Embedded newform 539.2.r.a.375.6

• Label: 539.2.r.a.375.6
• Level: $539$
• Weight: $2$
• Character: 539.375
• Analytic conductor: $4.304$
• Analytic rank: $0$
• Dimension: $276$
• 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			375.6
Character	\(\chi\)	\(=\)	539.375
Dual form			539.2.r.a.23.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.102537 - 1.36825i) q^{2} +(2.76390 + 0.852550i) q^{3} +(0.116054 - 0.0174923i) q^{4} +(1.55705 - 1.44473i) q^{5} +(0.883105 - 3.86914i) q^{6} +(-0.773931 + 2.53003i) q^{7} +(-0.646472 - 2.83238i) q^{8} +(4.43358 + 3.02276i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{2}{21}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.102537	−	1.36825i	−0.0725043	−	0.967502i	−0.908261	−	0.418403i	\(-0.862590\pi\)
							0.835757	−	0.549099i	\(-0.185029\pi\)
\(3\)	2.76390	+	0.852550i	1.59574	+	0.492220i	0.960074	−	0.279746i	\(-0.0902503\pi\)
							0.635664	+	0.771966i	\(0.280727\pi\)
\(5\)	1.55705	−	1.44473i	0.696332	−	0.646101i	−0.250393	−	0.968144i	\(-0.580560\pi\)
							0.946725	+	0.322043i	\(0.104369\pi\)
\(7\)	−0.773931	+	2.53003i	−0.292519	+	0.956260i
							\(11\)	−0.826239	+	0.563320i	−0.249120	+	0.169847i
\(13\)	−1.36381	−	0.656776i	−0.378253	−	0.182157i								0.235090	−	0.971974i	\(-0.424461\pi\)
							−0.613343	+	0.789817i	\(0.710176\pi\)
\(17\)	0.527426	+	1.34386i	0.127920	+	0.325934i	0.980457	−	0.196732i	\(-0.0630328\pi\)
							−0.852538	+	0.522666i	\(0.824938\pi\)
\(19\)	−3.26917	−	5.66236i	−0.749999	−	1.29904i	−0.947823	−	0.318798i	\(-0.896721\pi\)
							0.197824	−	0.980238i	\(-0.436612\pi\)
\(23\)	−2.47862	+	6.31543i	−0.516829	+	1.31686i	0.399610	+	0.916685i	\(0.369145\pi\)
							−0.916439	+	0.400174i	\(0.868950\pi\)
\(29\)	0.505242	−	0.633554i	0.0938211	−	0.117648i	−0.732705	−	0.680547i	\(-0.761742\pi\)
							0.826526	+	0.562899i	\(0.190314\pi\)
\(31\)	1.29421	−	2.24163i	0.232447	−	0.402609i	−0.726081	−	0.687609i	\(-0.758660\pi\)
							0.958528	+	0.285000i	\(0.0919936\pi\)
\(37\)	−5.72104	−	0.862308i	−0.940533	−	0.141763i	−0.339158	−	0.940729i	\(-0.610142\pi\)
							−0.601375	+	0.798967i	\(0.705380\pi\)
\(41\)	1.48984	+	6.52743i	0.232675	+	1.01941i	0.947411	+	0.320020i	\(0.103690\pi\)
							−0.714736	+	0.699394i	\(0.753453\pi\)
\(43\)	0.735936	−	3.22435i	0.112229	−	0.491708i	−0.887305	−	0.461184i	\(-0.847425\pi\)
							0.999534	−	0.0305249i	\(-0.00971789\pi\)
\(47\)	0.770534	+	10.2821i	0.112394	+	1.49979i	0.713518	+	0.700637i	\(0.247101\pi\)
							−0.601124	+	0.799156i	\(0.705280\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.375.6	yes	276
49.23	even	21	inner	539.2.r.a.23.6	✓	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.23.6	✓	276	49.23	even	21	inner
539.2.r.a.375.6	yes	276	1.1	even	1	trivial