Embedded newform 287.2.j.a.81.10

• Label: 287.2.j.a.81.10
• Level: $287$
• Weight: $2$
• Character: 287.81
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			81.10
Character	\(\chi\)	\(=\)	287.81
Dual form			287.2.j.a.163.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.655113 - 1.13469i) q^{2} +(0.913309 + 0.527299i) q^{3} +(0.141655 - 0.245353i) q^{4} +(1.55927 + 2.70074i) q^{5} -1.38176i q^{6} +(2.26438 + 1.36842i) q^{7} -2.99165 q^{8} +(-0.943911 - 1.63490i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.655113	−	1.13469i	−0.463235	−	0.802346i	0.535885	−	0.844291i	\(-0.319978\pi\)
							−0.999120	+	0.0419448i	\(0.986645\pi\)
\(3\)	0.913309	+	0.527299i	0.527299	+	0.304436i	0.739916	−	0.672699i	\(-0.234865\pi\)
							−0.212617	+	0.977136i	\(0.568199\pi\)
\(5\)	1.55927	+	2.70074i	0.697329	+	1.20781i	0.969389	+	0.245529i	\(0.0789616\pi\)
							−0.272060	+	0.962280i	\(0.587705\pi\)
\(7\)	2.26438	+	1.36842i	0.855856	+	0.517215i
							\(11\)	1.52324	+	0.879443i	0.459274	+	0.265162i	0.711739	−	0.702444i	\(-0.247908\pi\)
−0.252465	+	0.967606i	\(0.581241\pi\)
\(13\)			6.25700i			1.73538i	0.497107	+	0.867689i	\(0.334396\pi\)
							−0.497107	+	0.867689i	\(0.665604\pi\)
\(17\)	−0.280425	−	0.161903i	−0.0680130	−	0.0392673i	0.465608	−	0.884991i	\(-0.345836\pi\)
							−0.533621	+	0.845724i	\(0.679169\pi\)
\(19\)	2.26417	−	1.30722i	0.519436	−	0.299897i	−0.217268	−	0.976112i	\(-0.569714\pi\)
							0.736704	+	0.676215i	\(0.236381\pi\)
\(23\)	−4.04537	−	7.00678i	−0.843517	−	1.46101i	−0.886903	−	0.461956i	\(-0.847148\pi\)
							0.0433856	−	0.999058i	\(-0.486186\pi\)
\(29\)		−	7.15785i		−	1.32918i	−0.747209	−	0.664590i	\(-0.768606\pi\)
							0.747209	−	0.664590i	\(-0.231394\pi\)
\(31\)	1.35645	−	2.34944i	0.243626	−	0.421972i	−0.718119	−	0.695921i	\(-0.754996\pi\)
							0.961744	+	0.273948i	\(0.0883298\pi\)
\(37\)	−1.72869	−	2.99417i	−0.284194	−	0.492239i	0.688219	−	0.725503i	\(-0.258393\pi\)
							−0.972413	+	0.233264i	\(0.925059\pi\)
\(41\)	6.31531	+	1.05681i	0.986286	+	0.165045i
							\(43\)	2.15821			0.329124			0.164562	−	0.986367i	\(-0.447379\pi\)
0.164562	+	0.986367i	\(0.447379\pi\)
\(47\)	−8.51729	+	4.91746i	−1.24237	+	0.717285i	−0.969577	−	0.244787i	\(-0.921282\pi\)
							−0.272797	+	0.962072i	\(0.587949\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.j.a.81.10	yes	52
7.2	even	3	inner	287.2.j.a.163.9	yes	52
41.40	even	2	inner	287.2.j.a.81.9	✓	52
287.163	even	6	inner	287.2.j.a.163.10	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.j.a.81.9	✓	52	41.40	even	2	inner
287.2.j.a.81.10	yes	52	1.1	even	1	trivial
287.2.j.a.163.9	yes	52	7.2	even	3	inner
287.2.j.a.163.10	yes	52	287.163	even	6	inner