Embedded newform 177.3.g.a.10.10

• Label: 177.3.g.a.10.10
• 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			10.10
Character	\(\chi\)	\(=\)	177.10
Dual form			177.3.g.a.124.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0935508 + 0.0372741i) q^{2} +(1.72190 + 0.187268i) q^{3} +(-2.89662 - 2.74382i) q^{4} +(-5.55524 + 6.54013i) q^{5} +(0.154105 + 0.0817011i) q^{6} +(5.52684 + 3.32539i) q^{7} +(-0.337844 - 0.730238i) q^{8} +(2.92986 + 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0935508	+	0.0372741i	0.0467754	+	0.0186370i	0.393409	−	0.919364i	\(-0.371296\pi\)
							−0.346633	+	0.938001i	\(0.612675\pi\)
\(3\)	1.72190	+	0.187268i	0.573966	+	0.0624225i
							\(5\)	−5.55524	+	6.54013i	−1.11105	+	1.30803i	−0.164169	+	0.986432i	\(0.552494\pi\)
−0.946878	+	0.321594i	\(0.895782\pi\)
\(7\)	5.52684	+	3.32539i	0.789549	+	0.475056i	0.852351	−	0.522970i	\(-0.175176\pi\)
							−0.0628021	+	0.998026i	\(0.520004\pi\)
\(11\)	−21.1839	−	1.14856i	−1.92581	−	0.104414i	−0.948572	−	0.316561i	\(-0.897472\pi\)
							−0.977238	+	0.212147i	\(0.931954\pi\)
\(13\)	2.95937	+	13.4445i	0.227644	+	1.03419i	0.942313	+	0.334733i	\(0.108646\pi\)
							−0.714670	+	0.699462i	\(0.753423\pi\)
\(17\)	−7.74117	+	4.65771i	−0.455363	+	0.273983i	−0.724702	−	0.689063i	\(-0.758022\pi\)
							0.269339	+	0.963045i	\(0.413195\pi\)
\(19\)	−3.57727	+	12.8842i	−0.188278	+	0.678114i	0.807786	+	0.589475i	\(0.200666\pi\)
							−0.996064	+	0.0886389i	\(0.971748\pi\)
\(23\)	−1.14011	+	0.773015i	−0.0495701	+	0.0336094i	−0.585720	−	0.810513i	\(-0.699188\pi\)
							0.536150	+	0.844123i	\(0.319878\pi\)
\(29\)	6.28124	+	15.7647i	0.216594	+	0.543611i	0.996621	−	0.0821353i	\(-0.0261740\pi\)
							−0.780027	+	0.625746i	\(0.784795\pi\)
\(31\)	50.5330	−	14.0304i	1.63010	−	0.452594i	0.672356	−	0.740228i	\(-0.265283\pi\)
							0.957740	+	0.287634i	\(0.0928687\pi\)
\(37\)	1.84597	−	3.99000i	0.0498911	−	0.107838i	−0.881027	−	0.473066i	\(-0.843147\pi\)
							0.930918	+	0.365229i	\(0.119009\pi\)
\(41\)	35.1720	−	51.8749i	0.857855	−	1.26524i	−0.104796	−	0.994494i	\(-0.533419\pi\)
							0.962650	−	0.270747i	\(-0.0872708\pi\)
\(43\)	−61.8600	+	3.35395i	−1.43861	+	0.0779989i	−0.756853	−	0.653585i	\(-0.773264\pi\)
							−0.681752	+	0.731584i	\(0.738782\pi\)
\(47\)	−50.9521	+	43.2791i	−1.08409	+	0.920831i	−0.997155	−	0.0753830i	\(-0.975982\pi\)
							−0.0869317	+	0.996214i	\(0.527706\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.10.10	✓	560
59.6	odd	58	inner	177.3.g.a.124.10	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.10	✓	560	1.1	even	1	trivial
177.3.g.a.124.10	yes	560	59.6	odd	58	inner