Embedded newform 547.2.f.a.46.5

• Label: 547.2.f.a.46.5
• Level: $547$
• Weight: $2$
• Character: 547.46
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $528$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.f (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(44\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			46.5
Character	\(\chi\)	\(=\)	547.46
Dual form			547.2.f.a.440.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.268734 + 2.21322i) q^{2} +2.12681 q^{3} +(-2.88424 - 0.710902i) q^{4} +(-0.947553 - 2.49849i) q^{5} +(-0.571546 + 4.70710i) q^{6} +(-0.292661 - 2.41028i) q^{7} +(0.767309 - 2.02323i) q^{8} +1.52333 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 8 q^{2} - 46 q^{3} - 50 q^{4} - 9 q^{5} - 21 q^{6} - 3 q^{7} + 14 q^{8} + 466 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.268734	+	2.21322i	−0.190023	+	1.56498i	0.515871	+	0.856666i	\(0.327468\pi\)
							−0.705895	+	0.708317i	\(0.749455\pi\)
\(3\)	2.12681			1.22792			0.613958	−	0.789339i	\(-0.289577\pi\)
							0.613958	+	0.789339i	\(0.289577\pi\)
\(5\)	−0.947553	−	2.49849i	−0.423759	−	1.11736i	−0.961634	−	0.274336i	\(-0.911542\pi\)
							0.537875	−	0.843024i	\(-0.319227\pi\)
\(7\)	−0.292661	−	2.41028i	−0.110615	−	0.911000i	−0.937346	−	0.348401i	\(-0.886725\pi\)
							0.826730	−	0.562599i	\(-0.190198\pi\)
\(11\)	3.08888	−	4.47501i	0.931332	−	1.34927i	−0.00568484	−	0.999984i	\(-0.501810\pi\)
							0.937017	−	0.349284i	\(-0.113575\pi\)
\(13\)	5.00401			1.38786			0.693931	−	0.720041i	\(-0.255877\pi\)
							0.693931	+	0.720041i	\(0.255877\pi\)
\(17\)	2.08996	+	3.02783i	0.506890	+	0.734357i	0.989979	−	0.141214i	\(-0.0451006\pi\)
							−0.483089	+	0.875571i	\(0.660485\pi\)
\(19\)	−0.664423	+	0.348716i	−0.152429	+	0.0800010i	−0.539213	−	0.842169i	\(-0.681278\pi\)
							0.386784	+	0.922170i	\(0.373586\pi\)
\(23\)	−2.57618	+	2.28230i	−0.537171	+	0.475892i	−0.887547	−	0.460717i	\(-0.847592\pi\)
							0.350376	+	0.936609i	\(0.386054\pi\)
\(29\)	2.52027	+	2.23277i	0.468003	+	0.414614i	0.863887	−	0.503686i	\(-0.168023\pi\)
							−0.395884	+	0.918301i	\(0.629562\pi\)
\(31\)	−1.24372	−	0.652757i	−0.223380	−	0.117239i	0.349321	−	0.937003i	\(-0.386412\pi\)
							−0.572701	+	0.819764i	\(0.694104\pi\)
\(37\)	−1.60042	−	1.41785i	−0.263107	−	0.233093i	0.521247	−	0.853406i	\(-0.325467\pi\)
							−0.784354	+	0.620313i	\(0.787006\pi\)
\(41\)	−1.45260			−0.226858			−0.113429	−	0.993546i	\(-0.536183\pi\)
							−0.113429	+	0.993546i	\(0.536183\pi\)
\(43\)	10.8007	+	2.66214i	1.64710	+	0.405973i	0.950310	−	0.311304i	\(-0.100766\pi\)
							0.696788	+	0.717277i	\(0.254612\pi\)
\(47\)	−9.04495	−	8.01313i	−1.31934	−	1.16884i	−0.973020	−	0.230719i	\(-0.925892\pi\)
							−0.346322	−	0.938116i	\(-0.612569\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.f.a.46.5	✓	528
547.440	even	13	inner	547.2.f.a.440.5	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.f.a.46.5	✓	528	1.1	even	1	trivial
547.2.f.a.440.5	yes	528	547.440	even	13	inner