Embedded newform 287.2.j.a.81.7

• Label: 287.2.j.a.81.7
• 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			81.7
Character	\(\chi\)	\(=\)	287.81
Dual form			287.2.j.a.163.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.749549 - 1.29826i) q^{2} +(-1.70608 - 0.985006i) q^{3} +(-0.123649 + 0.214166i) q^{4} +(-0.142717 - 0.247192i) q^{5} +2.95324i q^{6} +(-1.33336 + 2.28520i) q^{7} -2.62747 q^{8} +(0.440474 + 0.762924i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.749549	−	1.29826i	−0.530012	−	0.918007i	−0.999387	−	0.0350082i	\(-0.988854\pi\)
							0.469376	−	0.882999i	\(-0.344479\pi\)
\(3\)	−1.70608	−	0.985006i	−0.985006	−	0.568694i	−0.0812283	−	0.996696i	\(-0.525884\pi\)
							−0.903778	+	0.428002i	\(0.859218\pi\)
\(5\)	−0.142717	−	0.247192i	−0.0638248	−	0.110548i	0.832347	−	0.554254i	\(-0.186997\pi\)
							−0.896172	+	0.443707i	\(0.853663\pi\)
\(7\)	−1.33336	+	2.28520i	−0.503964	+	0.863725i
							\(11\)	−1.18518	−	0.684265i	−0.357346	−	0.206314i	0.310570	−	0.950551i	\(-0.399480\pi\)
−0.667916	+	0.744237i	\(0.732813\pi\)
\(13\)			3.32843i			0.923139i	0.887104	+	0.461570i	\(0.152714\pi\)
							−0.887104	+	0.461570i	\(0.847286\pi\)
\(17\)	−2.19842	−	1.26926i	−0.533194	−	0.307840i	0.209122	−	0.977890i	\(-0.432939\pi\)
							−0.742316	+	0.670050i	\(0.766273\pi\)
\(19\)	−0.848282	+	0.489756i	−0.194609	+	0.112358i	−0.594139	−	0.804363i	\(-0.702507\pi\)
							0.399529	+	0.916720i	\(0.369174\pi\)
\(23\)	2.25865	+	3.91210i	0.470962	+	0.815729i	0.999448	−	0.0332121i	\(-0.0105737\pi\)
							−0.528487	+	0.848942i	\(0.677240\pi\)
\(29\)		−	3.58448i		−	0.665622i	−0.942994	−	0.332811i	\(-0.892003\pi\)
							0.942994	−	0.332811i	\(-0.107997\pi\)
\(31\)	−3.48543	+	6.03694i	−0.626002	+	1.08427i	0.362344	+	0.932044i	\(0.381976\pi\)
							−0.988346	+	0.152223i	\(0.951357\pi\)
\(37\)	0.705885	+	1.22263i	0.116047	+	0.200999i	0.918198	−	0.396122i	\(-0.129644\pi\)
							−0.802151	+	0.597121i	\(0.796311\pi\)
\(41\)	−6.30974	−	1.08956i	−0.985416	−	0.170160i
							\(43\)	−10.3990			−1.58583			−0.792916	−	0.609332i	\(-0.791438\pi\)
−0.792916	+	0.609332i	\(0.791438\pi\)
\(47\)	0.909869	−	0.525313i	0.132718	−	0.0766248i	−0.432171	−	0.901792i	\(-0.642252\pi\)
							0.564889	+	0.825167i	\(0.308919\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.j.a.81.7	✓	52
7.2	even	3	inner	287.2.j.a.163.8	yes	52
41.40	even	2	inner	287.2.j.a.81.8	yes	52
287.163	even	6	inner	287.2.j.a.163.7	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.j.a.81.7	✓	52	1.1	even	1	trivial
287.2.j.a.81.8	yes	52	41.40	even	2	inner
287.2.j.a.163.7	yes	52	287.163	even	6	inner
287.2.j.a.163.8	yes	52	7.2	even	3	inner