Embedded newform 177.3.g.a.10.19

• Label: 177.3.g.a.10.19
• 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			10.19
Character	\(\chi\)	\(=\)	177.10
Dual form			177.3.g.a.124.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.27139 + 1.30344i) q^{2} +(-1.72190 - 0.187268i) q^{3} +(6.09906 + 5.77734i) q^{4} +(-5.84142 + 6.87706i) q^{5} +(-5.38891 - 2.85702i) q^{6} +(-0.280020 - 0.168482i) q^{7} +(6.50743 + 14.0656i) q^{8} +(2.92986 + 0.644911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 560 q + 40 q^{4} + 8 q^{7} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.27139	+	1.30344i	1.63570	+	0.651721i	0.993396	−	0.114733i	\(-0.0366013\pi\)
							0.642299	+	0.766454i	\(0.277981\pi\)
\(3\)	−1.72190	−	0.187268i	−0.573966	−	0.0624225i
							\(5\)	−5.84142	+	6.87706i	−1.16828	+	1.37541i	−0.255582	+	0.966787i	\(0.582267\pi\)
−0.912703	+	0.408624i	\(0.866009\pi\)
\(7\)	−0.280020	−	0.168482i	−0.0400028	−	0.0240689i	0.495412	−	0.868658i	\(-0.335017\pi\)
							−0.535415	+	0.844589i	\(0.679845\pi\)
\(11\)	5.71778	+	0.310009i	0.519798	+	0.0281827i	0.312172	−	0.950026i	\(-0.398943\pi\)
							0.207626	+	0.978208i	\(0.433426\pi\)
\(13\)	0.0412895	+	0.187580i	0.00317612	+	0.0144292i	0.978182	−	0.207750i	\(-0.0666140\pi\)
							−0.975006	+	0.222179i	\(0.928683\pi\)
\(17\)	23.2666	−	13.9990i	1.36862	−	0.823472i	0.374038	−	0.927414i	\(-0.377973\pi\)
							0.994583	+	0.103942i	\(0.0331455\pi\)
\(19\)	−7.91732	+	28.5156i	−0.416701	+	1.50082i	0.396060	+	0.918224i	\(0.370377\pi\)
							−0.812761	+	0.582597i	\(0.802037\pi\)
\(23\)	24.1397	−	16.3671i	1.04955	−	0.711615i	0.0908041	−	0.995869i	\(-0.471056\pi\)
							0.958749	+	0.284254i	\(0.0917460\pi\)
\(29\)	14.1671	+	35.5567i	0.488521	+	1.22609i	0.942237	+	0.334948i	\(0.108719\pi\)
							−0.453716	+	0.891146i	\(0.649902\pi\)
\(31\)	0.724865	−	0.201258i	0.0233828	−	0.00649219i	−0.255817	−	0.966725i	\(-0.582345\pi\)
							0.279200	+	0.960233i	\(0.409931\pi\)
\(37\)	13.8200	−	29.8715i	0.373514	−	0.807338i	−0.626105	−	0.779739i	\(-0.715352\pi\)
							0.999619	−	0.0275990i	\(-0.00878616\pi\)
\(41\)	−3.61368	+	5.32977i	−0.0881384	+	0.129994i	−0.869202	−	0.494457i	\(-0.835367\pi\)
							0.781064	+	0.624451i	\(0.214677\pi\)
\(43\)	−64.4827	+	3.49615i	−1.49960	+	0.0813058i	−0.785553	−	0.618795i	\(-0.787621\pi\)
							−0.714044	+	0.700101i	\(0.753138\pi\)
\(47\)	16.2465	−	13.7999i	0.345670	−	0.293615i	−0.457686	−	0.889114i	\(-0.651322\pi\)
							0.803356	+	0.595499i	\(0.203046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.g.a.10.19	✓	560
59.6	odd	58	inner	177.3.g.a.124.19	yes	560

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.g.a.10.19	✓	560	1.1	even	1	trivial
177.3.g.a.124.19	yes	560	59.6	odd	58	inner