Embedded newform 287.3.o.a.139.2

• Label: 287.3.o.a.139.2
• 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			139.2
Character	\(\chi\)	\(=\)	287.139
Dual form			287.3.o.a.223.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19816 + 3.68755i) q^{2} +4.62584i q^{3} +(-8.92636 - 6.48538i) q^{4} +(1.55146 - 2.13541i) q^{5} +(-17.0580 - 5.54248i) q^{6} +(-3.63473 - 5.98237i) q^{7} +(22.0631 - 16.0298i) q^{8} -12.3984 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{2} - 110 q^{4} - 3 q^{7} - 2 q^{8} - 648 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19816	+	3.68755i	−0.599079	+	1.84377i	−0.0658026	+	0.997833i	\(0.520961\pi\)
							−0.533276	+	0.845941i	\(0.679039\pi\)
\(3\)			4.62584i			1.54195i	0.636867	+	0.770973i	\(0.280230\pi\)
							−0.636867	+	0.770973i	\(0.719770\pi\)
\(5\)	1.55146	−	2.13541i	0.310293	−	0.427082i	−0.625180	−	0.780481i	\(-0.714974\pi\)
							0.935473	+	0.353399i	\(0.114974\pi\)
\(7\)	−3.63473	−	5.98237i	−0.519248	−	0.854624i
							\(11\)	−6.87615	+	4.99582i	−0.625105	+	0.454165i	−0.854701	−	0.519121i	\(-0.826259\pi\)
0.229596	+	0.973286i	\(0.426259\pi\)
\(13\)	−0.674968	−	0.219310i	−0.0519206	−	0.0168700i	0.282942	−	0.959137i	\(-0.408690\pi\)
							−0.334862	+	0.942267i	\(0.608690\pi\)
\(17\)	−16.6407	−	22.9039i	−0.978863	−	1.34729i	−0.937440	−	0.348147i	\(-0.886811\pi\)
							−0.0414226	−	0.999142i	\(-0.513189\pi\)
\(19\)	27.3537	−	8.88775i	1.43967	−	0.467776i	0.517874	−	0.855457i	\(-0.326724\pi\)
							0.921794	+	0.387681i	\(0.126724\pi\)
\(23\)	−1.49402	+	4.59811i	−0.0649572	+	0.199918i	−0.978268	−	0.207347i	\(-0.933517\pi\)
							0.913310	+	0.407264i	\(0.133517\pi\)
\(29\)	−31.4504	−	22.8500i	−1.08450	−	0.787933i	−0.106035	−	0.994362i	\(-0.533815\pi\)
							−0.978461	+	0.206430i	\(0.933815\pi\)
\(31\)	−16.5358	−	22.7596i	−0.533414	−	0.734181i	0.454232	−	0.890883i	\(-0.349914\pi\)
							−0.987646	+	0.156702i	\(0.949914\pi\)
\(37\)	−29.5458	−	21.4663i	−0.798535	−	0.580170i	0.111949	−	0.993714i	\(-0.464291\pi\)
							−0.910484	+	0.413544i	\(0.864291\pi\)
\(41\)	−23.7792	−	33.3999i	−0.579980	−	0.814631i
							\(43\)	−2.05762	+	6.33270i	−0.0478516	+	0.147272i	−0.972127	−	0.234453i	\(-0.924670\pi\)
0.924276	+	0.381725i	\(0.124670\pi\)
\(47\)	−33.2070	−	10.7896i	−0.706532	−	0.229566i	−0.0663578	−	0.997796i	\(-0.521138\pi\)
							−0.640174	+	0.768230i	\(0.721138\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.o.a.139.2	yes	216
7.6	odd	2	inner	287.3.o.a.139.1	✓	216
41.18	even	5	inner	287.3.o.a.223.2	yes	216
287.223	odd	10	inner	287.3.o.a.223.1	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.o.a.139.1	✓	216	7.6	odd	2	inner
287.3.o.a.139.2	yes	216	1.1	even	1	trivial
287.3.o.a.223.1	yes	216	287.223	odd	10	inner
287.3.o.a.223.2	yes	216	41.18	even	5	inner