Embedded newform 547.2.j.a.11.13

• Label: 547.2.j.a.11.13
• Level: $547$
• Weight: $2$
• Character: 547.11
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.j (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			11.13
Character	\(\chi\)	\(=\)	547.11
Dual form			547.2.j.a.199.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41327 - 0.602138i) q^{2} +1.23661 q^{3} +(0.249311 + 0.259560i) q^{4} +(0.850668 - 0.138247i) q^{5} +(-1.74767 - 0.744612i) q^{6} +(-2.23146 - 1.67683i) q^{7} +(0.893433 + 2.35579i) q^{8} -1.47078 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 25 q^{2} - 74 q^{3} + 21 q^{4} - 27 q^{5} - 75 q^{6} - 28 q^{7} + 4 q^{8} + 1010 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41327	−	0.602138i	−0.999332	−	0.425776i	−0.170551	−	0.985349i	\(-0.554555\pi\)
							−0.828781	+	0.559573i	\(0.810965\pi\)
\(3\)	1.23661			0.713960			0.356980	−	0.934112i	\(-0.383806\pi\)
							0.356980	+	0.934112i	\(0.383806\pi\)
\(5\)	0.850668	−	0.138247i	0.380430	−	0.0618259i	0.0328107	−	0.999462i	\(-0.489554\pi\)
							0.347620	+	0.937636i	\(0.386990\pi\)
\(7\)	−2.23146	−	1.67683i	−0.843411	−	0.633782i	0.0886585	−	0.996062i	\(-0.471742\pi\)
							−0.932069	+	0.362280i	\(0.881998\pi\)
\(11\)	5.50543	−	0.444444i	1.65995	−	0.134005i	0.785416	−	0.618969i	\(-0.212449\pi\)
							0.874534	+	0.484964i	\(0.161167\pi\)
\(13\)	0.285366	−	0.494268i	0.0791462	−	0.137085i	−0.823736	−	0.566974i	\(-0.808114\pi\)
							0.902882	+	0.429889i	\(0.141447\pi\)
\(17\)	−3.08883	−	6.50958i	−0.749152	−	1.57880i	−0.813970	−	0.580907i	\(-0.802698\pi\)
							0.0648183	−	0.997897i	\(-0.479353\pi\)
\(19\)	−0.0470550	−	1.16766i	−0.0107952	−	0.267879i	−0.995569	−	0.0940370i	\(-0.970023\pi\)
							0.984774	−	0.173842i	\(-0.0556182\pi\)
\(23\)	0.285716	−	1.39953i	0.0595758	−	0.291822i	−0.939077	−	0.343706i	\(-0.888318\pi\)
							0.998653	+	0.0518844i	\(0.0165227\pi\)
\(29\)	3.64583	−	3.22992i	0.677013	−	0.599782i	−0.252930	−	0.967485i	\(-0.581394\pi\)
							0.929943	+	0.367703i	\(0.119856\pi\)
\(31\)	6.67466	−	3.50313i	1.19880	−	0.629181i	0.257349	−	0.966319i	\(-0.417151\pi\)
							0.941455	+	0.337138i	\(0.109459\pi\)
\(37\)	1.07847	+	5.28272i	0.177300	+	0.868473i	0.966960	+	0.254928i	\(0.0820519\pi\)
							−0.789660	+	0.613545i	\(0.789743\pi\)
\(41\)	−2.26477	+	3.92269i	−0.353697	+	0.612622i	−0.986894	−	0.161369i	\(-0.948409\pi\)
							0.633197	+	0.773991i	\(0.281742\pi\)
\(43\)	2.99985	−	10.3567i	0.457472	−	1.57938i	−0.318061	−	0.948070i	\(-0.603032\pi\)
							0.775533	−	0.631307i	\(-0.217481\pi\)
\(47\)	−7.58078	−	2.53084i	−1.10577	−	0.369161i	−0.295575	−	0.955320i	\(-0.595511\pi\)
							−0.810196	+	0.586159i	\(0.800639\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.j.a.11.13	✓	1080
547.199	even	39	inner	547.2.j.a.199.13	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.j.a.11.13	✓	1080	1.1	even	1	trivial
547.2.j.a.199.13	yes	1080	547.199	even	39	inner