Embedded newform 177.2.e.a.133.1

• Label: 177.2.e.a.133.1
• Level: $177$
• Weight: $2$
• Character: 177.133
• 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			133.1
Character	\(\chi\)	\(=\)	177.133
Dual form			177.2.e.a.4.1

$q$-expansion

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

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{28}{29}\right)\)	\(1\)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.a.133.1	yes	140
3.2	odd	2		531.2.i.c.487.5		140
59.4	even	29	inner	177.2.e.a.4.1	✓	140
177.122	odd	58		531.2.i.c.181.5		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.4.1	✓	140	59.4	even	29	inner
177.2.e.a.133.1	yes	140	1.1	even	1	trivial
531.2.i.c.181.5		140	177.122	odd	58
531.2.i.c.487.5		140	3.2	odd	2