Embedded newform 177.4.e.a.4.13

• Label: 177.4.e.a.4.13
• Level: $177$
• Weight: $4$
• Character: 177.4
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(15\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.a.133.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.96175 + 0.430866i) q^{2} +(-1.94216 + 2.28649i) q^{3} +(7.69682 + 1.69420i) q^{4} +(-13.8169 - 10.5033i) q^{5} +(-8.67951 + 8.22167i) q^{6} +(-7.10268 + 17.8264i) q^{7} +(-0.449126 - 0.151328i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 2 q^{2} + 45 q^{3} - 64 q^{4} + 14 q^{5} + 6 q^{6} + 6 q^{7} + 18 q^{8} - 135 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.96175	+	0.430866i	1.40069	+	0.152334i	0.777080	−	0.629402i	\(-0.216700\pi\)
							0.623609	+	0.781736i	\(0.285666\pi\)
\(3\)	−1.94216	+	2.28649i	−0.373769	+	0.440034i
							\(5\)	−13.8169	−	10.5033i	−1.23582	−	0.939446i	−0.236339	−	0.971671i	\(-0.575948\pi\)
−0.999481	+	0.0322245i	\(0.989741\pi\)
\(7\)	−7.10268	+	17.8264i	−0.383509	+	0.962535i	0.602839	+	0.797863i	\(0.294036\pi\)
							−0.986348	+	0.164672i	\(0.947343\pi\)
\(11\)	−26.5402	+	12.2788i	−0.727469	+	0.336563i	−0.748427	−	0.663217i	\(-0.769190\pi\)
							0.0209578	+	0.999780i	\(0.493328\pi\)
\(13\)	−12.9395	+	78.9277i	−0.276060	+	1.68389i	0.376706	+	0.926333i	\(0.377057\pi\)
							−0.652766	+	0.757560i	\(0.726391\pi\)
\(17\)	−42.5864	−	106.884i	−0.607572	−	1.52489i	−0.834516	−	0.550983i	\(-0.814253\pi\)
							0.226944	−	0.973908i	\(-0.427126\pi\)
\(19\)	42.2050	+	62.2477i	0.509604	+	0.751610i	0.992257	−	0.124202i	\(-0.0396370\pi\)
							−0.482653	+	0.875812i	\(0.660327\pi\)
\(23\)	4.46007	−	82.2612i	0.0404343	−	0.745767i	−0.906080	−	0.423106i	\(-0.860940\pi\)
							0.946514	−	0.322661i	\(-0.104577\pi\)
\(29\)	82.5447	−	8.97728i	0.528557	−	0.0574841i	0.160050	−	0.987109i	\(-0.448834\pi\)
							0.368507	+	0.929625i	\(0.379869\pi\)
\(31\)	82.8028	−	122.125i	0.479736	−	0.707558i	−0.508403	−	0.861119i	\(-0.669764\pi\)
							0.988139	+	0.153562i	\(0.0490743\pi\)
\(37\)	−7.24884	+	2.44242i	−0.0322081	+	0.0108522i	−0.335359	−	0.942090i	\(-0.608858\pi\)
							0.303151	+	0.952942i	\(0.401961\pi\)
\(41\)	18.8950	+	348.498i	0.0719733	+	1.32747i	0.781993	+	0.623287i	\(0.214203\pi\)
							−0.710020	+	0.704182i	\(0.751314\pi\)
\(43\)	−125.852	−	58.2253i	−0.446331	−	0.206495i	0.183832	−	0.982958i	\(-0.441150\pi\)
							−0.630163	+	0.776463i	\(0.717012\pi\)
\(47\)	71.0147	−	53.9840i	0.220395	−	0.167540i	−0.489151	−	0.872199i	\(-0.662693\pi\)
							0.709546	+	0.704659i	\(0.248900\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.a.4.13	✓	420
59.15	even	29	inner	177.4.e.a.133.13	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.a.4.13	✓	420	1.1	even	1	trivial
177.4.e.a.133.13	yes	420	59.15	even	29	inner