Embedded newform 177.2.e.a.133.3

• Label: 177.2.e.a.133.3
• 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			133.3
Character	\(\chi\)	\(=\)	177.133
Dual form			177.2.e.a.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.247015 - 0.0268645i) q^{2} +(0.647386 + 0.762162i) q^{3} +(-1.89295 + 0.416669i) q^{4} +(-2.60143 + 1.97756i) q^{5} +(0.180389 + 0.170874i) q^{6} +(0.505225 + 1.26802i) q^{7} +(-0.927322 + 0.312451i) q^{8} +(-0.161782 + 0.986827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - q^{2} - 5 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 9 q^{8} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.247015	−	0.0268645i	0.174666	−	0.0189961i	−0.0203681	−	0.999793i	\(-0.506484\pi\)
							0.195034	+	0.980796i	\(0.437518\pi\)
\(3\)	0.647386	+	0.762162i	0.373769	+	0.440034i
							\(5\)	−2.60143	+	1.97756i	−1.16340	+	0.884391i	−0.994856	−	0.101304i	\(-0.967699\pi\)
−0.168541	+	0.985695i	\(0.553906\pi\)
\(7\)	0.505225	+	1.26802i	0.190957	+	0.479266i	0.993016	−	0.117979i	\(-0.0376415\pi\)
							−0.802059	+	0.597245i	\(0.796262\pi\)
\(11\)	0.272923	+	0.126267i	0.0822893	+	0.0380711i	0.460602	−	0.887607i	\(-0.347634\pi\)
							−0.378313	+	0.925678i	\(0.623496\pi\)
\(13\)	0.501045	+	3.05624i	0.138965	+	0.847648i	0.960661	+	0.277722i	\(0.0895796\pi\)
							−0.821697	+	0.569925i	\(0.806972\pi\)
\(17\)	−0.182367	+	0.457707i	−0.0442306	+	0.111010i	−0.949432	−	0.313972i	\(-0.898340\pi\)
							0.905202	+	0.424982i	\(0.139720\pi\)
\(19\)	3.88962	−	5.73677i	0.892341	−	1.31610i	−0.0558615	−	0.998439i	\(-0.517791\pi\)
							0.948203	−	0.317666i	\(-0.102899\pi\)
\(23\)	0.233982	+	4.31554i	0.0487886	+	0.899852i	0.915818	+	0.401594i	\(0.131544\pi\)
							−0.867029	+	0.498258i	\(0.833973\pi\)
\(29\)	−4.09825	−	0.445711i	−0.761025	−	0.0827665i	−0.280627	−	0.959817i	\(-0.590542\pi\)
							−0.480398	+	0.877050i	\(0.659508\pi\)
\(31\)	3.49687	+	5.15750i	0.628056	+	0.926314i	0.999998	−	0.00194229i	\(-0.000618251\pi\)
							−0.371942	+	0.928256i	\(0.621308\pi\)
\(37\)	4.02986	+	1.35782i	0.662505	+	0.223224i	0.630412	−	0.776261i	\(-0.282886\pi\)
							0.0320935	+	0.999485i	\(0.489783\pi\)
\(41\)	−0.372998	+	6.87955i	−0.0582525	+	1.07440i	0.812588	+	0.582839i	\(0.198058\pi\)
							−0.870840	+	0.491566i	\(0.836425\pi\)
\(43\)	1.23791	−	0.572717i	0.188779	−	0.0873386i	−0.323222	−	0.946323i	\(-0.604766\pi\)
							0.512002	+	0.858984i	\(0.328904\pi\)
\(47\)	−7.70904	−	5.86026i	−1.12448	−	0.854807i	−0.133798	−	0.991009i	\(-0.542717\pi\)
							−0.990681	+	0.136202i	\(0.956510\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.a.4.3	✓	140	59.4	even	29	inner
177.2.e.a.133.3	yes	140	1.1	even	1	trivial
531.2.i.c.181.3		140	177.122	odd	58
531.2.i.c.487.3		140	3.2	odd	2