Embedded newform 287.2.j.a.81.5

• Label: 287.2.j.a.81.5
• 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			81.5
Character	\(\chi\)	\(=\)	287.81
Dual form			287.2.j.a.163.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02851 - 1.78143i) q^{2} +(-1.38029 - 0.796913i) q^{3} +(-1.11566 + 1.93238i) q^{4} +(-2.03101 - 3.51781i) q^{5} +3.27853i q^{6} +(2.61591 - 0.396236i) q^{7} +0.475827 q^{8} +(-0.229860 - 0.398129i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02851	−	1.78143i	−0.727265	−	1.25966i	−0.958035	−	0.286652i	\(-0.907458\pi\)
							0.230769	−	0.973008i	\(-0.425876\pi\)
\(3\)	−1.38029	−	0.796913i	−0.796913	−	0.460098i	0.0454777	−	0.998965i	\(-0.485519\pi\)
							−0.842391	+	0.538868i	\(0.818852\pi\)
\(5\)	−2.03101	−	3.51781i	−0.908294	−	1.57321i	−0.816433	−	0.577440i	\(-0.804052\pi\)
							−0.0918611	−	0.995772i	\(-0.529282\pi\)
\(7\)	2.61591	−	0.396236i	0.988722	−	0.149763i
							\(11\)	−3.36214	−	1.94113i	−1.01372	−	0.585274i	−0.101445	−	0.994841i	\(-0.532346\pi\)
−0.912280	+	0.409567i	\(0.865680\pi\)
\(13\)		−	0.721943i		−	0.200231i	−0.994976	−	0.100115i	\(-0.968079\pi\)
							0.994976	−	0.100115i	\(-0.0319212\pi\)
\(17\)	6.12798	+	3.53799i	1.48625	+	0.858089i	0.999877	−	0.0156603i	\(-0.00498502\pi\)
							0.486376	+	0.873749i	\(0.338318\pi\)
\(19\)	1.92864	−	1.11350i	0.442461	−	0.255455i	−0.262180	−	0.965019i	\(-0.584442\pi\)
							0.704641	+	0.709564i	\(0.251108\pi\)
\(23\)	−1.49544	−	2.59018i	−0.311821	−	0.540089i	0.666936	−	0.745115i	\(-0.267605\pi\)
							−0.978757	+	0.205026i	\(0.934272\pi\)
\(29\)			3.88310i			0.721073i	0.932745	+	0.360536i	\(0.117406\pi\)
							−0.932745	+	0.360536i	\(0.882594\pi\)
\(31\)	3.90458	−	6.76293i	0.701283	−	1.21466i	−0.266733	−	0.963770i	\(-0.585944\pi\)
							0.968016	−	0.250888i	\(-0.0807225\pi\)
\(37\)	−0.435496	−	0.754302i	−0.0715951	−	0.124006i	0.828005	−	0.560720i	\(-0.189476\pi\)
							−0.899601	+	0.436714i	\(0.856142\pi\)
\(41\)	3.38197	−	5.43712i	0.528175	−	0.849136i
							\(43\)	−7.06067			−1.07674			−0.538371	−	0.842708i	\(-0.680960\pi\)
−0.538371	+	0.842708i	\(0.680960\pi\)
\(47\)	−11.1082	+	6.41334i	−1.62030	+	0.935481i	−0.633462	+	0.773774i	\(0.718367\pi\)
							−0.986839	+	0.161708i	\(0.948300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.j.a.81.5	✓	52
7.2	even	3	inner	287.2.j.a.163.6	yes	52
41.40	even	2	inner	287.2.j.a.81.6	yes	52
287.163	even	6	inner	287.2.j.a.163.5	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.j.a.81.5	✓	52	1.1	even	1	trivial
287.2.j.a.81.6	yes	52	41.40	even	2	inner
287.2.j.a.163.5	yes	52	287.163	even	6	inner
287.2.j.a.163.6	yes	52	7.2	even	3	inner