Embedded newform 177.2.f.a.23.13

• Label: 177.2.f.a.23.13
• Level: $177$
• Weight: $2$
• Character: 177.23
• 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			23.13
Character	\(\chi\)	\(=\)	177.23
Dual form			177.2.f.a.77.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.703454 - 0.666347i) q^{2} +(-0.733345 + 1.56914i) q^{3} +(-0.0574486 + 1.05958i) q^{4} +(-0.383857 - 0.0629301i) q^{5} +(0.529719 + 1.59248i) q^{6} +(-0.729572 + 1.37612i) q^{7} +(1.92021 + 2.26064i) q^{8} +(-1.92441 - 2.30144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{15}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.703454	−	0.666347i	0.497417	−	0.471179i	−0.397378	−	0.917655i	\(-0.630080\pi\)
							0.894796	+	0.446476i	\(0.147321\pi\)
\(3\)	−0.733345	+	1.56914i	−0.423397	+	0.905944i
							\(5\)	−0.383857	−	0.0629301i	−0.171666	−	0.0281432i	0.0753359	−	0.997158i	\(-0.475997\pi\)
−0.247002	+	0.969015i	\(0.579445\pi\)
\(7\)	−0.729572	+	1.37612i	−0.275752	+	0.520124i	−0.982246	−	0.187595i	\(-0.939931\pi\)
							0.706494	+	0.707719i	\(0.250276\pi\)
\(11\)	2.37748	−	0.258567i	0.716837	−	0.0779607i	0.257571	−	0.966259i	\(-0.417078\pi\)
							0.459266	+	0.888299i	\(0.348112\pi\)
\(13\)	−2.83962	+	6.13774i	−0.787569	+	1.70230i	−0.0787687	+	0.996893i	\(0.525099\pi\)
							−0.708800	+	0.705409i	\(0.750763\pi\)
\(17\)	5.86001	−	3.10678i	1.42126	−	0.753505i	0.432307	−	0.901726i	\(-0.357700\pi\)
							0.988954	+	0.148221i	\(0.0473548\pi\)
\(19\)	4.92825	−	2.96523i	1.13062	−	0.680270i	0.177877	−	0.984053i	\(-0.443077\pi\)
							0.952741	+	0.303782i	\(0.0982496\pi\)
\(23\)	−2.59272	−	6.50724i	−0.540620	−	1.35685i	−0.904924	−	0.425574i	\(-0.860072\pi\)
							0.364304	−	0.931280i	\(-0.381307\pi\)
\(29\)	−1.89426	+	1.99975i	−0.351756	+	0.371344i	−0.877729	−	0.479157i	\(-0.840943\pi\)
							0.525973	+	0.850501i	\(0.323701\pi\)
\(31\)	4.19065	−	6.96492i	0.752663	−	1.25094i	−0.209325	−	0.977846i	\(-0.567127\pi\)
							0.961989	−	0.273090i	\(-0.0880457\pi\)
\(37\)	3.34683	+	2.84282i	0.550215	+	0.467357i	0.878849	−	0.477101i	\(-0.158312\pi\)
							−0.328634	+	0.944457i	\(0.606588\pi\)
\(41\)	4.48996	+	1.78897i	0.701215	+	0.279389i	0.693372	−	0.720580i	\(-0.256124\pi\)
							0.00784256	+	0.999969i	\(0.497504\pi\)
\(43\)	−0.462789	+	4.25528i	−0.0705747	+	0.648924i	0.904412	+	0.426660i	\(0.140310\pi\)
							−0.974987	+	0.222263i	\(0.928656\pi\)
\(47\)	0.416590	+	2.54109i	0.0607659	+	0.370656i	0.999651	+	0.0264244i	\(0.00841214\pi\)
							−0.938885	+	0.344231i	\(0.888140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.23.13	yes	504
3.2	odd	2	inner	177.2.f.a.23.6	✓	504
59.18	odd	58	inner	177.2.f.a.77.6	yes	504
177.77	even	58	inner	177.2.f.a.77.13	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.23.6	✓	504	3.2	odd	2	inner
177.2.f.a.23.13	yes	504	1.1	even	1	trivial
177.2.f.a.77.6	yes	504	59.18	odd	58	inner
177.2.f.a.77.13	yes	504	177.77	even	58	inner