Embedded newform 547.2.e.a.9.5

• Label: 547.2.e.a.9.5
• 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			9.5
Character	\(\chi\)	\(=\)	547.9
Dual form			547.2.e.a.304.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.512100 + 2.24366i) q^{2} +(-1.76126 - 0.848179i) q^{3} +(-2.96982 - 1.43019i) q^{4} +(1.48262 + 1.85914i) q^{5} +(2.80497 - 3.51732i) q^{6} +(1.07224 - 0.516363i) q^{7} +(1.85996 - 2.33231i) q^{8} +(0.512167 + 0.642236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{2} - q^{3} - 44 q^{4} - 3 q^{5} - 10 q^{6} - 18 q^{7} - q^{8} - 35 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.512100	+	2.24366i	−0.362110	+	1.58651i	0.385720	+	0.922616i	\(0.373953\pi\)
							−0.747830	+	0.663890i	\(0.768904\pi\)
\(3\)	−1.76126	−	0.848179i	−1.01687	−	0.489696i	−0.150236	−	0.988650i	\(-0.548003\pi\)
							−0.866629	+	0.498954i	\(0.833718\pi\)
\(5\)	1.48262	+	1.85914i	0.663046	+	0.831433i	0.993671	−	0.112329i	\(-0.0358311\pi\)
							−0.330625	+	0.943762i	\(0.607260\pi\)
\(7\)	1.07224	−	0.516363i	0.405268	−	0.195167i	−0.220134	−	0.975470i	\(-0.570649\pi\)
							0.625402	+	0.780303i	\(0.284935\pi\)
\(11\)	4.02103			1.21239			0.606194	−	0.795317i	\(-0.292696\pi\)
							0.606194	+	0.795317i	\(0.292696\pi\)
\(13\)	0.108638	+	0.475973i	0.0301307	+	0.132011i	0.987756	−	0.156005i	\(-0.0498615\pi\)
							−0.957626	+	0.288016i	\(0.907004\pi\)
\(17\)	4.49042	+	5.63081i	1.08909	+	1.36567i	0.925326	+	0.379171i	\(0.123791\pi\)
							0.163760	+	0.986500i	\(0.447638\pi\)
\(19\)	−0.0168701	−	0.0739128i	−0.00387027	−	0.0169568i	0.972956	−	0.230991i	\(-0.0741968\pi\)
							−0.976826	+	0.214034i	\(0.931340\pi\)
\(23\)	0.901675	+	3.95050i	0.188012	+	0.823735i	0.977663	+	0.210179i	\(0.0674046\pi\)
							−0.789651	+	0.613557i	\(0.789738\pi\)
\(29\)	−4.64499	+	5.82464i	−0.862553	+	1.08161i	0.133340	+	0.991070i	\(0.457430\pi\)
							−0.995893	+	0.0905372i	\(0.971142\pi\)
\(31\)	−0.992683	−	0.478051i	−0.178291	−	0.0858604i	0.342609	−	0.939478i	\(-0.388689\pi\)
							−0.520900	+	0.853618i	\(0.674404\pi\)
\(37\)	−5.78715	−	7.25685i	−0.951401	−	1.19302i	−0.981108	−	0.193461i	\(-0.938029\pi\)
							0.0297070	−	0.999559i	\(-0.490543\pi\)
\(41\)	−11.1564			−1.74234			−0.871170	−	0.490981i	\(-0.836639\pi\)
							−0.871170	+	0.490981i	\(0.836639\pi\)
\(43\)	2.10996	−	1.01610i	0.321766	−	0.154954i	−0.266027	−	0.963965i	\(-0.585711\pi\)
							0.587793	+	0.809011i	\(0.299997\pi\)
\(47\)	4.04201			0.589588			0.294794	−	0.955561i	\(-0.404749\pi\)
							0.294794	+	0.955561i	\(0.404749\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.e.a.9.5	✓	264
547.304	even	7	inner	547.2.e.a.304.5	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.e.a.9.5	✓	264	1.1	even	1	trivial
547.2.e.a.304.5	yes	264	547.304	even	7	inner