Embedded newform 177.2.e.a.22.4

• Label: 177.2.e.a.22.4
• 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			22.4
Character	\(\chi\)	\(=\)	177.22
Dual form			177.2.e.a.169.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.205084 + 1.25096i) q^{2} +(0.267528 + 0.963550i) q^{3} +(0.372473 - 0.125501i) q^{4} +(1.50420 - 2.21853i) q^{5} +(-1.15049 + 0.532275i) q^{6} +(1.03095 + 0.226929i) q^{7} +(1.42095 + 2.68019i) q^{8} +(-0.856857 + 0.515554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.205084	+	1.25096i	0.145016	+	0.884560i	0.954673	+	0.297656i	\(0.0962050\pi\)
							−0.809657	+	0.586904i	\(0.800347\pi\)
\(3\)	0.267528	+	0.963550i	0.154458	+	0.556306i
							\(5\)	1.50420	−	2.21853i	0.672698	−	0.992156i	−0.326203	−	0.945300i	\(-0.605769\pi\)
0.998902	−	0.0468562i	\(-0.0149202\pi\)
\(7\)	1.03095	+	0.226929i	0.389662	+	0.0857710i	0.405480	−	0.914104i	\(-0.367104\pi\)
							−0.0158185	+	0.999875i	\(0.505035\pi\)
\(11\)	−3.22993	−	2.45533i	−0.973860	−	0.740309i	−0.00815954	−	0.999967i	\(-0.502597\pi\)
							−0.965701	+	0.259657i	\(0.916390\pi\)
\(13\)	−0.227291	−	0.136757i	−0.0630393	−	0.0379295i	0.483685	−	0.875242i	\(-0.339298\pi\)
							−0.546725	+	0.837313i	\(0.684126\pi\)
\(17\)	−6.62522	+	1.45832i	−1.60685	+	0.353695i	−0.925644	−	0.378395i	\(-0.876476\pi\)
							−0.681208	+	0.732090i	\(0.738545\pi\)
\(19\)	−0.399908	−	0.0434926i	−0.0917451	−	0.00997788i	0.0621313	−	0.998068i	\(-0.480210\pi\)
							−0.153876	+	0.988090i	\(0.549176\pi\)
\(23\)	−4.63800	+	5.46027i	−0.967089	+	1.13855i	0.0232295	+	0.999730i	\(0.492605\pi\)
							−0.990318	+	0.138815i	\(0.955671\pi\)
\(29\)	1.42331	−	8.68183i	0.264303	−	1.61217i	−0.437103	−	0.899412i	\(-0.643995\pi\)
							0.701405	−	0.712763i	\(-0.252556\pi\)
\(31\)	9.46072	−	1.02892i	1.69920	−	0.184799i	0.793454	−	0.608630i	\(-0.208281\pi\)
							0.905741	+	0.423831i	\(0.139315\pi\)
\(37\)	0.879831	−	1.65954i	0.144643	−	0.272826i	−0.800596	−	0.599205i	\(-0.795484\pi\)
							0.945239	+	0.326378i	\(0.105828\pi\)
\(41\)	−2.64570	−	3.11476i	−0.413189	−	0.486443i	0.515717	−	0.856759i	\(-0.327525\pi\)
							−0.928906	+	0.370315i	\(0.879250\pi\)
\(43\)	−3.42236	+	2.60161i	−0.521904	+	0.396741i	−0.832691	−	0.553738i	\(-0.813201\pi\)
							0.310786	+	0.950480i	\(0.399408\pi\)
\(47\)	3.05223	+	4.50170i	0.445213	+	0.656641i	0.982393	−	0.186827i	\(-0.0598205\pi\)
							−0.537179	+	0.843468i	\(0.680510\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.22.4	✓	140
3.2	odd	2		531.2.i.c.199.2		140
59.51	even	29	inner	177.2.e.a.169.4	yes	140
177.110	odd	58		531.2.i.c.523.2		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.22.4	✓	140	1.1	even	1	trivial
177.2.e.a.169.4	yes	140	59.51	even	29	inner
531.2.i.c.199.2		140	3.2	odd	2
531.2.i.c.523.2		140	177.110	odd	58