Embedded newform 177.4.e.b.4.8

• Label: 177.4.e.b.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	177.4
Dual form			177.4.e.b.133.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.253308 + 0.0275489i) q^{2} +(1.94216 - 2.28649i) q^{3} +(-7.74956 - 1.70581i) q^{4} +(7.78313 + 5.91658i) q^{5} +(0.554954 - 0.525681i) q^{6} +(-0.440801 + 1.10633i) q^{7} +(-3.84774 - 1.29645i) q^{8} +(-1.45604 - 8.88144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q + 2 q^{2} - 45 q^{3} - 56 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} + 66 q^{8} - 135 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.253308	+	0.0275489i	0.0895579	+	0.00974000i	0.152788	−	0.988259i	\(-0.451175\pi\)
							−0.0632304	+	0.997999i	\(0.520140\pi\)
\(3\)	1.94216	−	2.28649i	0.373769	−	0.440034i
							\(5\)	7.78313	+	5.91658i	0.696145	+	0.529195i	0.892337	−	0.451370i	\(-0.149065\pi\)
−0.196192	+	0.980565i	\(0.562858\pi\)
\(7\)	−0.440801	+	1.10633i	−0.0238010	+	0.0597361i	−0.940397	−	0.340079i	\(-0.889546\pi\)
							0.916596	+	0.399815i	\(0.130926\pi\)
\(11\)	39.4402	−	18.2470i	1.08106	−	0.500152i	0.203234	−	0.979130i	\(-0.434855\pi\)
							0.877827	+	0.478978i	\(0.158993\pi\)
\(13\)	11.3974	−	69.5213i	0.243160	−	1.48321i	−0.529457	−	0.848337i	\(-0.677604\pi\)
							0.772617	−	0.634873i	\(-0.218948\pi\)
\(17\)	37.7436	+	94.7292i	0.538480	+	1.35148i	0.906713	+	0.421749i	\(0.138584\pi\)
							−0.368233	+	0.929734i	\(0.620037\pi\)
\(19\)	−70.9646	−	104.665i	−0.856863	−	1.26378i	−0.963020	−	0.269429i	\(-0.913165\pi\)
							0.106158	−	0.994349i	\(-0.466145\pi\)
\(23\)	10.8706	−	200.497i	0.0985516	−	1.81768i	−0.364344	−	0.931264i	\(-0.618707\pi\)
							0.462896	−	0.886413i	\(-0.346810\pi\)
\(29\)	149.707	−	16.2817i	0.958619	−	0.104256i	0.384587	−	0.923089i	\(-0.374344\pi\)
							0.574032	+	0.818833i	\(0.305378\pi\)
\(31\)	−32.1492	+	47.4166i	−0.186264	+	0.274718i	−0.909344	−	0.416046i	\(-0.863415\pi\)
							0.723080	+	0.690764i	\(0.242726\pi\)
\(37\)	−80.7959	+	27.2233i	−0.358994	+	0.120959i	−0.493018	−	0.870019i	\(-0.664106\pi\)
							0.134025	+	0.990978i	\(0.457210\pi\)
\(41\)	−7.42076	−	136.868i	−0.0282665	−	0.521345i	−0.978041	−	0.208411i	\(-0.933171\pi\)
							0.949775	−	0.312934i	\(-0.101312\pi\)
\(43\)	438.803	+	203.012i	1.55621	+	0.719978i	0.994239	−	0.107181i	\(-0.0341825\pi\)
							0.561968	+	0.827159i	\(0.310045\pi\)
\(47\)	−228.521	+	173.717i	−0.709218	+	0.539134i	−0.896430	−	0.443186i	\(-0.853848\pi\)
							0.187212	+	0.982320i	\(0.440055\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.e.b.4.8	✓	420
59.15	even	29	inner	177.4.e.b.133.8	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.e.b.4.8	✓	420	1.1	even	1	trivial
177.4.e.b.133.8	yes	420	59.15	even	29	inner