Embedded newform 177.3.h.a.5.4

• Label: 177.3.h.a.5.4
• Level: $177$
• Weight: $3$
• Character: 177.5
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00783 + 2.99114i) q^{2} +(-2.99774 - 0.116447i) q^{3} +(-4.74683 - 3.60844i) q^{4} +(-2.80678 - 1.11832i) q^{5} +(3.36953 - 8.84930i) q^{6} +(-12.2974 + 5.68939i) q^{7} +(5.12741 - 3.47647i) q^{8} +(8.97288 + 0.698155i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{3}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00783	+	2.99114i	−0.503916	+	1.49557i	0.330162	+	0.943924i	\(0.392897\pi\)
							−0.834078	+	0.551646i	\(0.814000\pi\)
\(3\)	−2.99774	−	0.116447i	−0.999246	−	0.0388156i
							\(5\)	−2.80678	−	1.11832i	−0.561356	−	0.223665i	0.0721618	−	0.997393i	\(-0.477010\pi\)
−0.633517	+	0.773728i	\(0.718390\pi\)
\(7\)	−12.2974	+	5.68939i	−1.75677	+	0.812769i	−0.772753	+	0.634707i	\(0.781121\pi\)
							−0.984019	+	0.178062i	\(0.943017\pi\)
\(11\)	15.0724	+	4.18482i	1.37022	+	0.380439i	0.873225	−	0.487317i	\(-0.162024\pi\)
							0.496991	+	0.867756i	\(0.334438\pi\)
\(13\)	9.63264	−	18.1691i	0.740973	−	1.39762i	−0.170121	−	0.985423i	\(-0.554416\pi\)
							0.911094	−	0.412199i	\(-0.135239\pi\)
\(17\)	−5.45020	+	11.7804i	−0.320600	+	0.692966i	−0.999134	−	0.0416030i	\(-0.986754\pi\)
							0.678534	+	0.734569i	\(0.262616\pi\)
\(19\)	8.96329	−	1.97297i	0.471752	−	0.103841i	0.0272698	−	0.999628i	\(-0.491319\pi\)
							0.444482	+	0.895788i	\(0.353388\pi\)
\(23\)	−0.699848	−	0.114734i	−0.0304282	−	0.00498845i	0.146549	−	0.989203i	\(-0.453184\pi\)
							−0.176977	+	0.984215i	\(0.556632\pi\)
\(29\)	−5.03571	−	14.9455i	−0.173645	−	0.515361i	0.825124	−	0.564951i	\(-0.191105\pi\)
							−0.998770	+	0.0495903i	\(0.984208\pi\)
\(31\)	−14.1807	−	3.12141i	−0.457443	−	0.100691i	−0.0197295	−	0.999805i	\(-0.506281\pi\)
							−0.437713	+	0.899115i	\(0.644212\pi\)
\(37\)	20.7324	−	30.5780i	0.560336	−	0.826433i	−0.436994	−	0.899464i	\(-0.643957\pi\)
							0.997330	+	0.0730310i	\(0.0232672\pi\)
\(41\)	−49.6344	+	8.13715i	−1.21060	+	0.198467i	−0.733072	−	0.680152i	\(-0.761914\pi\)
							−0.477524	+	0.878619i	\(0.658466\pi\)
\(43\)	−4.03175	−	14.5211i	−0.0937616	−	0.337699i	0.901760	−	0.432238i	\(-0.142276\pi\)
							−0.995521	+	0.0945393i	\(0.969862\pi\)
\(47\)	−28.7363	+	11.4496i	−0.611410	+	0.243608i	−0.655228	−	0.755431i	\(-0.727427\pi\)
							0.0438175	+	0.999040i	\(0.486048\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.5.4	✓	1064
3.2	odd	2	inner	177.3.h.a.5.35	yes	1064
59.12	even	29	inner	177.3.h.a.71.35	yes	1064
177.71	odd	58	inner	177.3.h.a.71.4	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.4	✓	1064	1.1	even	1	trivial
177.3.h.a.5.35	yes	1064	3.2	odd	2	inner
177.3.h.a.71.4	yes	1064	177.71	odd	58	inner
177.3.h.a.71.35	yes	1064	59.12	even	29	inner