Embedded newform 177.2.f.a.11.1

• Label: 177.2.f.a.11.1
• Level: $177$
• Weight: $2$
• Character: 177.11
• 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			11.1
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.53253 + 0.557453i) q^{2} +(1.58594 - 0.696273i) q^{3} +(4.28782 - 1.98376i) q^{4} +(2.64049 + 0.733128i) q^{5} +(-3.62830 + 2.64742i) q^{6} +(0.280620 - 0.265818i) q^{7} +(-5.62441 + 4.27556i) q^{8} +(2.03041 - 2.20849i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{25}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.53253	+	0.557453i	−1.79077	+	0.394179i	−0.981223	−	0.192875i	\(-0.938219\pi\)
							−0.809548	+	0.587053i	\(0.800288\pi\)
\(3\)	1.58594	−	0.696273i	0.915642	−	0.401994i
							\(5\)	2.64049	+	0.733128i	1.18086	+	0.327865i	0.801849	−	0.597526i	\(-0.203850\pi\)
0.379014	+	0.925391i	\(0.376263\pi\)
\(7\)	0.280620	−	0.265818i	0.106064	−	0.100470i	−0.632830	−	0.774291i	\(-0.718107\pi\)
							0.738894	+	0.673821i	\(0.235348\pi\)
\(11\)	−3.20904	−	3.77798i	−0.967562	−	1.13910i	−0.990242	−	0.139360i	\(-0.955496\pi\)
							0.0226795	−	0.999743i	\(-0.492780\pi\)
\(13\)	0.864286	+	2.56511i	0.239710	+	0.711433i	0.998213	+	0.0597634i	\(0.0190346\pi\)
							−0.758503	+	0.651670i	\(0.774069\pi\)
\(17\)	−1.81589	+	1.91701i	−0.440418	+	0.464944i	−0.907978	−	0.419018i	\(-0.862375\pi\)
							0.467560	+	0.883961i	\(0.345133\pi\)
\(19\)	0.241801	+	0.606874i	0.0554729	+	0.139226i	0.954101	−	0.299484i	\(-0.0968146\pi\)
							−0.898628	+	0.438711i	\(0.855435\pi\)
\(23\)	8.96389	−	0.974881i	1.86910	−	0.203277i	0.897680	−	0.440648i	\(-0.145251\pi\)
							0.971419	+	0.237371i	\(0.0762856\pi\)
\(29\)	−1.52907	+	6.94662i	−0.283941	+	1.28996i	0.591372	+	0.806399i	\(0.298586\pi\)
							−0.875313	+	0.483557i	\(0.839345\pi\)
\(31\)	−7.14408	−	2.84646i	−1.28311	−	0.511239i	−0.373862	−	0.927484i	\(-0.621967\pi\)
							−0.909252	+	0.416245i	\(0.863346\pi\)
\(37\)	−1.40641	+	1.85011i	−0.231213	+	0.304155i	−0.897039	−	0.441952i	\(-0.854286\pi\)
							0.665826	+	0.746107i	\(0.268079\pi\)
\(41\)	1.11595	−	10.2610i	0.174282	−	1.60250i	−0.501338	−	0.865251i	\(-0.667159\pi\)
							0.675621	−	0.737249i	\(-0.263876\pi\)
\(43\)	0.862720	+	0.732801i	0.131564	+	0.111751i	0.710752	−	0.703442i	\(-0.248355\pi\)
							−0.579189	+	0.815193i	\(0.696631\pi\)
\(47\)	2.34919	+	8.46103i	0.342665	+	1.23417i	0.911357	+	0.411617i	\(0.135036\pi\)
							−0.568692	+	0.822551i	\(0.692550\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.11.1	✓	504
3.2	odd	2	inner	177.2.f.a.11.18	yes	504
59.43	odd	58	inner	177.2.f.a.161.18	yes	504
177.161	even	58	inner	177.2.f.a.161.1	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.11.1	✓	504	1.1	even	1	trivial
177.2.f.a.11.18	yes	504	3.2	odd	2	inner
177.2.f.a.161.1	yes	504	177.161	even	58	inner
177.2.f.a.161.18	yes	504	59.43	odd	58	inner