Embedded newform 539.2.r.a.23.10

• Label: 539.2.r.a.23.10
• 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			23.10
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.a.375.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0405728 + 0.541406i) q^{2} +(-0.866143 + 0.267170i) q^{3} +(1.68619 + 0.254152i) q^{4} +(2.31163 + 2.14488i) q^{5} +(-0.109505 - 0.479775i) q^{6} +(1.75943 + 1.97596i) q^{7} +(-0.447637 + 1.96122i) q^{8} +(-1.79989 + 1.22715i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0405728	+	0.541406i	−0.0286893	+	0.382832i	0.964315	+	0.264756i	\(0.0852915\pi\)
							−0.993005	+	0.118076i	\(0.962328\pi\)
\(3\)	−0.866143	+	0.267170i	−0.500068	+	0.154250i	−0.534520	−	0.845156i	\(-0.679508\pi\)
							0.0344521	+	0.999406i	\(0.489031\pi\)
\(5\)	2.31163	+	2.14488i	1.03379	+	0.959221i	0.999192	−	0.0401800i	\(-0.0127931\pi\)
							0.0346021	+	0.999401i	\(0.488984\pi\)
\(7\)	1.75943	+	1.97596i	0.665003	+	0.746841i
							\(11\)	−0.826239	−	0.563320i	−0.249120	−	0.169847i
\(13\)	−2.95425	+	1.42269i	−0.819362	+	0.394584i								−0.796115	−	0.605146i	\(-0.793115\pi\)
							−0.0232475	+	0.999730i	\(0.507401\pi\)
\(17\)	1.90511	−	4.85413i	0.462056	−	1.17730i	−0.490094	−	0.871669i	\(-0.663038\pi\)
							0.952150	−	0.305630i	\(-0.0988671\pi\)
\(19\)	0.143389	−	0.248357i	0.0328956	−	0.0569769i	−0.849109	−	0.528218i	\(-0.822860\pi\)
							0.882005	+	0.471241i	\(0.156194\pi\)
\(23\)	−2.39467	−	6.10152i	−0.499323	−	1.27226i	−0.929270	−	0.369401i	\(-0.879563\pi\)
							0.429947	−	0.902854i	\(-0.358532\pi\)
\(29\)	1.50963	+	1.89301i	0.280331	+	0.351523i	0.901984	−	0.431769i	\(-0.142110\pi\)
							−0.621654	+	0.783292i	\(0.713539\pi\)
\(31\)	−4.30764	−	7.46105i	−0.773675	−	1.34004i	−0.935536	−	0.353231i	\(-0.885083\pi\)
							0.161861	−	0.986814i	\(-0.448250\pi\)
\(37\)	2.62503	−	0.395660i	0.431552	−	0.0650460i	0.0703257	−	0.997524i	\(-0.477596\pi\)
							0.361227	+	0.932478i	\(0.382358\pi\)
\(41\)	−0.157937	+	0.691969i	−0.0246657	+	0.108067i	−0.985762	−	0.168145i	\(-0.946222\pi\)
							0.961097	+	0.276213i	\(0.0890794\pi\)
\(43\)	2.15790	+	9.45437i	0.329077	+	1.44178i	0.820894	+	0.571080i	\(0.193476\pi\)
							−0.491818	+	0.870698i	\(0.663667\pi\)
\(47\)	−0.247728	+	3.30570i	−0.0361349	+	0.482187i	0.949553	+	0.313606i	\(0.101537\pi\)
							−0.985688	+	0.168580i	\(0.946082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.23.10	✓	276
49.32	even	21	inner	539.2.r.a.375.10	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.23.10	✓	276	1.1	even	1	trivial
539.2.r.a.375.10	yes	276	49.32	even	21	inner