Embedded newform 287.2.bc.a.2.8

• Label: 287.2.bc.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.326002 + 1.53372i) q^{2} +(-2.91195 - 0.780254i) q^{3} +(-0.418925 - 0.186518i) q^{4} +(0.701171 - 0.0736960i) q^{5} +(2.14599 - 4.21175i) q^{6} +(2.48429 + 0.910098i) q^{7} +(-1.42064 + 1.95534i) q^{8} +(5.27257 + 3.04412i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.326002	+	1.53372i	−0.230518	+	1.08450i	0.698827	+	0.715291i	\(0.253706\pi\)
							−0.929345	+	0.369212i	\(0.879628\pi\)
\(3\)	−2.91195	−	0.780254i	−1.68121	−	0.450480i	−0.713115	−	0.701047i	\(-0.752716\pi\)
							−0.968099	+	0.250567i	\(0.919383\pi\)
\(5\)	0.701171	−	0.0736960i	0.313573	−	0.0329579i	0.0535652	−	0.998564i	\(-0.482941\pi\)
							0.260008	+	0.965606i	\(0.416275\pi\)
\(7\)	2.48429	+	0.910098i	0.938975	+	0.343985i
							\(11\)	1.11047	+	1.37132i	0.334820	+	0.413468i	0.916554	−	0.399911i	\(-0.130959\pi\)
−0.581734	+	0.813379i	\(0.697626\pi\)
\(13\)	−5.36641	−	2.73432i	−1.48838	−	0.758365i	−0.494532	−	0.869159i	\(-0.664661\pi\)
							−0.993843	+	0.110794i	\(0.964661\pi\)
\(17\)	−4.01634	+	3.25237i	−0.974106	+	0.788816i	−0.977447	−	0.211179i	\(-0.932270\pi\)
							0.00334120	+	0.999994i	\(0.498936\pi\)
\(19\)	−0.344848	+	6.58010i	−0.0791136	+	1.50958i	0.614320	+	0.789057i	\(0.289431\pi\)
							−0.693433	+	0.720521i	\(0.743903\pi\)
\(23\)	4.99860	+	1.06249i	1.04228	+	0.221544i	0.697086	−	0.716988i	\(-0.254480\pi\)
							0.345195	+	0.938531i	\(0.387813\pi\)
\(29\)	−1.16786	+	7.37360i	−0.216867	+	1.36924i	0.603480	+	0.797378i	\(0.293780\pi\)
							−0.820347	+	0.571866i	\(0.806220\pi\)
\(31\)	0.132451	−	1.26019i	0.0237889	−	0.226337i	−0.976168	−	0.217018i	\(-0.930367\pi\)
							0.999957	−	0.00931875i	\(-0.00296629\pi\)
\(37\)	−0.565173	−	5.37726i	−0.0929139	−	0.884016i	−0.937357	−	0.348371i	\(-0.886735\pi\)
							0.844443	−	0.535645i	\(-0.179932\pi\)
\(41\)	5.55638	+	3.18224i	0.867761	+	0.496982i
							\(43\)	2.97190	−	0.965630i	0.453211	−	0.147257i	−0.0735116	−	0.997294i	\(-0.523421\pi\)
0.526723	+	0.850037i	\(0.323421\pi\)
\(47\)	1.62437	+	2.50131i	0.236938	+	0.364853i	0.937059	−	0.349172i	\(-0.113537\pi\)
							−0.700120	+	0.714025i	\(0.746870\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.8	✓	416
7.4	even	3	inner	287.2.bc.a.207.8	yes	416
41.21	even	20	inner	287.2.bc.a.226.8	yes	416
287.144	even	60	inner	287.2.bc.a.144.8	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.8	✓	416	1.1	even	1	trivial
287.2.bc.a.144.8	yes	416	287.144	even	60	inner
287.2.bc.a.207.8	yes	416	7.4	even	3	inner
287.2.bc.a.226.8	yes	416	41.21	even	20	inner