Embedded newform 177.2.e.b.22.2

• Label: 177.2.e.b.22.2
• Level: $177$
• Weight: $2$
• Character: 177.22
• 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			22.2
Character	\(\chi\)	\(=\)	177.22
Dual form			177.2.e.b.169.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0772031 - 0.470918i) q^{2} +(-0.267528 - 0.963550i) q^{3} +(1.67950 - 0.565890i) q^{4} +(-0.369128 + 0.544423i) q^{5} +(-0.433099 + 0.200373i) q^{6} +(3.18087 + 0.700162i) q^{7} +(-0.843204 - 1.59045i) q^{8} +(-0.856857 + 0.515554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{13}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0772031	−	0.470918i	−0.0545908	−	0.332989i	−0.999975	−	0.00703195i	\(-0.997762\pi\)
							0.945384	−	0.325957i	\(-0.105687\pi\)
\(3\)	−0.267528	−	0.963550i	−0.154458	−	0.556306i
							\(5\)	−0.369128	+	0.544423i	−0.165079	+	0.243473i	−0.901227	−	0.433347i	\(-0.857332\pi\)
0.736148	+	0.676821i	\(0.236643\pi\)
\(7\)	3.18087	+	0.700162i	1.20225	+	0.264636i	0.770550	−	0.637379i	\(-0.219981\pi\)
							0.431704	+	0.902015i	\(0.357912\pi\)
\(11\)	−1.46239	−	1.11168i	−0.440927	−	0.335184i	0.361042	−	0.932550i	\(-0.382421\pi\)
							−0.801969	+	0.597365i	\(0.796214\pi\)
\(13\)	−2.36213	−	1.42125i	−0.655137	−	0.394183i	0.148825	−	0.988864i	\(-0.452451\pi\)
							−0.803962	+	0.594681i	\(0.797278\pi\)
\(17\)	4.71674	−	1.03823i	1.14398	−	0.251809i	0.397747	−	0.917495i	\(-0.369792\pi\)
							0.746232	+	0.665686i	\(0.231861\pi\)
\(19\)	−3.87004	−	0.420892i	−0.887847	−	0.0965592i	−0.347183	−	0.937798i	\(-0.612862\pi\)
							−0.540664	+	0.841238i	\(0.681827\pi\)
\(23\)	−5.38869	+	6.34405i	−1.12362	+	1.32283i	−0.183097	+	0.983095i	\(0.558612\pi\)
							−0.940522	+	0.339732i	\(0.889664\pi\)
\(29\)	0.0667431	−	0.407115i	0.0123939	−	0.0755994i	−0.979905	−	0.199463i	\(-0.936080\pi\)
							0.992299	+	0.123864i	\(0.0395285\pi\)
\(31\)	4.84278	−	0.526684i	0.869788	−	0.0945952i	0.337669	−	0.941265i	\(-0.390361\pi\)
							0.532119	+	0.846670i	\(0.321396\pi\)
\(37\)	−2.50140	+	4.71815i	−0.411228	+	0.775659i	−0.999454	−	0.0330379i	\(-0.989482\pi\)
							0.588226	+	0.808697i	\(0.299827\pi\)
\(41\)	4.27135	+	5.02862i	0.667073	+	0.785338i	0.986760	−	0.162184i	\(-0.0518539\pi\)
							−0.319688	+	0.947523i	\(0.603578\pi\)
\(43\)	−6.45524	+	4.90714i	−0.984414	+	0.748332i	−0.967849	−	0.251530i	\(-0.919066\pi\)
							−0.0165650	+	0.999863i	\(0.505273\pi\)
\(47\)	−0.180009	−	0.265493i	−0.0262570	−	0.0387261i	0.814334	−	0.580397i	\(-0.197102\pi\)
							−0.840591	+	0.541670i	\(0.817792\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.22.2	✓	140
3.2	odd	2		531.2.i.b.199.4		140
59.51	even	29	inner	177.2.e.b.169.2	yes	140
177.110	odd	58		531.2.i.b.523.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.22.2	✓	140	1.1	even	1	trivial
177.2.e.b.169.2	yes	140	59.51	even	29	inner
531.2.i.b.199.4		140	3.2	odd	2
531.2.i.b.523.4		140	177.110	odd	58