Embedded newform 177.3.h.a.71.15

• Label: 177.3.h.a.71.15
• Level: $177$
• Weight: $3$
• Character: 177.71
• Analytic conductor: $4.823$
• Analytic rank: $0$
• Dimension: $1064$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.82290067918\)
Analytic rank:	\(0\)
Dimension:	\(1064\)
Relative dimension:	\(38\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			71.15
Character	\(\chi\)	\(=\)	177.71
Dual form			177.3.h.a.5.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.322077 - 0.955892i) q^{2} +(-0.589478 + 2.94152i) q^{3} +(2.37438 - 1.80495i) q^{4} +(5.46847 - 2.17884i) q^{5} +(3.00163 - 0.383919i) q^{6} +(3.90866 + 1.80834i) q^{7} +(-5.82961 - 3.95258i) q^{8} +(-8.30503 - 3.46792i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1064 q - 29 q^{3} + 18 q^{4} - 21 q^{6} - 46 q^{7} - 49 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{26}{29}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.322077	−	0.955892i	−0.161039	−	0.477946i	0.836508	−	0.547955i	\(-0.184594\pi\)
							−0.997546	+	0.0700094i	\(0.977697\pi\)
\(3\)	−0.589478	+	2.94152i	−0.196493	+	0.980505i
							\(5\)	5.46847	−	2.17884i	1.09369	−	0.435767i	0.247633	−	0.968854i	\(-0.420347\pi\)
0.846061	+	0.533087i	\(0.178968\pi\)
\(7\)	3.90866	+	1.80834i	0.558380	+	0.258334i	0.678718	−	0.734399i	\(-0.262536\pi\)
							−0.120338	+	0.992733i	\(0.538398\pi\)
\(11\)	0.609271	−	0.169163i	0.0553883	−	0.0153785i	−0.239724	−	0.970841i	\(-0.577057\pi\)
							0.295112	+	0.955463i	\(0.404643\pi\)
\(13\)	5.83434	+	11.0047i	0.448795	+	0.846517i	0.999927	+	0.0120991i	\(0.00385137\pi\)
							−0.551132	+	0.834418i	\(0.685804\pi\)
\(17\)	−10.1910	−	22.0274i	−0.599468	−	1.29573i	−0.935316	−	0.353812i	\(-0.884885\pi\)
							0.335849	−	0.941916i	\(-0.390977\pi\)
\(19\)	35.3427	+	7.77951i	1.86014	+	0.409448i	0.995648	−	0.0931959i	\(-0.0297083\pi\)
							0.864494	+	0.502644i	\(0.167639\pi\)
\(23\)	27.2467	−	4.46686i	1.18464	−	0.194211i	0.462921	−	0.886400i	\(-0.346801\pi\)
							0.721717	+	0.692188i	\(0.243353\pi\)
\(29\)	−8.57574	+	25.4519i	−0.295715	+	0.877651i	0.692070	+	0.721830i	\(0.256699\pi\)
							−0.987785	+	0.155821i	\(0.950198\pi\)
\(31\)	−47.8950	+	10.5425i	−1.54500	+	0.340080i	−0.904024	−	0.427481i	\(-0.859401\pi\)
							−0.640975	+	0.767561i	\(0.721470\pi\)
\(37\)	25.6167	+	37.7818i	0.692343	+	1.02113i	0.997626	+	0.0688681i	\(0.0219388\pi\)
							−0.305283	+	0.952262i	\(0.598751\pi\)
\(41\)	−20.9437	−	3.43355i	−0.510823	−	0.0837452i	−0.0991422	−	0.995073i	\(-0.531610\pi\)
							−0.411681	+	0.911328i	\(0.635058\pi\)
\(43\)	12.4033	−	44.6728i	0.288450	−	1.03890i	−0.667316	−	0.744774i	\(-0.732557\pi\)
							0.955766	−	0.294128i	\(-0.0950291\pi\)
\(47\)	−44.4828	−	17.7236i	−0.946442	−	0.377097i	−0.154706	−	0.987960i	\(-0.549443\pi\)
							−0.791736	+	0.610863i	\(0.790822\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.3.h.a.71.15	yes	1064
3.2	odd	2	inner	177.3.h.a.71.24	yes	1064
59.5	even	29	inner	177.3.h.a.5.24	yes	1064
177.5	odd	58	inner	177.3.h.a.5.15	✓	1064

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.h.a.5.15	✓	1064	177.5	odd	58	inner
177.3.h.a.5.24	yes	1064	59.5	even	29	inner
177.3.h.a.71.15	yes	1064	1.1	even	1	trivial
177.3.h.a.71.24	yes	1064	3.2	odd	2	inner