Embedded newform 177.4.f.a.89.1

• Label: 177.4.f.a.89.1
• Level: $177$
• Weight: $4$
• Character: 177.89
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			89.1
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.301320 - 5.55751i) q^{2} +(4.73924 + 2.13064i) q^{3} +(-22.8421 + 2.48423i) q^{4} +(-5.05728 - 15.0095i) q^{5} +(10.4130 - 26.9804i) q^{6} +(-9.02219 + 13.3067i) q^{7} +(13.4855 + 82.2577i) q^{8} +(17.9208 + 20.1952i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{57}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.301320	−	5.55751i	−0.106533	−	1.96488i	−0.231823	−	0.972758i	\(-0.574469\pi\)
							0.125290	−	0.992120i	\(-0.460014\pi\)
\(3\)	4.73924	+	2.13064i	0.912067	+	0.410041i
							\(5\)	−5.05728	−	15.0095i	−0.452337	−	1.34249i	−0.895107	−	0.445852i	\(-0.852901\pi\)
0.442770	−	0.896635i	\(-0.353996\pi\)
\(7\)	−9.02219	+	13.3067i	−0.487153	+	0.718496i	−0.989237	−	0.146322i	\(-0.953256\pi\)
							0.502084	+	0.864819i	\(0.332567\pi\)
\(11\)	−44.8647	−	9.87548i	−1.22975	−	0.270688i	−0.447858	−	0.894105i	\(-0.647813\pi\)
							−0.781890	+	0.623417i	\(0.785744\pi\)
\(13\)	11.3151	+	9.61111i	0.241403	+	0.205049i	0.759918	−	0.650019i	\(-0.225239\pi\)
							−0.518515	+	0.855069i	\(0.673515\pi\)
\(17\)	−82.8482	+	56.1725i	−1.18198	+	0.801402i	−0.983889	−	0.178782i	\(-0.942784\pi\)
							−0.198090	+	0.980184i	\(0.563474\pi\)
\(19\)	−53.3838	−	100.693i	−0.644583	−	1.21581i	−0.963418	−	0.268003i	\(-0.913636\pi\)
							0.318835	−	0.947810i	\(-0.396708\pi\)
\(23\)	43.3282	+	41.0427i	0.392807	+	0.372087i	0.858460	−	0.512881i	\(-0.171422\pi\)
							−0.465653	+	0.884968i	\(0.654180\pi\)
\(29\)	5.35273	+	0.290216i	0.0342751	+	0.00185834i	0.0712685	−	0.997457i	\(-0.477295\pi\)
							−0.0369934	+	0.999316i	\(0.511778\pi\)
\(31\)	6.85013	+	3.63171i	0.0396877	+	0.0210411i	0.488134	−	0.872769i	\(-0.337678\pi\)
							−0.448446	+	0.893810i	\(0.648023\pi\)
\(37\)	−84.8826	−	13.9158i	−0.377152	−	0.0618309i	−0.0297752	−	0.999557i	\(-0.509479\pi\)
							−0.347377	+	0.937726i	\(0.612927\pi\)
\(41\)	−15.8347	−	16.7164i	−0.0603160	−	0.0636749i	0.695139	−	0.718875i	\(-0.255343\pi\)
							−0.755455	+	0.655200i	\(0.772584\pi\)
\(43\)	−42.2032	−	191.731i	−0.149673	−	0.679970i	−0.990368	−	0.138461i	\(-0.955784\pi\)
							0.840695	−	0.541509i	\(-0.182147\pi\)
\(47\)	−366.658	−	123.541i	−1.13793	−	0.383412i	−0.313616	−	0.949550i	\(-0.601540\pi\)
							−0.824311	+	0.566138i	\(0.808437\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.1	yes	1624
3.2	odd	2	inner	177.4.f.a.89.58	yes	1624
59.2	odd	58	inner	177.4.f.a.2.58	yes	1624
177.2	even	58	inner	177.4.f.a.2.1	✓	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.1	✓	1624	177.2	even	58	inner
177.4.f.a.2.58	yes	1624	59.2	odd	58	inner
177.4.f.a.89.1	yes	1624	1.1	even	1	trivial
177.4.f.a.89.58	yes	1624	3.2	odd	2	inner