Embedded newform 177.3.h.a.5.9

• Label: 177.3.h.a.5.9
• 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			5.9
Character	\(\chi\)	\(=\)	177.5
Dual form			177.3.h.a.71.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.869361 + 2.58017i) q^{2} +(2.78047 + 1.12649i) q^{3} +(-2.71713 - 2.06551i) q^{4} +(3.05628 + 1.21773i) q^{5} +(-5.32377 + 6.19477i) q^{6} +(9.38908 - 4.34385i) q^{7} +(-1.32265 + 0.896782i) q^{8} +(6.46204 + 6.26435i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.869361	+	2.58017i	−0.434681	+	1.29009i	0.476897	+	0.878959i	\(0.341761\pi\)
							−0.911578	+	0.411127i	\(0.865135\pi\)
\(3\)	2.78047	+	1.12649i	0.926824	+	0.375497i
							\(5\)	3.05628	+	1.21773i	0.611257	+	0.243547i	0.655162	−	0.755489i	\(-0.272601\pi\)
−0.0439048	+	0.999036i	\(0.513980\pi\)
\(7\)	9.38908	−	4.34385i	1.34130	−	0.620550i	0.387742	−	0.921768i	\(-0.373255\pi\)
							0.953555	+	0.301217i	\(0.0973930\pi\)
\(11\)	1.49552	+	0.415228i	0.135956	+	0.0377480i	0.334840	−	0.942275i	\(-0.391318\pi\)
							−0.198884	+	0.980023i	\(0.563732\pi\)
\(13\)	0.591291	−	1.11529i	0.0454840	−	0.0857918i	−0.859728	−	0.510752i	\(-0.829367\pi\)
							0.905212	+	0.424960i	\(0.139712\pi\)
\(17\)	−1.27613	+	2.75830i	−0.0750662	+	0.162253i	−0.941475	−	0.337082i	\(-0.890560\pi\)
							0.866409	+	0.499335i	\(0.166422\pi\)
\(19\)	−8.93462	+	1.96666i	−0.470243	+	0.103508i	−0.443769	−	0.896141i	\(-0.646359\pi\)
							−0.0264739	+	0.999650i	\(0.508428\pi\)
\(23\)	−11.6324	−	1.90703i	−0.505755	−	0.0829143i	−0.0964973	−	0.995333i	\(-0.530764\pi\)
							−0.409257	+	0.912419i	\(0.634212\pi\)
\(29\)	−15.6030	−	46.3080i	−0.538034	−	1.59683i	−0.779489	−	0.626416i	\(-0.784521\pi\)
							0.241455	−	0.970412i	\(-0.422375\pi\)
\(31\)	−28.5689	−	6.28848i	−0.921576	−	0.202854i	−0.271250	−	0.962509i	\(-0.587437\pi\)
							−0.650327	+	0.759655i	\(0.725368\pi\)
\(37\)	−20.1495	+	29.7183i	−0.544580	+	0.803196i	−0.996022	−	0.0891045i	\(-0.971599\pi\)
							0.451442	+	0.892301i	\(0.350910\pi\)
\(41\)	−47.6813	+	7.81695i	−1.16296	+	0.190657i	−0.712192	−	0.701985i	\(-0.752297\pi\)
							−0.450766	+	0.892642i	\(0.648849\pi\)
\(43\)	13.6358	+	49.1119i	0.317113	+	1.14214i	0.934643	+	0.355589i	\(0.115720\pi\)
							−0.617530	+	0.786547i	\(0.711867\pi\)
\(47\)	40.7721	−	16.2451i	0.867491	−	0.345640i	0.106430	−	0.994320i	\(-0.466058\pi\)
							0.761061	+	0.648680i	\(0.224679\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.5.9	✓	1064
3.2	odd	2	inner	177.3.h.a.5.30	yes	1064
59.12	even	29	inner	177.3.h.a.71.30	yes	1064
177.71	odd	58	inner	177.3.h.a.71.9	yes	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.9	✓	1064	1.1	even	1	trivial
177.3.h.a.5.30	yes	1064	3.2	odd	2	inner
177.3.h.a.71.9	yes	1064	177.71	odd	58	inner
177.3.h.a.71.30	yes	1064	59.12	even	29	inner