Embedded newform 539.2.r.a.23.19

• Label: 539.2.r.a.23.19
• 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			23.19
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.a.375.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.143761 - 1.91836i) q^{2} +(-2.52291 + 0.778215i) q^{3} +(-1.68177 - 0.253486i) q^{4} +(-1.84824 - 1.71492i) q^{5} +(1.13020 + 4.95172i) q^{6} +(-0.883304 - 2.49395i) q^{7} +(0.128093 - 0.561211i) q^{8} +(3.28074 - 2.23677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.143761	−	1.91836i	0.101654	−	1.35648i	−0.680201	−	0.733025i	\(-0.738108\pi\)
							0.781856	−	0.623459i	\(-0.214273\pi\)
\(3\)	−2.52291	+	0.778215i	−1.45660	+	0.449302i	−0.919122	−	0.393974i	\(-0.871100\pi\)
							−0.537481	+	0.843276i	\(0.680624\pi\)
\(5\)	−1.84824	−	1.71492i	−0.826558	−	0.766933i	0.148494	−	0.988913i	\(-0.452558\pi\)
							−0.975051	+	0.221980i	\(0.928748\pi\)
\(7\)	−0.883304	−	2.49395i	−0.333858	−	0.942624i
							\(11\)	−0.826239	−	0.563320i	−0.249120	−	0.169847i
\(13\)	−0.964449	+	0.464454i	−0.267490	+	0.128816i								−0.562822	−	0.826578i	\(-0.690284\pi\)
							0.295331	+	0.955395i	\(0.404570\pi\)
\(17\)	−1.84047	+	4.68945i	−0.446380	+	1.13736i	0.513783	+	0.857920i	\(0.328244\pi\)
							−0.960164	+	0.279439i	\(0.909852\pi\)
\(19\)	0.714539	−	1.23762i	0.163926	−	0.283929i	−0.772347	−	0.635201i	\(-0.780917\pi\)
							0.936274	+	0.351272i	\(0.114251\pi\)
\(23\)	1.16316	+	2.96368i	0.242535	+	0.617969i	0.999365	−	0.0356430i	\(-0.0113479\pi\)
							−0.756829	+	0.653612i	\(0.773253\pi\)
\(29\)	2.61934	+	3.28454i	0.486398	+	0.609924i	0.963101	−	0.269140i	\(-0.0867396\pi\)
							−0.476703	+	0.879065i	\(0.658168\pi\)
\(31\)	1.96972	+	3.41166i	0.353772	+	0.612751i	0.986907	−	0.161290i	\(-0.0515655\pi\)
							−0.633135	+	0.774042i	\(0.718232\pi\)
\(37\)	2.87164	−	0.432830i	0.472094	−	0.0711568i	0.0913137	−	0.995822i	\(-0.470893\pi\)
							0.380781	+	0.924665i	\(0.375655\pi\)
\(41\)	−2.01710	+	8.83747i	−0.315017	+	1.38018i	0.531156	+	0.847274i	\(0.321758\pi\)
							−0.846174	+	0.532907i	\(0.821099\pi\)
\(43\)	−2.27074	−	9.94878i	−0.346285	−	1.51717i	−0.785539	−	0.618812i	\(-0.787614\pi\)
							0.439254	−	0.898363i	\(-0.355243\pi\)
\(47\)	−0.321975	+	4.29646i	−0.0469649	+	0.626703i	0.923210	+	0.384296i	\(0.125556\pi\)
							−0.970175	+	0.242407i	\(0.922063\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.23.19	✓	276
49.32	even	21	inner	539.2.r.a.375.19	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.23.19	✓	276	1.1	even	1	trivial
539.2.r.a.375.19	yes	276	49.32	even	21	inner