Embedded newform 539.2.p.a.76.8

• Label: 539.2.p.a.76.8
• 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			76.8
Character	\(\chi\)	\(=\)	539.76
Dual form			539.2.p.a.461.8

$q$-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	\(-0.838036\pi\)
							0.163635	−	0.986521i	\(-0.447678\pi\)
\(3\)	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
							\(11\)	1.70888	+	2.84249i	0.515245	+	0.857043i
\(13\)	5.99187	−	2.88553i	1.66185	−	0.800303i								0.663191	−	0.748451i	\(-0.269202\pi\)
							0.998655	−	0.0518519i	\(-0.0165124\pi\)
\(17\)	2.03434	+	2.55098i	0.493400	+	0.618704i	0.964726	−	0.263255i	\(-0.0847959\pi\)
							−0.471326	+	0.881959i	\(0.656225\pi\)
\(19\)	−2.79305			−0.640770			−0.320385	−	0.947287i	\(-0.603812\pi\)
							−0.320385	+	0.947287i	\(0.603812\pi\)
\(23\)	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	−0.348004	−	0.937493i	\(-0.613140\pi\)
							0.836550	+	0.547891i	\(0.184569\pi\)
\(31\)			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	\(0.182619\pi\)
							−0.992208	+	0.124590i	\(0.960238\pi\)
\(43\)	−2.09678	+	0.478577i	−0.319756	+	0.0729822i	−0.379387	−	0.925238i	\(-0.623865\pi\)
							0.0596304	+	0.998221i	\(0.481008\pi\)
\(47\)	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.76.8	✓	324
11.10	odd	2	inner	539.2.p.a.76.47	yes	324
49.20	odd	14	inner	539.2.p.a.461.47	yes	324
539.461	even	14	inner	539.2.p.a.461.8	yes	324

    

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