Embedded newform 177.4.f.a.89.20

• Label: 177.4.f.a.89.20
• 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			89.20
Character	\(\chi\)	\(=\)	177.89
Dual form			177.4.f.a.2.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.100013 - 1.84463i) q^{2} +(2.00907 + 4.79204i) q^{3} +(4.56043 - 0.495977i) q^{4} +(6.12126 + 18.1673i) q^{5} +(8.63862 - 4.18527i) q^{6} +(1.32430 - 1.95320i) q^{7} +(-3.76193 - 22.9468i) q^{8} +(-18.9273 + 19.2551i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1624 q - 21 q^{3} - 278 q^{4} - 29 q^{6} - 42 q^{7} - 25 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.100013	−	1.84463i	−0.0353600	−	0.652177i	−0.961468	−	0.274918i	\(-0.911349\pi\)
							0.926108	−	0.377259i	\(-0.123133\pi\)
\(3\)	2.00907	+	4.79204i	0.386646	+	0.922228i
							\(5\)	6.12126	+	18.1673i	0.547502	+	1.62493i	0.761551	+	0.648105i	\(0.224438\pi\)
−0.214048	+	0.976823i	\(0.568665\pi\)
\(7\)	1.32430	−	1.95320i	0.0715057	−	0.105463i	−0.790255	−	0.612779i	\(-0.790052\pi\)
							0.861760	+	0.507315i	\(0.169362\pi\)
\(11\)	34.4557	+	7.58428i	0.944436	+	0.207886i	0.660389	−	0.750924i	\(-0.270392\pi\)
							0.284048	+	0.958810i	\(0.408323\pi\)
\(13\)	−2.61388	−	2.22025i	−0.0557661	−	0.0473681i	0.619070	−	0.785336i	\(-0.287510\pi\)
							−0.674836	+	0.737968i	\(0.735786\pi\)
\(17\)	−95.2636	+	64.5903i	−1.35911	+	0.921498i	−0.999931	−	0.0117719i	\(-0.996253\pi\)
							−0.359177	+	0.933270i	\(0.616942\pi\)
\(19\)	−31.0271	−	58.5233i	−0.374637	−	0.706640i	0.622471	−	0.782643i	\(-0.286129\pi\)
							−0.997108	+	0.0760032i	\(0.975784\pi\)
\(23\)	−26.8145	−	25.4000i	−0.243096	−	0.230272i	0.556445	−	0.830884i	\(-0.312165\pi\)
							−0.799541	+	0.600612i	\(0.794924\pi\)
\(29\)	269.497	+	14.6117i	1.72566	+	0.0935628i	0.890081	−	0.455802i	\(-0.150648\pi\)
							0.835583	+	0.549365i	\(0.185130\pi\)
\(31\)	285.949	+	151.601i	1.65671	+	0.878332i	0.991265	+	0.131885i	\(0.0421029\pi\)
							0.665445	+	0.746447i	\(0.268242\pi\)
\(37\)	217.413	+	35.6431i	0.966015	+	0.158370i	0.624086	−	0.781356i	\(-0.285472\pi\)
							0.341929	+	0.939726i	\(0.388920\pi\)
\(41\)	−272.365	−	287.533i	−1.03747	−	1.09525i	−0.995369	−	0.0961324i	\(-0.969353\pi\)
							−0.0421032	−	0.999113i	\(-0.513406\pi\)
\(43\)	−56.7474	−	257.806i	−0.201253	−	0.914304i	−0.963645	−	0.267187i	\(-0.913906\pi\)
							0.762391	−	0.647116i	\(-0.224025\pi\)
\(47\)	−70.1368	−	23.6318i	−0.217670	−	0.0733416i	0.208354	−	0.978054i	\(-0.433189\pi\)
							−0.426024	+	0.904712i	\(0.640086\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.f.a.89.20	yes	1624
3.2	odd	2	inner	177.4.f.a.89.39	yes	1624
59.2	odd	58	inner	177.4.f.a.2.39	yes	1624
177.2	even	58	inner	177.4.f.a.2.20	✓	1624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.20	✓	1624	177.2	even	58	inner
177.4.f.a.2.39	yes	1624	59.2	odd	58	inner
177.4.f.a.89.20	yes	1624	1.1	even	1	trivial
177.4.f.a.89.39	yes	1624	3.2	odd	2	inner