Embedded newform 177.2.f.a.11.18

• Label: 177.2.f.a.11.18
• 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			11.18
Character	\(\chi\)	\(=\)	177.11
Dual form			177.2.f.a.161.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.53253 - 0.557453i) q^{2} +(-1.28060 - 1.16622i) q^{3} +(4.28782 - 1.98376i) q^{4} +(-2.64049 - 0.733128i) q^{5} +(-3.89327 - 2.23961i) q^{6} +(0.280620 - 0.265818i) q^{7} +(5.62441 - 4.27556i) q^{8} +(0.279869 + 2.98692i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

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	0.809548	−	0.587053i	\(-0.199712\pi\)
							0.981223	+	0.192875i	\(0.0617811\pi\)
\(3\)	−1.28060	−	1.16622i	−0.739354	−	0.673317i
							\(5\)	−2.64049	−	0.733128i	−1.18086	−	0.327865i	−0.379014	−	0.925391i	\(-0.623737\pi\)
−0.801849	+	0.597526i	\(0.796150\pi\)
\(7\)	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	\(0.0445044\pi\)
							−0.0226795	+	0.999743i	\(0.507220\pi\)
\(13\)	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	−0.467560	−	0.883961i	\(-0.654867\pi\)
							0.907978	+	0.419018i	\(0.137625\pi\)
\(19\)	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	−0.971419	−	0.237371i	\(-0.923714\pi\)
							−0.897680	+	0.440648i	\(0.854749\pi\)
\(29\)	1.52907	−	6.94662i	0.283941	−	1.28996i	−0.591372	−	0.806399i	\(-0.701414\pi\)
							0.875313	−	0.483557i	\(-0.160655\pi\)
\(31\)	−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	\(0.332841\pi\)
							−0.675621	+	0.737249i	\(0.736124\pi\)
\(43\)	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	\(-0.864964\pi\)
							0.568692	−	0.822551i	\(-0.307450\pi\)

(See \(a_n\) instead)

Twists

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

    

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