Embedded newform 177.2.e.b.4.3

• Label: 177.2.e.b.4.3
• Level: $177$
• Weight: $2$
• Character: 177.4
• 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			4.3
Character	\(\chi\)	\(=\)	177.4
Dual form			177.2.e.b.133.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.233737 - 0.0254205i) q^{2} +(-0.647386 + 0.762162i) q^{3} +(-1.89925 - 0.418057i) q^{4} +(1.40408 + 1.06736i) q^{5} +(0.170693 - 0.161689i) q^{6} +(-0.853505 + 2.14214i) q^{7} +(0.878915 + 0.296141i) q^{8} +(-0.161782 - 0.986827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.233737	−	0.0254205i	−0.165277	−	0.0179750i	0.0251062	−	0.999685i	\(-0.492008\pi\)
							−0.190383	+	0.981710i	\(0.560973\pi\)
\(3\)	−0.647386	+	0.762162i	−0.373769	+	0.440034i
							\(5\)	1.40408	+	1.06736i	0.627926	+	0.477337i	0.870086	−	0.492900i	\(-0.164063\pi\)
−0.242161	+	0.970236i	\(0.577856\pi\)
\(7\)	−0.853505	+	2.14214i	−0.322594	+	0.809651i	0.674833	+	0.737970i	\(0.264216\pi\)
							−0.997428	+	0.0716807i	\(0.977164\pi\)
\(11\)	−5.19354	+	2.40279i	−1.56591	+	0.724468i	−0.995308	−	0.0967576i	\(-0.969153\pi\)
							−0.570604	+	0.821226i	\(0.693291\pi\)
\(13\)	−0.785665	+	4.79234i	−0.217904	+	1.32916i	0.620542	+	0.784173i	\(0.286913\pi\)
							−0.838446	+	0.544984i	\(0.816536\pi\)
\(17\)	−0.277543	−	0.696579i	−0.0673139	−	0.168945i	0.891471	−	0.453077i	\(-0.149674\pi\)
							−0.958785	+	0.284132i	\(0.908295\pi\)
\(19\)	3.13134	+	4.61839i	0.718379	+	1.05953i	0.995128	+	0.0985880i	\(0.0314326\pi\)
							−0.276749	+	0.960942i	\(0.589257\pi\)
\(23\)	0.481286	−	8.87680i	0.100355	−	1.85094i	−0.324754	−	0.945799i	\(-0.605281\pi\)
							0.425109	−	0.905142i	\(-0.360236\pi\)
\(29\)	5.02130	−	0.546099i	0.932432	−	0.101408i	0.370723	−	0.928743i	\(-0.379110\pi\)
							0.561708	+	0.827335i	\(0.310144\pi\)
\(31\)	−3.40419	+	5.02081i	−0.611411	+	0.901764i	−0.999805	−	0.0197293i	\(-0.993720\pi\)
							0.388394	+	0.921493i	\(0.373030\pi\)
\(37\)	10.2336	−	3.44811i	1.68240	−	0.566867i	0.694166	−	0.719815i	\(-0.255773\pi\)
							0.988234	+	0.152949i	\(0.0488769\pi\)
\(41\)	0.0293230	+	0.540831i	0.00457948	+	0.0844636i	0.999930	−	0.0118587i	\(-0.00377483\pi\)
							−0.995350	+	0.0963222i	\(0.969292\pi\)
\(43\)	−1.82029	−	0.842155i	−0.277591	−	0.128427i	0.276154	−	0.961113i	\(-0.410940\pi\)
							−0.553746	+	0.832686i	\(0.686802\pi\)
\(47\)	−4.79551	+	3.64545i	−0.699497	+	0.531744i	−0.893392	−	0.449279i	\(-0.851681\pi\)
							0.193895	+	0.981022i	\(0.437888\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.4.3	✓	140
3.2	odd	2		531.2.i.b.181.3		140
59.15	even	29	inner	177.2.e.b.133.3	yes	140
177.74	odd	58		531.2.i.b.487.3		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.4.3	✓	140	1.1	even	1	trivial
177.2.e.b.133.3	yes	140	59.15	even	29	inner
531.2.i.b.181.3		140	3.2	odd	2
531.2.i.b.487.3		140	177.74	odd	58