Embedded newform 539.2.r.b.144.16

• Label: 539.2.r.b.144.16
• Level: $539$
• Weight: $2$
• Character: 539.144
• 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			144.16
Character	\(\chi\)	\(=\)	539.144
Dual form			539.2.r.b.408.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.401920 - 1.02407i) q^{2} +(-0.172926 + 2.30753i) q^{3} +(0.578914 + 0.537154i) q^{4} +(1.24420 + 0.848282i) q^{5} +(2.29358 + 1.10453i) q^{6} +(1.33485 + 2.28433i) q^{7} +(2.76511 - 1.33161i) q^{8} +(-2.32831 - 0.350936i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{11}{21}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.401920	−	1.02407i	0.284200	−	0.724130i	−0.715475	−	0.698638i	\(-0.753790\pi\)
							0.999675	−	0.0254918i	\(-0.00811516\pi\)
\(3\)	−0.172926	+	2.30753i	−0.0998387	+	1.33225i	0.692166	+	0.721739i	\(0.256657\pi\)
							−0.792004	+	0.610516i	\(0.790962\pi\)
\(5\)	1.24420	+	0.848282i	0.556424	+	0.379363i	0.808637	−	0.588308i	\(-0.200206\pi\)
							−0.252213	+	0.967672i	\(0.581158\pi\)
\(7\)	1.33485	+	2.28433i	0.504527	+	0.863396i
							\(11\)	−0.988831	+	0.149042i	−0.298144	+	0.0449379i
\(13\)	0.486778	+	0.610401i	0.135008	+	0.169295i								0.844740	−	0.535177i	\(-0.179755\pi\)
							−0.709732	+	0.704472i	\(0.751184\pi\)
\(17\)	−5.82973	−	1.79823i	−1.41392	−	0.436136i	−0.508649	−	0.860974i	\(-0.669855\pi\)
							−0.905269	+	0.424838i	\(0.860331\pi\)
\(19\)	−3.36104	−	5.82150i	−0.771077	−	1.33554i	−0.936973	−	0.349401i	\(-0.886385\pi\)
							0.165897	−	0.986143i	\(-0.446948\pi\)
\(23\)	6.73459	−	2.07735i	1.40426	−	0.433157i	0.502193	−	0.864755i	\(-0.332527\pi\)
							0.902066	+	0.431599i	\(0.142050\pi\)
\(29\)	−0.149520	−	0.655089i	−0.0277651	−	0.121647i	0.959146	−	0.282911i	\(-0.0913001\pi\)
							−0.986911	+	0.161264i	\(0.948443\pi\)
\(31\)	−3.19704	+	5.53744i	−0.574205	+	0.994553i	0.421922	+	0.906632i	\(0.361356\pi\)
							−0.996127	+	0.0879208i	\(0.971978\pi\)
\(37\)	4.08161	−	3.78718i	0.671012	−	0.622608i	−0.269246	−	0.963071i	\(-0.586774\pi\)
							0.940258	+	0.340463i	\(0.110584\pi\)
\(41\)	10.1990	−	4.91158i	1.59282	−	0.767061i	0.593530	−	0.804812i	\(-0.297734\pi\)
							0.999287	+	0.0377510i	\(0.0120194\pi\)
\(43\)	−1.21174	−	0.583541i	−0.184788	−	0.0889892i	0.339202	−	0.940713i	\(-0.389843\pi\)
							−0.523990	+	0.851724i	\(0.675557\pi\)
\(47\)	−4.99346	+	12.7231i	−0.728372	+	1.85586i	−0.290216	+	0.956961i	\(0.593727\pi\)
							−0.438156	+	0.898899i	\(0.644368\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.144.16	✓	276
49.16	even	21	inner	539.2.r.b.408.16	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.144.16	✓	276	1.1	even	1	trivial
539.2.r.b.408.16	yes	276	49.16	even	21	inner