Embedded newform 539.2.r.a.23.14

• Label: 539.2.r.a.23.14
• Level: $539$
• Weight: $2$
• Character: 539.23
• Analytic conductor: $4.304$
• Analytic rank: $0$
• Dimension: $276$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.r (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.30393666895\)
Analytic rank:	\(0\)
Dimension:	\(276\)
Relative dimension:	\(23\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			23.14
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.a.375.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0301718 - 0.402615i) q^{2} +(-2.74875 + 0.847876i) q^{3} +(1.81647 + 0.273789i) q^{4} +(2.87831 + 2.67068i) q^{5} +(0.258433 + 1.13227i) q^{6} +(0.132111 - 2.64245i) q^{7} +(0.344721 - 1.51032i) q^{8} +(4.35800 - 2.97123i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0301718	−	0.402615i	0.0213347	−	0.284692i	−0.976325	−	0.216311i	\(-0.930598\pi\)
							0.997659	−	0.0683810i	\(-0.0217834\pi\)
\(3\)	−2.74875	+	0.847876i	−1.58699	+	0.489521i	−0.957729	−	0.287673i	\(-0.907118\pi\)
							−0.629261	+	0.777194i	\(0.716642\pi\)
\(5\)	2.87831	+	2.67068i	1.28722	+	1.19437i	0.969198	+	0.246282i	\(0.0792089\pi\)
							0.318021	+	0.948084i	\(0.396982\pi\)
\(7\)	0.132111	−	2.64245i	0.0499332	−	0.998753i
							\(11\)	−0.826239	−	0.563320i	−0.249120	−	0.169847i
\(13\)	4.56264	−	2.19725i	1.26545	−	0.609408i								0.323838	−	0.946113i	\(-0.395027\pi\)
							0.941610	+	0.336705i	\(0.109312\pi\)
\(17\)	−0.457321	+	1.16524i	−0.110917	+	0.282611i	−0.975545	−	0.219802i	\(-0.929459\pi\)
							0.864628	+	0.502413i	\(0.167554\pi\)
\(19\)	−1.23097	+	2.13210i	−0.282403	+	0.489137i	−0.971976	−	0.235080i	\(-0.924465\pi\)
							0.689573	+	0.724216i	\(0.257798\pi\)
\(23\)	−2.37820	−	6.05956i	−0.495889	−	1.26351i	−0.931622	−	0.363429i	\(-0.881606\pi\)
							0.435733	−	0.900076i	\(-0.356489\pi\)
\(29\)	−0.986513	−	1.23705i	−0.183191	−	0.229714i	0.681753	−	0.731582i	\(-0.261218\pi\)
							−0.864944	+	0.501868i	\(0.832646\pi\)
\(31\)	4.28009	+	7.41333i	0.768727	+	1.33147i	0.938254	+	0.345948i	\(0.112443\pi\)
							−0.169527	+	0.985526i	\(0.554224\pi\)
\(37\)	6.87474	−	1.03620i	1.13020	−	0.170350i	0.442794	−	0.896623i	\(-0.353987\pi\)
							0.687406	+	0.726273i	\(0.258749\pi\)
\(41\)	0.734637	−	3.21866i	0.114731	−	0.502670i	−0.884609	−	0.466334i	\(-0.845574\pi\)
							0.999340	−	0.0363354i	\(-0.0115685\pi\)
\(43\)	1.92317	+	8.42594i	0.293280	+	1.28494i	0.879930	+	0.475103i	\(0.157589\pi\)
							−0.586650	+	0.809841i	\(0.699554\pi\)
\(47\)	−0.187050	+	2.49601i	−0.0272841	+	0.364080i	0.966805	+	0.255515i	\(0.0822449\pi\)
							−0.994089	+	0.108566i	\(0.965374\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.23.14	✓	276
49.32	even	21	inner	539.2.r.a.375.14	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.23.14	✓	276	1.1	even	1	trivial
539.2.r.a.375.14	yes	276	49.32	even	21	inner