Embedded newform 287.2.j.a.81.9

• Label: 287.2.j.a.81.9
• 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			81.9
Character	\(\chi\)	\(=\)	287.81
Dual form			287.2.j.a.163.9

$q$-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	0.212617	−	0.977136i	\(-0.431801\pi\)
							−0.739916	+	0.672699i	\(0.765135\pi\)
\(5\)	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	0.252465	−	0.967606i	\(-0.418759\pi\)
−0.711739	+	0.702444i	\(0.752092\pi\)
\(13\)		−	6.25700i		−	1.73538i	−0.497107	−	0.867689i	\(-0.665604\pi\)
							0.497107	−	0.867689i	\(-0.334396\pi\)
\(17\)	0.280425	+	0.161903i	0.0680130	+	0.0392673i	0.533621	−	0.845724i	\(-0.320831\pi\)
							−0.465608	+	0.884991i	\(0.654164\pi\)
\(19\)	−2.26417	+	1.30722i	−0.519436	+	0.299897i	−0.736704	−	0.676215i	\(-0.763619\pi\)
							0.217268	+	0.976112i	\(0.430286\pi\)
\(23\)	−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	\(0.231394\pi\)
							−0.747209	+	0.664590i	\(0.768606\pi\)
\(31\)	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	0.272797	−	0.962072i	\(-0.412051\pi\)
							0.969577	+	0.244787i	\(0.0787180\pi\)

(See \(a_n\) instead)

Twists

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

    

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