Embedded newform 539.2.p.a.76.16

• Label: 539.2.p.a.76.16
• Level: $539$
• Weight: $2$
• Character: 539.76
• Analytic conductor: $4.304$
• Analytic rank: $0$
• Dimension: $324$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.30393666895\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(54\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			76.16
Character	\(\chi\)	\(=\)	539.76
Dual form			539.2.p.a.461.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.627090 - 1.30217i) q^{2} +(-1.71706 - 0.391908i) q^{3} +(-0.0554150 + 0.0694883i) q^{4} +(-2.96698 - 0.677193i) q^{5} +(0.566423 + 2.48166i) q^{6} +(-2.62859 - 0.300822i) q^{7} +(-2.69288 - 0.614633i) q^{8} +(0.0918039 + 0.0442104i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 14 q^{3} + 42 q^{4} - 14 q^{5} + 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.627090	−	1.30217i	−0.443420	−	0.920771i	−0.996171	−	0.0874244i	\(-0.972136\pi\)
							0.552751	−	0.833346i	\(-0.313578\pi\)
\(3\)	−1.71706	−	0.391908i	−0.991346	−	0.226268i	−0.304058	−	0.952653i	\(-0.598342\pi\)
							−0.687288	+	0.726385i	\(0.741199\pi\)
\(5\)	−2.96698	−	0.677193i	−1.32687	−	0.302850i	−0.500384	−	0.865804i	\(-0.666808\pi\)
							−0.826488	+	0.562954i	\(0.809665\pi\)
\(7\)	−2.62859	−	0.300822i	−0.993515	−	0.113700i
							\(11\)	−2.48978	−	2.19112i	−0.750697	−	0.660647i
\(13\)	5.81713	−	2.80138i	1.61338	−	0.776964i								0.613463	−	0.789723i	\(-0.289776\pi\)
							0.999919	+	0.0127599i	\(0.00406170\pi\)
\(17\)	−0.153301	−	0.192233i	−0.0371809	−	0.0466234i	0.762893	−	0.646524i	\(-0.223778\pi\)
							−0.800074	+	0.599901i	\(0.795207\pi\)
\(19\)	2.60892			0.598527			0.299263	−	0.954171i	\(-0.403259\pi\)
							0.299263	+	0.954171i	\(0.403259\pi\)
\(23\)	−3.91656	+	4.91121i	−0.816660	+	1.02406i	0.182506	+	0.983205i	\(0.441579\pi\)
							−0.999165	+	0.0408539i	\(0.986992\pi\)
\(29\)	1.98857	−	1.58583i	0.369268	−	0.294481i	−0.421221	−	0.906958i	\(-0.638398\pi\)
							0.790488	+	0.612477i	\(0.209827\pi\)
\(31\)			1.34716i			0.241957i	0.992655	+	0.120978i	\(0.0386032\pi\)
							−0.992655	+	0.120978i	\(0.961397\pi\)
\(37\)	−7.26747	−	9.11312i	−1.19477	−	1.49819i	−0.821254	−	0.570563i	\(-0.806725\pi\)
							−0.373511	−	0.927626i	\(-0.621846\pi\)
\(41\)	1.23720	−	5.42051i	0.193217	−	0.846541i	−0.781643	−	0.623726i	\(-0.785618\pi\)
							0.974861	−	0.222815i	\(-0.0715247\pi\)
\(43\)	−0.148882	+	0.0339813i	−0.0227043	+	0.00518210i	−0.233858	−	0.972271i	\(-0.575135\pi\)
							0.211154	+	0.977453i	\(0.432278\pi\)
\(47\)	3.36265	+	6.98262i	0.490493	+	1.01852i	0.988482	+	0.151336i	\(0.0483576\pi\)
							−0.497989	+	0.867183i	\(0.665928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.p.a.76.16	✓	324
11.10	odd	2	inner	539.2.p.a.76.39	yes	324
49.20	odd	14	inner	539.2.p.a.461.39	yes	324
539.461	even	14	inner	539.2.p.a.461.16	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.p.a.76.16	✓	324	1.1	even	1	trivial
539.2.p.a.76.39	yes	324	11.10	odd	2	inner
539.2.p.a.461.16	yes	324	539.461	even	14	inner
539.2.p.a.461.39	yes	324	49.20	odd	14	inner