Embedded newform 177.2.f.a.101.17

• Label: 177.2.f.a.101.17
• Level: $177$
• Weight: $2$
• Character: 177.101
• 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			101.17
Character	\(\chi\)	\(=\)	177.101
Dual form			177.2.f.a.170.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.27485 - 1.88026i) q^{2} +(0.789934 - 1.54143i) q^{3} +(-1.16986 - 2.93613i) q^{4} +(-0.346794 + 0.749584i) q^{5} +(-1.89124 - 3.45037i) q^{6} +(0.990843 + 3.56869i) q^{7} +(-2.57490 - 0.566779i) q^{8} +(-1.75201 - 2.43525i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.27485	−	1.88026i	0.901453	−	1.32954i	−0.0423738	−	0.999102i	\(-0.513492\pi\)
							0.943826	−	0.330441i	\(-0.107198\pi\)
\(3\)	0.789934	−	1.54143i	0.456069	−	0.889945i
							\(5\)	−0.346794	+	0.749584i	−0.155091	+	0.335224i	−0.969509	−	0.245054i	\(-0.921194\pi\)
0.814418	+	0.580278i	\(0.197056\pi\)
\(7\)	0.990843	+	3.56869i	0.374503	+	1.34884i	0.875428	+	0.483348i	\(0.160579\pi\)
							−0.500925	+	0.865491i	\(0.667007\pi\)
\(11\)	−1.13865	−	1.07858i	−0.343315	−	0.325205i	0.496542	−	0.868013i	\(-0.334603\pi\)
							−0.839857	+	0.542807i	\(0.817361\pi\)
\(13\)	−0.644952	+	5.93024i	−0.178878	+	1.64475i	0.469138	+	0.883125i	\(0.344565\pi\)
							−0.648015	+	0.761627i	\(0.724401\pi\)
\(17\)	−2.93746	−	0.815581i	−0.712438	−	0.197807i	−0.107644	−	0.994189i	\(-0.534331\pi\)
							−0.604794	+	0.796382i	\(0.706744\pi\)
\(19\)	−2.13518	−	1.62312i	−0.489845	−	0.372370i	0.330900	−	0.943666i	\(-0.392648\pi\)
							−0.820744	+	0.571296i	\(0.806441\pi\)
\(23\)	−0.526705	−	0.993471i	−0.109826	−	0.207153i	0.822507	−	0.568755i	\(-0.192575\pi\)
							−0.932333	+	0.361602i	\(0.882230\pi\)
\(29\)	5.88183	−	3.98798i	1.09223	−	0.740550i	0.124418	−	0.992230i	\(-0.460293\pi\)
							0.967810	+	0.251680i	\(0.0809832\pi\)
\(31\)	−2.13570	−	2.80946i	−0.383583	−	0.504595i	0.563265	−	0.826276i	\(-0.309545\pi\)
							−0.946848	+	0.321682i	\(0.895752\pi\)
\(37\)	−1.21090	−	5.50115i	−0.199070	−	0.904384i	−0.965162	−	0.261653i	\(-0.915732\pi\)
							0.766092	−	0.642731i	\(-0.222199\pi\)
\(41\)	−7.37426	−	3.90959i	−1.15167	−	0.610575i	−0.220578	−	0.975369i	\(-0.570794\pi\)
							−0.931088	+	0.364795i	\(0.881139\pi\)
\(43\)	2.77467	+	2.92919i	0.423133	+	0.446697i	0.902353	−	0.430999i	\(-0.141839\pi\)
							−0.479219	+	0.877695i	\(0.659080\pi\)
\(47\)	4.74976	−	2.19747i	0.692824	−	0.320534i	−0.0416878	−	0.999131i	\(-0.513273\pi\)
							0.734511	+	0.678596i	\(0.237411\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.101.17	yes	504
3.2	odd	2	inner	177.2.f.a.101.2	✓	504
59.52	odd	58	inner	177.2.f.a.170.2	yes	504
177.170	even	58	inner	177.2.f.a.170.17	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.101.2	✓	504	3.2	odd	2	inner
177.2.f.a.101.17	yes	504	1.1	even	1	trivial
177.2.f.a.170.2	yes	504	59.52	odd	58	inner
177.2.f.a.170.17	yes	504	177.170	even	58	inner