Embedded newform 539.2.r.b.23.13

• Label: 539.2.r.b.23.13
• 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			23.13
Character	\(\chi\)	\(=\)	539.23
Dual form			539.2.r.b.375.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.00841924 - 0.112347i) q^{2} +(2.88830 - 0.890922i) q^{3} +(1.96511 + 0.296193i) q^{4} +(-1.46461 - 1.35896i) q^{5} +(-0.0757751 - 0.331992i) q^{6} +(-2.03943 - 1.68544i) q^{7} +(0.0999604 - 0.437955i) q^{8} +(5.06980 - 3.45653i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.00841924	−	0.112347i	0.00595330	−	0.0794413i	−0.993436	−	0.114391i	\(-0.963508\pi\)
							0.999389	+	0.0349501i	\(0.0111272\pi\)
\(3\)	2.88830	−	0.890922i	1.66756	−	0.514374i	0.689782	−	0.724018i	\(-0.257707\pi\)
							0.977778	+	0.209644i	\(0.0672304\pi\)
\(5\)	−1.46461	−	1.35896i	−0.654993	−	0.607745i	0.281023	−	0.959701i	\(-0.409326\pi\)
							−0.936017	+	0.351956i	\(0.885517\pi\)
\(7\)	−2.03943	−	1.68544i	−0.770833	−	0.637037i
							\(11\)	0.826239	+	0.563320i	0.249120	+	0.169847i
\(13\)	−4.57892	+	2.20509i	−1.26996	+	0.611582i								−0.942792	−	0.333382i	\(-0.891810\pi\)
							−0.327172	+	0.944965i	\(0.606096\pi\)
\(17\)	−0.986832	+	2.51441i	−0.239342	+	0.609834i	−0.999189	−	0.0402686i	\(-0.987179\pi\)
							0.759847	+	0.650102i	\(0.225274\pi\)
\(19\)	4.01754	−	6.95858i	0.921686	−	1.59641i	0.124880	−	0.992172i	\(-0.460145\pi\)
							0.796806	−	0.604235i	\(-0.206521\pi\)
\(23\)	2.60707	+	6.64271i	0.543612	+	1.38510i	0.893819	+	0.448428i	\(0.148016\pi\)
							−0.350207	+	0.936672i	\(0.613889\pi\)
\(29\)	2.32062	+	2.90997i	0.430929	+	0.540368i	0.949127	−	0.314893i	\(-0.101969\pi\)
							−0.518198	+	0.855261i	\(0.673397\pi\)
\(31\)	2.14913	+	3.72241i	0.385996	+	0.668565i	0.991907	−	0.126968i	\(-0.0405245\pi\)
							−0.605911	+	0.795533i	\(0.707191\pi\)
\(37\)	−9.89927	+	1.49208i	−1.62743	+	0.245296i	−0.898398	−	0.439183i	\(-0.855268\pi\)
							−0.729033	+	0.684479i	\(0.760030\pi\)
\(41\)	−1.59553	+	6.99045i	−0.249179	+	1.09173i	0.683196	+	0.730235i	\(0.260589\pi\)
							−0.932376	+	0.361491i	\(0.882268\pi\)
\(43\)	0.927777	+	4.06486i	0.141485	+	0.619885i	0.995091	+	0.0989658i	\(0.0315534\pi\)
							−0.853606	+	0.520919i	\(0.825589\pi\)
\(47\)	0.433027	−	5.77834i	0.0631635	−	0.842858i	−0.872195	−	0.489158i	\(-0.837304\pi\)
							0.935359	−	0.353700i	\(-0.115077\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.23.13	✓	276
49.32	even	21	inner	539.2.r.b.375.13	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.23.13	✓	276	1.1	even	1	trivial
539.2.r.b.375.13	yes	276	49.32	even	21	inner