Embedded newform 547.2.o.a.4.7

• Label: 547.2.o.a.4.7
• 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			4.7
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.14370 - 0.0246701i) q^{2} +(-0.0167644 + 0.0734497i) q^{3} +(2.59538 + 0.0597440i) q^{4} +(0.940362 - 1.76693i) q^{5} +(0.0377499 - 0.157041i) q^{6} +(-0.448666 - 1.10560i) q^{7} +(-1.27711 - 0.0441071i) q^{8} +(2.69779 + 1.29919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.14370	−	0.0246701i	−1.51583	−	0.0174444i	−0.750407	−	0.660976i	\(-0.770142\pi\)
							−0.765419	+	0.643532i	\(0.777468\pi\)
\(3\)	−0.0167644	+	0.0734497i	−0.00967895	+	0.0424062i	−0.979537	−	0.201265i	\(-0.935495\pi\)
							0.969858	+	0.243671i	\(0.0783518\pi\)
\(5\)	0.940362	−	1.76693i	0.420543	−	0.790194i	−0.579144	−	0.815225i	\(-0.696613\pi\)
							0.999687	+	0.0250308i	\(0.00796839\pi\)
\(7\)	−0.448666	−	1.10560i	−0.169580	−	0.417877i	0.817954	−	0.575283i	\(-0.195108\pi\)
							−0.987534	+	0.157406i	\(0.949687\pi\)
\(11\)	1.89559	+	1.97351i	0.571541	+	0.595037i	0.942745	−	0.333514i	\(-0.108235\pi\)
							−0.371204	+	0.928551i	\(0.621055\pi\)
\(13\)	0.0532230	+	0.00802208i	0.0147614	+	0.00222492i	0.156419	−	0.987691i	\(-0.450005\pi\)
							−0.141657	+	0.989916i	\(0.545243\pi\)
\(17\)	−0.318701	+	1.89222i	−0.0772964	+	0.458931i	0.920393	+	0.390994i	\(0.127869\pi\)
							−0.997690	+	0.0679367i	\(0.978358\pi\)
\(19\)	5.10191	−	2.75303i	1.17046	−	0.631589i	0.230566	−	0.973057i	\(-0.425942\pi\)
							0.939893	+	0.341468i	\(0.110924\pi\)
\(23\)	−4.67650	−	1.09573i	−0.975117	−	0.228475i	−0.290096	−	0.956998i	\(-0.593687\pi\)
							−0.685022	+	0.728523i	\(0.740207\pi\)
\(29\)	−5.89967	−	3.36107i	−1.09554	−	0.624134i	−0.163521	−	0.986540i	\(-0.552285\pi\)
							−0.932021	+	0.362406i	\(0.881956\pi\)
\(31\)	7.85577	+	2.82519i	1.41094	+	0.507419i	0.929422	−	0.369019i	\(-0.120306\pi\)
							0.481515	+	0.876438i	\(0.340087\pi\)
\(37\)	−0.102687	+	0.0828610i	−0.0168816	+	0.0136223i	−0.636382	−	0.771374i	\(-0.719570\pi\)
							0.619500	+	0.784996i	\(0.287335\pi\)
\(41\)	0.741425	+	1.28419i	0.115791	+	0.200556i	0.918096	−	0.396359i	\(-0.129726\pi\)
							−0.802305	+	0.596915i	\(0.796393\pi\)
\(43\)	−0.0245740	+	0.250427i	−0.00374749	+	0.0381897i	−0.996910	−	0.0785464i	\(-0.974972\pi\)
							0.993163	+	0.116736i	\(0.0372432\pi\)
\(47\)	−1.44559	+	0.116700i	−0.210861	+	0.0170225i	−0.185446	−	0.982655i	\(-0.559373\pi\)
							−0.0254153	+	0.999677i	\(0.508091\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.7	✓	6480
547.137	even	273	inner	547.2.o.a.137.7	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.7	✓	6480	1.1	even	1	trivial
547.2.o.a.137.7	yes	6480	547.137	even	273	inner