Embedded newform 547.2.o.a.4.16

• Label: 547.2.o.a.4.16
• 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			4.16
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.956173 - 0.0110038i) q^{2} +(-0.143432 + 0.628415i) q^{3} +(-1.08532 - 0.0249835i) q^{4} +(-1.82781 + 3.43443i) q^{5} +(0.144060 - 0.599296i) q^{6} +(0.431500 + 1.06330i) q^{7} +(2.94882 + 0.101842i) q^{8} +(2.32857 + 1.12138i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.956173	−	0.0110038i	−0.676117	−	0.00778087i	−0.327228	−	0.944945i	\(-0.606115\pi\)
							−0.348888	+	0.937164i	\(0.613441\pi\)
\(3\)	−0.143432	+	0.628415i	−0.0828103	+	0.362816i	−0.999307	−	0.0372270i	\(-0.988148\pi\)
							0.916497	+	0.400043i	\(0.131005\pi\)
\(5\)	−1.82781	+	3.43443i	−0.817422	+	1.53592i	0.0266190	+	0.999646i	\(0.491526\pi\)
							−0.844041	+	0.536279i	\(0.819829\pi\)
\(7\)	0.431500	+	1.06330i	0.163092	+	0.401890i	0.986103	−	0.166135i	\(-0.0531288\pi\)
							−0.823011	+	0.568025i	\(0.807708\pi\)
\(11\)	0.710739	+	0.739958i	0.214296	+	0.223106i	0.819671	−	0.572835i	\(-0.194156\pi\)
							−0.605375	+	0.795940i	\(0.706977\pi\)
\(13\)	0.0967855	+	0.0145881i	0.0268435	+	0.00404600i	0.162450	−	0.986717i	\(-0.448060\pi\)
							−0.135607	+	0.990763i	\(0.543298\pi\)
\(17\)	−0.853878	+	5.06971i	−0.207096	+	1.22959i	0.667454	+	0.744651i	\(0.267384\pi\)
							−0.874550	+	0.484935i	\(0.838843\pi\)
\(19\)	−0.515000	+	0.277898i	−0.118149	+	0.0637542i	−0.532885	−	0.846188i	\(-0.678892\pi\)
							0.414736	+	0.909942i	\(0.363874\pi\)
\(23\)	−2.17880	−	0.510505i	−0.454312	−	0.106448i	−0.00532482	−	0.999986i	\(-0.501695\pi\)
							−0.448987	+	0.893538i	\(0.648215\pi\)
\(29\)	−5.77655	−	3.29092i	−1.07268	−	0.611109i	−0.146891	−	0.989153i	\(-0.546927\pi\)
							−0.925788	+	0.378044i	\(0.876597\pi\)
\(31\)	−2.73209	−	0.982546i	−0.490697	−	0.176471i	0.0813592	−	0.996685i	\(-0.474074\pi\)
							−0.572056	+	0.820214i	\(0.693854\pi\)
\(37\)	−2.75678	+	2.22453i	−0.453212	+	0.365710i	−0.827372	−	0.561654i	\(-0.810165\pi\)
							0.374160	+	0.927364i	\(0.377931\pi\)
\(41\)	3.51155	+	6.08219i	0.548413	+	0.949879i	0.998384	+	0.0568354i	\(0.0181010\pi\)
							−0.449971	+	0.893043i	\(0.648566\pi\)
\(43\)	0.581048	−	5.92130i	0.0886089	−	0.902990i	−0.842609	−	0.538526i	\(-0.818981\pi\)
							0.931218	−	0.364464i	\(-0.118748\pi\)
\(47\)	10.6410	−	0.859028i	1.55214	−	0.125302i	0.725577	−	0.688141i	\(-0.241573\pi\)
							0.826567	+	0.562838i	\(0.190291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.16	✓	6480
547.137	even	273	inner	547.2.o.a.137.16	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.16	✓	6480	1.1	even	1	trivial
547.2.o.a.137.16	yes	6480	547.137	even	273	inner