Embedded newform 287.2.j.a.81.12

• Label: 287.2.j.a.81.12
• 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			81.12
Character	\(\chi\)	\(=\)	287.81
Dual form			287.2.j.a.163.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.158705 - 0.274885i) q^{2} +(0.453525 + 0.261843i) q^{3} +(0.949625 - 1.64480i) q^{4} +(-0.708472 - 1.22711i) q^{5} -0.166223i q^{6} +(-1.98670 - 1.74729i) q^{7} -1.23766 q^{8} +(-1.36288 - 2.36057i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.158705	−	0.274885i	−0.112221	−	0.194373i	0.804444	−	0.594028i	\(-0.202463\pi\)
							−0.916666	+	0.399655i	\(0.869130\pi\)
\(3\)	0.453525	+	0.261843i	0.261843	+	0.151175i	0.625175	−	0.780485i	\(-0.285028\pi\)
							−0.363332	+	0.931660i	\(0.618361\pi\)
\(5\)	−0.708472	−	1.22711i	−0.316838	−	0.548780i	0.662988	−	0.748630i	\(-0.269288\pi\)
							−0.979827	+	0.199850i	\(0.935955\pi\)
\(7\)	−1.98670	−	1.74729i	−0.750902	−	0.660413i
							\(11\)	−2.12636	−	1.22766i	−0.641123	−	0.370152i	0.143924	−	0.989589i	\(-0.454028\pi\)
−0.785047	+	0.619436i	\(0.787361\pi\)
\(13\)			4.70377i			1.30459i	0.757965	+	0.652295i	\(0.226194\pi\)
							−0.757965	+	0.652295i	\(0.773806\pi\)
\(17\)	2.83108	+	1.63453i	0.686638	+	0.396431i	0.802351	−	0.596852i	\(-0.203582\pi\)
							−0.115713	+	0.993283i	\(0.536915\pi\)
\(19\)	3.56920	−	2.06068i	0.818830	−	0.472752i	−0.0311826	−	0.999514i	\(-0.509927\pi\)
							0.850013	+	0.526762i	\(0.176594\pi\)
\(23\)	2.59094	+	4.48763i	0.540248	+	0.935736i	0.998889	+	0.0471152i	\(0.0150028\pi\)
							−0.458642	+	0.888621i	\(0.651664\pi\)
\(29\)			2.09808i			0.389604i	0.980843	+	0.194802i	\(0.0624064\pi\)
							−0.980843	+	0.194802i	\(0.937594\pi\)
\(31\)	5.07571	−	8.79138i	0.911624	−	1.57898i	0.0998536	−	0.995002i	\(-0.468163\pi\)
							0.811770	−	0.583977i	\(-0.198504\pi\)
\(37\)	1.25243	+	2.16927i	0.205898	+	0.356625i	0.950418	−	0.310974i	\(-0.100655\pi\)
							−0.744521	+	0.667599i	\(0.767322\pi\)
\(41\)	0.564046	−	6.37823i	0.0880891	−	0.996113i
							\(43\)	9.23450			1.40825			0.704124	−	0.710077i	\(-0.251340\pi\)
0.704124	+	0.710077i	\(0.251340\pi\)
\(47\)	6.77934	−	3.91405i	0.988868	−	0.570923i	0.0839322	−	0.996471i	\(-0.473252\pi\)
							0.904936	+	0.425548i	\(0.139919\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.j.a.81.12	yes	52
7.2	even	3	inner	287.2.j.a.163.11	yes	52
41.40	even	2	inner	287.2.j.a.81.11	✓	52
287.163	even	6	inner	287.2.j.a.163.12	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.j.a.81.11	✓	52	41.40	even	2	inner
287.2.j.a.81.12	yes	52	1.1	even	1	trivial
287.2.j.a.163.11	yes	52	7.2	even	3	inner
287.2.j.a.163.12	yes	52	287.163	even	6	inner