Embedded newform 287.3.x.a.31.8

• Label: 287.3.x.a.31.8
• Level: $287$
• Weight: $3$
• Character: 287.31
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.x (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			31.8
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.327450 + 3.11548i) q^{2} +(-1.62561 + 2.81565i) q^{3} +(-5.68637 - 1.20868i) q^{4} +(-3.32056 - 2.98985i) q^{5} +(-8.23977 - 5.98654i) q^{6} +(-6.84936 - 1.44437i) q^{7} +(1.75545 - 5.40272i) q^{8} +(-0.785238 - 1.36007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.327450	+	3.11548i	−0.163725	+	1.55774i	0.536553	+	0.843867i	\(0.319726\pi\)
							−0.700277	+	0.713871i	\(0.746940\pi\)
\(3\)	−1.62561	+	2.81565i	−0.541871	+	0.938548i	0.456926	+	0.889505i	\(0.348951\pi\)
							−0.998797	+	0.0490434i	\(0.984383\pi\)
\(5\)	−3.32056	−	2.98985i	−0.664112	−	0.597969i	0.266562	−	0.963818i	\(-0.414112\pi\)
							−0.930674	+	0.365848i	\(0.880779\pi\)
\(7\)	−6.84936	−	1.44437i	−0.978481	−	0.206339i
							\(11\)	−0.0800575	+	0.0720841i	−0.00727796	+	0.00655310i	−0.672762	−	0.739859i	\(-0.734892\pi\)
0.665484	+	0.746412i	\(0.268225\pi\)
\(13\)	0.305462	+	0.221931i	0.0234971	+	0.0170716i	0.599472	−	0.800396i	\(-0.295377\pi\)
							−0.575975	+	0.817468i	\(0.695377\pi\)
\(17\)	5.49984	+	6.10819i	0.323520	+	0.359305i	0.882863	−	0.469631i	\(-0.155613\pi\)
							−0.559343	+	0.828936i	\(0.688946\pi\)
\(19\)	27.9943	+	12.4639i	1.47338	+	0.655992i	0.977218	−	0.212237i	\(-0.0680747\pi\)
							0.496164	+	0.868229i	\(0.334741\pi\)
\(23\)	4.01333	−	38.1843i	0.174493	−	1.66019i	−0.460494	−	0.887663i	\(-0.652328\pi\)
							0.634987	−	0.772523i	\(-0.281006\pi\)
\(29\)	−30.4407	+	9.89079i	−1.04968	+	0.341062i	−0.782543	−	0.622597i	\(-0.786078\pi\)
							−0.267137	+	0.963659i	\(0.586078\pi\)
\(31\)	1.44129	−	1.29775i	0.0464933	−	0.0418628i	−0.645556	−	0.763713i	\(-0.723375\pi\)
							0.692050	+	0.721850i	\(0.256708\pi\)
\(37\)	5.25559	−	5.83692i	0.142043	−	0.157755i	−0.667925	−	0.744228i	\(-0.732818\pi\)
							0.809968	+	0.586474i	\(0.199484\pi\)
\(41\)	−28.7705	+	29.2106i	−0.701719	+	0.712454i
							\(43\)	−52.7938	−	38.3569i	−1.22776	−	0.892022i	−0.231042	−	0.972944i	\(-0.574214\pi\)
−0.996720	+	0.0809218i	\(0.974214\pi\)
\(47\)	1.23897	−	11.7880i	0.0263611	−	0.250809i	−0.973403	−	0.229099i	\(-0.926422\pi\)
							0.999764	−	0.0217104i	\(-0.00691116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.8	✓	432
7.5	odd	6	inner	287.3.x.a.236.47	yes	432
41.4	even	10	inner	287.3.x.a.45.47	yes	432
287.250	odd	30	inner	287.3.x.a.250.8	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.8	✓	432	1.1	even	1	trivial
287.3.x.a.45.47	yes	432	41.4	even	10	inner
287.3.x.a.236.47	yes	432	7.5	odd	6	inner
287.3.x.a.250.8	yes	432	287.250	odd	30	inner