Embedded newform 177.2.f.a.2.15

• Label: 177.2.f.a.2.15
• 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			2.15
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.106315 - 1.96087i) q^{2} +(0.106118 + 1.72880i) q^{3} +(-1.84543 - 0.200702i) q^{4} +(-1.09139 + 3.23912i) q^{5} +(3.40123 - 0.0242854i) q^{6} +(2.29285 + 3.38170i) q^{7} +(0.0456507 - 0.278457i) q^{8} +(-2.97748 + 0.366912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.106315	−	1.96087i	0.0751762	−	1.38654i	−0.681052	−	0.732235i	\(-0.738477\pi\)
							0.756228	−	0.654308i	\(-0.227040\pi\)
\(3\)	0.106118	+	1.72880i	0.0612671	+	0.998121i
							\(5\)	−1.09139	+	3.23912i	−0.488083	+	1.44858i	0.367064	+	0.930196i	\(0.380363\pi\)
−0.855148	+	0.518385i	\(0.826534\pi\)
\(7\)	2.29285	+	3.38170i	0.866616	+	1.27816i	0.959278	+	0.282465i	\(0.0911521\pi\)
							−0.0926619	+	0.995698i	\(0.529538\pi\)
\(11\)	−2.51296	+	0.553145i	−0.757686	+	0.166779i	−0.576981	−	0.816758i	\(-0.695769\pi\)
							−0.180706	+	0.983537i	\(0.557838\pi\)
\(13\)	4.63008	−	3.93283i	1.28415	−	1.09077i	0.293274	−	0.956029i	\(-0.405255\pi\)
							0.990881	−	0.134742i	\(-0.0430205\pi\)
\(17\)	−0.197309	−	0.133779i	−0.0478545	−	0.0324462i	0.537025	−	0.843566i	\(-0.319548\pi\)
							−0.584880	+	0.811120i	\(0.698858\pi\)
\(19\)	0.250238	−	0.471999i	0.0574085	−	0.108284i	−0.853147	−	0.521670i	\(-0.825309\pi\)
							0.910556	+	0.413386i	\(0.135654\pi\)
\(23\)	2.54412	−	2.40991i	0.530485	−	0.502502i	−0.374917	−	0.927059i	\(-0.622329\pi\)
							0.905401	+	0.424557i	\(0.139570\pi\)
\(29\)	7.46082	−	0.404514i	1.38544	−	0.0751164i	0.653728	−	0.756730i	\(-0.273204\pi\)
							0.731713	+	0.681613i	\(0.238721\pi\)
\(31\)	1.17735	−	0.624190i	0.211458	−	0.112108i	−0.359313	−	0.933217i	\(-0.616989\pi\)
							0.570771	+	0.821109i	\(0.306644\pi\)
\(37\)	0.814370	−	0.133509i	0.133882	−	0.0219488i	−0.0944685	−	0.995528i	\(-0.530115\pi\)
							0.228350	+	0.973579i	\(0.426667\pi\)
\(41\)	2.71678	−	2.86807i	0.424290	−	0.447918i	−0.478443	−	0.878119i	\(-0.658799\pi\)
							0.902733	+	0.430201i	\(0.141557\pi\)
\(43\)	−1.21573	+	5.52310i	−0.185397	+	0.842265i	0.788457	+	0.615089i	\(0.210880\pi\)
							−0.973854	+	0.227175i	\(0.927051\pi\)
\(47\)	2.62013	−	0.882823i	0.382185	−	0.128773i	−0.121650	−	0.992573i	\(-0.538819\pi\)
							0.503835	+	0.863800i	\(0.331922\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.2.15	yes	504
3.2	odd	2	inner	177.2.f.a.2.4	✓	504
59.30	odd	58	inner	177.2.f.a.89.4	yes	504
177.89	even	58	inner	177.2.f.a.89.15	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.2.4	✓	504	3.2	odd	2	inner
177.2.f.a.2.15	yes	504	1.1	even	1	trivial
177.2.f.a.89.4	yes	504	59.30	odd	58	inner
177.2.f.a.89.15	yes	504	177.89	even	58	inner