Embedded newform 177.2.e.b.28.2

• Label: 177.2.e.b.28.2
• Level: $177$
• Weight: $2$
• Character: 177.28
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.e (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(5\) over \(\Q(\zeta_{29})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{29}]$

Embedding invariants

Embedding label			28.2
Character	\(\chi\)	\(=\)	177.28
Dual form			177.2.e.b.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.715981 - 1.35048i) q^{2} +(0.725995 + 0.687699i) q^{3} +(-0.188804 + 0.278465i) q^{4} +(-3.13782 - 0.690687i) q^{5} +(0.408928 - 1.47283i) q^{6} +(-2.11659 - 1.60899i) q^{7} +(-2.52792 - 0.274928i) q^{8} +(0.0541389 + 0.998533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + q^{2} + 5 q^{3} - q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 3 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{10}{29}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.715981	−	1.35048i	−0.506275	−	0.954937i	−0.996520	−	0.0833518i	\(-0.973437\pi\)
							0.490245	−	0.871585i	\(-0.336907\pi\)
\(3\)	0.725995	+	0.687699i	0.419154	+	0.397043i
							\(5\)	−3.13782	−	0.690687i	−1.40328	−	0.308885i	−0.552106	−	0.833774i	\(-0.686176\pi\)
−0.851171	+	0.524889i	\(0.824107\pi\)
\(7\)	−2.11659	−	1.60899i	−0.799995	−	0.608141i	0.123239	−	0.992377i	\(-0.460672\pi\)
							−0.923235	+	0.384236i	\(0.874465\pi\)
\(11\)	2.12945	−	5.34452i	0.642054	−	1.61143i	−0.141377	−	0.989956i	\(-0.545153\pi\)
							0.783431	−	0.621479i	\(-0.213468\pi\)
\(13\)	0.173941	−	3.20816i	0.0482426	−	0.889783i	−0.869825	−	0.493360i	\(-0.835769\pi\)
							0.918068	−	0.396423i	\(-0.129749\pi\)
\(17\)	−1.99184	+	1.51416i	−0.483092	+	0.367237i	−0.818192	−	0.574946i	\(-0.805023\pi\)
							0.335100	+	0.942183i	\(0.391230\pi\)
\(19\)	5.97573	+	2.01346i	1.37093	+	0.461919i	0.905943	−	0.423400i	\(-0.139164\pi\)
							0.464984	+	0.885319i	\(0.346060\pi\)
\(23\)	0.758390	+	0.456308i	0.158135	+	0.0951468i	0.592422	−	0.805628i	\(-0.298172\pi\)
							−0.434287	+	0.900775i	\(0.642999\pi\)
\(29\)	−1.90564	+	3.59442i	−0.353868	+	0.667466i	−0.994990	−	0.0999726i	\(-0.968124\pi\)
							0.641122	+	0.767439i	\(0.278469\pi\)
\(31\)	−4.32032	+	1.45569i	−0.775953	+	0.261449i	−0.679292	−	0.733868i	\(-0.737713\pi\)
							−0.0966614	+	0.995317i	\(0.530816\pi\)
\(37\)	2.35133	−	0.255722i	0.386556	−	0.0420405i	0.0872221	−	0.996189i	\(-0.472201\pi\)
							0.299334	+	0.954148i	\(0.403236\pi\)
\(41\)	10.8057	−	6.50154i	1.68756	−	1.01537i	0.754050	−	0.656817i	\(-0.228098\pi\)
							0.933509	−	0.358553i	\(-0.116730\pi\)
\(43\)	−2.62049	−	6.57694i	−0.399621	−	1.00297i	−0.981801	−	0.189914i	\(-0.939179\pi\)
							0.582179	−	0.813060i	\(-0.302200\pi\)
\(47\)	5.95006	−	1.30971i	0.867906	−	0.191041i	0.241370	−	0.970433i	\(-0.422403\pi\)
							0.626536	+	0.779393i	\(0.284472\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.e.b.28.2	yes	140
3.2	odd	2		531.2.i.b.28.4		140
59.19	even	29	inner	177.2.e.b.19.2	✓	140
177.137	odd	58		531.2.i.b.19.4		140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.e.b.19.2	✓	140	59.19	even	29	inner
177.2.e.b.28.2	yes	140	1.1	even	1	trivial
531.2.i.b.19.4		140	177.137	odd	58
531.2.i.b.28.4		140	3.2	odd	2