Embedded newform 287.2.j.a.81.20

• Label: 287.2.j.a.81.20
• Level: $287$
• Weight: $2$
• Character: 287.81
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			81.20
Character	\(\chi\)	\(=\)	287.81
Dual form			287.2.j.a.163.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.649153 + 1.12437i) q^{2} +(0.800607 + 0.462230i) q^{3} +(0.157202 - 0.272281i) q^{4} +(-1.62887 - 2.82129i) q^{5} +1.20023i q^{6} +(2.18878 - 1.48635i) q^{7} +3.00480 q^{8} +(-1.07269 - 1.85795i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.649153	+	1.12437i	0.459020	+	0.795046i	0.998909	−	0.0466897i	\(-0.0148672\pi\)
							−0.539889	+	0.841736i	\(0.681534\pi\)
\(3\)	0.800607	+	0.462230i	0.462230	+	0.266869i	0.712982	−	0.701183i	\(-0.247344\pi\)
							−0.250751	+	0.968052i	\(0.580678\pi\)
\(5\)	−1.62887	−	2.82129i	−0.728454	−	1.26172i	−0.957536	−	0.288313i	\(-0.906906\pi\)
							0.229082	−	0.973407i	\(-0.426428\pi\)
\(7\)	2.18878	−	1.48635i	0.827282	−	0.561788i
							\(11\)	0.320547	+	0.185068i	0.0966486	+	0.0558001i	0.547545	−	0.836776i	\(-0.315562\pi\)
−0.450897	+	0.892576i	\(0.648896\pi\)
\(13\)			4.50664i			1.24992i	0.780657	+	0.624959i	\(0.214884\pi\)
							−0.780657	+	0.624959i	\(0.785116\pi\)
\(17\)	2.36362	+	1.36464i	0.573263	+	0.330973i	0.758451	−	0.651730i	\(-0.225956\pi\)
							−0.185189	+	0.982703i	\(0.559290\pi\)
\(19\)	−6.50468	+	3.75548i	−1.49228	+	0.861566i	−0.999961	−	0.00885034i	\(-0.997183\pi\)
							−0.492316	+	0.870417i	\(0.663849\pi\)
\(23\)	−0.0445877	−	0.0772281i	−0.00929717	−	0.0161032i	0.861339	−	0.508030i	\(-0.169626\pi\)
							−0.870637	+	0.491927i	\(0.836293\pi\)
\(29\)		−	3.57090i		−	0.663099i	−0.943438	−	0.331549i	\(-0.892429\pi\)
							0.943438	−	0.331549i	\(-0.107571\pi\)
\(31\)	−4.13934	+	7.16954i	−0.743447	+	1.28769i	0.207470	+	0.978241i	\(0.433477\pi\)
							−0.950917	+	0.309447i	\(0.899856\pi\)
\(37\)	2.96124	+	5.12901i	0.486824	+	0.843205i	0.999885	−	0.0151476i	\(-0.00482182\pi\)
							−0.513061	+	0.858352i	\(0.671488\pi\)
\(41\)	3.85181	+	5.11503i	0.601552	+	0.798834i
							\(43\)	2.91871			0.445099			0.222549	−	0.974921i	\(-0.428562\pi\)
0.222549	+	0.974921i	\(0.428562\pi\)
\(47\)	2.24736	−	1.29752i	0.327811	−	0.189262i	−0.327058	−	0.945004i	\(-0.606057\pi\)
							0.654869	+	0.755742i	\(0.272724\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.j.a.81.20	yes	52
7.2	even	3	inner	287.2.j.a.163.19	yes	52
41.40	even	2	inner	287.2.j.a.81.19	✓	52
287.163	even	6	inner	287.2.j.a.163.20	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.j.a.81.19	✓	52	41.40	even	2	inner
287.2.j.a.81.20	yes	52	1.1	even	1	trivial
287.2.j.a.163.19	yes	52	7.2	even	3	inner
287.2.j.a.163.20	yes	52	287.163	even	6	inner