032i	−0.831401	−	0.555673i	\(-0.812461\pi\)
							0.823941	+	0.566676i	\(0.191771\pi\)
\(43\)	32.0863	−	1.73967i	0.746194	−	0.0404574i	0.322879	−	0.946440i	\(-0.395349\pi\)
							0.423315	+	0.905983i	\(0.360867\pi\)
\(47\)	−65.4692	+	55.6101i	−1.39296	+	1.18319i	−0.433569	+	0.901120i	\(0.642746\pi\)
							−0.959393	+	0.282072i	\(0.908978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.10.3	✓	560
59.6	odd	58	inner	177.3.g.a.124.3	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.3	✓	560	1.1	even	1	trivial
177.3.g.a.124.3	yes	560	59.6	odd	58	inner