Embedded newform 287.3.o.a.139.3

• Label: 287.3.o.a.139.3
• Level: $287$
• Weight: $3$
• Character: 287.139
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.o (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			139.3
Character	\(\chi\)	\(=\)	287.139
Dual form			287.3.o.a.223.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08729 + 3.34632i) q^{2} -0.232377i q^{3} +(-6.77961 - 4.92568i) q^{4} +(1.40348 - 1.93173i) q^{5} +(0.777610 + 0.252661i) q^{6} +(-6.87163 + 1.33444i) q^{7} +(12.4681 - 9.05858i) q^{8} +8.94600 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{2} - 110 q^{4} - 3 q^{7} - 2 q^{8} - 648 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08729	+	3.34632i	−0.543643	+	1.67316i	0.180552	+	0.983565i	\(0.442212\pi\)
							−0.724195	+	0.689596i	\(0.757788\pi\)
\(3\)		−	0.232377i		−	0.0774591i	−0.999250	−	0.0387296i	\(-0.987669\pi\)
							0.999250	−	0.0387296i	\(-0.0123311\pi\)
\(5\)	1.40348	−	1.93173i	0.280697	−	0.386346i	−0.645268	−	0.763956i	\(-0.723254\pi\)
							0.925964	+	0.377611i	\(0.123254\pi\)
\(7\)	−6.87163	+	1.33444i	−0.981661	+	0.190635i
							\(11\)	15.0798	−	10.9561i	1.37089	−	0.996009i	0.373222	−	0.927742i	\(-0.378253\pi\)
0.997667	−	0.0682673i	\(-0.0217471\pi\)
\(13\)	−13.2324	−	4.29948i	−1.01788	−	0.330729i	−0.247892	−	0.968788i	\(-0.579738\pi\)
							−0.769988	+	0.638058i	\(0.779738\pi\)
\(17\)	17.4186	+	23.9746i	1.02462	+	1.41027i	0.908911	+	0.416989i	\(0.136915\pi\)
							0.115711	+	0.993283i	\(0.463085\pi\)
\(19\)	14.9508	−	4.85780i	0.786882	−	0.255673i	0.112106	−	0.993696i	\(-0.464240\pi\)
							0.674776	+	0.738023i	\(0.264240\pi\)
\(23\)	2.51124	−	7.72879i	0.109184	−	0.336034i	−0.881506	−	0.472174i	\(-0.843470\pi\)
							0.990690	+	0.136139i	\(0.0434695\pi\)
\(29\)	7.02615	+	5.10479i	0.242281	+	0.176027i	0.702299	−	0.711882i	\(-0.252157\pi\)
							−0.460018	+	0.887910i	\(0.652157\pi\)
\(31\)	−2.41937	−	3.32998i	−0.0780442	−	0.107419i	0.768210	−	0.640198i	\(-0.221148\pi\)
							−0.846254	+	0.532779i	\(0.821148\pi\)
\(37\)	28.8425	+	20.9553i	0.779526	+	0.566359i	0.904837	−	0.425759i	\(-0.139993\pi\)
							−0.125311	+	0.992118i	\(0.539993\pi\)
\(41\)	−16.7708	−	37.4131i	−0.409043	−	0.912515i
							\(43\)	7.02222	−	21.6122i	0.163307	−	0.502609i	−0.835600	−	0.549338i	\(-0.814880\pi\)
0.998908	+	0.0467294i	\(0.0148799\pi\)
\(47\)	9.23578	+	3.00089i	0.196506	+	0.0638486i	0.405616	−	0.914043i	\(-0.367057\pi\)
							−0.209111	+	0.977892i	\(0.567057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.o.a.139.3	✓	216
7.6	odd	2	inner	287.3.o.a.139.4	yes	216
41.18	even	5	inner	287.3.o.a.223.3	yes	216
287.223	odd	10	inner	287.3.o.a.223.4	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.o.a.139.3	✓	216	1.1	even	1	trivial
287.3.o.a.139.4	yes	216	7.6	odd	2	inner
287.3.o.a.223.3	yes	216	41.18	even	5	inner
287.3.o.a.223.4	yes	216	287.223	odd	10	inner