Embedded newform 177.2.e.a.16.1

• Label: 177.2.e.a.16.1
• Level: $177$
• Weight: $2$
• Character: 177.16
• 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			16.1
Character	\(\chi\)	\(=\)	177.16
Dual form			177.2.e.a.166.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19928 - 0.484098i) q^{2} +(-0.161782 - 0.986827i) q^{3} +(2.78734 + 1.28956i) q^{4} +(0.0503762 + 0.181439i) q^{5} +(-0.121917 + 2.24863i) q^{6} +(2.06658 + 1.95757i) q^{7} +(-1.92037 - 1.45983i) q^{8} +(-0.947653 + 0.319302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{2}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19928	−	0.484098i	−1.55513	−	0.342309i	−0.647493	−	0.762072i	\(-0.724182\pi\)
							−0.907634	+	0.419763i	\(0.862113\pi\)
\(3\)	−0.161782	−	0.986827i	−0.0934049	−	0.569745i
							\(5\)	0.0503762	+	0.181439i	0.0225289	+	0.0811418i	0.973960	−	0.226720i	\(-0.0728003\pi\)
−0.951431	+	0.307862i	\(0.900386\pi\)
\(7\)	2.06658	+	1.95757i	0.781094	+	0.739892i	0.970263	−	0.242053i	\(-0.0778207\pi\)
							−0.189169	+	0.981945i	\(0.560579\pi\)
\(11\)	0.782051	−	0.920701i	0.235797	−	0.277602i	−0.631518	−	0.775361i	\(-0.717568\pi\)
							0.867315	+	0.497759i	\(0.165844\pi\)
\(13\)	2.31365	+	0.779560i	0.641691	+	0.216211i	0.621290	−	0.783581i	\(-0.286609\pi\)
							0.0204015	+	0.999792i	\(0.493506\pi\)
\(17\)	2.03588	−	1.92849i	0.493774	−	0.467728i	−0.399826	−	0.916591i	\(-0.630930\pi\)
							0.893600	+	0.448863i	\(0.148171\pi\)
\(19\)	1.85300	−	4.65068i	0.425108	−	1.06694i	−0.547977	−	0.836493i	\(-0.684602\pi\)
							0.973085	−	0.230447i	\(-0.0740187\pi\)
\(23\)	4.29536	+	0.467148i	0.895644	+	0.0974072i	0.544350	−	0.838858i	\(-0.316776\pi\)
							0.351294	+	0.936265i	\(0.385742\pi\)
\(29\)	−3.96069	+	0.871814i	−0.735482	+	0.161892i	−0.566889	−	0.823794i	\(-0.691853\pi\)
							−0.168592	+	0.985686i	\(0.553922\pi\)
\(31\)	−0.590642	−	1.48240i	−0.106082	−	0.266247i	0.866348	−	0.499441i	\(-0.166461\pi\)
							−0.972430	+	0.233195i	\(0.925082\pi\)
\(37\)	−9.19319	+	6.98848i	−1.51135	+	1.14890i	−0.565299	+	0.824886i	\(0.691239\pi\)
							−0.946053	+	0.324013i	\(0.894968\pi\)
\(41\)	7.16053	−	0.778754i	1.11829	−	0.121621i	0.469746	−	0.882801i	\(-0.344345\pi\)
							0.648540	+	0.761180i	\(0.275380\pi\)
\(43\)	1.09890	+	1.29372i	0.167580	+	0.197291i	0.839541	−	0.543296i	\(-0.182824\pi\)
							−0.671961	+	0.740586i	\(0.734548\pi\)
\(47\)	−0.710095	+	2.55753i	−0.103578	+	0.373054i	−0.997112	−	0.0759407i	\(-0.975804\pi\)
							0.893534	+	0.448995i	\(0.148218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.16.1	✓	140
3.2	odd	2		531.2.i.c.370.5		140
59.48	even	29	inner	177.2.e.a.166.1	yes	140
177.107	odd	58		531.2.i.c.343.5		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.16.1	✓	140	1.1	even	1	trivial
177.2.e.a.166.1	yes	140	59.48	even	29	inner
531.2.i.c.343.5		140	177.107	odd	58
531.2.i.c.370.5		140	3.2	odd	2