Embedded newform 177.2.f.a.11.2

• Label: 177.2.f.a.11.2
• Level: $177$
• Weight: $2$
• Character: 177.11
• 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			11.2
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.17818 + 0.479453i) q^{2} +(-1.34798 + 1.08764i) q^{3} +(2.69942 - 1.24889i) q^{4} +(2.10236 + 0.583716i) q^{5} +(2.41467 - 3.01535i) q^{6} +(0.124510 - 0.117942i) q^{7} +(-1.72995 + 1.31507i) q^{8} +(0.634098 - 2.93222i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.17818	+	0.479453i	−1.54020	+	0.339024i	−0.902305	−	0.431097i	\(-0.858127\pi\)
							−0.637897	+	0.770122i	\(0.720196\pi\)
\(3\)	−1.34798	+	1.08764i	−0.778256	+	0.627947i
							\(5\)	2.10236	+	0.583716i	0.940202	+	0.261046i	0.703629	−	0.710568i	\(-0.251562\pi\)
0.236573	+	0.971614i	\(0.423976\pi\)
\(7\)	0.124510	−	0.117942i	0.0470602	−	0.0445778i	−0.663808	−	0.747903i	\(-0.731061\pi\)
							0.710868	+	0.703325i	\(0.248302\pi\)
\(11\)	3.16160	+	3.72212i	0.953258	+	1.12226i	0.992403	+	0.123028i	\(0.0392606\pi\)
							−0.0391453	+	0.999234i	\(0.512464\pi\)
\(13\)	−0.472507	−	1.40235i	−0.131050	−	0.388942i	0.861960	−	0.506977i	\(-0.169237\pi\)
							−0.993009	+	0.118035i	\(0.962341\pi\)
\(17\)	−4.34724	+	4.58933i	−1.05436	+	1.11308i	−0.0609886	+	0.998138i	\(0.519425\pi\)
							−0.993373	+	0.114937i	\(0.963333\pi\)
\(19\)	2.44877	+	6.14594i	0.561786	+	1.40998i	0.885864	+	0.463946i	\(0.153567\pi\)
							−0.324078	+	0.946030i	\(0.605054\pi\)
\(23\)	−1.31230	+	0.142721i	−0.273634	+	0.0297595i	−0.243907	−	0.969799i	\(-0.578429\pi\)
							−0.0297267	+	0.999558i	\(0.509464\pi\)
\(29\)	1.17684	−	5.34644i	0.218534	−	0.992808i	−0.731786	−	0.681534i	\(-0.761313\pi\)
							0.950320	−	0.311274i	\(-0.100756\pi\)
\(31\)	4.65896	+	1.85630i	0.836774	+	0.333401i	0.748861	−	0.662727i	\(-0.230601\pi\)
							0.0879131	+	0.996128i	\(0.471980\pi\)
\(37\)	−0.190852	+	0.251061i	−0.0313758	+	0.0412742i	−0.811513	−	0.584334i	\(-0.801356\pi\)
							0.780138	+	0.625608i	\(0.215149\pi\)
\(41\)	−0.377726	+	3.47313i	−0.0589909	+	0.542412i	0.926934	+	0.375224i	\(0.122434\pi\)
							−0.985925	+	0.167188i	\(0.946531\pi\)
\(43\)	−0.826594	−	0.702115i	−0.126054	−	0.107072i	0.582142	−	0.813087i	\(-0.302215\pi\)
							−0.708196	+	0.706016i	\(0.750491\pi\)
\(47\)	1.77635	+	6.39785i	0.259108	+	0.933222i	0.972586	+	0.232544i	\(0.0747051\pi\)
							−0.713478	+	0.700678i	\(0.752881\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.11.2	✓	504
3.2	odd	2	inner	177.2.f.a.11.17	yes	504
59.43	odd	58	inner	177.2.f.a.161.17	yes	504
177.161	even	58	inner	177.2.f.a.161.2	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.11.2	✓	504	1.1	even	1	trivial
177.2.f.a.11.17	yes	504	3.2	odd	2	inner
177.2.f.a.161.2	yes	504	177.161	even	58	inner
177.2.f.a.161.17	yes	504	59.43	odd	58	inner