Embedded newform 177.2.e.a.4.1

• Label: 177.2.e.a.4.1
• 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			4.1
Character	\(\chi\)	\(=\)	177.4
Dual form			177.2.e.a.133.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29327 - 0.249408i) q^{2} +(0.647386 - 0.762162i) q^{3} +(3.24363 + 0.713978i) q^{4} +(-1.11360 - 0.846538i) q^{5} +(-1.67472 + 1.58638i) q^{6} +(0.605271 - 1.51911i) q^{7} +(-2.88838 - 0.973208i) q^{8} +(-0.161782 - 0.986827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140q - q^{2} - 5q^{3} - 9q^{4} - 2q^{5} - q^{6} - 2q^{7} - 9q^{8} - 5q^{9} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29327	−	0.249408i	−1.62159	−	0.176358i	−0.748229	−	0.663440i	\(-0.769096\pi\)
							−0.873356	+	0.487082i	\(0.838061\pi\)
\(3\)	0.647386	−	0.762162i	0.373769	−	0.440034i
							\(5\)	−1.11360	−	0.846538i	−0.498018	−	0.378583i	0.325799	−	0.945439i	\(-0.394367\pi\)
−0.823817	+	0.566856i	\(0.808160\pi\)
\(7\)	0.605271	−	1.51911i	0.228771	−	0.574171i	−0.769119	−	0.639106i	\(-0.779305\pi\)
							0.997890	+	0.0649346i	\(0.0206839\pi\)
\(11\)	−0.0689026	+	0.0318778i	−0.0207749	+	0.00961150i	−0.430249	−	0.902710i	\(-0.641574\pi\)
							0.409474	+	0.912322i	\(0.365712\pi\)
\(13\)	0.0422595	−	0.257771i	0.0117207	−	0.0714929i	−0.980319	−	0.197422i	\(-0.936743\pi\)
							0.992039	+	0.125929i	\(0.0401913\pi\)
\(17\)	−2.01049	−	5.04594i	−0.487614	−	1.22382i	−0.942781	−	0.333413i	\(-0.891800\pi\)
							0.455166	−	0.890406i	\(-0.349580\pi\)
\(19\)	−2.27132	−	3.34995i	−0.521077	−	0.768531i	0.472543	−	0.881307i	\(-0.343336\pi\)
							−0.993620	+	0.112776i	\(0.964026\pi\)
\(23\)	0.190332	−	3.51047i	0.0396870	−	0.731984i	−0.909198	−	0.416363i	\(-0.863304\pi\)
							0.948885	−	0.315621i	\(-0.102213\pi\)
\(29\)	2.87022	−	0.312155i	0.532987	−	0.0579658i	0.162331	−	0.986736i	\(-0.448099\pi\)
							0.370655	+	0.928771i	\(0.379133\pi\)
\(31\)	−3.62843	+	5.35154i	−0.651686	+	0.961165i	0.348033	+	0.937482i	\(0.386850\pi\)
							−0.999720	+	0.0236829i	\(0.992461\pi\)
\(37\)	4.27614	−	1.44080i	0.702993	−	0.236866i	0.0549717	−	0.998488i	\(-0.482493\pi\)
							0.648022	+	0.761622i	\(0.275597\pi\)
\(41\)	0.625949	+	11.5450i	0.0977568	+	1.80302i	0.478239	+	0.878230i	\(0.341275\pi\)
							−0.380482	+	0.924788i	\(0.624242\pi\)
\(43\)	−1.54395	−	0.714308i	−0.235450	−	0.108931i	0.298617	−	0.954373i	\(-0.403475\pi\)
							−0.534067	+	0.845442i	\(0.679337\pi\)
\(47\)	10.5238	−	7.99999i	1.53506	−	1.16692i	0.606871	−	0.794801i	\(-0.292425\pi\)
							0.928185	−	0.372119i	\(-0.121369\pi\)