Embedded newform 287.2.bc.a.2.3

• Label: 287.2.bc.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.512813 + 2.41260i) q^{2} +(-2.70362 - 0.724432i) q^{3} +(-3.73056 - 1.66095i) q^{4} +(-1.04456 + 0.109788i) q^{5} +(3.13421 - 6.15124i) q^{6} +(-2.52935 - 0.776139i) q^{7} +(3.02074 - 4.15770i) q^{8} +(4.18667 + 2.41718i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.512813	+	2.41260i	−0.362614	+	1.70596i	0.297436	+	0.954742i	\(0.403869\pi\)
							−0.660050	+	0.751222i	\(0.729465\pi\)
\(3\)	−2.70362	−	0.724432i	−1.56093	−	0.418251i	−0.627975	−	0.778233i	\(-0.716116\pi\)
							−0.932959	+	0.359982i	\(0.882783\pi\)
\(5\)	−1.04456	+	0.109788i	−0.467142	+	0.0490986i	−0.335176	−	0.942156i	\(-0.608796\pi\)
							−0.131966	+	0.991254i	\(0.542129\pi\)
\(7\)	−2.52935	−	0.776139i	−0.956004	−	0.293353i
							\(11\)	1.47921	+	1.82667i	0.445998	+	0.550762i	0.949719	−	0.313105i	\(-0.101369\pi\)
−0.503720	+	0.863867i	\(0.668036\pi\)
\(13\)	3.27372	+	1.66804i	0.907966	+	0.462632i	0.844624	−	0.535360i	\(-0.179824\pi\)
							0.0633420	+	0.997992i	\(0.479824\pi\)
\(17\)	−0.960046	+	0.777430i	−0.232845	+	0.188555i	−0.738641	−	0.674099i	\(-0.764532\pi\)
							0.505796	+	0.862653i	\(0.331199\pi\)
\(19\)	0.426633	−	8.14064i	0.0978763	−	1.86759i	−0.305021	−	0.952346i	\(-0.598663\pi\)
							0.402897	−	0.915245i	\(-0.368003\pi\)
\(23\)	5.30010	+	1.12657i	1.10515	+	0.234906i	0.724131	−	0.689662i	\(-0.242241\pi\)
							0.381016	+	0.924568i	\(0.375574\pi\)
\(29\)	0.441325	−	2.78641i	0.0819519	−	0.517424i	−0.912227	−	0.409685i	\(-0.865639\pi\)
							0.994179	−	0.107739i	\(-0.0343612\pi\)
\(31\)	0.633282	−	6.02528i	0.113741	−	1.08217i	−0.777575	−	0.628790i	\(-0.783551\pi\)
							0.891316	−	0.453382i	\(-0.149783\pi\)
\(37\)	0.375167	+	3.56948i	0.0616771	+	0.586819i	0.981094	+	0.193534i	\(0.0619948\pi\)
							−0.919417	+	0.393285i	\(0.871338\pi\)
\(41\)	5.86111	−	2.57826i	0.915351	−	0.402657i
							\(43\)	−2.15100	+	0.698904i	−0.328025	+	0.106582i	−0.468400	−	0.883517i	\(-0.655169\pi\)
0.140375	+	0.990098i	\(0.455169\pi\)
\(47\)	−6.42506	−	9.89373i	−0.937192	−	1.44315i	−0.896028	−	0.443997i	\(-0.853560\pi\)
							−0.0411635	−	0.999152i	\(-0.513106\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.3	✓	416
7.4	even	3	inner	287.2.bc.a.207.3	yes	416
41.21	even	20	inner	287.2.bc.a.226.3	yes	416
287.144	even	60	inner	287.2.bc.a.144.3	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.3	✓	416	1.1	even	1	trivial
287.2.bc.a.144.3	yes	416	287.144	even	60	inner
287.2.bc.a.207.3	yes	416	7.4	even	3	inner
287.2.bc.a.226.3	yes	416	41.21	even	20	inner