Embedded newform 177.2.f.a.101.8

• Label: 177.2.f.a.101.8
• Level: $177$
• Weight: $2$
• Character: 177.101
• 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			101.8
Character	\(\chi\)	\(=\)	177.101
Dual form			177.2.f.a.170.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.369271 + 0.544634i) q^{2} +(0.293354 - 1.70703i) q^{3} +(0.580011 + 1.45572i) q^{4} +(-1.46523 + 3.16705i) q^{5} +(0.821377 + 0.790126i) q^{6} +(0.554487 + 1.99708i) q^{7} +(-2.29228 - 0.504569i) q^{8} +(-2.82789 - 1.00153i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{11}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.369271	+	0.544634i	−0.261114	+	0.385114i	−0.935536	−	0.353232i	\(-0.885083\pi\)
							0.674422	+	0.738346i	\(0.264393\pi\)
\(3\)	0.293354	−	1.70703i	0.169368	−	0.985553i
							\(5\)	−1.46523	+	3.16705i	−0.655272	+	1.41635i	0.239975	+	0.970779i	\(0.422861\pi\)
−0.895248	+	0.445569i	\(0.853001\pi\)
\(7\)	0.554487	+	1.99708i	0.209576	+	0.754825i	0.991348	+	0.131261i	\(0.0419025\pi\)
							−0.781772	+	0.623565i	\(0.785684\pi\)
\(11\)	2.57838	+	2.44237i	0.777410	+	0.736402i	0.969531	−	0.244969i	\(-0.0787779\pi\)
							−0.192121	+	0.981371i	\(0.561537\pi\)
\(13\)	0.370611	−	3.40771i	0.102789	−	0.945130i	−0.823244	−	0.567688i	\(-0.807838\pi\)
							0.926033	−	0.377442i	\(-0.123196\pi\)
\(17\)	2.77090	+	0.769338i	0.672043	+	0.186592i	0.586755	−	0.809765i	\(-0.300405\pi\)
							0.0852879	+	0.996356i	\(0.472819\pi\)
\(19\)	5.15631	+	3.91972i	1.18294	+	0.899246i	0.996531	−	0.0832262i	\(-0.0265224\pi\)
							0.186408	+	0.982473i	\(0.440316\pi\)
\(23\)	−3.83071	−	7.22549i	−0.798758	−	1.50662i	−0.861027	−	0.508559i	\(-0.830178\pi\)
							0.0622690	−	0.998059i	\(-0.480166\pi\)
\(29\)	4.32051	−	2.92938i	0.802299	−	0.543972i	−0.0897214	−	0.995967i	\(-0.528598\pi\)
							0.892021	+	0.451994i	\(0.149287\pi\)
\(31\)	−2.30637	−	3.03398i	−0.414236	−	0.544918i	0.540900	−	0.841087i	\(-0.318084\pi\)
							−0.955136	+	0.296169i	\(0.904291\pi\)
\(37\)	1.23453	+	5.60852i	0.202955	+	0.922036i	0.962437	+	0.271506i	\(0.0875218\pi\)
							−0.759481	+	0.650529i	\(0.774547\pi\)
\(41\)	1.75235	+	0.929039i	0.273672	+	0.145092i	0.599592	−	0.800306i	\(-0.295329\pi\)
							−0.325921	+	0.945397i	\(0.605674\pi\)
\(43\)	1.47547	+	1.55764i	0.225008	+	0.237538i	0.828718	−	0.559666i	\(-0.189070\pi\)
							−0.603711	+	0.797203i	\(0.706312\pi\)
\(47\)	−1.65146	+	0.764047i	−0.240890	+	0.111448i	−0.536620	−	0.843824i	\(-0.680299\pi\)
							0.295730	+	0.955272i	\(0.404437\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.101.8	✓	504
3.2	odd	2	inner	177.2.f.a.101.11	yes	504
59.52	odd	58	inner	177.2.f.a.170.11	yes	504
177.170	even	58	inner	177.2.f.a.170.8	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.101.8	✓	504	1.1	even	1	trivial
177.2.f.a.101.11	yes	504	3.2	odd	2	inner
177.2.f.a.170.8	yes	504	177.170	even	58	inner
177.2.f.a.170.11	yes	504	59.52	odd	58	inner