Embedded newform 177.2.f.a.8.3

• Label: 177.2.f.a.8.3
• Level: $177$
• Weight: $2$
• Character: 177.8
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			8.3
Character	\(\chi\)	\(=\)	177.8
Dual form			177.2.f.a.155.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.317307 + 1.93549i) q^{2} +(1.61303 + 0.630969i) q^{3} +(-1.75012 - 0.589685i) q^{4} +(2.92507 - 1.98325i) q^{5} +(-1.73306 + 2.92180i) q^{6} +(-2.73852 + 0.602794i) q^{7} +(-0.140747 + 0.265477i) q^{8} +(2.20376 + 2.03555i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.317307	+	1.93549i	−0.224370	+	1.36860i	0.598844	+	0.800866i	\(0.295627\pi\)
							−0.823214	+	0.567731i	\(0.807821\pi\)
\(3\)	1.61303	+	0.630969i	0.931286	+	0.364290i
							\(5\)	2.92507	−	1.98325i	1.30813	−	0.886936i	0.310166	−	0.950682i	\(-0.399615\pi\)
0.997966	+	0.0637465i	\(0.0203049\pi\)
\(7\)	−2.73852	+	0.602794i	−1.03506	+	0.227835i	−0.699837	−	0.714303i	\(-0.746744\pi\)
							−0.335226	+	0.942138i	\(0.608813\pi\)
\(11\)	2.01160	−	1.52918i	0.606521	−	0.461065i	−0.256285	−	0.966601i	\(-0.582498\pi\)
							0.862805	+	0.505536i	\(0.168705\pi\)
\(13\)	−2.03098	−	3.37551i	−0.563292	−	0.936199i	−0.999342	−	0.0362761i	\(-0.988450\pi\)
							0.436050	−	0.899923i	\(-0.356377\pi\)
\(17\)	−1.04883	+	4.76487i	−0.254378	+	1.15565i	0.660151	+	0.751133i	\(0.270492\pi\)
							−0.914530	+	0.404519i	\(0.867439\pi\)
\(19\)	−5.48029	+	0.596018i	−1.25727	+	0.136736i	−0.712392	−	0.701782i	\(-0.752388\pi\)
							−0.544874	+	0.838518i	\(0.683422\pi\)
\(23\)	−5.39194	−	6.34788i	−1.12430	−	1.32362i	−0.940164	−	0.340723i	\(-0.889328\pi\)
							−0.184133	−	0.982901i	\(-0.558948\pi\)
\(29\)	4.96592	−	0.814121i	0.922148	−	0.151179i	0.318037	−	0.948078i	\(-0.396976\pi\)
							0.604112	+	0.796900i	\(0.293528\pi\)
\(31\)	−0.0516452	+	0.474870i	−0.00927576	+	0.0852892i	−0.997861	−	0.0653749i	\(-0.979176\pi\)
							0.988585	+	0.150664i	\(0.0481412\pi\)
\(37\)	4.35883	−	2.31091i	0.716587	−	0.379911i	−0.0698691	−	0.997556i	\(-0.522258\pi\)
							0.786457	+	0.617646i	\(0.211913\pi\)
\(41\)	5.04670	+	4.28671i	0.788162	+	0.669471i	0.948212	−	0.317638i	\(-0.102890\pi\)
							−0.160050	+	0.987109i	\(0.551166\pi\)
\(43\)	−1.09884	+	1.44550i	−0.167571	+	0.220437i	−0.872212	−	0.489128i	\(-0.837315\pi\)
							0.704641	+	0.709564i	\(0.251108\pi\)
\(47\)	−2.14776	+	3.16771i	−0.313283	+	0.462058i	−0.951434	−	0.307853i	\(-0.900389\pi\)
							0.638151	+	0.769912i	\(0.279700\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.8.3	✓	504
3.2	odd	2	inner	177.2.f.a.8.16	yes	504
59.37	odd	58	inner	177.2.f.a.155.16	yes	504
177.155	even	58	inner	177.2.f.a.155.3	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.8.3	✓	504	1.1	even	1	trivial
177.2.f.a.8.16	yes	504	3.2	odd	2	inner
177.2.f.a.155.3	yes	504	177.155	even	58	inner
177.2.f.a.155.16	yes	504	59.37	odd	58	inner