Embedded newform 287.2.s.a.16.10

• Label: 287.2.s.a.16.10
• Level: $287$
• Weight: $2$
• Character: 287.16
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.s (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.10
Character	\(\chi\)	\(=\)	287.16
Dual form			287.2.s.a.18.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.693484 - 0.770192i) q^{2} +(0.967334 + 1.67547i) q^{3} +(0.0967811 - 0.920811i) q^{4} +(-2.15281 + 0.958491i) q^{5} +(0.619605 - 1.90695i) q^{6} +(-2.14904 + 1.54325i) q^{7} +(-2.45324 + 1.78238i) q^{8} +(-0.371471 + 0.643406i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.693484	−	0.770192i	−0.490368	−	0.544608i	0.446275	−	0.894896i	\(-0.352750\pi\)
							−0.936642	+	0.350288i	\(0.886084\pi\)
\(3\)	0.967334	+	1.67547i	0.558491	+	0.967334i	0.997623	+	0.0689116i	\(0.0219527\pi\)
							−0.439132	+	0.898422i	\(0.644714\pi\)
\(5\)	−2.15281	+	0.958491i	−0.962764	+	0.428650i	−0.827069	−	0.562101i	\(-0.809993\pi\)
							−0.135695	+	0.990751i	\(0.543327\pi\)
\(7\)	−2.14904	+	1.54325i	−0.812260	+	0.583295i
							\(11\)	5.63697	+	2.50974i	1.69961	+	0.756716i	0.999071	+	0.0431028i	\(0.0137243\pi\)
0.700540	+	0.713613i	\(0.252942\pi\)
\(13\)	−2.18131	+	6.71337i	−0.604986	+	1.86195i	−0.108096	+	0.994140i	\(0.534475\pi\)
							−0.496890	+	0.867814i	\(0.665525\pi\)
\(17\)	0.809709	+	0.360506i	0.196383	+	0.0874355i	0.502572	−	0.864535i	\(-0.332387\pi\)
							−0.306189	+	0.951971i	\(0.599054\pi\)
\(19\)	−3.05404	−	0.649156i	−0.700645	−	0.148927i	−0.156203	−	0.987725i	\(-0.549925\pi\)
							−0.544442	+	0.838798i	\(0.683259\pi\)
\(23\)	0.0805582	+	0.0894690i	0.0167976	+	0.0186556i	0.751485	−	0.659750i	\(-0.229338\pi\)
							−0.734688	+	0.678406i	\(0.762671\pi\)
\(29\)	−0.532672	−	0.387009i	−0.0989148	−	0.0718658i	0.537229	−	0.843437i	\(-0.319471\pi\)
							−0.636143	+	0.771571i	\(0.719471\pi\)
\(31\)	−9.57064	−	4.26112i	−1.71894	−	0.765321i	−0.997331	−	0.0730181i	\(-0.976737\pi\)
							−0.721607	−	0.692302i	\(-0.756596\pi\)
\(37\)	6.57442	−	2.92712i	1.08083	−	0.481216i	0.212477	−	0.977166i	\(-0.431847\pi\)
							0.868351	+	0.495951i	\(0.165180\pi\)
\(41\)	2.79106	+	5.76281i	0.435890	+	0.900000i
							\(43\)	0.931608	−	2.86719i	0.142069	−	0.437243i	−0.854554	−	0.519363i	\(-0.826169\pi\)
0.996622	+	0.0821202i	\(0.0261692\pi\)
\(47\)	4.78951	+	5.31929i	0.698622	+	0.775899i	0.983156	−	0.182767i	\(-0.0585054\pi\)
							−0.284534	+	0.958666i	\(0.591839\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.s.a.16.10	✓	208
7.4	even	3	inner	287.2.s.a.221.17	yes	208
41.18	even	5	inner	287.2.s.a.100.17	yes	208
287.18	even	15	inner	287.2.s.a.18.10	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.10	✓	208	1.1	even	1	trivial
287.2.s.a.18.10	yes	208	287.18	even	15	inner
287.2.s.a.100.17	yes	208	41.18	even	5	inner
287.2.s.a.221.17	yes	208	7.4	even	3	inner