Embedded newform 287.2.w.a.3.6

• Label: 287.2.w.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67812 - 0.449651i) q^{2} +(-1.39067 + 1.06710i) q^{3} +(0.881854 + 0.509138i) q^{4} +(0.769944 + 0.206306i) q^{5} +(2.81353 - 1.16540i) q^{6} +(-2.27419 - 1.35207i) q^{7} +(1.20602 + 1.20602i) q^{8} +(0.0188054 - 0.0701826i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67812	−	0.449651i	−1.18661	−	0.317951i	−0.389065	−	0.921210i	\(-0.627202\pi\)
							−0.797546	+	0.603259i	\(0.793869\pi\)
\(3\)	−1.39067	+	1.06710i	−0.802903	+	0.616089i	−0.926522	−	0.376241i	\(-0.877217\pi\)
							0.123619	+	0.992330i	\(0.460550\pi\)
\(5\)	0.769944	+	0.206306i	0.344329	+	0.0922627i	0.426839	−	0.904327i	\(-0.359627\pi\)
							−0.0825101	+	0.996590i	\(0.526294\pi\)
\(7\)	−2.27419	−	1.35207i	−0.859561	−	0.511033i
							\(11\)	−2.17857	+	1.67168i	−0.656865	+	0.504030i	−0.882579	−	0.470164i	\(-0.844195\pi\)
0.225715	+	0.974193i	\(0.427528\pi\)
\(13\)	2.46662	+	1.02171i	0.684116	+	0.283370i	0.697546	−	0.716540i	\(-0.254275\pi\)
							−0.0134302	+	0.999910i	\(0.504275\pi\)
\(17\)	0.992667	−	7.54005i	0.240757	−	1.82873i	−0.262894	−	0.964825i	\(-0.584677\pi\)
							0.503651	−	0.863907i	\(-0.331990\pi\)
\(19\)	−1.98985	−	1.52686i	−0.456502	−	0.350286i	0.354772	−	0.934953i	\(-0.384559\pi\)
							−0.811274	+	0.584667i	\(0.801225\pi\)
\(23\)	3.62746	−	2.09432i	0.756379	−	0.436695i	−0.0716153	−	0.997432i	\(-0.522815\pi\)
							0.827994	+	0.560737i	\(0.189482\pi\)
\(29\)	3.14099	−	7.58303i	0.583268	−	1.40813i	−0.306566	−	0.951849i	\(-0.599180\pi\)
							0.889834	−	0.456284i	\(-0.150820\pi\)
\(31\)	−1.95738	+	3.39029i	−0.351557	+	0.608914i	−0.986522	−	0.163626i	\(-0.947681\pi\)
							0.634966	+	0.772540i	\(0.281014\pi\)
\(37\)	−5.76623	−	9.98740i	−0.947962	−	1.64192i	−0.749709	−	0.661768i	\(-0.769806\pi\)
							−0.198253	−	0.980151i	\(-0.563527\pi\)
\(41\)	0.0515157	+	6.40292i	0.00804540	+	0.999968i
							\(43\)	1.60146	−	1.60146i	0.244221	−	0.244221i	−0.574373	−	0.818594i	\(-0.694754\pi\)
0.818594	+	0.574373i	\(0.194754\pi\)
\(47\)	3.49352	+	2.68068i	0.509583	+	0.391017i	0.831304	−	0.555818i	\(-0.187595\pi\)
							−0.321721	+	0.946834i	\(0.604261\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.6	✓	208
7.5	odd	6	inner	287.2.w.a.208.21	yes	208
41.14	odd	8	inner	287.2.w.a.178.21	yes	208
287.96	even	24	inner	287.2.w.a.96.6	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.6	✓	208	1.1	even	1	trivial
287.2.w.a.96.6	yes	208	287.96	even	24	inner
287.2.w.a.178.21	yes	208	41.14	odd	8	inner
287.2.w.a.208.21	yes	208	7.5	odd	6	inner