Embedded newform 177.3.g.a.10.11

• Label: 177.3.g.a.10.11
• 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			10.11
Character	\(\chi\)	\(=\)	177.10
Dual form			177.3.g.a.124.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.140866 + 0.0561262i) q^{2} +(1.72190 + 0.187268i) q^{3} +(-2.88729 - 2.73499i) q^{4} +(1.82294 - 2.14613i) q^{5} +(0.232047 + 0.123023i) q^{6} +(-8.41764 - 5.06473i) q^{7} +(-0.507898 - 1.09780i) q^{8} +(2.92986 + 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.140866	+	0.0561262i	0.0704331	+	0.0280631i	0.405089	−	0.914277i	\(-0.367241\pi\)
							−0.334656	+	0.942341i	\(0.608620\pi\)
\(3\)	1.72190	+	0.187268i	0.573966	+	0.0624225i
							\(5\)	1.82294	−	2.14613i	0.364587	−	0.429225i	−0.549029	−	0.835803i	\(-0.685002\pi\)
0.913616	+	0.406578i	\(0.133278\pi\)
\(7\)	−8.41764	−	5.06473i	−1.20252	−	0.723532i	−0.233944	−	0.972250i	\(-0.575163\pi\)
							−0.968576	+	0.248718i	\(0.919991\pi\)
\(11\)	−7.27840	−	0.394624i	−0.661673	−	0.0358749i	−0.279755	−	0.960071i	\(-0.590253\pi\)
							−0.381918	+	0.924196i	\(0.624736\pi\)
\(13\)	−2.70845	−	12.3046i	−0.208342	−	0.946507i	−0.958465	−	0.285209i	\(-0.907937\pi\)
							0.750123	−	0.661298i	\(-0.229994\pi\)
\(17\)	−2.95697	+	1.77915i	−0.173940	+	0.104656i	−0.599861	−	0.800104i	\(-0.704778\pi\)
							0.425922	+	0.904760i	\(0.359950\pi\)
\(19\)	2.43835	−	8.78214i	0.128334	−	0.462218i	−0.871256	−	0.490828i	\(-0.836694\pi\)
							0.999591	+	0.0286102i	\(0.00910814\pi\)
\(23\)	32.9657	−	22.3513i	1.43329	−	0.971796i	0.435872	−	0.900009i	\(-0.356440\pi\)
							0.997420	−	0.0717869i	\(-0.0228702\pi\)
\(29\)	−18.1509	−	45.5554i	−0.625893	−	1.57087i	−0.809018	−	0.587784i	\(-0.800000\pi\)
							0.183124	−	0.983090i	\(-0.441379\pi\)
\(31\)	33.0183	−	9.16750i	1.06511	−	0.295726i	0.309601	−	0.950867i	\(-0.399805\pi\)
							0.755507	+	0.655141i	\(0.227391\pi\)
\(37\)	−22.3197	+	48.2433i	−0.603236	+	1.30387i	0.329787	+	0.944055i	\(0.393023\pi\)
							−0.933023	+	0.359817i	\(0.882839\pi\)
\(41\)	−5.08728	+	7.50318i	−0.124080	+	0.183004i	−0.884656	−	0.466245i	\(-0.845607\pi\)
							0.760576	+	0.649249i	\(0.224917\pi\)
\(43\)	39.5683	−	2.14533i	0.920194	−	0.0498915i	0.412048	−	0.911162i	\(-0.364814\pi\)
							0.508146	+	0.861271i	\(0.330331\pi\)
\(47\)	37.9147	−	32.2050i	0.806695	−	0.685213i	−0.145952	−	0.989292i	\(-0.546624\pi\)
							0.952647	+	0.304079i	\(0.0983486\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.10.11	✓	560
59.6	odd	58	inner	177.3.g.a.124.11	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.11	✓	560	1.1	even	1	trivial
177.3.g.a.124.11	yes	560	59.6	odd	58	inner