Embedded newform 177.2.f.a.65.12

• Label: 177.2.f.a.65.12
• Level: $177$
• Weight: $2$
• Character: 177.65
• 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			65.12
Character	\(\chi\)	\(=\)	177.65
Dual form			177.2.f.a.128.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.283536 + 0.711623i) q^{2} +(1.61797 + 0.618213i) q^{3} +(1.02598 - 0.971857i) q^{4} +(-1.06627 + 0.905694i) q^{5} +(0.0188181 + 1.32667i) q^{6} +(-2.13028 + 1.28174i) q^{7} +(2.37295 + 1.09785i) q^{8} +(2.23563 + 2.00049i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{51}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.283536	+	0.711623i	0.200491	+	0.503193i	0.994503	−	0.104708i	\(-0.0333909\pi\)
							−0.794012	+	0.607901i	\(0.792012\pi\)
\(3\)	1.61797	+	0.618213i	0.934133	+	0.356925i
							\(5\)	−1.06627	+	0.905694i	−0.476848	+	0.405039i	−0.853318	−	0.521390i	\(-0.825414\pi\)
0.376470	+	0.926429i	\(0.377138\pi\)
\(7\)	−2.13028	+	1.28174i	−0.805169	+	0.484454i	−0.857691	−	0.514166i	\(-0.828102\pi\)
							0.0525220	+	0.998620i	\(0.483274\pi\)
\(11\)	−0.330818	−	6.10158i	−0.0997453	−	1.83969i	−0.438396	−	0.898782i	\(-0.644453\pi\)
							0.338650	−	0.940912i	\(-0.390030\pi\)
\(13\)	−0.904760	+	4.11036i	−0.250935	+	1.14001i	0.667553	+	0.744562i	\(0.267342\pi\)
							−0.918488	+	0.395448i	\(0.870589\pi\)
\(17\)	0.896296	−	1.48966i	0.217384	−	0.361295i	−0.728906	−	0.684613i	\(-0.759971\pi\)
							0.946290	+	0.323319i	\(0.104799\pi\)
\(19\)	−1.94489	−	7.00485i	−0.446188	−	1.60702i	−0.754881	−	0.655862i	\(-0.772305\pi\)
							0.308693	−	0.951162i	\(-0.400108\pi\)
\(23\)	−2.65976	+	3.92285i	−0.554598	+	0.817971i	−0.996882	−	0.0789037i	\(-0.974858\pi\)
							0.442284	+	0.896875i	\(0.354168\pi\)
\(29\)	−7.17351	−	2.85819i	−1.33209	−	0.530752i	−0.408017	−	0.912974i	\(-0.633780\pi\)
							−0.924070	+	0.382222i	\(0.875159\pi\)
\(31\)	2.76803	+	0.768539i	0.497152	+	0.138034i	0.507049	−	0.861917i	\(-0.330736\pi\)
							−0.00989658	+	0.999951i	\(0.503150\pi\)
\(37\)	−1.90203	−	4.11117i	−0.312692	−	0.675872i	0.685968	−	0.727632i	\(-0.259379\pi\)
							−0.998660	+	0.0517600i	\(0.983517\pi\)
\(41\)	3.35824	−	2.27694i	0.524469	−	0.355599i	−0.270042	−	0.962849i	\(-0.587038\pi\)
							0.794511	+	0.607250i	\(0.207727\pi\)
\(43\)	−5.23689	−	0.283936i	−0.798618	−	0.0432998i	−0.349688	−	0.936866i	\(-0.613712\pi\)
							−0.448931	+	0.893567i	\(0.648195\pi\)
\(47\)	4.44143	−	5.22885i	0.647849	−	0.762707i	−0.335970	−	0.941873i	\(-0.609064\pi\)
							0.983819	+	0.179166i	\(0.0573399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.65.12	yes	504
3.2	odd	2	inner	177.2.f.a.65.7	✓	504
59.10	odd	58	inner	177.2.f.a.128.7	yes	504
177.128	even	58	inner	177.2.f.a.128.12	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.65.7	✓	504	3.2	odd	2	inner
177.2.f.a.65.12	yes	504	1.1	even	1	trivial
177.2.f.a.128.7	yes	504	59.10	odd	58	inner
177.2.f.a.128.12	yes	504	177.128	even	58	inner