Embedded newform 177.2.f.a.2.1

• Label: 177.2.f.a.2.1
• Level: $177$
• Weight: $2$
• Character: 177.2
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.139199 + 2.56737i) q^{2} +(1.16564 + 1.28112i) q^{3} +(-4.58374 - 0.498512i) q^{4} +(-0.00781348 + 0.0231896i) q^{5} +(-3.45137 + 2.81431i) q^{6} +(0.224941 + 0.331763i) q^{7} +(1.08599 - 6.62424i) q^{8} +(-0.282552 + 2.98666i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.139199	+	2.56737i	−0.0984284	+	1.81541i	0.366892	+	0.930263i	\(0.380422\pi\)
							−0.465321	+	0.885142i	\(0.654061\pi\)
\(3\)	1.16564	+	1.28112i	0.672984	+	0.739657i
							\(5\)	−0.00781348	+	0.0231896i	−0.00349430	+	0.0103707i	−0.949386	−	0.314111i	\(-0.898293\pi\)
0.945892	+	0.324482i	\(0.105190\pi\)
\(7\)	0.224941	+	0.331763i	0.0850196	+	0.125395i	0.867821	−	0.496877i	\(-0.165520\pi\)
							−0.782801	+	0.622272i	\(0.786210\pi\)
\(11\)	−3.87063	+	0.851990i	−1.16704	+	0.256885i	−0.755899	−	0.654688i	\(-0.772800\pi\)
							−0.411139	+	0.911573i	\(0.634869\pi\)
\(13\)	3.03016	−	2.57384i	0.840415	−	0.713855i	−0.119896	−	0.992786i	\(-0.538256\pi\)
							0.960311	+	0.278931i	\(0.0899802\pi\)
\(17\)	5.70560	+	3.86849i	1.38381	+	0.938247i	0.999889	+	0.0148819i	\(0.00473723\pi\)
							0.383922	+	0.923365i	\(0.374573\pi\)
\(19\)	0.706627	−	1.33284i	0.162111	−	0.305774i	−0.789128	−	0.614229i	\(-0.789467\pi\)
							0.951239	+	0.308455i	\(0.0998119\pi\)
\(23\)	4.64450	−	4.39951i	0.968446	−	0.917361i	−0.0281665	−	0.999603i	\(-0.508967\pi\)
							0.996612	+	0.0822427i	\(0.0262083\pi\)
\(29\)	−5.95527	+	0.322885i	−1.10587	+	0.0599583i	−0.598014	−	0.801486i	\(-0.704043\pi\)
							−0.507852	+	0.861444i	\(0.669560\pi\)
\(31\)	2.12454	−	1.12636i	0.381578	−	0.202300i	−0.266568	−	0.963816i	\(-0.585890\pi\)
							0.648146	+	0.761516i	\(0.275545\pi\)
\(37\)	−5.89562	+	0.966538i	−0.969234	+	0.158898i	−0.625547	−	0.780187i	\(-0.715124\pi\)
							−0.343687	+	0.939084i	\(0.611676\pi\)
\(41\)	−6.13713	+	6.47889i	−0.958459	+	1.01183i	0.0414603	+	0.999140i	\(0.486799\pi\)
							−0.999919	+	0.0126928i	\(0.995960\pi\)
\(43\)	0.943671	−	4.28714i	0.143908	−	0.653783i	−0.848334	−	0.529462i	\(-0.822394\pi\)
							0.992242	−	0.124321i	\(-0.0396751\pi\)
\(47\)	3.42447	−	1.15384i	0.499510	−	0.168305i	−0.0582576	−	0.998302i	\(-0.518554\pi\)
							0.557768	+	0.829997i	\(0.311658\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.2.1	✓	504
3.2	odd	2	inner	177.2.f.a.2.18	yes	504
59.30	odd	58	inner	177.2.f.a.89.18	yes	504
177.89	even	58	inner	177.2.f.a.89.1	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.2.1	✓	504	1.1	even	1	trivial
177.2.f.a.2.18	yes	504	3.2	odd	2	inner
177.2.f.a.89.1	yes	504	177.89	even	58	inner
177.2.f.a.89.18	yes	504	59.30	odd	58	inner