Embedded newform 177.4.f.a.89.8

• Label: 177.4.f.a.89.8
• Level: $177$
• Weight: $4$
• Character: 177.89
• Analytic conductor: $10.443$
• Analytic rank: $0$
• Dimension: $1624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4433380710\)
Analytic rank:	\(0\)
Dimension:	\(1624\)
Relative dimension:	\(58\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			89.8
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.238865 - 4.40561i) q^{2} +(-4.39778 + 2.76759i) q^{3} +(-11.3992 + 1.23974i) q^{4} +(-6.88505 - 20.4341i) q^{5} +(13.2434 + 18.7138i) q^{6} +(12.3388 - 18.1984i) q^{7} +(2.47433 + 15.0928i) q^{8} +(11.6809 - 24.3425i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{57}{58}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.238865	−	4.40561i	−0.0844516	−	1.55762i	−0.667146	−	0.744927i	\(-0.732484\pi\)
							0.582695	−	0.812691i	\(-0.301998\pi\)
\(3\)	−4.39778	+	2.76759i	−0.846352	+	0.532624i
							\(5\)	−6.88505	−	20.4341i	−0.615817	−	1.82768i	−0.558793	−	0.829307i	\(-0.688735\pi\)
−0.0570245	−	0.998373i	\(-0.518161\pi\)
\(7\)	12.3388	−	18.1984i	0.666235	−	0.982623i	−0.332977	−	0.942935i	\(-0.608053\pi\)
							0.999212	−	0.0396880i	\(-0.0126364\pi\)
\(11\)	−46.5935	−	10.2560i	−1.27713	−	0.281119i	−0.475916	−	0.879491i	\(-0.657884\pi\)
							−0.801218	+	0.598372i	\(0.795815\pi\)
\(13\)	15.2475	+	12.9514i	0.325300	+	0.276313i	0.795123	−	0.606448i	\(-0.207406\pi\)
							−0.469823	+	0.882761i	\(0.655682\pi\)
\(17\)	68.4735	−	46.4262i	0.976898	−	0.662353i	0.0354006	−	0.999373i	\(-0.488729\pi\)
							0.941498	+	0.337020i	\(0.109419\pi\)
\(19\)	27.3237	+	51.5379i	0.329920	+	0.622295i	0.991870	−	0.127252i	\(-0.0406158\pi\)
							−0.661950	+	0.749548i	\(0.730271\pi\)
\(23\)	4.92299	+	4.66330i	0.0446311	+	0.0422768i	0.709690	−	0.704514i	\(-0.248835\pi\)
							−0.665059	+	0.746791i	\(0.731594\pi\)
\(29\)	201.980	+	10.9511i	1.29334	+	0.0701227i	0.687921	−	0.725786i	\(-0.258524\pi\)
							0.605417	+	0.795908i	\(0.293006\pi\)
\(31\)	−73.5277	−	38.9819i	−0.425999	−	0.225850i	0.241593	−	0.970378i	\(-0.422330\pi\)
							−0.667592	+	0.744527i	\(0.732675\pi\)
\(37\)	327.232	+	53.6470i	1.45396	+	0.238365i	0.836384	−	0.548144i	\(-0.184665\pi\)
							0.617579	+	0.786509i	\(0.288114\pi\)
\(41\)	−282.573	−	298.308i	−1.07635	−	1.13629i	−0.990150	−	0.140013i	\(-0.955285\pi\)
							−0.0862024	−	0.996278i	\(-0.527473\pi\)
\(43\)	−74.2998	−	337.547i	−0.263503	−	1.19710i	−0.903453	−	0.428688i	\(-0.858976\pi\)
							0.639950	−	0.768417i	\(-0.278955\pi\)
\(47\)	−138.274	−	46.5899i	−0.429135	−	0.144592i	0.0964418	−	0.995339i	\(-0.469254\pi\)
							−0.525577	+	0.850746i	\(0.676150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.8	yes	1624
3.2	odd	2	inner	177.4.f.a.89.51	yes	1624
59.2	odd	58	inner	177.4.f.a.2.51	yes	1624
177.2	even	58	inner	177.4.f.a.2.8	✓	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.8	✓	1624	177.2	even	58	inner
177.4.f.a.2.51	yes	1624	59.2	odd	58	inner
177.4.f.a.89.8	yes	1624	1.1	even	1	trivial
177.4.f.a.89.51	yes	1624	3.2	odd	2	inner