Embedded newform 539.2.r.b.23.10

• Label: 539.2.r.b.23.10
• 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			23.10
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.b.375.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0264393 + 0.352807i) q^{2} +(-1.73623 + 0.535555i) q^{3} +(1.85389 + 0.279429i) q^{4} +(-1.35577 - 1.25797i) q^{5} +(-0.143043 - 0.626713i) q^{6} +(-2.45984 - 0.974260i) q^{7} +(-0.305054 + 1.33653i) q^{8} +(0.248947 - 0.169729i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0264393	+	0.352807i	−0.0186954	+	0.249473i	0.980024	+	0.198879i	\(0.0637302\pi\)
							−0.998719	+	0.0505931i	\(0.983889\pi\)
\(3\)	−1.73623	+	0.535555i	−1.00241	+	0.309203i	−0.752162	−	0.658979i	\(-0.770989\pi\)
							−0.250249	+	0.968181i	\(0.580513\pi\)
\(5\)	−1.35577	−	1.25797i	−0.606317	−	0.562580i	0.316124	−	0.948718i	\(-0.397618\pi\)
							−0.922441	+	0.386138i	\(0.873809\pi\)
\(7\)	−2.45984	−	0.974260i	−0.929732	−	0.368236i
							\(11\)	0.826239	+	0.563320i	0.249120	+	0.169847i
\(13\)	6.38778	−	3.07619i	1.77165	−	0.853182i								0.806610	−	0.591085i	\(-0.201300\pi\)
							0.965041	−	0.262098i	\(-0.0844142\pi\)
\(17\)	0.377614	−	0.962145i	0.0915848	−	0.233354i	−0.877740	−	0.479137i	\(-0.840950\pi\)
							0.969325	+	0.245783i	\(0.0790449\pi\)
\(19\)	1.47305	−	2.55140i	0.337941	−	0.585332i	−0.646104	−	0.763249i	\(-0.723603\pi\)
							0.984045	+	0.177918i	\(0.0569361\pi\)
\(23\)	−1.52203	−	3.87808i	−0.317366	−	0.808635i	−0.997166	−	0.0752375i	\(-0.976029\pi\)
							0.679800	−	0.733398i	\(-0.262067\pi\)
\(29\)	5.71970	+	7.17227i	1.06212	+	1.33186i	0.940694	+	0.339257i	\(0.110176\pi\)
							0.121427	+	0.992600i	\(0.461253\pi\)
\(31\)	−1.10237	−	1.90935i	−0.197991	−	0.342930i	0.749886	−	0.661567i	\(-0.230108\pi\)
							−0.947877	+	0.318637i	\(0.896775\pi\)
\(37\)	5.78432	−	0.871846i	0.950936	−	0.143331i	0.344787	−	0.938681i	\(-0.387951\pi\)
							0.606150	+	0.795350i	\(0.292713\pi\)
\(41\)	−0.0979156	+	0.428996i	−0.0152918	+	0.0669980i	−0.981998	−	0.188891i	\(-0.939511\pi\)
							0.966706	+	0.255889i	\(0.0823680\pi\)
\(43\)	−1.67509	−	7.33906i	−0.255449	−	1.11920i	−0.926057	−	0.377384i	\(-0.876824\pi\)
							0.670608	−	0.741812i	\(-0.266033\pi\)
\(47\)	0.435268	−	5.80824i	0.0634903	−	0.847219i	−0.871012	−	0.491262i	\(-0.836536\pi\)
							0.934502	−	0.355957i	\(-0.115845\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.23.10	✓	276
49.32	even	21	inner	539.2.r.b.375.10	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.23.10	✓	276	1.1	even	1	trivial
539.2.r.b.375.10	yes	276	49.32	even	21	inner