Embedded newform 287.3.x.a.31.9

• Label: 287.3.x.a.31.9
• 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			31.9
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.287755 + 2.73781i) q^{2} +(1.34904 - 2.33661i) q^{3} +(-3.50018 - 0.743987i) q^{4} +(-5.88685 - 5.30054i) q^{5} +(6.00898 + 4.36578i) q^{6} +(-0.00249787 + 7.00000i) q^{7} +(-0.358667 + 1.10386i) q^{8} +(0.860183 + 1.48988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.287755	+	2.73781i	−0.143877	+	1.36890i	0.649588	+	0.760287i	\(0.274942\pi\)
							−0.793465	+	0.608616i	\(0.791725\pi\)
\(3\)	1.34904	−	2.33661i	0.449680	−	0.778868i	−0.548685	−	0.836029i	\(-0.684871\pi\)
							0.998365	+	0.0571606i	\(0.0182047\pi\)
\(5\)	−5.88685	−	5.30054i	−1.17737	−	1.06011i	−0.997070	−	0.0764964i	\(-0.975627\pi\)
							−0.180300	−	0.983612i	\(-0.557707\pi\)
\(7\)	−0.00249787	+	7.00000i	−0.000356839	+	1.00000i
							\(11\)	4.96397	−	4.46958i	0.451270	−	0.406326i	−0.411908	−	0.911226i	\(-0.635137\pi\)
0.863178	+	0.504900i	\(0.168471\pi\)
\(13\)	19.0016	+	13.8055i	1.46166	+	1.06196i	0.982927	+	0.183997i	\(0.0589037\pi\)
							0.478733	+	0.877961i	\(0.341096\pi\)
\(17\)	2.41173	+	2.67850i	0.141866	+	0.157559i	0.809891	−	0.586581i	\(-0.199526\pi\)
							−0.668024	+	0.744140i	\(0.732860\pi\)
\(19\)	14.0056	+	6.23572i	0.737139	+	0.328196i	0.740734	−	0.671798i	\(-0.234478\pi\)
							−0.00359510	+	0.999994i	\(0.501144\pi\)
\(23\)	−3.55078	+	33.7834i	−0.154382	+	1.46885i	0.593404	+	0.804905i	\(0.297784\pi\)
							−0.747786	+	0.663940i	\(0.768883\pi\)
\(29\)	−16.3939	+	5.32669i	−0.565306	+	0.183679i	−0.577707	−	0.816244i	\(-0.696052\pi\)
							0.0124014	+	0.999923i	\(0.496052\pi\)
\(31\)	−10.7461	+	9.67581i	−0.346648	+	0.312123i	−0.824001	−	0.566589i	\(-0.808263\pi\)
							0.477353	+	0.878712i	\(0.341596\pi\)
\(37\)	8.10183	−	8.99799i	0.218968	−	0.243189i	−0.623645	−	0.781707i	\(-0.714349\pi\)
							0.842614	+	0.538518i	\(0.181016\pi\)
\(41\)	−21.5394	+	34.8863i	−0.525351	+	0.850886i
							\(43\)	20.0679	+	14.5802i	0.466696	+	0.339074i	0.796152	−	0.605097i	\(-0.206866\pi\)
−0.329457	+	0.944171i	\(0.606866\pi\)
\(47\)	5.43965	−	51.7548i	0.115737	−	1.10117i	−0.770342	−	0.637631i	\(-0.779914\pi\)
							0.886079	−	0.463534i	\(-0.153419\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.9	✓	432
7.5	odd	6	inner	287.3.x.a.236.46	yes	432
41.4	even	10	inner	287.3.x.a.45.46	yes	432
287.250	odd	30	inner	287.3.x.a.250.9	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.9	✓	432	1.1	even	1	trivial
287.3.x.a.45.46	yes	432	41.4	even	10	inner
287.3.x.a.236.46	yes	432	7.5	odd	6	inner
287.3.x.a.250.9	yes	432	287.250	odd	30	inner