Embedded newform 539.2.r.b.485.18

• Label: 539.2.r.b.485.18
• Level: $539$
• Weight: $2$
• Character: 539.485
• 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			485.18
Character	\(\chi\)	\(=\)	539.485
Dual form			539.2.r.b.529.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.68505 - 0.253980i) q^{2} +(-2.67752 - 1.82550i) q^{3} +(0.863741 - 0.266429i) q^{4} +(0.261864 - 3.49434i) q^{5} +(-4.97540 - 2.39603i) q^{6} +(-0.917110 - 2.48171i) q^{7} +(-1.68287 + 0.810429i) q^{8} +(2.74064 + 6.98304i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{4}{21}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.68505	−	0.253980i	1.19151	−	0.179591i	0.476800	−	0.879012i	\(-0.341797\pi\)
							0.714710	+	0.699421i	\(0.246559\pi\)
\(3\)	−2.67752	−	1.82550i	−1.54587	−	1.05396i	−0.971992	−	0.235013i	\(-0.924487\pi\)
							−0.573876	−	0.818942i	\(-0.694561\pi\)
\(5\)	0.261864	−	3.49434i	0.117109	−	1.56271i	−0.561039	−	0.827790i	\(-0.689598\pi\)
							0.678148	−	0.734925i	\(-0.262783\pi\)
\(7\)	−0.917110	−	2.48171i	−0.346635	−	0.938000i
							\(11\)	0.365341	−	0.930874i	0.110154	−	0.280669i
\(13\)	2.31409	+	2.90178i	0.641813	+	0.804808i								0.991228	−	0.132160i	\(-0.0421912\pi\)
							−0.349415	+	0.936968i	\(0.613620\pi\)
\(17\)	−3.86073	+	3.58223i	−0.936365	+	0.868819i	−0.991559	−	0.129656i	\(-0.958613\pi\)
							0.0551944	+	0.998476i	\(0.482422\pi\)
\(19\)	2.27170	−	3.93471i	0.521165	−	0.902684i	−0.478532	−	0.878070i	\(-0.658831\pi\)
							0.999697	−	0.0246138i	\(-0.00783560\pi\)
\(23\)	2.40117	+	2.22796i	0.500679	+	0.464562i	0.889621	−	0.456700i	\(-0.150969\pi\)
							−0.388941	+	0.921262i	\(0.627159\pi\)
\(29\)	−1.53865	−	6.74126i	−0.285720	−	1.25182i	−0.890336	−	0.455304i	\(-0.849530\pi\)
							0.604616	−	0.796517i	\(-0.293327\pi\)
\(31\)	−4.61643	−	7.99590i	−0.829136	−	1.43611i	−0.898717	−	0.438530i	\(-0.855499\pi\)
							0.0695806	−	0.997576i	\(-0.477834\pi\)
\(37\)	1.63352	+	0.503875i	0.268549	+	0.0828365i	0.426105	−	0.904674i	\(-0.359885\pi\)
							−0.157556	+	0.987510i	\(0.550361\pi\)
\(41\)	−2.59432	+	1.24936i	−0.405165	+	0.195117i	−0.625356	−	0.780339i	\(-0.715046\pi\)
							0.220191	+	0.975457i	\(0.429332\pi\)
\(43\)	−5.17383	−	2.49158i	−0.789001	−	0.379963i	−0.00442097	−	0.999990i	\(-0.501407\pi\)
							−0.784580	+	0.620027i	\(0.787122\pi\)
\(47\)	3.50811	−	0.528763i	0.511711	−	0.0771280i	0.111889	−	0.993721i	\(-0.464310\pi\)
							0.399822	+	0.916593i	\(0.369072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.485.18	✓	276
49.39	even	21	inner	539.2.r.b.529.18	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.485.18	✓	276	1.1	even	1	trivial
539.2.r.b.529.18	yes	276	49.39	even	21	inner