Embedded newform 539.2.r.a.23.17

• Label: 539.2.r.a.23.17
• 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			23.17
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.a.375.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0905072 - 1.20773i) q^{2} +(2.16284 - 0.667146i) q^{3} +(0.527231 + 0.0794673i) q^{4} +(-1.64013 - 1.52182i) q^{5} +(-0.609983 - 2.67251i) q^{6} +(-2.55293 + 0.694643i) q^{7} +(0.682693 - 2.99107i) q^{8} +(1.75406 - 1.19589i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0905072	−	1.20773i	0.0639982	−	0.853997i	−0.869160	−	0.494531i	\(-0.835340\pi\)
							0.933158	−	0.359466i	\(-0.117041\pi\)
\(3\)	2.16284	−	0.667146i	1.24871	−	0.385177i	0.401219	−	0.915982i	\(-0.368586\pi\)
							0.847494	+	0.530805i	\(0.178110\pi\)
\(5\)	−1.64013	−	1.52182i	−0.733487	−	0.680576i	0.222187	−	0.975004i	\(-0.428680\pi\)
							−0.955674	+	0.294428i	\(0.904871\pi\)
\(7\)	−2.55293	+	0.694643i	−0.964918	+	0.262550i
							\(11\)	−0.826239	−	0.563320i	−0.249120	−	0.169847i
\(13\)	3.06414	−	1.47561i	0.849839	−	0.409261i								0.0423210	−	0.999104i	\(-0.486525\pi\)
							0.807518	+	0.589843i	\(0.200811\pi\)
\(17\)	2.58695	−	6.59145i	0.627428	−	1.59866i	−0.163357	−	0.986567i	\(-0.552232\pi\)
							0.790785	−	0.612094i	\(-0.209673\pi\)
\(19\)	−2.04369	+	3.53977i	−0.468854	+	0.812079i	−0.999366	−	0.0355986i	\(-0.988666\pi\)
							0.530512	+	0.847677i	\(0.322000\pi\)
\(23\)	−0.199795	−	0.509069i	−0.0416601	−	0.106148i	0.908529	−	0.417821i	\(-0.137206\pi\)
							−0.950190	+	0.311673i	\(0.899111\pi\)
\(29\)	0.698353	+	0.875707i	0.129681	+	0.162615i	0.842433	−	0.538802i	\(-0.181123\pi\)
							−0.712752	+	0.701416i	\(0.752551\pi\)
\(31\)	4.96339	+	8.59684i	0.891451	+	1.54404i	0.838136	+	0.545461i	\(0.183645\pi\)
							0.0533147	+	0.998578i	\(0.483021\pi\)
\(37\)	9.59360	−	1.44600i	1.57718	−	0.237721i	0.698668	−	0.715446i	\(-0.253776\pi\)
							0.878510	+	0.477724i	\(0.158538\pi\)
\(41\)	0.310205	−	1.35910i	0.0484459	−	0.212255i	−0.944911	−	0.327326i	\(-0.893852\pi\)
							0.993357	+	0.115071i	\(0.0367095\pi\)
\(43\)	0.391182	+	1.71388i	0.0596547	+	0.261364i	0.995957	−	0.0898306i	\(-0.0286326\pi\)
							−0.936302	+	0.351195i	\(0.885775\pi\)
\(47\)	−0.952651	+	12.7122i	−0.138958	+	1.85427i	0.297718	+	0.954654i	\(0.403774\pi\)
							−0.436677	+	0.899618i	\(0.643845\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.23.17	✓	276
49.32	even	21	inner	539.2.r.a.375.17	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.23.17	✓	276	1.1	even	1	trivial
539.2.r.a.375.17	yes	276	49.32	even	21	inner