Embedded newform 287.2.bc.a.2.10

• Label: 287.2.bc.a.2.10
• Level: $287$
• Weight: $2$
• Character: 287.2
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $416$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.bc (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(26\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.180323 + 0.848351i) q^{2} +(0.677072 + 0.181421i) q^{3} +(1.13991 + 0.507520i) q^{4} +(-0.215015 + 0.0225990i) q^{5} +(-0.276000 + 0.541680i) q^{6} +(-0.905986 + 2.48580i) q^{7} +(-1.65568 + 2.27885i) q^{8} +(-2.17256 - 1.25433i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.180323	+	0.848351i	−0.127507	+	0.599875i	0.867273	+	0.497833i	\(0.165871\pi\)
							−0.994780	+	0.102042i	\(0.967463\pi\)
\(3\)	0.677072	+	0.181421i	0.390908	+	0.104743i	0.448919	−	0.893573i	\(-0.351809\pi\)
							−0.0580112	+	0.998316i	\(0.518476\pi\)
\(5\)	−0.215015	+	0.0225990i	−0.0961575	+	0.0101066i	−0.152485	−	0.988306i	\(-0.548728\pi\)
							0.0563275	+	0.998412i	\(0.482061\pi\)
\(7\)	−0.905986	+	2.48580i	−0.342431	+	0.939543i
							\(11\)	2.88487	+	3.56252i	0.869821	+	1.07414i	0.996615	+	0.0822104i	\(0.0261979\pi\)
−0.126794	+	0.991929i	\(0.540469\pi\)
\(13\)	−1.15967	−	0.590882i	−0.321635	−	0.163881i	0.285717	−	0.958314i	\(-0.407768\pi\)
							−0.607352	+	0.794433i	\(0.707768\pi\)
\(17\)	5.14806	−	4.16882i	1.24859	−	1.01109i	0.249501	−	0.968375i	\(-0.419733\pi\)
							0.999087	−	0.0427122i	\(-0.0135999\pi\)
\(19\)	−0.0473788	+	0.904041i	−0.0108694	+	0.207401i	0.987796	+	0.155754i	\(0.0497806\pi\)
							−0.998665	+	0.0516476i	\(0.983553\pi\)
\(23\)	5.73634	+	1.21930i	1.19611	+	0.254241i	0.762577	−	0.646897i	\(-0.223934\pi\)
							0.433532	+	0.901138i	\(0.357267\pi\)
\(29\)	1.10516	−	6.97773i	0.205224	−	1.29573i	−0.642907	−	0.765944i	\(-0.722272\pi\)
							0.848131	−	0.529787i	\(-0.177728\pi\)
\(31\)	−0.151875	+	1.44500i	−0.0272776	+	0.259529i	0.972381	+	0.233399i	\(0.0749849\pi\)
							−0.999659	+	0.0261296i	\(0.991682\pi\)
\(37\)	−0.0616126	−	0.586205i	−0.0101291	−	0.0963715i	0.988289	−	0.152591i	\(-0.0487619\pi\)
							−0.998418	+	0.0562200i	\(0.982095\pi\)
\(41\)	6.40306	+	0.0283739i	0.999990	+	0.00443126i
							\(43\)	0.590668	−	0.191920i	0.0900761	−	0.0292675i	−0.263632	−	0.964623i	\(-0.584921\pi\)
0.353708	+	0.935356i	\(0.384921\pi\)
\(47\)	−3.20031	−	4.92805i	−0.466813	−	0.718830i	0.524346	−	0.851505i	\(-0.324310\pi\)
							−0.991160	+	0.132676i	\(0.957643\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.10	✓	416
7.4	even	3	inner	287.2.bc.a.207.10	yes	416
41.21	even	20	inner	287.2.bc.a.226.10	yes	416
287.144	even	60	inner	287.2.bc.a.144.10	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.10	✓	416	1.1	even	1	trivial
287.2.bc.a.144.10	yes	416	287.144	even	60	inner
287.2.bc.a.207.10	yes	416	7.4	even	3	inner
287.2.bc.a.226.10	yes	416	41.21	even	20	inner