Embedded newform 287.2.bc.a.2.1

• Label: 287.2.bc.a.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.568103 + 2.67272i) q^{2} +(0.0505447 + 0.0135434i) q^{3} +(-4.99358 - 2.22329i) q^{4} +(-0.549935 + 0.0578005i) q^{5} +(-0.0649123 + 0.127398i) q^{6} +(1.14537 - 2.38498i) q^{7} +(5.56692 - 7.66221i) q^{8} +(-2.59570 - 1.49863i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.568103	+	2.67272i	−0.401710	+	1.88990i	0.0518228	+	0.998656i	\(0.483497\pi\)
							−0.453533	+	0.891240i	\(0.649836\pi\)
\(3\)	0.0505447	+	0.0135434i	0.0291820	+	0.00781929i	0.273381	−	0.961906i	\(-0.411858\pi\)
							−0.244199	+	0.969725i	\(0.578525\pi\)
\(5\)	−0.549935	+	0.0578005i	−0.245939	+	0.0258492i	−0.226696	−	0.973966i	\(-0.572792\pi\)
							−0.0192429	+	0.999815i	\(0.506126\pi\)
\(7\)	1.14537	−	2.38498i	0.432908	−	0.901438i
							\(11\)	−1.78418	−	2.20328i	−0.537950	−	0.664313i	0.433612	−	0.901100i	\(-0.357239\pi\)
−0.971562	+	0.236787i	\(0.923906\pi\)
\(13\)	−5.51374	−	2.80939i	−1.52924	−	0.779185i	−0.531541	−	0.847033i	\(-0.678387\pi\)
							−0.997696	+	0.0678476i	\(0.978387\pi\)
\(17\)	1.52248	−	1.23288i	0.369255	−	0.299017i	−0.426675	−	0.904405i	\(-0.640315\pi\)
							0.795931	+	0.605388i	\(0.206982\pi\)
\(19\)	−0.338765	+	6.46402i	−0.0777180	+	1.48295i	0.630078	+	0.776532i	\(0.283023\pi\)
							−0.707796	+	0.706417i	\(0.750310\pi\)
\(23\)	0.854356	+	0.181599i	0.178146	+	0.0378660i	0.296121	−	0.955151i	\(-0.404307\pi\)
							−0.117975	+	0.993017i	\(0.537640\pi\)
\(29\)	−0.246949	+	1.55918i	−0.0458573	+	0.289532i	−0.999950	−	0.00997924i	\(-0.996823\pi\)
							0.954093	+	0.299511i	\(0.0968235\pi\)
\(31\)	0.587629	−	5.59092i	0.105541	−	1.00416i	−0.805711	−	0.592309i	\(-0.798216\pi\)
							0.911252	−	0.411849i	\(-0.135117\pi\)
\(37\)	0.468551	+	4.45797i	0.0770294	+	0.732886i	0.963065	+	0.269269i	\(0.0867820\pi\)
							−0.886036	+	0.463617i	\(0.846551\pi\)
\(41\)	2.98453	−	5.66503i	0.466105	−	0.884729i
							\(43\)	−7.97716	+	2.59194i	−1.21651	+	0.395267i	−0.845808	−	0.533487i	\(-0.820881\pi\)
−0.370697	+	0.928754i	\(0.620881\pi\)
\(47\)	−4.21352	−	6.48825i	−0.614605	−	0.946408i	−0.999723	−	0.0235411i	\(-0.992506\pi\)
							0.385118	−	0.922867i	\(-0.374161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.1	✓	416
7.4	even	3	inner	287.2.bc.a.207.1	yes	416
41.21	even	20	inner	287.2.bc.a.226.1	yes	416
287.144	even	60	inner	287.2.bc.a.144.1	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.1	✓	416	1.1	even	1	trivial
287.2.bc.a.144.1	yes	416	287.144	even	60	inner
287.2.bc.a.207.1	yes	416	7.4	even	3	inner
287.2.bc.a.226.1	yes	416	41.21	even	20	inner