Embedded newform 547.2.o.a.4.4

• Label: 547.2.o.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.52384 - 0.0290448i) q^{2} +(-0.715155 + 3.13330i) q^{3} +(4.36948 + 0.100583i) q^{4} +(1.84693 - 3.47035i) q^{5} +(1.89595 - 7.88719i) q^{6} +(1.46224 + 3.60324i) q^{7} +(-5.97995 - 0.206528i) q^{8} +(-6.60321 - 3.17994i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.52384	−	0.0290448i	−1.78463	−	0.0205378i	−0.887121	−	0.461538i	\(-0.847298\pi\)
							−0.897507	+	0.441000i	\(0.854624\pi\)
\(3\)	−0.715155	+	3.13330i	−0.412895	+	1.80901i	0.157357	+	0.987542i	\(0.449703\pi\)
							−0.570252	+	0.821470i	\(0.693154\pi\)
\(5\)	1.84693	−	3.47035i	0.825970	−	1.55199i	−0.00786110	−	0.999969i	\(-0.502502\pi\)
							0.833832	−	0.552019i	\(-0.186142\pi\)
\(7\)	1.46224	+	3.60324i	0.552675	+	1.36190i	0.904757	+	0.425929i	\(0.140053\pi\)
							−0.352082	+	0.935969i	\(0.614526\pi\)
\(11\)	−0.949980	−	0.989034i	−0.286430	−	0.298205i	0.562406	−	0.826861i	\(-0.309876\pi\)
							−0.848836	+	0.528656i	\(0.822696\pi\)
\(13\)	4.50664	+	0.679267i	1.24992	+	0.188395i	0.740460	−	0.672101i	\(-0.234608\pi\)
							0.509458	+	0.860496i	\(0.329846\pi\)
\(17\)	−0.0472716	+	0.280665i	−0.0114651	+	0.0680713i	−0.991256	−	0.131953i	\(-0.957875\pi\)
							0.979791	+	0.200025i	\(0.0641022\pi\)
\(19\)	5.32700	−	2.87449i	1.22210	−	0.659454i	0.269300	−	0.963056i	\(-0.413208\pi\)
							0.952799	+	0.303602i	\(0.0981894\pi\)
\(23\)	0.898632	+	0.210554i	0.187378	+	0.0439036i	0.320757	−	0.947162i	\(-0.396063\pi\)
							−0.133379	+	0.991065i	\(0.542583\pi\)
\(29\)	2.05440	+	1.17040i	0.381492	+	0.217337i	0.673679	−	0.739024i	\(-0.264713\pi\)
							−0.292188	+	0.956361i	\(0.594383\pi\)
\(31\)	5.72464	+	2.05877i	1.02818	+	0.369765i	0.797534	−	0.603273i	\(-0.206137\pi\)
							0.230642	+	0.973039i	\(0.425917\pi\)
\(37\)	3.19129	−	2.57515i	0.524645	−	0.423352i	−0.328905	−	0.944363i	\(-0.606680\pi\)
							0.853550	+	0.521012i	\(0.174445\pi\)
\(41\)	2.29658	+	3.97779i	0.358666	+	0.621227i	0.987738	−	0.156120i	\(-0.0498985\pi\)
							−0.629073	+	0.777347i	\(0.716565\pi\)
\(43\)	−0.576621	+	5.87619i	−0.0879338	+	0.896111i	0.844689	+	0.535258i	\(0.179785\pi\)
							−0.932623	+	0.360853i	\(0.882486\pi\)
\(47\)	−3.18885	+	0.257431i	−0.465142	+	0.0375501i	−0.310816	−	0.950470i	\(-0.600602\pi\)
							−0.154325	+	0.988020i	\(0.549320\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.4	✓	6480
547.137	even	273	inner	547.2.o.a.137.4	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.4	✓	6480	1.1	even	1	trivial
547.2.o.a.137.4	yes	6480	547.137	even	273	inner