Embedded newform 287.3.m.a.85.9

• Label: 287.3.m.a.85.9
• Level: $287$
• Weight: $3$
• Character: 287.85
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.m (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(42\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			85.9
Character	\(\chi\)	\(=\)	287.85
Dual form			287.3.m.a.260.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.74186 + 1.74186i) q^{2} +(1.60775 - 3.88144i) q^{3} -2.06815i q^{4} +(-6.92683 - 6.92683i) q^{5} +(3.96046 + 9.56139i) q^{6} +(1.01249 - 2.44436i) q^{7} +(-3.36502 - 3.36502i) q^{8} +(-6.11678 - 6.11678i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 8 q^{2} - 16 q^{3} + 24 q^{6} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.74186	+	1.74186i	−0.870930	+	0.870930i	−0.992574	−	0.121644i	\(-0.961183\pi\)
							0.121644	+	0.992574i	\(0.461183\pi\)
\(3\)	1.60775	−	3.88144i	0.535915	−	1.29381i	−0.391637	−	0.920120i	\(-0.628091\pi\)
							0.927552	−	0.373694i	\(-0.121909\pi\)
\(5\)	−6.92683	−	6.92683i	−1.38537	−	1.38537i	−0.834775	−	0.550591i	\(-0.814403\pi\)
							−0.550591	−	0.834775i	\(-0.685597\pi\)
\(7\)	1.01249	−	2.44436i	0.144641	−	0.349194i
							\(11\)	2.47289	+	1.02430i	0.224808	+	0.0931185i	0.492244	−	0.870457i	\(-0.336177\pi\)
−0.267437	+	0.963575i	\(0.586177\pi\)
\(13\)	−6.28727	+	15.1788i	−0.483636	+	1.16760i	0.474234	+	0.880399i	\(0.342725\pi\)
							−0.957870	+	0.287202i	\(0.907275\pi\)
\(17\)	−5.44881	−	13.1546i	−0.320518	−	0.773800i	−0.999224	−	0.0393895i	\(-0.987459\pi\)
							0.678705	−	0.734411i	\(-0.262541\pi\)
\(19\)	−3.25747	−	7.86422i	−0.171446	−	0.413907i	0.814679	−	0.579912i	\(-0.196913\pi\)
							−0.986125	+	0.166005i	\(0.946913\pi\)
\(23\)			29.9538i			1.30234i	0.758932	+	0.651170i	\(0.225721\pi\)
							−0.758932	+	0.651170i	\(0.774279\pi\)
\(29\)	−13.4830	+	32.5508i	−0.464930	+	1.12244i	0.501418	+	0.865205i	\(0.332812\pi\)
							−0.966349	+	0.257236i	\(0.917188\pi\)
\(31\)		−	5.39996i		−	0.174192i	−0.996200	−	0.0870961i	\(-0.972241\pi\)
							0.996200	−	0.0870961i	\(-0.0277587\pi\)
\(37\)	−7.74836			−0.209415			−0.104708	−	0.994503i	\(-0.533391\pi\)
							−0.104708	+	0.994503i	\(0.533391\pi\)
\(41\)	−34.4188	−	22.2789i	−0.839482	−	0.543387i
							\(43\)	17.1671	−	17.1671i	0.399235	−	0.399235i	−0.478728	−	0.877963i	\(-0.658902\pi\)
0.877963	+	0.478728i	\(0.158902\pi\)
\(47\)	−27.6157	−	66.6702i	−0.587568	−	1.41851i	−0.885821	−	0.464027i	\(-0.846404\pi\)
							0.298253	−	0.954487i	\(-0.403596\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.m.a.85.9	✓	168
41.14	odd	8	inner	287.3.m.a.260.9	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.m.a.85.9	✓	168	1.1	even	1	trivial
287.3.m.a.260.9	yes	168	41.14	odd	8	inner