Embedded newform 177.2.f.a.11.4

• Label: 177.2.f.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78776 + 0.393517i) q^{2} +(-1.26030 - 1.18812i) q^{3} +(1.22610 - 0.567252i) q^{4} +(1.62977 + 0.452503i) q^{5} +(2.72067 + 1.62813i) q^{6} +(-1.02629 + 0.972157i) q^{7} +(0.945848 - 0.719015i) q^{8} +(0.176732 + 2.99479i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78776	+	0.393517i	−1.26414	+	0.278258i	−0.795946	−	0.605367i	\(-0.793026\pi\)
							−0.468194	+	0.883626i	\(0.655095\pi\)
\(3\)	−1.26030	−	1.18812i	−0.727637	−	0.685963i
							\(5\)	1.62977	+	0.452503i	0.728854	+	0.202365i	0.612083	−	0.790793i	\(-0.290332\pi\)
0.116771	+	0.993159i	\(0.462746\pi\)
\(7\)	−1.02629	+	0.972157i	−0.387902	+	0.367441i	−0.856656	−	0.515888i	\(-0.827462\pi\)
							0.468753	+	0.883329i	\(0.344703\pi\)
\(11\)	−3.21764	−	3.78810i	−0.970154	−	1.14215i	−0.989817	−	0.142347i	\(-0.954535\pi\)
							0.0196624	−	0.999807i	\(-0.493741\pi\)
\(13\)	−1.84920	−	5.48822i	−0.512875	−	1.52216i	−0.821066	−	0.570834i	\(-0.806620\pi\)
							0.308191	−	0.951325i	\(-0.400276\pi\)
\(17\)	5.14321	−	5.42962i	1.24741	−	1.31688i	0.317880	−	0.948131i	\(-0.397029\pi\)
							0.929531	−	0.368745i	\(-0.120212\pi\)
\(19\)	−0.166897	−	0.418880i	−0.0382888	−	0.0960976i	0.908589	−	0.417692i	\(-0.137161\pi\)
							−0.946878	+	0.321594i	\(0.895781\pi\)
\(23\)	−5.89022	+	0.640600i	−1.22820	+	0.133574i	−0.699123	−	0.715001i	\(-0.746426\pi\)
							−0.529074	+	0.848576i	\(0.677460\pi\)
\(29\)	0.131537	−	0.597577i	0.0244257	−	0.110967i	−0.962830	−	0.270107i	\(-0.912941\pi\)
							0.987256	+	0.159140i	\(0.0508720\pi\)
\(31\)	−1.33169	−	0.530594i	−0.239179	−	0.0952975i	0.247469	−	0.968896i	\(-0.420401\pi\)
							−0.486648	+	0.873598i	\(0.661780\pi\)
\(37\)	6.08052	−	7.99879i	0.999631	−	1.31499i	0.0509697	−	0.998700i	\(-0.483769\pi\)
							0.948662	−	0.316292i	\(-0.102438\pi\)
\(41\)	−0.316559	+	2.91071i	−0.0494382	+	0.454577i	0.943088	+	0.332544i	\(0.107907\pi\)
							−0.992526	+	0.122033i	\(0.961059\pi\)
\(43\)	2.57517	+	2.18737i	0.392709	+	0.333570i	0.821888	−	0.569649i	\(-0.192921\pi\)
							−0.429178	+	0.903220i	\(0.641197\pi\)
\(47\)	0.731593	+	2.63496i	0.106714	+	0.384348i	0.997548	−	0.0699793i	\(-0.0222933\pi\)
							−0.890835	+	0.454328i	\(0.849880\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.11.4	✓	504
3.2	odd	2	inner	177.2.f.a.11.15	yes	504
59.43	odd	58	inner	177.2.f.a.161.15	yes	504
177.161	even	58	inner	177.2.f.a.161.4	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.11.4	✓	504	1.1	even	1	trivial
177.2.f.a.11.15	yes	504	3.2	odd	2	inner
177.2.f.a.161.4	yes	504	177.161	even	58	inner
177.2.f.a.161.15	yes	504	59.43	odd	58	inner