Embedded newform 539.2.r.a.485.22

• Label: 539.2.r.a.485.22
• Level: $539$
• Weight: $2$
• Character: 539.485
• 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			485.22
Character	\(\chi\)	\(=\)	539.485
Dual form			539.2.r.a.529.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.36927 - 0.357110i) q^{2} +(-1.21368 - 0.827476i) q^{3} +(3.57477 - 1.10267i) q^{4} +(0.300934 - 4.01568i) q^{5} +(-3.17105 - 1.52710i) q^{6} +(-0.923926 + 2.47919i) q^{7} +(3.75833 - 1.80992i) q^{8} +(-0.307710 - 0.784033i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{4}{21}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.36927	−	0.357110i	1.67533	−	0.252515i	0.758464	−	0.651715i	\(-0.225950\pi\)
							0.916864	+	0.399200i	\(0.130712\pi\)
\(3\)	−1.21368	−	0.827476i	−0.700721	−	0.477743i	0.159806	−	0.987148i	\(-0.448913\pi\)
							−0.860527	+	0.509405i	\(0.829866\pi\)
\(5\)	0.300934	−	4.01568i	0.134582	−	1.79587i	−0.366450	−	0.930438i	\(-0.619427\pi\)
							0.501031	−	0.865429i	\(-0.332954\pi\)
\(7\)	−0.923926	+	2.47919i	−0.349211	+	0.937044i
							\(11\)	−0.365341	+	0.930874i	−0.110154	+	0.280669i
\(13\)	0.333079	+	0.417668i	0.0923795	+	0.115840i								0.825873	−	0.563856i	\(-0.190683\pi\)
							−0.733493	+	0.679697i	\(0.762111\pi\)
\(17\)	4.53980	−	4.21232i	1.10106	−	1.02164i	0.101423	−	0.994843i	\(-0.467661\pi\)
							0.999641	−	0.0267945i	\(-0.00852996\pi\)
\(19\)	0.349273	−	0.604958i	0.0801287	−	0.138787i	−0.823176	−	0.567786i	\(-0.807800\pi\)
							0.903305	+	0.428999i	\(0.141134\pi\)
\(23\)	2.43480	+	2.25917i	0.507692	+	0.471069i	0.891936	−	0.452163i	\(-0.149347\pi\)
							−0.384244	+	0.923232i	\(0.625538\pi\)
\(29\)	0.349488	+	1.53121i	0.0648982	+	0.284338i	0.996955	−	0.0779734i	\(-0.0248449\pi\)
							−0.932057	+	0.362311i	\(0.881988\pi\)
\(31\)	−1.73065	−	2.99757i	−0.310834	−	0.538380i	0.667709	−	0.744422i	\(-0.267275\pi\)
							−0.978543	+	0.206042i	\(0.933942\pi\)
\(37\)	8.38567	+	2.58664i	1.37860	+	0.425240i	0.893438	−	0.449187i	\(-0.148286\pi\)
							0.485157	+	0.874427i	\(0.338762\pi\)
\(41\)	4.39665	−	2.11731i	0.686641	−	0.330669i	−0.0578273	−	0.998327i	\(-0.518417\pi\)
							0.744468	+	0.667658i	\(0.232703\pi\)
\(43\)	8.32478	+	4.00900i	1.26952	+	0.611367i	0.942675	−	0.333712i	\(-0.108301\pi\)
							0.326842	+	0.945079i	\(0.394016\pi\)
\(47\)	−2.15546	+	0.324883i	−0.314406	+	0.0473891i	−0.304350	−	0.952560i	\(-0.598439\pi\)
							−0.0100564	+	0.999949i	\(0.503201\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.485.22	✓	276
49.39	even	21	inner	539.2.r.a.529.22	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.485.22	✓	276	1.1	even	1	trivial
539.2.r.a.529.22	yes	276	49.39	even	21	inner