Embedded newform 177.2.f.a.89.6

• Label: 177.2.f.a.89.6
• Level: $177$
• Weight: $2$
• Character: 177.89
• 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			89.6
Character	\(\chi\)	\(=\)	177.89
Dual form			177.2.f.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0767664 - 1.41587i) q^{2} +(-0.135861 - 1.72671i) q^{3} +(-0.0105295 + 0.00114515i) q^{4} +(-0.435891 - 1.29368i) q^{5} +(-2.43438 + 0.324915i) q^{6} +(-0.863119 + 1.27300i) q^{7} +(-0.456369 - 2.78373i) q^{8} +(-2.96308 + 0.469186i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{57}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0767664	−	1.41587i	−0.0542821	−	1.00117i	−0.891133	−	0.453743i	\(-0.850089\pi\)
							0.836851	−	0.547431i	\(-0.184394\pi\)
\(3\)	−0.135861	−	1.72671i	−0.0784393	−	0.996919i
							\(5\)	−0.435891	−	1.29368i	−0.194936	−	0.578551i	0.804954	−	0.593337i	\(-0.202190\pi\)
−0.999891	+	0.0147862i	\(0.995293\pi\)
\(7\)	−0.863119	+	1.27300i	−0.326228	+	0.481151i	−0.955069	−	0.296383i	\(-0.904219\pi\)
							0.628841	+	0.777534i	\(0.283530\pi\)
\(11\)	4.10679	+	0.903973i	1.23824	+	0.272558i	0.785376	−	0.619019i	\(-0.212470\pi\)
							0.452869	+	0.891577i	\(0.350401\pi\)
\(13\)	1.27734	+	1.08498i	0.354270	+	0.300919i	0.806794	−	0.590833i	\(-0.201201\pi\)
							−0.452525	+	0.891752i	\(0.649477\pi\)
\(17\)	−1.70949	+	1.15906i	−0.414612	+	0.281114i	−0.750606	−	0.660750i	\(-0.770238\pi\)
							0.335994	+	0.941864i	\(0.390928\pi\)
\(19\)	−0.106239	−	0.200388i	−0.0243729	−	0.0459722i	0.871027	−	0.491236i	\(-0.163455\pi\)
							−0.895399	+	0.445264i	\(0.853110\pi\)
\(23\)	−2.98538	−	2.82791i	−0.622496	−	0.589659i	0.310046	−	0.950722i	\(-0.399656\pi\)
							−0.932542	+	0.361062i	\(0.882414\pi\)
\(29\)	1.83130	+	0.0992904i	0.340065	+	0.0184378i	0.223381	−	0.974731i	\(-0.428291\pi\)
							0.116684	+	0.993169i	\(0.462774\pi\)
\(31\)	5.88013	+	3.11745i	1.05610	+	0.559910i	0.903593	−	0.428392i	\(-0.140920\pi\)
							0.152509	+	0.988302i	\(0.451265\pi\)
\(37\)	9.18142	+	1.50522i	1.50942	+	0.247456i	0.858942	−	0.512072i	\(-0.171122\pi\)
							0.650474	+	0.759528i	\(0.274570\pi\)
\(41\)	−0.553401	−	0.584218i	−0.0864267	−	0.0912395i	0.681347	−	0.731960i	\(-0.261394\pi\)
							−0.767774	+	0.640721i	\(0.778636\pi\)
\(43\)	−1.76942	−	8.03857i	−0.269834	−	1.22587i	−0.895245	−	0.445574i	\(-0.853000\pi\)
							0.625411	−	0.780296i	\(-0.284931\pi\)
\(47\)	−4.56389	−	1.53775i	−0.665712	−	0.224305i	−0.0338991	−	0.999425i	\(-0.510792\pi\)
							−0.631813	+	0.775121i	\(0.717689\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.89.6	yes	504
3.2	odd	2	inner	177.2.f.a.89.13	yes	504
59.2	odd	58	inner	177.2.f.a.2.13	yes	504
177.2	even	58	inner	177.2.f.a.2.6	✓	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.2.6	✓	504	177.2	even	58	inner
177.2.f.a.2.13	yes	504	59.2	odd	58	inner
177.2.f.a.89.6	yes	504	1.1	even	1	trivial
177.2.f.a.89.13	yes	504	3.2	odd	2	inner