Embedded newform 539.2.r.b.375.1

• Label: 539.2.r.b.375.1
• 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			375.1
Character	\(\chi\)	\(=\)	539.375
Dual form			539.2.r.b.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.209677 - 2.79795i) q^{2} +(2.93495 + 0.905312i) q^{3} +(-5.80688 + 0.875247i) q^{4} +(0.632033 - 0.586441i) q^{5} +(1.91762 - 8.40165i) q^{6} +(2.01885 + 1.71005i) q^{7} +(2.41777 + 10.5929i) q^{8} +(5.31562 + 3.62413i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.209677	−	2.79795i	−0.148264	−	1.97845i	−0.200037	−	0.979788i	\(-0.564106\pi\)
							0.0517726	−	0.998659i	\(-0.483513\pi\)
\(3\)	2.93495	+	0.905312i	1.69449	+	0.522682i	0.983561	−	0.180578i	\(-0.0577969\pi\)
							0.710933	+	0.703260i	\(0.248273\pi\)
\(5\)	0.632033	−	0.586441i	0.282654	−	0.262264i	−0.526086	−	0.850432i	\(-0.676341\pi\)
							0.808739	+	0.588167i	\(0.200150\pi\)
\(7\)	2.01885	+	1.71005i	0.763052	+	0.646337i
							\(11\)	0.826239	−	0.563320i	0.249120	−	0.169847i
\(13\)	0.970595	+	0.467414i	0.269195	+	0.129637i								0.563612	−	0.826040i	\(-0.309411\pi\)
							−0.294417	+	0.955677i	\(0.595126\pi\)
\(17\)	−2.38973	−	6.08894i	−0.579595	−	1.47679i	−0.856633	−	0.515926i	\(-0.827448\pi\)
							0.277038	−	0.960859i	\(-0.410647\pi\)
\(19\)	−1.58243	−	2.74085i	−0.363034	−	0.628794i	0.625424	−	0.780285i	\(-0.284926\pi\)
							−0.988459	+	0.151491i	\(0.951592\pi\)
\(23\)	0.951667	−	2.42481i	0.198436	−	0.505608i	−0.796670	−	0.604414i	\(-0.793407\pi\)
							0.995107	+	0.0988066i	\(0.0315025\pi\)
\(29\)	−2.49652	+	3.13054i	−0.463592	+	0.581326i	−0.957589	−	0.288137i	\(-0.906964\pi\)
							0.493997	+	0.869464i	\(0.335535\pi\)
\(31\)	−1.43219	+	2.48063i	−0.257229	+	0.445534i	−0.965499	−	0.260408i	\(-0.916143\pi\)
							0.708269	+	0.705942i	\(0.249476\pi\)
\(37\)	1.33847	+	0.201743i	0.220044	+	0.0331663i	0.258139	−	0.966108i	\(-0.416891\pi\)
							−0.0380949	+	0.999274i	\(0.512129\pi\)
\(41\)	−0.760144	−	3.33041i	−0.118715	−	0.520123i	−0.998960	−	0.0456032i	\(-0.985479\pi\)
							0.880245	−	0.474519i	\(-0.157378\pi\)
\(43\)	−0.734073	+	3.21618i	−0.111945	+	0.490463i	0.887609	+	0.460598i	\(0.152365\pi\)
							−0.999554	+	0.0298654i	\(0.990492\pi\)
\(47\)	−0.0781955	−	1.04345i	−0.0114060	−	0.152202i	−0.999999	−	0.00158899i	\(-0.999494\pi\)
							0.988593	−	0.150613i	\(-0.0481248\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.375.1	yes	276
49.23	even	21	inner	539.2.r.b.23.1	✓	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.23.1	✓	276	49.23	even	21	inner
539.2.r.b.375.1	yes	276	1.1	even	1	trivial