Embedded newform 177.2.e.a.16.4

• Label: 177.2.e.a.16.4
• 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			16.4
Character	\(\chi\)	\(=\)	177.16
Dual form			177.2.e.a.166.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.94094 + 0.427233i) q^{2} +(-0.161782 - 0.986827i) q^{3} +(1.76956 + 0.818686i) q^{4} +(0.704369 + 2.53691i) q^{5} +(0.107596 - 1.98449i) q^{6} +(0.381927 + 0.361781i) q^{7} +(-0.0794753 - 0.0604155i) q^{8} +(-0.947653 + 0.319302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.94094	+	0.427233i	1.37245	+	0.302099i	0.839184	−	0.543848i	\(-0.183033\pi\)
							0.533267	+	0.845947i	\(0.320964\pi\)
\(3\)	−0.161782	−	0.986827i	−0.0934049	−	0.569745i
							\(5\)	0.704369	+	2.53691i	0.315003	+	1.13454i	0.936369	+	0.351018i	\(0.114164\pi\)
−0.621365	+	0.783521i	\(0.713422\pi\)
\(7\)	0.381927	+	0.361781i	0.144355	+	0.136740i	0.756470	−	0.654029i	\(-0.226922\pi\)
							−0.612115	+	0.790769i	\(0.709681\pi\)
\(11\)	3.19732	−	3.76418i	0.964029	−	1.13494i	−0.0267756	−	0.999641i	\(-0.508524\pi\)
							0.990805	−	0.135301i	\(-0.0432002\pi\)
\(13\)	−4.57048	−	1.53997i	−1.26762	−	0.427112i	−0.396435	−	0.918063i	\(-0.629753\pi\)
							−0.871187	+	0.490951i	\(0.836649\pi\)
\(17\)	−2.16927	+	2.05484i	−0.526126	+	0.498373i	−0.904028	−	0.427473i	\(-0.859404\pi\)
							0.377903	+	0.925845i	\(0.376645\pi\)
\(19\)	−1.05669	+	2.65210i	−0.242422	+	0.608433i	−0.998964	−	0.0455122i	\(-0.985508\pi\)
							0.756542	+	0.653945i	\(0.226887\pi\)
\(23\)	−2.64614	−	0.287785i	−0.551758	−	0.0600073i	−0.172007	−	0.985096i	\(-0.555025\pi\)
							−0.379752	+	0.925088i	\(0.623991\pi\)
\(29\)	0.405480	−	0.0892529i	0.0752958	−	0.0165738i	−0.177163	−	0.984182i	\(-0.556692\pi\)
							0.252459	+	0.967608i	\(0.418761\pi\)
\(31\)	1.91847	+	4.81499i	0.344567	+	0.864798i	0.994493	+	0.104801i	\(0.0334205\pi\)
							−0.649926	+	0.759997i	\(0.725200\pi\)
\(37\)	−7.39693	+	5.62300i	−1.21605	+	0.924416i	−0.998664	−	0.0516705i	\(-0.983545\pi\)
							−0.217384	+	0.976086i	\(0.569752\pi\)
\(41\)	9.60209	−	1.04429i	1.49960	−	0.163091i	0.678804	−	0.734319i	\(-0.262499\pi\)
							0.820791	+	0.571229i	\(0.193533\pi\)
\(43\)	5.32065	+	6.26395i	0.811392	+	0.955244i	0.999639	−	0.0268776i	\(-0.00855645\pi\)
							−0.188247	+	0.982122i	\(0.560281\pi\)
\(47\)	1.37567	−	4.95471i	0.200662	−	0.722718i	−0.792893	−	0.609361i	\(-0.791426\pi\)
							0.993554	−	0.113357i	\(-0.0361603\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.16.4	✓	140
3.2	odd	2		531.2.i.c.370.2		140
59.48	even	29	inner	177.2.e.a.166.4	yes	140
177.107	odd	58		531.2.i.c.343.2		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.16.4	✓	140	1.1	even	1	trivial
177.2.e.a.166.4	yes	140	59.48	even	29	inner
531.2.i.c.343.2		140	177.107	odd	58
531.2.i.c.370.2		140	3.2	odd	2