Embedded newform 287.3.i.a.40.2

• Label: 287.3.i.a.40.2
• Level: $287$
• Weight: $3$
• Character: 287.40
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.i (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			40.2
Character	\(\chi\)	\(=\)	287.40
Dual form			287.3.i.a.122.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94902 + 3.37581i) q^{2} +(0.674249 + 1.16783i) q^{3} +(-5.59740 - 9.69497i) q^{4} +(4.70200 + 2.71470i) q^{5} -5.25651 q^{6} +(-1.91113 + 6.73406i) q^{7} +28.0457 q^{8} +(3.59078 - 6.21941i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 106 q^{4} - 6 q^{5} + 20 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94902	+	3.37581i	−0.974512	+	1.68791i	−0.292978	+	0.956119i	\(0.594646\pi\)
							−0.681535	+	0.731786i	\(0.738687\pi\)
\(3\)	0.674249	+	1.16783i	0.224750	+	0.389278i	0.956244	−	0.292570i	\(-0.0945103\pi\)
							−0.731495	+	0.681847i	\(0.761177\pi\)
\(5\)	4.70200	+	2.71470i	0.940400	+	0.542940i	0.890086	−	0.455793i	\(-0.150644\pi\)
							0.0503141	+	0.998733i	\(0.483978\pi\)
\(7\)	−1.91113	+	6.73406i	−0.273019	+	0.962009i
							\(11\)	16.8645	−	9.73672i	1.53314	−	0.885156i	0.533921	−	0.845535i	\(-0.320718\pi\)
0.999215	−	0.0396215i	\(-0.0126152\pi\)
\(13\)	23.3272			1.79440			0.897200	−	0.441625i	\(-0.145598\pi\)
							0.897200	+	0.441625i	\(0.145598\pi\)
\(17\)	−2.05974	−	3.56758i	−0.121161	−	0.209857i	0.799065	−	0.601245i	\(-0.205328\pi\)
							−0.920226	+	0.391388i	\(0.871995\pi\)
\(19\)	6.06005	−	10.4963i	0.318950	−	0.552438i	−0.661319	−	0.750105i	\(-0.730003\pi\)
							0.980269	+	0.197667i	\(0.0633365\pi\)
\(23\)	−11.4835	+	19.8900i	−0.499283	+	0.864784i	−1.00000		0.000827655i	\(-0.999737\pi\)
							0.500717	+	0.865611i	\(0.333070\pi\)
\(29\)		−	15.3530i		−	0.529413i	−0.964329	−	0.264706i	\(-0.914725\pi\)
							0.964329	−	0.264706i	\(-0.0852750\pi\)
\(31\)	−0.0644875	+	0.0372319i	−0.00208024	+	0.00120103i	−0.501040	−	0.865424i	\(-0.667049\pi\)
							0.498960	+	0.866625i	\(0.333716\pi\)
\(37\)	−17.4970	+	30.3057i	−0.472893	+	0.819074i	−0.999519	−	0.0310231i	\(-0.990123\pi\)
							0.526626	+	0.850097i	\(0.323457\pi\)
\(41\)	−22.0286	−	34.5795i	−0.537284	−	0.843401i
							\(43\)	−19.3390			−0.449743			−0.224872	−	0.974388i	\(-0.572196\pi\)
−0.224872	+	0.974388i	\(0.572196\pi\)
\(47\)	11.5315	−	19.9732i	0.245352	−	0.424962i	−0.716878	−	0.697198i	\(-0.754430\pi\)
							0.962231	+	0.272236i	\(0.0877631\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.i.a.40.2	yes	108
7.3	odd	6	inner	287.3.i.a.122.1	yes	108
41.40	even	2	inner	287.3.i.a.40.1	✓	108
287.122	odd	6	inner	287.3.i.a.122.2	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.i.a.40.1	✓	108	41.40	even	2	inner
287.3.i.a.40.2	yes	108	1.1	even	1	trivial
287.3.i.a.122.1	yes	108	7.3	odd	6	inner
287.3.i.a.122.2	yes	108	287.122	odd	6	inner