Embedded newform 177.2.f.a.83.6

• Label: 177.2.f.a.83.6
• Level: $177$
• Weight: $2$
• Character: 177.83
• 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			83.6
Character	\(\chi\)	\(=\)	177.83
Dual form			177.2.f.a.32.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.252641 - 0.909931i) q^{2} +(-1.39523 + 1.02632i) q^{3} +(0.949568 - 0.571336i) q^{4} +(-2.96288 + 0.160643i) q^{5} +(1.28637 + 1.01028i) q^{6} +(-0.330530 - 2.01614i) q^{7} +(-2.13097 - 2.01856i) q^{8} +(0.893347 - 2.86390i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{53}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.252641	−	0.909931i	−0.178644	−	0.643418i	−0.997580	−	0.0695326i	\(-0.977849\pi\)
							0.818936	−	0.573886i	\(-0.194565\pi\)
\(3\)	−1.39523	+	1.02632i	−0.805538	+	0.592544i
							\(5\)	−2.96288	+	0.160643i	−1.32504	+	0.0718416i	−0.703026	−	0.711165i	\(-0.748168\pi\)
−0.622014	+	0.783006i	\(0.713685\pi\)
\(7\)	−0.330530	−	2.01614i	−0.124929	−	0.762030i	−0.972769	−	0.231778i	\(-0.925546\pi\)
							0.847840	−	0.530252i	\(-0.177903\pi\)
\(11\)	−0.839863	−	1.58415i	−0.253228	−	0.477639i	0.724008	−	0.689791i	\(-0.242298\pi\)
							−0.977237	+	0.212152i	\(0.931953\pi\)
\(13\)	−5.22613	−	2.08228i	−1.44947	−	0.577521i	−0.493175	−	0.869930i	\(-0.664164\pi\)
							−0.956293	+	0.292410i	\(0.905543\pi\)
\(17\)	5.41207	+	0.887263i	1.31262	+	0.215193i	0.777125	−	0.629346i	\(-0.216677\pi\)
							0.535494	+	0.844539i	\(0.320126\pi\)
\(19\)	−2.57002	+	3.02566i	−0.589603	+	0.694134i	−0.973355	−	0.229302i	\(-0.926356\pi\)
							0.383753	+	0.923436i	\(0.374632\pi\)
\(23\)	−2.77697	−	2.11100i	−0.579038	−	0.440173i	0.274216	−	0.961668i	\(-0.411582\pi\)
							−0.853255	+	0.521495i	\(0.825375\pi\)
\(29\)	3.39577	+	0.942832i	0.630579	+	0.175079i	0.568090	−	0.822966i	\(-0.307682\pi\)
							0.0624889	+	0.998046i	\(0.480096\pi\)
\(31\)	4.87839	−	4.14374i	0.876184	−	0.744237i	−0.0916596	−	0.995790i	\(-0.529217\pi\)
							0.967844	+	0.251553i	\(0.0809413\pi\)
\(37\)	−0.945498	−	0.998150i	−0.155439	−	0.164095i	0.643625	−	0.765341i	\(-0.277430\pi\)
							−0.799064	+	0.601246i	\(0.794671\pi\)
\(41\)	2.99315	+	3.93742i	0.467452	+	0.614922i	0.968050	−	0.250758i	\(-0.0806797\pi\)
							−0.500598	+	0.865680i	\(0.666887\pi\)
\(43\)	0.467072	+	0.247626i	0.0712278	+	0.0377626i	0.503642	−	0.863913i	\(-0.331993\pi\)
							−0.432414	+	0.901675i	\(0.642338\pi\)
\(47\)	−0.0740635	+	1.36602i	−0.0108033	+	0.199255i	0.988149	+	0.153496i	\(0.0490532\pi\)
							−0.998953	+	0.0457585i	\(0.985430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.83.6	yes	504
3.2	odd	2	inner	177.2.f.a.83.13	yes	504
59.32	odd	58	inner	177.2.f.a.32.13	yes	504
177.32	even	58	inner	177.2.f.a.32.6	✓	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.32.6	✓	504	177.32	even	58	inner
177.2.f.a.32.13	yes	504	59.32	odd	58	inner
177.2.f.a.83.6	yes	504	1.1	even	1	trivial
177.2.f.a.83.13	yes	504	3.2	odd	2	inner