Embedded newform 287.3.x.a.31.2

• Label: 287.3.x.a.31.2
• 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			31.2
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.382108 + 3.63551i) q^{2} +(-0.296842 + 0.514145i) q^{3} +(-9.15836 - 1.94667i) q^{4} +(-3.78875 - 3.41141i) q^{5} +(-1.75576 - 1.27563i) q^{6} +(4.94220 - 4.95728i) q^{7} +(6.05813 - 18.6450i) q^{8} +(4.32377 + 7.48899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.382108	+	3.63551i	−0.191054	+	1.81776i	0.308223	+	0.951314i	\(0.400266\pi\)
							−0.499277	+	0.866443i	\(0.666401\pi\)
\(3\)	−0.296842	+	0.514145i	−0.0989473	+	0.171382i	−0.911249	−	0.411855i	\(-0.864881\pi\)
							0.812302	+	0.583237i	\(0.198214\pi\)
\(5\)	−3.78875	−	3.41141i	−0.757751	−	0.682282i	0.196762	−	0.980451i	\(-0.436957\pi\)
							−0.954513	+	0.298169i	\(0.903624\pi\)
\(7\)	4.94220	−	4.95728i	0.706029	−	0.708183i
							\(11\)	8.31485	−	7.48673i	0.755896	−	0.680612i	−0.198184	−	0.980165i	\(-0.563504\pi\)
0.954080	+	0.299553i	\(0.0968376\pi\)
\(13\)	−1.94297	−	1.41165i	−0.149459	−	0.108588i	0.510543	−	0.859852i	\(-0.329445\pi\)
							−0.660002	+	0.751264i	\(0.729445\pi\)
\(17\)	−8.85189	−	9.83102i	−0.520699	−	0.578295i	0.424236	−	0.905552i	\(-0.360543\pi\)
							−0.944936	+	0.327256i	\(0.893876\pi\)
\(19\)	8.12659	+	3.61819i	0.427715	+	0.190431i	0.609292	−	0.792946i	\(-0.291454\pi\)
							−0.181577	+	0.983377i	\(0.558120\pi\)
\(23\)	−0.699285	+	6.65325i	−0.0304037	+	0.289272i	0.968746	+	0.248053i	\(0.0797907\pi\)
							−0.999150	+	0.0412187i	\(0.986876\pi\)
\(29\)	44.4586	−	14.4455i	1.53306	−	0.498120i	0.583605	−	0.812038i	\(-0.301642\pi\)
							0.949450	+	0.313918i	\(0.101642\pi\)
\(31\)	−12.7538	+	11.4836i	−0.411412	+	0.370437i	−0.848691	−	0.528889i	\(-0.822609\pi\)
							0.437279	+	0.899326i	\(0.355942\pi\)
\(37\)	37.2225	−	41.3398i	1.00601	−	1.11729i	0.0129268	−	0.999916i	\(-0.495885\pi\)
							0.993088	−	0.117376i	\(-0.0374482\pi\)
\(41\)	23.3506	−	33.7009i	0.569528	−	0.821972i
							\(43\)	45.6792	+	33.1879i	1.06231	+	0.771812i	0.974514	−	0.224327i	\(-0.0720183\pi\)
0.0877939	+	0.996139i	\(0.472018\pi\)
\(47\)	−0.909713	+	8.65535i	−0.0193556	+	0.184156i	−0.999928	−	0.0120013i	\(-0.996180\pi\)
							0.980572	+	0.196158i	\(0.0628464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.2	✓	432
7.5	odd	6	inner	287.3.x.a.236.53	yes	432
41.4	even	10	inner	287.3.x.a.45.53	yes	432
287.250	odd	30	inner	287.3.x.a.250.2	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.2	✓	432	1.1	even	1	trivial
287.3.x.a.45.53	yes	432	41.4	even	10	inner
287.3.x.a.236.53	yes	432	7.5	odd	6	inner
287.3.x.a.250.2	yes	432	287.250	odd	30	inner