Embedded newform 287.3.ba.a.15.18

• Label: 287.3.ba.a.15.18
• Level: $287$
• Weight: $3$
• Character: 287.15
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.ba (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(42\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			15.18
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00230 - 0.158748i) q^{2} +(1.12918 - 0.467721i) q^{3} +(-2.82483 - 0.917843i) q^{4} +(-1.44168 - 2.82945i) q^{5} +(-1.20602 + 0.289540i) q^{6} +(-2.57265 - 0.617638i) q^{7} +(6.30235 + 3.21121i) q^{8} +(-5.30768 + 5.30768i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{37}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00230	−	0.158748i	−0.501148	−	0.0793741i	−0.0992576	−	0.995062i	\(-0.531647\pi\)
							−0.401891	+	0.915688i	\(0.631647\pi\)
\(3\)	1.12918	−	0.467721i	0.376393	−	0.155907i	−0.186463	−	0.982462i	\(-0.559703\pi\)
							0.562856	+	0.826555i	\(0.309703\pi\)
\(5\)	−1.44168	−	2.82945i	−0.288336	−	0.565891i	0.700719	−	0.713437i	\(-0.252863\pi\)
							−0.989055	+	0.147546i	\(0.952863\pi\)
\(7\)	−2.57265	−	0.617638i	−0.367521	−	0.0882341i
							\(11\)	−1.26978	+	1.08449i	−0.115434	+	0.0985903i	−0.705292	−	0.708917i	\(-0.749184\pi\)
0.589857	+	0.807507i	\(0.299184\pi\)
\(13\)	−0.934744	+	1.52536i	−0.0719034	+	0.117336i	−0.886571	−	0.462592i	\(-0.846919\pi\)
							0.814668	+	0.579928i	\(0.196919\pi\)
\(17\)	−0.0528027	+	0.670922i	−0.00310604	+	0.0394660i	−0.998275	−	0.0587107i	\(-0.981301\pi\)
							0.995169	+	0.0981767i	\(0.0313010\pi\)
\(19\)	8.10105	+	13.2197i	0.426371	+	0.695774i	0.991626	−	0.129142i	\(-0.0412222\pi\)
							−0.565255	+	0.824916i	\(0.691222\pi\)
\(23\)	24.2142	+	33.3280i	1.05279	+	1.44904i	0.886364	+	0.462988i	\(0.153223\pi\)
							0.166426	+	0.986054i	\(0.446777\pi\)
\(29\)	0.500076	+	6.35407i	0.0172440	+	0.219106i	0.999502	+	0.0315690i	\(0.0100504\pi\)
							−0.982258	+	0.187537i	\(0.939950\pi\)
\(31\)	−26.0614	+	8.46787i	−0.840691	+	0.273157i	−0.697542	−	0.716544i	\(-0.745723\pi\)
							−0.143149	+	0.989701i	\(0.545723\pi\)
\(37\)	−14.6397	+	45.0563i	−0.395667	+	1.21774i	0.532774	+	0.846258i	\(0.321150\pi\)
							−0.928441	+	0.371480i	\(0.878850\pi\)
\(41\)	−25.4879	+	32.1149i	−0.621657	+	0.783290i
							\(43\)	9.21477	+	1.45948i	0.214297	+	0.0339413i	0.262660	−	0.964889i	\(-0.415400\pi\)
−0.0483628	+	0.998830i	\(0.515400\pi\)
\(47\)	−10.9905	+	2.63858i	−0.233840	+	0.0561399i	−0.348671	−	0.937245i	\(-0.613367\pi\)
							0.114832	+	0.993385i	\(0.463367\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.18	✓	672
41.11	odd	40	inner	287.3.ba.a.134.18	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.18	✓	672	1.1	even	1	trivial
287.3.ba.a.134.18	yes	672	41.11	odd	40	inner