Embedded newform 287.2.bc.a.2.5

• Label: 287.2.bc.a.2.5
• 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			2.5
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.421114 + 1.98118i) q^{2} +(-0.975781 - 0.261460i) q^{3} +(-1.92066 - 0.855135i) q^{4} +(-1.74205 + 0.183096i) q^{5} +(0.928914 - 1.82310i) q^{6} +(0.892415 + 2.49070i) q^{7} +(0.121944 - 0.167842i) q^{8} +(-1.71429 - 0.989746i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.421114	+	1.98118i	−0.297772	+	1.40091i	0.533855	+	0.845576i	\(0.320743\pi\)
							−0.831628	+	0.555333i	\(0.812591\pi\)
\(3\)	−0.975781	−	0.261460i	−0.563367	−	0.150954i	−0.0341143	−	0.999418i	\(-0.510861\pi\)
							−0.529253	+	0.848464i	\(0.677528\pi\)
\(5\)	−1.74205	+	0.183096i	−0.779067	+	0.0818832i	−0.485710	−	0.874120i	\(-0.661439\pi\)
							−0.293357	+	0.956003i	\(0.594772\pi\)
\(7\)	0.892415	+	2.49070i	0.337301	+	0.941397i
							\(11\)	−2.52919	−	3.12329i	−0.762581	−	0.941709i	0.236975	−	0.971516i	\(-0.423844\pi\)
−0.999556	+	0.0298072i	\(0.990511\pi\)
\(13\)	0.674562	+	0.343707i	0.187090	+	0.0953270i	0.545026	−	0.838419i	\(-0.316520\pi\)
							−0.357937	+	0.933746i	\(0.616520\pi\)
\(17\)	1.54896	−	1.25433i	0.375679	−	0.304219i	−0.422822	−	0.906213i	\(-0.638960\pi\)
							0.798500	+	0.601994i	\(0.205627\pi\)
\(19\)	0.198171	−	3.78132i	0.0454635	−	0.867495i	−0.876782	−	0.480889i	\(-0.840314\pi\)
							0.922245	−	0.386606i	\(-0.126353\pi\)
\(23\)	−7.47467	−	1.58879i	−1.55858	−	0.331286i	−0.653631	−	0.756813i	\(-0.726755\pi\)
							−0.904946	+	0.425527i	\(0.860089\pi\)
\(29\)	0.0971026	−	0.613082i	0.0180315	−	0.113846i	−0.977031	−	0.213098i	\(-0.931645\pi\)
							0.995062	+	0.0992512i	\(0.0316447\pi\)
\(31\)	−0.885214	+	8.42225i	−0.158989	+	1.51268i	0.566279	+	0.824214i	\(0.308382\pi\)
							−0.725268	+	0.688467i	\(0.758284\pi\)
\(37\)	0.789273	+	7.50943i	0.129756	+	1.23454i	0.844654	+	0.535313i	\(0.179806\pi\)
							−0.714898	+	0.699229i	\(0.753527\pi\)
\(41\)	−3.92705	−	5.05750i	−0.613302	−	0.789849i
							\(43\)	−0.762846	+	0.247864i	−0.116333	+	0.0377989i	−0.366605	−	0.930377i	\(-0.619480\pi\)
0.250272	+	0.968176i	\(0.419480\pi\)
\(47\)	2.96810	+	4.57047i	0.432942	+	0.666672i	0.986052	−	0.166436i	\(-0.0532258\pi\)
							−0.553110	+	0.833108i	\(0.686559\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.5	✓	416
7.4	even	3	inner	287.2.bc.a.207.5	yes	416
41.21	even	20	inner	287.2.bc.a.226.5	yes	416
287.144	even	60	inner	287.2.bc.a.144.5	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.5	✓	416	1.1	even	1	trivial
287.2.bc.a.144.5	yes	416	287.144	even	60	inner
287.2.bc.a.207.5	yes	416	7.4	even	3	inner
287.2.bc.a.226.5	yes	416	41.21	even	20	inner