Embedded newform 177.4.f.a.101.31

• Label: 177.4.f.a.101.31
• 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			101.31
Character	\(\chi\)	\(=\)	177.101
Dual form			177.4.f.a.170.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.171155 - 0.252434i) q^{2} +(4.85521 + 1.85119i) q^{3} +(2.92668 + 7.34540i) q^{4} +(-2.63274 + 5.69057i) q^{5} +(1.29830 - 0.908782i) q^{6} +(4.19184 + 15.0976i) q^{7} +(4.73800 + 1.04291i) q^{8} +(20.1462 + 17.9759i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.171155	−	0.252434i	0.0605124	−	0.0892491i	−0.796229	−	0.604995i	\(-0.793175\pi\)
							0.856741	+	0.515746i	\(0.172485\pi\)
\(3\)	4.85521	+	1.85119i	0.934386	+	0.356262i
							\(5\)	−2.63274	+	5.69057i	−0.235479	+	0.508980i	−0.988904	−	0.148554i	\(-0.952538\pi\)
0.753425	+	0.657534i	\(0.228400\pi\)
\(7\)	4.19184	+	15.0976i	0.226338	+	0.815195i	0.986263	+	0.165184i	\(0.0528219\pi\)
							−0.759925	+	0.650011i	\(0.774764\pi\)
\(11\)	−43.2454	−	40.9642i	−1.18536	−	1.12283i	−0.989848	−	0.142132i	\(-0.954604\pi\)
							−0.195513	−	0.980701i	\(-0.562637\pi\)
\(13\)	−1.84753	+	16.9878i	−0.0394164	+	0.362428i	0.957774	+	0.287522i	\(0.0928313\pi\)
							−0.997191	+	0.0749063i	\(0.976134\pi\)
\(17\)	65.8419	+	18.2809i	0.939354	+	0.260810i	0.703270	−	0.710922i	\(-0.251722\pi\)
							0.236083	+	0.971733i	\(0.424136\pi\)
\(19\)	−80.5121	−	61.2037i	−0.972144	−	0.739005i	−0.00679791	−	0.999977i	\(-0.502164\pi\)
							−0.965346	+	0.260972i	\(0.915957\pi\)
\(23\)	−9.89376	−	18.6616i	−0.0896952	−	0.169183i	0.834606	−	0.550848i	\(-0.185696\pi\)
							−0.924301	+	0.381665i	\(0.875351\pi\)
\(29\)	234.221	−	158.806i	1.49978	−	1.01688i	0.512363	−	0.858769i	\(-0.328770\pi\)
							0.987422	−	0.158110i	\(-0.0505400\pi\)
\(31\)	−135.501	−	178.249i	−0.785056	−	1.03272i	−0.998444	−	0.0557559i	\(-0.982243\pi\)
							0.213389	−	0.976967i	\(-0.431550\pi\)
\(37\)	62.9632	+	286.045i	0.279759	+	1.27096i	0.881468	+	0.472244i	\(0.156556\pi\)
							−0.601709	+	0.798716i	\(0.705513\pi\)
\(41\)	383.370	+	203.250i	1.46030	+	0.774202i	0.993891	−	0.110367i	\(-0.0352025\pi\)
							0.466410	+	0.884569i	\(0.345547\pi\)
\(43\)	377.582	+	398.609i	1.33909	+	1.41366i	0.835262	+	0.549852i	\(0.185316\pi\)
							0.503825	+	0.863806i	\(0.331926\pi\)
\(47\)	−290.081	+	134.206i	−0.900269	+	0.416509i	−0.814740	−	0.579827i	\(-0.803120\pi\)
							−0.0855290	+	0.996336i	\(0.527258\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.101.31	yes	1624
3.2	odd	2	inner	177.4.f.a.101.28	✓	1624
59.52	odd	58	inner	177.4.f.a.170.28	yes	1624
177.170	even	58	inner	177.4.f.a.170.31	yes	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.101.28	✓	1624	3.2	odd	2	inner
177.4.f.a.101.31	yes	1624	1.1	even	1	trivial
177.4.f.a.170.28	yes	1624	59.52	odd	58	inner
177.4.f.a.170.31	yes	1624	177.170	even	58	inner