Embedded newform 177.2.f.a.56.16

• Label: 177.2.f.a.56.16
• Level: $177$
• Weight: $2$
• Character: 177.56
• 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			56.16
Character	\(\chi\)	\(=\)	177.56
Dual form			177.2.f.a.98.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.88661 - 0.872839i) q^{2} +(-1.71360 + 0.252163i) q^{3} +(1.50268 - 1.76909i) q^{4} +(1.06337 - 1.76733i) q^{5} +(-3.01279 + 1.97143i) q^{6} +(0.146905 - 2.70950i) q^{7} +(0.178597 - 0.643249i) q^{8} +(2.87283 - 0.864210i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{21}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.88661	−	0.872839i	1.33404	−	0.617190i	0.382279	−	0.924047i	\(-0.375139\pi\)
							0.951756	+	0.306856i	\(0.0992771\pi\)
\(3\)	−1.71360	+	0.252163i	−0.989346	+	0.145586i
							\(5\)	1.06337	−	1.76733i	0.475552	−	0.790373i	−0.522508	−	0.852634i	\(-0.675004\pi\)
0.998060	+	0.0622611i	\(0.0198311\pi\)
\(7\)	0.146905	−	2.70950i	0.0555247	−	1.02409i	−0.829487	−	0.558526i	\(-0.811367\pi\)
							0.885012	−	0.465568i	\(-0.154150\pi\)
\(11\)	0.130558	−	0.796366i	0.0393646	−	0.240113i	−0.959814	−	0.280637i	\(-0.909454\pi\)
							0.999179	+	0.0405236i	\(0.0129026\pi\)
\(13\)	3.87459	+	5.09693i	1.07462	+	1.41363i	0.903605	+	0.428366i	\(0.140911\pi\)
							0.171012	+	0.985269i	\(0.445296\pi\)
\(17\)	−6.60805	+	0.358278i	−1.60269	+	0.0868952i	−0.833644	−	0.552302i	\(-0.813750\pi\)
							−0.769043	+	0.639197i	\(0.779267\pi\)
\(19\)	−3.42483	+	3.24417i	−0.785710	+	0.744264i	−0.971170	−	0.238388i	\(-0.923381\pi\)
							0.185460	+	0.982652i	\(0.440622\pi\)
\(23\)	−1.86637	+	0.410819i	−0.389165	+	0.0856616i	−0.405243	−	0.914209i	\(-0.632813\pi\)
							0.0160781	+	0.999871i	\(0.494882\pi\)
\(29\)	2.17867	−	4.70912i	0.404569	−	0.874462i	−0.593028	−	0.805182i	\(-0.702068\pi\)
							0.997597	−	0.0692799i	\(-0.0220702\pi\)
\(31\)	2.95132	−	3.11567i	0.530073	−	0.559592i	−0.404877	−	0.914371i	\(-0.632686\pi\)
							0.934950	+	0.354780i	\(0.115444\pi\)
\(37\)	5.44042	−	1.51053i	0.894400	−	0.248329i	0.210230	−	0.977652i	\(-0.432579\pi\)
							0.684170	+	0.729323i	\(0.260165\pi\)
\(41\)	−1.42093	+	6.45537i	−0.221913	+	1.00816i	0.725519	+	0.688202i	\(0.241600\pi\)
							−0.947432	+	0.319957i	\(0.896331\pi\)
\(43\)	−4.07815	+	0.668578i	−0.621912	+	0.101957i	−0.464496	−	0.885575i	\(-0.653765\pi\)
							−0.157416	+	0.987532i	\(0.550316\pi\)
\(47\)	3.00532	−	1.80824i	0.438371	−	0.263759i	−0.279215	−	0.960229i	\(-0.590074\pi\)
							0.717587	+	0.696469i	\(0.245247\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.56.16	yes	504
3.2	odd	2	inner	177.2.f.a.56.3	✓	504
59.39	odd	58	inner	177.2.f.a.98.3	yes	504
177.98	even	58	inner	177.2.f.a.98.16	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.56.3	✓	504	3.2	odd	2	inner
177.2.f.a.56.16	yes	504	1.1	even	1	trivial
177.2.f.a.98.3	yes	504	59.39	odd	58	inner
177.2.f.a.98.16	yes	504	177.98	even	58	inner