Embedded newform 539.2.r.b.485.17

• Label: 539.2.r.b.485.17
• 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			485.17
Character	\(\chi\)	\(=\)	539.485
Dual form			539.2.r.b.529.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.52898 - 0.230457i) q^{2} +(-0.192895 - 0.131514i) q^{3} +(0.373531 - 0.115219i) q^{4} +(0.125063 - 1.66884i) q^{5} +(-0.325242 - 0.156628i) q^{6} +(-2.31351 - 1.28362i) q^{7} +(-2.24168 + 1.07954i) q^{8} +(-1.07611 - 2.74188i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q + 3 q^{3} + 20 q^{4} + 6 q^{5} - 8 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.52898	−	0.230457i	1.08115	−	0.162958i	0.415778	−	0.909466i	\(-0.363509\pi\)
							0.665376	+	0.746508i	\(0.268271\pi\)
\(3\)	−0.192895	−	0.131514i	−0.111368	−	0.0759296i	0.506355	−	0.862325i	\(-0.330992\pi\)
							−0.617723	+	0.786395i	\(0.711945\pi\)
\(5\)	0.125063	−	1.66884i	0.0559297	−	0.746330i	−0.896786	−	0.442465i	\(-0.854104\pi\)
							0.952715	−	0.303864i	\(-0.0982770\pi\)
\(7\)	−2.31351	−	1.28362i	−0.874424	−	0.485162i
							\(11\)	0.365341	−	0.930874i	0.110154	−	0.280669i
\(13\)	−2.04361	−	2.56260i	−0.566795	−	0.710738i								0.413003	−	0.910730i	\(-0.364480\pi\)
							−0.979798	+	0.199992i	\(0.935908\pi\)
\(17\)	5.22921	−	4.85200i	1.26827	−	1.17678i	0.292916	−	0.956138i	\(-0.405374\pi\)
							0.975354	−	0.220644i	\(-0.0708160\pi\)
\(19\)	−1.24720	+	2.16021i	−0.286126	+	0.495585i	−0.972882	−	0.231303i	\(-0.925701\pi\)
							0.686755	+	0.726889i	\(0.259034\pi\)
\(23\)	2.84103	+	2.63609i	0.592396	+	0.549663i	0.918374	−	0.395714i	\(-0.129503\pi\)
							−0.325977	+	0.945378i	\(0.605693\pi\)
\(29\)	−0.811241	−	3.55428i	−0.150644	−	0.660013i	−0.992699	−	0.120621i	\(-0.961511\pi\)
							0.842055	−	0.539392i	\(-0.181346\pi\)
\(31\)	3.09131	+	5.35431i	0.555216	+	0.961662i	0.997887	+	0.0649777i	\(0.0206976\pi\)
							−0.442671	+	0.896684i	\(0.645969\pi\)
\(37\)	−10.2658	−	3.16657i	−1.68768	−	0.520581i	−0.705532	−	0.708678i	\(-0.749292\pi\)
							−0.982151	+	0.188097i	\(0.939768\pi\)
\(41\)	5.51647	−	2.65659i	0.861527	−	0.414890i	0.0496840	−	0.998765i	\(-0.484179\pi\)
							0.811843	+	0.583875i	\(0.198464\pi\)
\(43\)	0.853228	+	0.410893i	0.130116	+	0.0626606i	0.497809	−	0.867287i	\(-0.334138\pi\)
							−0.367693	+	0.929947i	\(0.619852\pi\)
\(47\)	6.24949	−	0.941958i	0.911581	−	0.137399i	0.323526	−	0.946219i	\(-0.395132\pi\)
							0.588055	+	0.808821i	\(0.299894\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.b.485.17	✓	276
49.39	even	21	inner	539.2.r.b.529.17	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.b.485.17	✓	276	1.1	even	1	trivial
539.2.r.b.529.17	yes	276	49.39	even	21	inner