Embedded newform 177.3.g.a.10.4

• Label: 177.3.g.a.10.4
• Level: $177$
• Weight: $3$
• Character: 177.10
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(20\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			10.4
Character	\(\chi\)	\(=\)	177.10
Dual form			177.3.g.a.124.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.31503 - 0.922394i) q^{2} +(1.72190 + 0.187268i) q^{3} +(1.60459 + 1.51995i) q^{4} +(-0.698140 + 0.821913i) q^{5} +(-3.81352 - 2.02180i) q^{6} +(-4.78757 - 2.88058i) q^{7} +(1.87281 + 4.04801i) q^{8} +(2.92986 + 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{7}{58}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.31503	−	0.922394i	−1.15752	−	0.461197i	−0.289230	−	0.957260i	\(-0.593399\pi\)
							−0.868287	+	0.496063i	\(0.834779\pi\)
\(3\)	1.72190	+	0.187268i	0.573966	+	0.0624225i
							\(5\)	−0.698140	+	0.821913i	−0.139628	+	0.164383i	−0.827531	−	0.561420i	\(-0.810255\pi\)
0.687903	+	0.725802i	\(0.258531\pi\)
\(7\)	−4.78757	−	2.88058i	−0.683938	−	0.411512i	0.130747	−	0.991416i	\(-0.458263\pi\)
							−0.814685	+	0.579904i	\(0.803090\pi\)
\(11\)	−14.3132	−	0.776037i	−1.30120	−	0.0705488i	−0.609528	−	0.792765i	\(-0.708641\pi\)
							−0.691668	+	0.722216i	\(0.743124\pi\)
\(13\)	−0.277910	−	1.26256i	−0.0213777	−	0.0971199i	0.964724	−	0.263264i	\(-0.0847992\pi\)
							−0.986101	+	0.166145i	\(0.946868\pi\)
\(17\)	−14.0617	+	8.46063i	−0.827157	+	0.497684i	−0.865102	−	0.501596i	\(-0.832746\pi\)
							0.0379450	+	0.999280i	\(0.487919\pi\)
\(19\)	4.12045	−	14.8405i	0.216866	−	0.781079i	−0.772423	−	0.635108i	\(-0.780956\pi\)
							0.989289	−	0.145971i	\(-0.0466307\pi\)
\(23\)	−37.5540	+	25.4623i	−1.63278	+	1.10705i	−0.723989	+	0.689811i	\(0.757693\pi\)
							−0.908795	+	0.417243i	\(0.862996\pi\)
\(29\)	3.61980	+	9.08501i	0.124821	+	0.313276i	0.978086	−	0.208200i	\(-0.0667606\pi\)
							−0.853266	+	0.521477i	\(0.825381\pi\)
\(31\)	−52.3426	+	14.5329i	−1.68847	+	0.468802i	−0.973184	−	0.230028i	\(-0.926118\pi\)
							−0.715288	+	0.698830i	\(0.753705\pi\)
\(37\)	12.7192	−	27.4921i	0.343762	−	0.743030i	−0.656171	−	0.754613i	\(-0.727825\pi\)
							0.999933	+	0.0115824i	\(0.00368689\pi\)
\(41\)	−31.7822	+	46.8753i	−0.775177	+	1.14330i	0.210981	+	0.977490i	\(0.432334\pi\)
							−0.986158	+	0.165810i	\(0.946976\pi\)
\(43\)	−4.45720	+	0.241662i	−0.103656	+	0.00562005i	−0.105894	−	0.994377i	\(-0.533770\pi\)
							0.00223803	+	0.999997i	\(0.499288\pi\)
\(47\)	41.4091	−	35.1732i	0.881045	−	0.748366i	−0.0877732	−	0.996140i	\(-0.527975\pi\)
							0.968818	+	0.247775i	\(0.0796992\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.10.4	✓	560
59.6	odd	58	inner	177.3.g.a.124.4	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.4	✓	560	1.1	even	1	trivial
177.3.g.a.124.4	yes	560	59.6	odd	58	inner