Embedded newform 177.4.f.a.2.20

• Label: 177.4.f.a.2.20
• Level: $177$
• Weight: $4$
• Character: 177.2
• 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			2.20
Character	\(\chi\)	\(=\)	177.2
Dual form			177.4.f.a.89.20

$q$-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 \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.f.a.2.20	✓	1624	1.1	even	1	trivial
177.4.f.a.2.39	yes	1624	3.2	odd	2	inner
177.4.f.a.89.20	yes	1624	177.89	even	58	inner
177.4.f.a.89.39	yes	1624	59.30	odd	58	inner