Embedded newform 547.2.e.a.9.17

• Label: 547.2.e.a.9.17
• Level: $547$
• Weight: $2$
• Character: 547.9
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(44\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			9.17
Character	\(\chi\)	\(=\)	547.9
Dual form			547.2.e.a.304.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.217875 + 0.954575i) q^{2} +(-0.359677 - 0.173211i) q^{3} +(0.938195 + 0.451811i) q^{4} +(-0.0979486 - 0.122824i) q^{5} +(0.243708 - 0.305600i) q^{6} +(3.84057 - 1.84952i) q^{7} +(-1.85664 + 2.32816i) q^{8} +(-1.77110 - 2.22089i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{2} - q^{3} - 44 q^{4} - 3 q^{5} - 10 q^{6} - 18 q^{7} - q^{8} - 35 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.217875	+	0.954575i	−0.154061	+	0.674986i	0.837618	+	0.546256i	\(0.183947\pi\)
							−0.991680	+	0.128730i	\(0.958910\pi\)
\(3\)	−0.359677	−	0.173211i	−0.207659	−	0.100004i	0.327163	−	0.944968i	\(-0.393907\pi\)
							−0.534823	+	0.844964i	\(0.679622\pi\)
\(5\)	−0.0979486	−	0.122824i	−0.0438039	−	0.0549284i	0.759447	−	0.650569i	\(-0.225470\pi\)
							−0.803251	+	0.595641i	\(0.796898\pi\)
\(7\)	3.84057	−	1.84952i	1.45160	−	0.699054i	0.468728	−	0.883342i	\(-0.344712\pi\)
							0.982872	+	0.184288i	\(0.0589980\pi\)
\(11\)	5.40390			1.62934			0.814668	−	0.579928i	\(-0.196919\pi\)
							0.814668	+	0.579928i	\(0.196919\pi\)
\(13\)	−0.710245	−	3.11179i	−0.196986	−	0.863054i	−0.972718	−	0.231991i	\(-0.925476\pi\)
							0.775732	−	0.631063i	\(-0.217381\pi\)
\(17\)	−3.18832	−	3.99803i	−0.773281	−	0.969664i	0.226710	−	0.973962i	\(-0.427203\pi\)
							−0.999991	+	0.00429861i	\(0.998632\pi\)
\(19\)	−0.0399711	−	0.175125i	−0.00917000	−	0.0401764i	0.970136	−	0.242562i	\(-0.0779877\pi\)
							−0.979306	+	0.202385i	\(0.935131\pi\)
\(23\)	1.35512	+	5.93718i	0.282563	+	1.23799i	0.894495	+	0.447078i	\(0.147535\pi\)
							−0.611932	+	0.790910i	\(0.709607\pi\)
\(29\)	1.01182	−	1.26878i	0.187890	−	0.235606i	−0.678961	−	0.734175i	\(-0.737569\pi\)
							0.866850	+	0.498568i	\(0.166141\pi\)
\(31\)	5.44702	+	2.62315i	0.978314	+	0.471131i	0.853525	−	0.521052i	\(-0.174460\pi\)
							0.124789	+	0.992183i	\(0.460175\pi\)
\(37\)	4.99629	+	6.26515i	0.821385	+	1.02998i	0.998947	+	0.0458748i	\(0.0146075\pi\)
							−0.177562	+	0.984110i	\(0.556821\pi\)
\(41\)	−10.9348			−1.70773			−0.853863	−	0.520498i	\(-0.825746\pi\)
							−0.853863	+	0.520498i	\(0.825746\pi\)
\(43\)	−5.50135	+	2.64931i	−0.838947	+	0.404016i	−0.803463	−	0.595354i	\(-0.797012\pi\)
							−0.0354842	+	0.999370i	\(0.511297\pi\)
\(47\)	0.926355			0.135123			0.0675614	−	0.997715i	\(-0.478478\pi\)
							0.0675614	+	0.997715i	\(0.478478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.e.a.9.17	✓	264
547.304	even	7	inner	547.2.e.a.304.17	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.e.a.9.17	✓	264	1.1	even	1	trivial
547.2.e.a.304.17	yes	264	547.304	even	7	inner