Embedded newform 539.2.r.b.23.18

• Label: 539.2.r.b.23.18
• 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			23.18
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.b.375.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.126531 - 1.68844i) q^{2} +(-2.90212 + 0.895185i) q^{3} +(-0.857151 - 0.129195i) q^{4} +(-0.837978 - 0.777530i) q^{5} +(1.14426 + 5.01332i) q^{6} +(0.435612 + 2.60964i) q^{7} +(0.426939 - 1.87054i) q^{8} +(5.14222 - 3.50590i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.126531	−	1.68844i	0.0894709	−	1.19391i	−0.754053	−	0.656814i	\(-0.771904\pi\)
							0.843524	−	0.537092i	\(-0.180477\pi\)
\(3\)	−2.90212	+	0.895185i	−1.67554	+	0.516835i	−0.979547	−	0.201215i	\(-0.935511\pi\)
							−0.695992	+	0.718050i	\(0.745035\pi\)
\(5\)	−0.837978	−	0.777530i	−0.374755	−	0.347722i	0.470199	−	0.882560i	\(-0.344182\pi\)
							−0.844954	+	0.534838i	\(0.820373\pi\)
\(7\)	0.435612	+	2.60964i	0.164646	+	0.986353i
							\(11\)	0.826239	+	0.563320i	0.249120	+	0.169847i
\(13\)	3.41719	−	1.64563i	0.947759	−	0.456417i								0.104859	−	0.994487i	\(-0.466561\pi\)
							0.842900	+	0.538070i	\(0.180846\pi\)
\(17\)	−1.25596	+	3.20013i	−0.304614	+	0.776145i	0.693817	+	0.720152i	\(0.255928\pi\)
							−0.998431	+	0.0559933i	\(0.982167\pi\)
\(19\)	−0.589764	+	1.02150i	−0.135301	+	0.234348i	−0.925712	−	0.378228i	\(-0.876534\pi\)
							0.790411	+	0.612576i	\(0.209867\pi\)
\(23\)	−2.59464	−	6.61103i	−0.541020	−	1.37850i	−0.896180	−	0.443690i	\(-0.853669\pi\)
							0.355161	−	0.934805i	\(-0.384426\pi\)
\(29\)	−6.11422	−	7.66698i	−1.13538	−	1.42372i	−0.890976	−	0.454050i	\(-0.849979\pi\)
							−0.244405	−	0.969673i	\(-0.578593\pi\)
\(31\)	−3.23464	−	5.60256i	−0.580959	−	1.00625i	−0.995366	−	0.0961585i	\(-0.969344\pi\)
							0.414407	−	0.910092i	\(-0.363989\pi\)
\(37\)	10.9970	−	1.65752i	1.80789	−	0.272495i	0.843527	−	0.537086i	\(-0.180475\pi\)
							0.964361	+	0.264591i	\(0.0852370\pi\)
\(41\)	2.12498	−	9.31014i	0.331866	−	1.45400i	−0.483648	−	0.875263i	\(-0.660689\pi\)
							0.815514	−	0.578737i	\(-0.196454\pi\)
\(43\)	2.35393	+	10.3133i	0.358971	+	1.57276i	0.755762	+	0.654847i	\(0.227267\pi\)
							−0.396790	+	0.917909i	\(0.629876\pi\)
\(47\)	0.0999707	−	1.33402i	0.0145822	−	0.194586i	−0.985155	−	0.171667i	\(-0.945085\pi\)
							0.999737	−	0.0229196i	\(-0.00729616\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.23.18	✓	276
49.32	even	21	inner	539.2.r.b.375.18	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.23.18	✓	276	1.1	even	1	trivial
539.2.r.b.375.18	yes	276	49.32	even	21	inner