Embedded newform 177.4.f.a.89.10

• Label: 177.4.f.a.89.10
• 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			89.10
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.214445 - 3.95521i) q^{2} +(4.30093 - 2.91582i) q^{3} +(-7.64463 + 0.831403i) q^{4} +(-4.87721 - 14.4750i) q^{5} +(-12.4550 - 16.3858i) q^{6} +(8.97536 - 13.2377i) q^{7} +(-0.198845 - 1.21290i) q^{8} +(9.99602 - 25.0815i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.214445	−	3.95521i	−0.0758179	−	1.39838i	−0.750792	−	0.660538i	\(-0.770328\pi\)
							0.674975	−	0.737841i	\(-0.264155\pi\)
\(3\)	4.30093	−	2.91582i	0.827715	−	0.561149i
							\(5\)	−4.87721	−	14.4750i	−0.436231	−	1.29469i	−0.910209	−	0.414150i	\(-0.864079\pi\)
0.473978	−	0.880537i	\(-0.342818\pi\)
\(7\)	8.97536	−	13.2377i	0.484624	−	0.714766i	−0.504244	−	0.863561i	\(-0.668229\pi\)
							0.988868	+	0.148795i	\(0.0475393\pi\)
\(11\)	49.8206	+	10.9663i	1.36559	+	0.300589i	0.836495	−	0.547975i	\(-0.184601\pi\)
							0.529093	+	0.848564i	\(0.322532\pi\)
\(13\)	52.0156	+	44.1824i	1.10973	+	0.942615i	0.998688	−	0.0512004i	\(-0.0163047\pi\)
							0.111044	+	0.993816i	\(0.464581\pi\)
\(17\)	−66.4626	+	45.0628i	−0.948209	+	0.642901i	−0.934105	−	0.356999i	\(-0.883800\pi\)
							−0.0141040	+	0.999901i	\(0.504490\pi\)
\(19\)	22.2276	+	41.9257i	0.268387	+	0.506232i	0.980672	−	0.195658i	\(-0.0626843\pi\)
							−0.712285	+	0.701891i	\(0.752339\pi\)
\(23\)	80.4565	+	76.2125i	0.729406	+	0.690930i	0.959333	−	0.282278i	\(-0.0910899\pi\)
							−0.229926	+	0.973208i	\(0.573849\pi\)
\(29\)	−265.523	−	14.3962i	−1.70022	−	0.0921833i	−0.821710	−	0.569905i	\(-0.806980\pi\)
							−0.878511	+	0.477722i	\(0.841463\pi\)
\(31\)	−43.3830	−	23.0002i	−0.251349	−	0.133257i	0.337971	−	0.941157i	\(-0.390259\pi\)
							−0.589320	+	0.807900i	\(0.700604\pi\)
\(37\)	270.667	+	44.3736i	1.20263	+	0.197162i	0.729594	−	0.683880i	\(-0.239709\pi\)
							0.473038	+	0.881042i	\(0.343157\pi\)
\(41\)	77.4960	+	81.8116i	0.295191	+	0.311630i	0.856673	−	0.515859i	\(-0.172527\pi\)
							−0.561482	+	0.827489i	\(0.689769\pi\)
\(43\)	20.5742	+	93.4697i	0.0729661	+	0.331488i	0.998936	−	0.0461079i	\(-0.0146818\pi\)
							−0.925970	+	0.377596i	\(0.876751\pi\)
\(47\)	246.380	+	83.0151i	0.764643	+	0.257638i	0.674484	−	0.738289i	\(-0.264366\pi\)
							0.0901587	+	0.995927i	\(0.471263\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.10	yes	1624
3.2	odd	2	inner	177.4.f.a.89.49	yes	1624
59.2	odd	58	inner	177.4.f.a.2.49	yes	1624
177.2	even	58	inner	177.4.f.a.2.10	✓	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.10	✓	1624	177.2	even	58	inner
177.4.f.a.2.49	yes	1624	59.2	odd	58	inner
177.4.f.a.89.10	yes	1624	1.1	even	1	trivial
177.4.f.a.89.49	yes	1624	3.2	odd	2	inner