Embedded newform 547.2.e.a.9.10

• Label: 547.2.e.a.9.10
• 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			9.10
Character	\(\chi\)	\(=\)	547.9
Dual form			547.2.e.a.304.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.387202 + 1.69644i) q^{2} +(-2.32711 - 1.12068i) q^{3} +(-0.926057 - 0.445966i) q^{4} +(0.458209 + 0.574576i) q^{5} +(2.80223 - 3.51389i) q^{6} +(4.08231 - 1.96594i) q^{7} +(-1.05471 + 1.32256i) q^{8} +(2.28906 + 2.87040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{2} - q^{3} - 44 q^{4} - 3 q^{5} - 10 q^{6} - 18 q^{7} - q^{8} - 35 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.387202	+	1.69644i	−0.273793	+	1.19957i	0.631702	+	0.775211i	\(0.282357\pi\)
							−0.905496	+	0.424356i	\(0.860501\pi\)
\(3\)	−2.32711	−	1.12068i	−1.34356	−	0.647024i	−0.382651	−	0.923893i	\(-0.624989\pi\)
							−0.960908	+	0.276869i	\(0.910703\pi\)
\(5\)	0.458209	+	0.574576i	0.204918	+	0.256958i	0.873661	−	0.486534i	\(-0.161739\pi\)
							−0.668744	+	0.743493i	\(0.733168\pi\)
\(7\)	4.08231	−	1.96594i	1.54297	−	0.743054i	0.547379	−	0.836885i	\(-0.315626\pi\)
							0.995588	+	0.0938314i	\(0.0299115\pi\)
\(11\)	−6.09985			−1.83917			−0.919587	−	0.392886i	\(-0.871477\pi\)
							−0.919587	+	0.392886i	\(0.871477\pi\)
\(13\)	−1.03221	−	4.52239i	−0.286282	−	1.25428i	−0.889584	−	0.456772i	\(-0.849006\pi\)
							0.603302	−	0.797513i	\(-0.293851\pi\)
\(17\)	0.0536584	+	0.0672855i	0.0130141	+	0.0163191i	0.788296	−	0.615296i	\(-0.210964\pi\)
							−0.775282	+	0.631615i	\(0.782392\pi\)
\(19\)	0.0806612	+	0.353400i	0.0185049	+	0.0810754i	0.983337	−	0.181791i	\(-0.0581894\pi\)
							−0.964832	+	0.262866i	\(0.915332\pi\)
\(23\)	0.175893	+	0.770636i	0.0366762	+	0.160689i	0.989950	−	0.141420i	\(-0.0451667\pi\)
							−0.953274	+	0.302108i	\(0.902310\pi\)
\(29\)	5.42980	−	6.80875i	1.00829	−	1.26435i	0.0441294	−	0.999026i	\(-0.485949\pi\)
							0.964158	−	0.265327i	\(-0.0854800\pi\)
\(31\)	−5.43917	−	2.61937i	−0.976904	−	0.470452i	−0.123865	−	0.992299i	\(-0.539529\pi\)
							−0.853039	+	0.521847i	\(0.825243\pi\)
\(37\)	−0.494742	−	0.620387i	−0.0813351	−	0.101991i	0.739499	−	0.673158i	\(-0.235062\pi\)
							−0.820834	+	0.571167i	\(0.806491\pi\)
\(41\)	7.89018			1.23224			0.616119	−	0.787653i	\(-0.288704\pi\)
							0.616119	+	0.787653i	\(0.288704\pi\)
\(43\)	4.32411	−	2.08238i	0.659420	−	0.317560i	−0.0740771	−	0.997253i	\(-0.523601\pi\)
							0.733497	+	0.679693i	\(0.237887\pi\)
\(47\)	−4.51992			−0.659298			−0.329649	−	0.944104i	\(-0.606930\pi\)
							−0.329649	+	0.944104i	\(0.606930\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.e.a.9.10	✓	264
547.304	even	7	inner	547.2.e.a.304.10	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.e.a.9.10	✓	264	1.1	even	1	trivial
547.2.e.a.304.10	yes	264	547.304	even	7	inner