Embedded newform 177.3.h.a.104.13

• Label: 177.3.h.a.104.13
• Level: $177$
• Weight: $3$
• Character: 177.104
• 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			104.13
Character	\(\chi\)	\(=\)	177.104
Dual form			177.3.h.a.80.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.891508 - 1.48170i) q^{2} +(-0.506593 + 2.95692i) q^{3} +(0.472992 - 0.892157i) q^{4} +(1.05004 + 9.65499i) q^{5} +(4.83289 - 1.88550i) q^{6} +(-1.35411 + 0.456253i) q^{7} +(-8.65033 + 0.469007i) q^{8} +(-8.48673 - 2.99591i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{24}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.891508	−	1.48170i	−0.445754	−	0.740849i	0.549774	−	0.835313i	\(-0.314714\pi\)
							−0.995528	+	0.0944643i	\(0.969886\pi\)
\(3\)	−0.506593	+	2.95692i	−0.168864	+	0.985639i
							\(5\)	1.05004	+	9.65499i	0.210009	+	1.93100i	0.341905	+	0.939735i	\(0.388928\pi\)
−0.131896	+	0.991264i	\(0.542106\pi\)
\(7\)	−1.35411	+	0.456253i	−0.193444	+	0.0651790i	−0.414356	−	0.910115i	\(-0.635993\pi\)
							0.220912	+	0.975294i	\(0.429097\pi\)
\(11\)	−12.7458	−	8.64189i	−1.15871	−	0.785626i	−0.178570	−	0.983927i	\(-0.557147\pi\)
							−0.980141	+	0.198301i	\(0.936458\pi\)
\(13\)	−9.09167	−	8.61209i	−0.699359	−	0.662468i	0.252993	−	0.967468i	\(-0.418585\pi\)
							−0.952352	+	0.305000i	\(0.901344\pi\)
\(17\)	−1.14428	+	3.39610i	−0.0673105	+	0.199770i	−0.976030	−	0.217638i	\(-0.930165\pi\)
							0.908719	+	0.417408i	\(0.137061\pi\)
\(19\)	3.10314	+	18.9283i	0.163323	+	0.996228i	0.933547	+	0.358454i	\(0.116696\pi\)
							−0.770224	+	0.637774i	\(0.779856\pi\)
\(23\)	24.8642	−	6.90351i	1.08105	−	0.300152i	0.319057	−	0.947736i	\(-0.396634\pi\)
							0.761995	+	0.647583i	\(0.224220\pi\)
\(29\)	−17.7971	+	29.5790i	−0.613693	+	1.01997i	0.381719	+	0.924278i	\(0.375332\pi\)
							−0.995412	+	0.0956867i	\(0.969495\pi\)
\(31\)	−1.76368	+	10.7580i	−0.0568930	+	0.347032i	0.943005	+	0.332778i	\(0.107986\pi\)
							−0.999898	+	0.0142546i	\(0.995462\pi\)
\(37\)	0.114216	−	2.10659i	0.00308692	−	0.0569349i	−0.996584	−	0.0825843i	\(-0.973683\pi\)
							0.999671	+	0.0256495i	\(0.00816537\pi\)
\(41\)	44.0373	+	12.2269i	1.07408	+	0.298217i	0.759162	−	0.650902i	\(-0.225609\pi\)
							0.314918	+	0.949119i	\(0.398023\pi\)
\(43\)	34.3766	+	50.7017i	0.799456	+	1.17911i	0.980671	+	0.195666i	\(0.0626868\pi\)
							−0.181215	+	0.983444i	\(0.558003\pi\)
\(47\)	−1.22085	+	11.2255i	−0.0259755	+	0.238841i	0.973951	+	0.226757i	\(0.0728122\pi\)
							−0.999927	+	0.0120844i	\(0.996153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.104.13	yes	1064
3.2	odd	2	inner	177.3.h.a.104.26	yes	1064
59.21	even	29	inner	177.3.h.a.80.26	yes	1064
177.80	odd	58	inner	177.3.h.a.80.13	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.80.13	✓	1064	177.80	odd	58	inner
177.3.h.a.80.26	yes	1064	59.21	even	29	inner
177.3.h.a.104.13	yes	1064	1.1	even	1	trivial
177.3.h.a.104.26	yes	1064	3.2	odd	2	inner