Embedded newform 177.2.f.a.11.15

• Label: 177.2.f.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.78776 - 0.393517i) q^{2} +(1.57370 - 0.723511i) q^{3} +(1.22610 - 0.567252i) q^{4} +(-1.62977 - 0.452503i) q^{5} +(2.52869 - 1.91274i) q^{6} +(-1.02629 + 0.972157i) q^{7} +(-0.945848 + 0.719015i) q^{8} +(1.95306 - 2.27718i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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	0.468194	−	0.883626i	\(-0.344905\pi\)
							0.795946	+	0.605367i	\(0.206974\pi\)
\(3\)	1.57370	−	0.723511i	0.908576	−	0.417719i
							\(5\)	−1.62977	−	0.452503i	−0.728854	−	0.202365i	−0.116771	−	0.993159i	\(-0.537254\pi\)
−0.612083	+	0.790793i	\(0.709668\pi\)
\(7\)	−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	\(0.0454650\pi\)
							−0.0196624	+	0.999807i	\(0.506259\pi\)
\(13\)	−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	\(0.602971\pi\)
							−0.929531	+	0.368745i	\(0.879788\pi\)
\(19\)	−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	0.529074	−	0.848576i	\(-0.322540\pi\)
							0.699123	+	0.715001i	\(0.253574\pi\)
\(29\)	−0.131537	+	0.597577i	−0.0244257	+	0.110967i	−0.987256	−	0.159140i	\(-0.949128\pi\)
							0.962830	+	0.270107i	\(0.0870590\pi\)
\(31\)	−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	\(-0.892093\pi\)
							0.992526	−	0.122033i	\(-0.0389414\pi\)
\(43\)	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	0.890835	−	0.454328i	\(-0.150120\pi\)
							−0.997548	+	0.0699793i	\(0.977707\pi\)

(See \(a_n\) instead)

Twists

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

    

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