Embedded newform 539.2.r.a.144.11

• Label: 539.2.r.a.144.11
• 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			144.11
Character	\(\chi\)	\(=\)	539.144
Dual form			539.2.r.a.408.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0544039 + 0.138619i) q^{2} +(0.211417 - 2.82116i) q^{3} +(1.44985 + 1.34526i) q^{4} +(-2.51857 - 1.71713i) q^{5} +(0.379564 + 0.182788i) q^{6} +(-0.507933 - 2.59654i) q^{7} +(-0.533688 + 0.257010i) q^{8} +(-4.94776 - 0.745754i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0544039	+	0.138619i	−0.0384694	+	0.0980184i	−0.948817	−	0.315825i	\(-0.897719\pi\)
							0.910348	+	0.413844i	\(0.135814\pi\)
\(3\)	0.211417	−	2.82116i	0.122062	−	1.62880i	−0.514443	−	0.857525i	\(-0.672001\pi\)
							0.636504	−	0.771273i	\(-0.280380\pi\)
\(5\)	−2.51857	−	1.71713i	−1.12634	−	0.767925i	−0.150977	−	0.988537i	\(-0.548242\pi\)
							−0.975362	+	0.220612i	\(0.929194\pi\)
\(7\)	−0.507933	−	2.59654i	−0.191980	−	0.981399i
							\(11\)	0.988831	−	0.149042i	0.298144	−	0.0449379i
\(13\)	1.39662	+	1.75131i	0.387353	+	0.485726i								0.936831	−	0.349783i	\(-0.113745\pi\)
							−0.549477	+	0.835509i	\(0.685173\pi\)
\(17\)	−2.68588	−	0.828484i	−0.651421	−	0.200937i	−0.0486065	−	0.998818i	\(-0.515478\pi\)
							−0.602815	+	0.797881i	\(0.705954\pi\)
\(19\)	−3.58226	−	6.20466i	−0.821827	−	1.42345i	−0.904320	−	0.426854i	\(-0.859622\pi\)
							0.0824934	−	0.996592i	\(-0.473712\pi\)
\(23\)	−1.63284	+	0.503664i	−0.340471	+	0.105021i	−0.460275	−	0.887776i	\(-0.652249\pi\)
							0.119805	+	0.992797i	\(0.461773\pi\)
\(29\)	1.03537	+	4.53626i	0.192264	+	0.842363i	0.975388	+	0.220497i	\(0.0707680\pi\)
							−0.783124	+	0.621866i	\(0.786375\pi\)
\(31\)	−0.113858	+	0.197207i	−0.0204494	+	0.0354194i	−0.876069	−	0.482186i	\(-0.839843\pi\)
							0.855620	+	0.517605i	\(0.173176\pi\)
\(37\)	4.64654	−	4.31136i	0.763886	−	0.708783i	−0.198610	−	0.980079i	\(-0.563643\pi\)
							0.962496	+	0.271296i	\(0.0874522\pi\)
\(41\)	−4.78564	+	2.30465i	−0.747392	+	0.359925i	−0.768498	−	0.639852i	\(-0.778996\pi\)
							0.0211059	+	0.999777i	\(0.493281\pi\)
\(43\)	8.95378	+	4.31191i	1.36544	+	0.657561i	0.965842	−	0.259130i	\(-0.0834359\pi\)
							0.399597	+	0.916691i	\(0.369150\pi\)
\(47\)	3.04853	−	7.76752i	0.444673	−	1.13301i	−0.516310	−	0.856402i	\(-0.672695\pi\)
							0.960983	−	0.276607i	\(-0.0892100\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.144.11	✓	276
49.16	even	21	inner	539.2.r.a.408.11	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.144.11	✓	276	1.1	even	1	trivial
539.2.r.a.408.11	yes	276	49.16	even	21	inner