Embedded newform 803.2.bk.a.3.16

• Label: 803.2.bk.a.3.16
• Level: $803$
• Weight: $2$
• Character: 803.3
• Analytic conductor: $6.412$
• Analytic rank: $0$
• Dimension: $1152$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 803 = 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	803.bk (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41198728231\)
Analytic rank:	\(0\)
Dimension:	\(1152\)
Relative dimension:	\(72\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			3.16
Character	\(\chi\)	\(=\)	803.3
Dual form			803.2.bk.a.268.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.184259 - 1.75311i) q^{2} +(-1.10662 + 0.359563i) q^{3} +(-1.08315 + 0.230232i) q^{4} +(2.40651 + 2.97179i) q^{5} +(0.834260 + 1.87378i) q^{6} +(0.587833 + 1.15369i) q^{7} +(-0.486248 - 1.49652i) q^{8} +(-1.33173 + 0.967555i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 2 q^{2} + 134 q^{4} - 8 q^{5} - 42 q^{6} - 8 q^{7} - 64 q^{8} + 272 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/803\mathbb{Z}\right)^\times\).

\(n\)	\(78\)	\(585\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.184259	−	1.75311i	−0.130291	−	1.23964i	−0.842898	−	0.538074i	\(-0.819152\pi\)
							0.712607	−	0.701564i	\(-0.247514\pi\)
\(3\)	−1.10662	+	0.359563i	−0.638908	+	0.207594i	−0.610517	−	0.792003i	\(-0.709038\pi\)
							−0.0283910	+	0.999597i	\(0.509038\pi\)
\(5\)	2.40651	+	2.97179i	1.07622	+	1.32903i	0.941074	+	0.338200i	\(0.109818\pi\)
							0.135149	+	0.990825i	\(0.456849\pi\)
\(7\)	0.587833	+	1.15369i	0.222180	+	0.436053i	0.975010	−	0.222162i	\(-0.0713115\pi\)
							−0.752830	+	0.658216i	\(0.771312\pi\)
\(11\)	−2.67158	+	1.96537i	−0.805510	+	0.592582i
							\(13\)	1.77205	−	2.18830i	0.491478	−	0.606925i	−0.469709	−	0.882821i	\(-0.655641\pi\)
0.961188	+	0.275896i	\(0.0889745\pi\)
\(17\)	2.56490	−	0.406240i	0.622080	−	0.0985277i	0.162565	−	0.986698i	\(-0.448023\pi\)
							0.459515	+	0.888170i	\(0.348023\pi\)
\(19\)	−0.153873	+	0.723914i	−0.0353008	+	0.166077i	−0.992267	−	0.124123i	\(-0.960388\pi\)
							0.956966	+	0.290200i	\(0.0937218\pi\)
\(23\)	−3.25034	+	1.87658i	−0.677743	+	0.391295i	−0.799004	−	0.601326i	\(-0.794639\pi\)
							0.121261	+	0.992621i	\(0.461306\pi\)
\(29\)	−0.460713	+	8.79092i	−0.0855522	+	1.63243i	0.532595	+	0.846370i	\(0.321217\pi\)
							−0.618147	+	0.786063i	\(0.712116\pi\)
\(31\)	−5.72298	+	7.06729i	−1.02788	+	1.26932i	−0.0654299	+	0.997857i	\(0.520842\pi\)
							−0.962448	+	0.271466i	\(0.912491\pi\)
\(37\)	5.24027	−	1.11385i	0.861496	−	0.183117i	0.244081	−	0.969755i	\(-0.421514\pi\)
							0.617414	+	0.786638i	\(0.288180\pi\)
\(41\)	−1.86565	+	2.07201i	−0.291366	+	0.323594i	−0.871001	−	0.491282i	\(-0.836528\pi\)
							0.579635	+	0.814876i	\(0.303195\pi\)
\(43\)	7.87099	+	7.87099i	1.20032	+	1.20032i	0.974070	+	0.226245i	\(0.0726450\pi\)
							0.226245	+	0.974070i	\(0.427355\pi\)
\(47\)	−9.52789	+	0.499335i	−1.38979	+	0.0728356i	−0.732480	−	0.680789i	\(-0.761637\pi\)
							−0.657305	+	0.753624i	\(0.728304\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	803.2.bk.a.3.16	✓	1152
11.4	even	5	inner	803.2.bk.a.587.57	yes	1152
73.49	even	12	inner	803.2.bk.a.487.57	yes	1152
803.268	even	60	inner	803.2.bk.a.268.16	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
803.2.bk.a.3.16	✓	1152	1.1	even	1	trivial
803.2.bk.a.268.16	yes	1152	803.268	even	60	inner
803.2.bk.a.487.57	yes	1152	73.49	even	12	inner
803.2.bk.a.587.57	yes	1152	11.4	even	5	inner