Embedded newform 539.2.r.a.485.15

• Label: 539.2.r.a.485.15
• 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			485.15
Character	\(\chi\)	\(=\)	539.485
Dual form			539.2.r.a.529.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.872695 - 0.131538i) q^{2} +(-0.136328 - 0.0929467i) q^{3} +(-1.16685 + 0.359926i) q^{4} +(-0.243569 + 3.25020i) q^{5} +(-0.131199 - 0.0631819i) q^{6} +(2.64144 + 0.151040i) q^{7} +(-2.56127 + 1.23344i) q^{8} +(-1.08608 - 2.76728i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.872695	−	0.131538i	0.617089	−	0.0930111i	0.166943	−	0.985966i	\(-0.446610\pi\)
							0.450145	+	0.892955i	\(0.351372\pi\)
\(3\)	−0.136328	−	0.0929467i	−0.0787089	−	0.0536628i	0.523326	−	0.852132i	\(-0.324691\pi\)
							−0.602035	+	0.798470i	\(0.705643\pi\)
\(5\)	−0.243569	+	3.25020i	−0.108927	+	1.45353i	0.628343	+	0.777936i	\(0.283733\pi\)
							−0.737270	+	0.675598i	\(0.763886\pi\)
\(7\)	2.64144	+	0.151040i	0.998369	+	0.0570878i
							\(11\)	−0.365341	+	0.930874i	−0.110154	+	0.280669i
\(13\)	−0.308323	−	0.386625i	−0.0855134	−	0.107230i								0.737234	−	0.675637i	\(-0.236131\pi\)
							−0.822748	+	0.568407i	\(0.807560\pi\)
\(17\)	−5.07563	+	4.70950i	−1.23102	+	1.14222i	−0.246096	+	0.969245i	\(0.579148\pi\)
							−0.984926	+	0.172976i	\(0.944662\pi\)
\(19\)	−4.17969	+	7.23944i	−0.958887	+	1.66084i	−0.233676	+	0.972315i	\(0.575075\pi\)
							−0.725211	+	0.688526i	\(0.758258\pi\)
\(23\)	4.10584	+	3.80966i	0.856126	+	0.794369i	0.980222	−	0.197901i	\(-0.0634125\pi\)
							−0.124096	+	0.992270i	\(0.539603\pi\)
\(29\)	0.426216	+	1.86738i	0.0791464	+	0.346763i	0.998960	−	0.0455923i	\(-0.0145175\pi\)
							−0.919814	+	0.392355i	\(0.871660\pi\)
\(31\)	−1.92267	−	3.33016i	−0.345321	−	0.598114i	0.640091	−	0.768299i	\(-0.278897\pi\)
							−0.985412	+	0.170186i	\(0.945563\pi\)
\(37\)	−4.14073	−	1.27725i	−0.680732	−	0.209978i	−0.0649429	−	0.997889i	\(-0.520687\pi\)
							−0.615789	+	0.787911i	\(0.711163\pi\)
\(41\)	10.8630	−	5.23134i	1.69652	−	0.816999i	0.702027	−	0.712150i	\(-0.252279\pi\)
							0.994488	−	0.104848i	\(-0.0334357\pi\)
\(43\)	−1.78510	−	0.859658i	−0.272225	−	0.131097i	0.292791	−	0.956176i	\(-0.405416\pi\)
							−0.565016	+	0.825080i	\(0.691130\pi\)
\(47\)	4.06289	−	0.612382i	0.592634	−	0.0893251i	0.154124	−	0.988052i	\(-0.450745\pi\)
							0.438510	+	0.898726i	\(0.355506\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.485.15	✓	276
49.39	even	21	inner	539.2.r.a.529.15	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.485.15	✓	276	1.1	even	1	trivial
539.2.r.a.529.15	yes	276	49.39	even	21	inner