Embedded newform 287.2.w.a.3.10

• Label: 287.2.w.a.3.10
• Level: $287$
• Weight: $2$
• Character: 287.3
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.w (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			3.10
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.866669 - 0.232223i) q^{2} +(2.43423 - 1.86785i) q^{3} +(-1.03486 - 0.597479i) q^{4} +(-3.67938 - 0.985886i) q^{5} +(-2.54343 + 1.05352i) q^{6} +(-1.65100 + 2.06741i) q^{7} +(2.02703 + 2.02703i) q^{8} +(1.66016 - 6.19579i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.866669	−	0.232223i	−0.612827	−	0.164207i	−0.0609624	−	0.998140i	\(-0.519417\pi\)
							−0.551865	+	0.833933i	\(0.686084\pi\)
\(3\)	2.43423	−	1.86785i	1.40540	−	1.07840i	0.420122	−	0.907468i	\(-0.361987\pi\)
							0.985282	−	0.170937i	\(-0.0546794\pi\)
\(5\)	−3.67938	−	0.985886i	−1.64547	−	0.440902i	−0.687128	−	0.726536i	\(-0.741129\pi\)
							−0.958339	+	0.285635i	\(0.907796\pi\)
\(7\)	−1.65100	+	2.06741i	−0.624019	+	0.781409i
							\(11\)	−0.406387	+	0.311831i	−0.122530	+	0.0940207i	−0.668209	−	0.743974i	\(-0.732939\pi\)
0.545679	+	0.837995i	\(0.316272\pi\)
\(13\)	−3.49460	−	1.44751i	−0.969228	−	0.401467i	−0.158803	−	0.987310i	\(-0.550764\pi\)
							−0.810425	+	0.585843i	\(0.800764\pi\)
\(17\)	0.716880	−	5.44524i	0.173869	−	1.32067i	−0.652609	−	0.757695i	\(-0.726325\pi\)
							0.826477	−	0.562970i	\(-0.190341\pi\)
\(19\)	−0.562502	−	0.431623i	−0.129047	−	0.0990210i	0.542221	−	0.840236i	\(-0.317583\pi\)
							−0.671268	+	0.741215i	\(0.734250\pi\)
\(23\)	−2.62205	+	1.51384i	−0.546735	+	0.315657i	−0.747804	−	0.663920i	\(-0.768892\pi\)
							0.201069	+	0.979577i	\(0.435558\pi\)
\(29\)	2.34041	−	5.65024i	0.434603	−	1.04922i	−0.543183	−	0.839615i	\(-0.682781\pi\)
							0.977785	−	0.209609i	\(-0.0672191\pi\)
\(31\)	−2.57660	+	4.46281i	−0.462772	+	0.801544i	−0.999098	−	0.0424667i	\(-0.986478\pi\)
							0.536326	+	0.844011i	\(0.319812\pi\)
\(37\)	1.73732	+	3.00912i	0.285613	+	0.494696i	0.972758	−	0.231824i	\(-0.0744695\pi\)
							−0.687145	+	0.726521i	\(0.741136\pi\)
\(41\)	−2.21429	−	6.00807i	−0.345814	−	0.938303i
							\(43\)	3.15660	−	3.15660i	0.481378	−	0.481378i	−0.424194	−	0.905571i	\(-0.639442\pi\)
0.905571	+	0.424194i	\(0.139442\pi\)
\(47\)	−6.57806	−	5.04752i	−0.959509	−	0.736257i	0.00508773	−	0.999987i	\(-0.498381\pi\)
							−0.964597	+	0.263730i	\(0.915047\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.10	✓	208
7.5	odd	6	inner	287.2.w.a.208.17	yes	208
41.14	odd	8	inner	287.2.w.a.178.17	yes	208
287.96	even	24	inner	287.2.w.a.96.10	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.10	✓	208	1.1	even	1	trivial
287.2.w.a.96.10	yes	208	287.96	even	24	inner
287.2.w.a.178.17	yes	208	41.14	odd	8	inner
287.2.w.a.208.17	yes	208	7.5	odd	6	inner