Embedded newform 177.3.h.a.107.38

• Label: 177.3.h.a.107.38
• Level: $177$
• Weight: $3$
• Character: 177.107
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			107.38
Character	\(\chi\)	\(=\)	177.107
Dual form			177.3.h.a.134.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.852292 + 3.87200i) q^{2} +(1.07260 + 2.80170i) q^{3} +(-10.6357 + 4.92060i) q^{4} +(7.10645 + 1.97310i) q^{5} +(-9.93402 + 6.54098i) q^{6} +(2.41863 - 2.29105i) q^{7} +(-18.5200 - 24.3626i) q^{8} +(-6.69905 + 6.01022i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{27}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.852292	+	3.87200i	0.426146	+	1.93600i	0.342541	+	0.939503i	\(0.388712\pi\)
							0.0836051	+	0.996499i	\(0.473357\pi\)
\(3\)	1.07260	+	2.80170i	0.357534	+	0.933900i
							\(5\)	7.10645	+	1.97310i	1.42129	+	0.394619i	0.891311	−	0.453392i	\(-0.149786\pi\)
0.529979	+	0.848011i	\(0.322200\pi\)
\(7\)	2.41863	−	2.29105i	0.345519	−	0.327293i	−0.495185	−	0.868787i	\(-0.664900\pi\)
							0.840704	+	0.541494i	\(0.182141\pi\)
\(11\)	−0.555239	+	0.471624i	−0.0504763	+	0.0428750i	−0.672269	−	0.740307i	\(-0.734680\pi\)
							0.621793	+	0.783182i	\(0.286404\pi\)
\(13\)	11.8234	−	3.98378i	0.909495	−	0.306445i	0.174611	−	0.984637i	\(-0.444133\pi\)
							0.734884	+	0.678193i	\(0.237237\pi\)
\(17\)	7.69490	−	8.12341i	0.452641	−	0.477847i	−0.459234	−	0.888315i	\(-0.651876\pi\)
							0.911875	+	0.410468i	\(0.134635\pi\)
\(19\)	5.68316	+	14.2637i	0.299114	+	0.750719i	0.999323	+	0.0367814i	\(0.0117105\pi\)
							−0.700210	+	0.713937i	\(0.746910\pi\)
\(23\)	−4.28228	−	39.3749i	−0.186186	−	1.71195i	−0.598676	−	0.800991i	\(-0.704306\pi\)
							0.412490	−	0.910962i	\(-0.364659\pi\)
\(29\)	−3.36437	+	15.2845i	−0.116013	+	0.527051i	0.882388	+	0.470522i	\(0.155934\pi\)
							−0.998401	+	0.0565292i	\(0.981997\pi\)
\(31\)	−4.78852	+	12.0183i	−0.154468	+	0.387686i	−0.985762	−	0.168147i	\(-0.946222\pi\)
							0.831294	+	0.555834i	\(0.187601\pi\)
\(37\)	−19.0748	−	14.5003i	−0.515534	−	0.391899i	0.314806	−	0.949156i	\(-0.398061\pi\)
							−0.830340	+	0.557257i	\(0.811854\pi\)
\(41\)	2.59235	−	23.8362i	0.0632280	−	0.581372i	−0.919076	−	0.394080i	\(-0.871063\pi\)
							0.982304	−	0.187292i	\(-0.0599711\pi\)
\(43\)	−17.9571	+	21.1408i	−0.417608	+	0.491646i	−0.930234	−	0.366966i	\(-0.880397\pi\)
							0.512627	+	0.858612i	\(0.328672\pi\)
\(47\)	−44.8106	+	12.4416i	−0.953416	+	0.264715i	−0.709201	−	0.705006i	\(-0.750944\pi\)
							−0.244215	+	0.969721i	\(0.578530\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.107.38	yes	1064
3.2	odd	2	inner	177.3.h.a.107.1	✓	1064
59.16	even	29	inner	177.3.h.a.134.1	yes	1064
177.134	odd	58	inner	177.3.h.a.134.38	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.107.1	✓	1064	3.2	odd	2	inner
177.3.h.a.107.38	yes	1064	1.1	even	1	trivial
177.3.h.a.134.1	yes	1064	59.16	even	29	inner
177.3.h.a.134.38	yes	1064	177.134	odd	58	inner