Embedded newform 539.2.r.a.23.4

• Label: 539.2.r.a.23.4
• 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			23.4
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.a.375.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.156649 + 2.09034i) q^{2} +(1.86160 - 0.574229i) q^{3} +(-2.36732 - 0.356817i) q^{4} +(1.22307 + 1.13484i) q^{5} +(0.908715 + 3.98134i) q^{6} +(2.06946 + 1.64844i) q^{7} +(0.183810 - 0.805326i) q^{8} +(0.657114 - 0.448012i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.156649	+	2.09034i	−0.110768	+	1.47809i	0.614112	+	0.789219i	\(0.289514\pi\)
							−0.724880	+	0.688875i	\(0.758105\pi\)
\(3\)	1.86160	−	0.574229i	1.07480	−	0.331531i	0.293673	−	0.955906i	\(-0.405122\pi\)
							0.781125	+	0.624375i	\(0.214646\pi\)
\(5\)	1.22307	+	1.13484i	0.546972	+	0.507515i	0.904550	−	0.426368i	\(-0.140207\pi\)
							−0.357578	+	0.933883i	\(0.616398\pi\)
\(7\)	2.06946	+	1.64844i	0.782181	+	0.623051i
							\(11\)	−0.826239	−	0.563320i	−0.249120	−	0.169847i
\(13\)	0.132189	−	0.0636591i	0.0366627	−	0.0176558i								−0.415463	−	0.909610i	\(-0.636380\pi\)
							0.452125	+	0.891954i	\(0.350666\pi\)
\(17\)	2.36387	−	6.02304i	0.573323	−	1.46080i	−0.290435	−	0.956895i	\(-0.593800\pi\)
							0.863758	−	0.503907i	\(-0.168105\pi\)
\(19\)	0.956199	−	1.65619i	0.219367	−	0.379955i	−0.735247	−	0.677799i	\(-0.762934\pi\)
							0.954615	+	0.297844i	\(0.0962674\pi\)
\(23\)	3.19763	+	8.14743i	0.666752	+	1.69886i	0.714161	+	0.699982i	\(0.246808\pi\)
							−0.0474086	+	0.998876i	\(0.515096\pi\)
\(29\)	−4.60473	−	5.77414i	−0.855076	−	1.07223i	−0.996608	−	0.0822979i	\(-0.973774\pi\)
							0.141532	−	0.989934i	\(-0.454797\pi\)
\(31\)	2.28762	+	3.96227i	0.410868	+	0.711644i	0.994985	−	0.100025i	\(-0.0318924\pi\)
							−0.584117	+	0.811670i	\(0.698559\pi\)
\(37\)	−9.69162	+	1.46078i	−1.59329	+	0.240150i	−0.884939	−	0.465707i	\(-0.845800\pi\)
							−0.708354	+	0.705857i	\(0.750562\pi\)
\(41\)	−1.09333	+	4.79020i	−0.170750	+	0.748104i	0.814942	+	0.579543i	\(0.196769\pi\)
							−0.985691	+	0.168561i	\(0.946088\pi\)
\(43\)	−1.92086	−	8.41586i	−0.292929	−	1.28341i	−0.880427	−	0.474181i	\(-0.842744\pi\)
							0.587498	−	0.809225i	\(-0.300113\pi\)
\(47\)	−0.297895	+	3.97514i	−0.0434525	+	0.579834i	0.932404	+	0.361419i	\(0.117708\pi\)
							−0.975856	+	0.218415i	\(0.929911\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.23.4	✓	276
49.32	even	21	inner	539.2.r.a.375.4	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.23.4	✓	276	1.1	even	1	trivial
539.2.r.a.375.4	yes	276	49.32	even	21	inner