Embedded newform 177.3.g.a.124.10

• Label: 177.3.g.a.124.10
• Level: $177$
• Weight: $3$
• Character: 177.124
• 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			124.10
Character	\(\chi\)	\(=\)	177.124
Dual form			177.3.g.a.10.10

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{51}{58}\right)\)	\(1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.10	✓	560	59.10	odd	58	inner
177.3.g.a.124.10	yes	560	1.1	even	1	trivial