Embedded newform 539.2.p.a.76.47

• Label: 539.2.p.a.76.47
• 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			76.47
Character	\(\chi\)	\(=\)	539.76
Dual form			539.2.p.a.461.47

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00364 + 2.08409i) q^{2} +(2.98209 + 0.680642i) q^{3} +(-2.08914 + 2.61970i) q^{4} +(-0.377041 - 0.0860571i) q^{5} +(1.57444 + 6.89805i) q^{6} +(1.63031 - 2.08377i) q^{7} +(-3.04609 - 0.695251i) q^{8} +(5.72667 + 2.75782i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 14 q^{3} + 42 q^{4} - 14 q^{5} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00364	+	2.08409i	0.709683	+	1.47367i	0.873318	+	0.487151i	\(0.161964\pi\)
							−0.163635	+	0.986521i	\(0.552322\pi\)
\(3\)	2.98209	+	0.680642i	1.72171	+	0.392969i	0.965303	−	0.261133i	\(-0.0840961\pi\)
							0.756407	+	0.654102i	\(0.226953\pi\)
\(5\)	−0.377041	−	0.0860571i	−0.168618	−	0.0384859i	0.137378	−	0.990519i	\(-0.456132\pi\)
							−0.305996	+	0.952033i	\(0.598989\pi\)
\(7\)	1.63031	−	2.08377i	0.616198	−	0.787591i
							\(11\)	−1.15688	−	3.10832i	−0.348813	−	0.937192i
\(13\)	−5.99187	+	2.88553i	−1.66185	+	0.800303i								−0.663191	+	0.748451i	\(0.730798\pi\)
							−0.998655	+	0.0518519i	\(0.983488\pi\)
\(17\)	−2.03434	−	2.55098i	−0.493400	−	0.618704i	0.471326	−	0.881959i	\(-0.343775\pi\)
							−0.964726	+	0.263255i	\(0.915204\pi\)
\(19\)	2.79305			0.640770			0.320385	−	0.947287i	\(-0.396188\pi\)
							0.320385	+	0.947287i	\(0.396188\pi\)
\(23\)	1.63464	−	2.04978i	0.340847	−	0.427408i	−0.581634	−	0.813450i	\(-0.697587\pi\)
							0.922481	+	0.386042i	\(0.126158\pi\)
\(29\)	−2.63090	+	2.09807i	−0.488546	+	0.389603i	−0.836550	−	0.547891i	\(-0.815431\pi\)
							0.348004	+	0.937493i	\(0.386860\pi\)
\(31\)			0.666310i			0.119673i	0.998208	+	0.0598364i	\(0.0190579\pi\)
							−0.998208	+	0.0598364i	\(0.980942\pi\)
\(37\)	−4.21315	−	5.28312i	−0.692638	−	0.868540i	0.303812	−	0.952732i	\(-0.401741\pi\)
							−0.996450	+	0.0841919i	\(0.973169\pi\)
\(41\)	0.975305	−	4.27309i	0.152317	−	0.667345i	−0.839891	−	0.542755i	\(-0.817381\pi\)
							0.992208	−	0.124590i	\(-0.0397616\pi\)
\(43\)	2.09678	−	0.478577i	0.319756	−	0.0729822i	−0.0596304	−	0.998221i	\(-0.518992\pi\)
							0.379387	+	0.925238i	\(0.376135\pi\)
\(47\)	5.05720	+	10.5014i	0.737669	+	1.53179i	0.843338	+	0.537384i	\(0.180587\pi\)
							−0.105669	+	0.994401i	\(0.533698\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.p.a.76.47	yes	324
11.10	odd	2	inner	539.2.p.a.76.8	✓	324
49.20	odd	14	inner	539.2.p.a.461.8	yes	324
539.461	even	14	inner	539.2.p.a.461.47	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.p.a.76.8	✓	324	11.10	odd	2	inner
539.2.p.a.76.47	yes	324	1.1	even	1	trivial
539.2.p.a.461.8	yes	324	49.20	odd	14	inner
539.2.p.a.461.47	yes	324	539.461	even	14	inner