Embedded newform 539.2.p.a.461.39

• Label: 539.2.p.a.461.39
• Level: $539$
• Weight: $2$
• Character: 539.461
• 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			461.39
Character	\(\chi\)	\(=\)	539.461
Dual form			539.2.p.a.76.39

$q$-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 \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)	\(-1\)

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	−0.552751	−	0.833346i	\(-0.686422\pi\)
							0.996171	−	0.0874244i	\(-0.0278636\pi\)
\(3\)	−1.71706	+	0.391908i	−0.991346	+	0.226268i	−0.687288	−	0.726385i	\(-0.741199\pi\)
							−0.304058	+	0.952653i	\(0.598342\pi\)
\(5\)	−2.96698	+	0.677193i	−1.32687	+	0.302850i	−0.826488	−	0.562954i	\(-0.809665\pi\)
							−0.500384	+	0.865804i	\(0.666808\pi\)
\(7\)	2.62859	−	0.300822i	0.993515	−	0.113700i
							\(11\)	0.160731	−	3.31273i	0.0484623	−	0.998825i
\(13\)	−5.81713	−	2.80138i	−1.61338	−	0.776964i								−0.613463	−	0.789723i	\(-0.710224\pi\)
							−0.999919	+	0.0127599i	\(0.995938\pi\)
\(17\)	0.153301	−	0.192233i	0.0371809	−	0.0466234i	−0.762893	−	0.646524i	\(-0.776222\pi\)
							0.800074	+	0.599901i	\(0.204793\pi\)
\(19\)	−2.60892			−0.598527			−0.299263	−	0.954171i	\(-0.596741\pi\)
							−0.299263	+	0.954171i	\(0.596741\pi\)
\(23\)	−3.91656	−	4.91121i	−0.816660	−	1.02406i	−0.999165	−	0.0408539i	\(-0.986992\pi\)
							0.182506	−	0.983205i	\(-0.441579\pi\)
\(29\)	−1.98857	−	1.58583i	−0.369268	−	0.294481i	0.421221	−	0.906958i	\(-0.361602\pi\)
							−0.790488	+	0.612477i	\(0.790173\pi\)
\(31\)		−	1.34716i		−	0.241957i	−0.992655	−	0.120978i	\(-0.961397\pi\)
							0.992655	−	0.120978i	\(-0.0386032\pi\)
\(37\)	−7.26747	+	9.11312i	−1.19477	+	1.49819i	−0.373511	+	0.927626i	\(0.621846\pi\)
							−0.821254	+	0.570563i	\(0.806725\pi\)
\(41\)	−1.23720	−	5.42051i	−0.193217	−	0.846541i	−0.974861	−	0.222815i	\(-0.928475\pi\)
							0.781643	−	0.623726i	\(-0.214382\pi\)
\(43\)	0.148882	+	0.0339813i	0.0227043	+	0.00518210i	0.233858	−	0.972271i	\(-0.424865\pi\)
							−0.211154	+	0.977453i	\(0.567722\pi\)
\(47\)	3.36265	−	6.98262i	0.490493	−	1.01852i	−0.497989	−	0.867183i	\(-0.665928\pi\)
							0.988482	−	0.151336i	\(-0.0483576\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.p.a.76.16	✓	324	49.27	odd	14	inner
539.2.p.a.76.39	yes	324	539.76	even	14	inner
539.2.p.a.461.16	yes	324	11.10	odd	2	inner
539.2.p.a.461.39	yes	324	1.1	even	1	trivial