Embedded newform 287.3.k.a.124.6

• Label: 287.3.k.a.124.6
• Level: $287$
• Weight: $3$
• Character: 287.124
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(54\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			124.6
Character	\(\chi\)	\(=\)	287.124
Dual form			287.3.k.a.206.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68897 + 2.92538i) q^{2} +(3.96120 - 2.28700i) q^{3} +(-3.70522 - 6.41763i) q^{4} +(5.00681 + 2.89068i) q^{5} +15.4507i q^{6} +(-1.27365 + 6.88315i) q^{7} +11.5203 q^{8} +(5.96073 - 10.3243i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 6 q^{3} - 106 q^{4} - 6 q^{5} - 4 q^{7} - 4 q^{8} + 164 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68897	+	2.92538i	−0.844484	+	1.46269i	0.0415851	+	0.999135i	\(0.486759\pi\)
							−0.886069	+	0.463554i	\(0.846574\pi\)
\(3\)	3.96120	−	2.28700i	1.32040	−	0.762333i	0.336607	−	0.941645i	\(-0.390720\pi\)
							0.983792	+	0.179312i	\(0.0573872\pi\)
\(5\)	5.00681	+	2.89068i	1.00136	+	0.578137i	0.908651	−	0.417556i	\(-0.137113\pi\)
							0.0927112	+	0.995693i	\(0.470447\pi\)
\(7\)	−1.27365	+	6.88315i	−0.181950	+	0.983308i
							\(11\)	9.08878	+	15.7422i	0.826253	+	1.43111i	0.900958	+	0.433906i	\(0.142865\pi\)
−0.0747053	+	0.997206i	\(0.523802\pi\)
\(13\)		−	24.5298i		−	1.88691i	−0.331507	−	0.943453i	\(-0.607557\pi\)
							0.331507	−	0.943453i	\(-0.392443\pi\)
\(17\)	−6.24931	+	3.60804i	−0.367606	+	0.212238i	−0.672412	−	0.740177i	\(-0.734742\pi\)
							0.304806	+	0.952415i	\(0.401408\pi\)
\(19\)	10.2029	+	5.89063i	0.536993	+	0.310033i	0.743859	−	0.668336i	\(-0.232993\pi\)
							−0.206866	+	0.978369i	\(0.566327\pi\)
\(23\)	−11.4927	+	19.9060i	−0.499685	+	0.865479i	−1.00000		0.000364123i	\(-0.999884\pi\)
							0.500315	+	0.865843i	\(0.333217\pi\)
\(29\)	45.9724			1.58526			0.792628	−	0.609705i	\(-0.208712\pi\)
							0.792628	+	0.609705i	\(0.208712\pi\)
\(31\)	−18.0366	+	10.4134i	−0.581826	+	0.335917i	−0.761859	−	0.647743i	\(-0.775713\pi\)
							0.180033	+	0.983661i	\(0.442380\pi\)
\(37\)	−0.895374	+	1.55083i	−0.0241993	+	0.0419144i	−0.877871	−	0.478896i	\(-0.841037\pi\)
							0.853672	+	0.520811i	\(0.174370\pi\)
\(41\)		−	6.40312i		−	0.156174i
							\(43\)	−27.6068			−0.642017			−0.321009	−	0.947076i	\(-0.604022\pi\)
−0.321009	+	0.947076i	\(0.604022\pi\)
\(47\)	31.0044	+	17.9004i	0.659669	+	0.380860i	0.792151	−	0.610325i	\(-0.208961\pi\)
							−0.132482	+	0.991185i	\(0.542295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.k.a.124.6	✓	108
7.3	odd	6	inner	287.3.k.a.206.6	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.k.a.124.6	✓	108	1.1	even	1	trivial
287.3.k.a.206.6	yes	108	7.3	odd	6	inner