Embedded newform 177.2.e.b.76.2

• Label: 177.2.e.b.76.2
• Level: $177$
• Weight: $2$
• Character: 177.76
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			76.2
Character	\(\chi\)	\(=\)	177.76
Dual form			177.2.e.b.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.492250 + 0.726014i) q^{2} +(-0.0541389 - 0.998533i) q^{3} +(0.455490 + 1.14319i) q^{4} +(-3.82749 - 1.77078i) q^{5} +(0.751599 + 0.452222i) q^{6} +(-1.35341 - 4.87453i) q^{7} +(-2.76749 - 0.609171i) q^{8} +(-0.994138 + 0.108119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{20}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.492250	+	0.726014i	−0.348073	+	0.513370i	−0.960918	−	0.276834i	\(-0.910715\pi\)
							0.612845	+	0.790203i	\(0.290025\pi\)
\(3\)	−0.0541389	−	0.998533i	−0.0312571	−	0.576504i
							\(5\)	−3.82749	−	1.77078i	−1.71170	−	0.791919i	−0.995577	−	0.0939499i	\(-0.970051\pi\)
−0.716128	−	0.697969i	\(-0.754087\pi\)
\(7\)	−1.35341	−	4.87453i	−0.511540	−	1.84240i	−0.538294	−	0.842757i	\(-0.680931\pi\)
							0.0267548	−	0.999642i	\(-0.491483\pi\)
\(11\)	1.54377	+	1.46233i	0.465463	+	0.440910i	0.884138	−	0.467226i	\(-0.154747\pi\)
							−0.418675	+	0.908136i	\(0.637505\pi\)
\(13\)	1.12276	+	0.122107i	0.311398	+	0.0338665i	0.262484	−	0.964936i	\(-0.415458\pi\)
							0.0489140	+	0.998803i	\(0.484424\pi\)
\(17\)	0.0667612	−	0.240452i	0.0161920	−	0.0583182i	−0.955013	−	0.296565i	\(-0.904159\pi\)
							0.971205	+	0.238247i	\(0.0765727\pi\)
\(19\)	−2.30089	−	1.74909i	−0.527860	−	0.401269i	0.307018	−	0.951704i	\(-0.400669\pi\)
							−0.834878	+	0.550435i	\(0.814462\pi\)
\(23\)	−2.78087	−	5.24528i	−0.579852	−	1.09372i	−0.983856	−	0.178960i	\(-0.942727\pi\)
							0.404004	−	0.914757i	\(-0.367618\pi\)
\(29\)	1.49567	+	2.20594i	0.277738	+	0.409633i	0.940816	−	0.338917i	\(-0.110061\pi\)
							−0.663078	+	0.748550i	\(0.730750\pi\)
\(31\)	1.38038	−	1.04934i	0.247923	−	0.188466i	−0.473831	−	0.880616i	\(-0.657129\pi\)
							0.721754	+	0.692150i	\(0.243336\pi\)
\(37\)	−7.66527	+	1.68725i	−1.26016	+	0.277383i	−0.794328	−	0.607489i	\(-0.792177\pi\)
							−0.465835	+	0.884872i	\(0.654246\pi\)
\(41\)	2.32120	−	4.37824i	0.362510	−	0.683767i	−0.633429	−	0.773801i	\(-0.718353\pi\)
							0.995939	+	0.0900346i	\(0.0286978\pi\)
\(43\)	1.98662	−	1.88183i	0.302957	−	0.286976i	−0.521085	−	0.853505i	\(-0.674473\pi\)
							0.824042	+	0.566529i	\(0.191714\pi\)
\(47\)	5.44805	−	2.52054i	0.794680	−	0.367658i	0.0198301	−	0.999803i	\(-0.493687\pi\)
							0.774850	+	0.632145i	\(0.217825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.76.2	yes	140
3.2	odd	2		531.2.i.b.253.4		140
59.7	even	29	inner	177.2.e.b.7.2	✓	140
177.125	odd	58		531.2.i.b.361.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.7.2	✓	140	59.7	even	29	inner
177.2.e.b.76.2	yes	140	1.1	even	1	trivial
531.2.i.b.253.4		140	3.2	odd	2
531.2.i.b.361.4		140	177.125	odd	58