Embedded newform 287.2.l.a.27.6

• Label: 287.2.l.a.27.6
• Level: $287$
• Weight: $2$
• Character: 287.27
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.l (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			27.6
Character	\(\chi\)	\(=\)	287.27
Dual form			287.2.l.a.202.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55293 - 1.55293i) q^{2} +(0.699693 + 1.68921i) q^{3} +2.82318i q^{4} +(-1.29416 - 1.29416i) q^{5} +(1.53665 - 3.70980i) q^{6} +(0.387642 - 2.61720i) q^{7} +(1.27834 - 1.27834i) q^{8} +(-0.242535 + 0.242535i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 8 q^{2} - 4 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55293	−	1.55293i	−1.09809	−	1.09809i	−0.994634	−	0.103453i	\(-0.967011\pi\)
							−0.103453	−	0.994634i	\(-0.532989\pi\)
\(3\)	0.699693	+	1.68921i	0.403968	+	0.975265i	0.986693	+	0.162596i	\(0.0519866\pi\)
							−0.582725	+	0.812670i	\(0.698013\pi\)
\(5\)	−1.29416	−	1.29416i	−0.578765	−	0.578765i	0.355798	−	0.934563i	\(-0.384209\pi\)
							−0.934563	+	0.355798i	\(0.884209\pi\)
\(7\)	0.387642	−	2.61720i	0.146515	−	0.989208i
							\(11\)	2.00222	+	4.83380i	0.603693	+	1.45744i	0.869753	+	0.493488i	\(0.164278\pi\)
−0.266059	+	0.963957i	\(0.585722\pi\)
\(13\)	−2.00127	−	4.83150i	−0.555053	−	1.34002i	−0.913642	−	0.406520i	\(-0.866742\pi\)
							0.358589	−	0.933496i	\(-0.383258\pi\)
\(17\)	1.28980	−	3.11386i	0.312823	−	0.755221i	−0.686775	−	0.726870i	\(-0.740974\pi\)
							0.999598	−	0.0283513i	\(-0.00902572\pi\)
\(19\)	1.62994	−	3.93502i	0.373933	−	0.902755i	−0.619142	−	0.785279i	\(-0.712520\pi\)
							0.993076	−	0.117476i	\(-0.0374803\pi\)
\(23\)		−	7.59275i		−	1.58320i	−0.611041	−	0.791599i	\(-0.709249\pi\)
							0.611041	−	0.791599i	\(-0.290751\pi\)
\(29\)	−8.42197	+	3.48849i	−1.56392	+	0.647797i	−0.985765	−	0.168129i	\(-0.946227\pi\)
							−0.578156	+	0.815926i	\(0.696227\pi\)
\(31\)	1.65081			0.296494			0.148247	−	0.988950i	\(-0.452637\pi\)
							0.148247	+	0.988950i	\(0.452637\pi\)
\(37\)	8.06048			1.32513			0.662567	−	0.749002i	\(-0.269467\pi\)
							0.662567	+	0.749002i	\(0.269467\pi\)
\(41\)	3.02640	−	5.64278i	0.472644	−	0.881254i
							\(43\)	1.73178	+	1.73178i	0.264093	+	0.264093i	0.826715	−	0.562621i	\(-0.190207\pi\)
−0.562621	+	0.826715i	\(0.690207\pi\)
\(47\)	−3.38737	+	8.17784i	−0.494099	+	1.19286i	0.458517	+	0.888686i	\(0.348381\pi\)
							−0.952616	+	0.304175i	\(0.901619\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.l.a.27.6	yes	104
7.6	odd	2	inner	287.2.l.a.27.5	✓	104
41.38	odd	8	inner	287.2.l.a.202.5	yes	104
287.202	even	8	inner	287.2.l.a.202.6	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.l.a.27.5	✓	104	7.6	odd	2	inner
287.2.l.a.27.6	yes	104	1.1	even	1	trivial
287.2.l.a.202.5	yes	104	41.38	odd	8	inner
287.2.l.a.202.6	yes	104	287.202	even	8	inner