Embedded newform 547.2.o.a.4.18

• Label: 547.2.o.a.4.18
• Level: $547$
• Weight: $2$
• Character: 547.4
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $6480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.o (of order \(273\), degree \(144\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(6480\)
Relative dimension:	\(45\) over \(\Q(\zeta_{273})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{273}]$

Embedding invariants

Embedding label			4.18
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.866936 - 0.00997685i) q^{2} +(-0.129306 + 0.566525i) q^{3} +(-1.24799 - 0.0287280i) q^{4} +(-0.554911 + 1.04267i) q^{5} +(0.117752 - 0.489851i) q^{6} +(-1.27973 - 3.15351i) q^{7} +(2.81460 + 0.0972069i) q^{8} +(2.39868 + 1.15514i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/547\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{273}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.866936	−	0.00997685i	−0.613016	−	0.00705470i	−0.295555	−	0.955326i	\(-0.595504\pi\)
							−0.317462	+	0.948271i	\(0.602830\pi\)
\(3\)	−0.129306	+	0.566525i	−0.0746546	+	0.327083i	−0.998441	−	0.0558262i	\(-0.982221\pi\)
							0.923786	+	0.382909i	\(0.125078\pi\)
\(5\)	−0.554911	+	1.04267i	−0.248164	+	0.466297i	−0.975504	−	0.219980i	\(-0.929401\pi\)
							0.727341	+	0.686277i	\(0.240756\pi\)
\(7\)	−1.27973	−	3.15351i	−0.483693	−	1.19191i	−0.951380	−	0.308020i	\(-0.900334\pi\)
							0.467686	−	0.883894i	\(-0.345088\pi\)
\(11\)	0.239237	+	0.249073i	0.0721328	+	0.0750982i	0.756291	−	0.654236i	\(-0.227010\pi\)
							−0.684158	+	0.729334i	\(0.739830\pi\)
\(13\)	−2.45005	−	0.369285i	−0.679521	−	0.102421i	−0.199793	−	0.979838i	\(-0.564027\pi\)
							−0.479729	+	0.877417i	\(0.659265\pi\)
\(17\)	0.407576	−	2.41989i	0.0988517	−	0.586910i	−0.892009	−	0.452018i	\(-0.850704\pi\)
							0.990860	−	0.134892i	\(-0.0430687\pi\)
\(19\)	−2.85687	+	1.54159i	−0.655411	+	0.353665i	−0.768436	−	0.639927i	\(-0.778965\pi\)
							0.113024	+	0.993592i	\(0.463946\pi\)
\(23\)	6.10011	+	1.42929i	1.27196	+	0.298027i	0.808714	−	0.588202i	\(-0.200164\pi\)
							0.463247	+	0.886229i	\(0.346684\pi\)
\(29\)	6.65475	+	3.79124i	1.23576	+	0.704015i	0.965907	−	0.258888i	\(-0.0833561\pi\)
							0.269848	+	0.962903i	\(0.413026\pi\)
\(31\)	6.18467	+	2.22421i	1.11080	+	0.399479i	0.828580	−	0.559870i	\(-0.189149\pi\)
							0.282220	+	0.959350i	\(0.408929\pi\)
\(37\)	3.50964	−	2.83203i	0.576981	−	0.465583i	−0.294746	−	0.955576i	\(-0.595235\pi\)
							0.871726	+	0.489993i	\(0.163001\pi\)
\(41\)	3.46652	+	6.00418i	0.541379	+	0.937695i	0.998825	+	0.0484583i	\(0.0154308\pi\)
							−0.457447	+	0.889237i	\(0.651236\pi\)
\(43\)	−0.970693	+	9.89208i	−0.148029	+	1.50853i	0.575253	+	0.817976i	\(0.304904\pi\)
							−0.723282	+	0.690553i	\(0.757367\pi\)
\(47\)	−9.52245	+	0.768732i	−1.38899	+	0.112131i	−0.752215	−	0.658917i	\(-0.771015\pi\)
							−0.636777	+	0.771048i	\(0.719733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.18	✓	6480
547.137	even	273	inner	547.2.o.a.137.18	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.18	✓	6480	1.1	even	1	trivial
547.2.o.a.137.18	yes	6480	547.137	even	273	inner