Embedded newform 287.3.m.a.85.14

• Label: 287.3.m.a.85.14
• 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			85.14
Character	\(\chi\)	\(=\)	287.85
Dual form			287.3.m.a.260.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20352 + 1.20352i) q^{2} +(2.13250 - 5.14832i) q^{3} +1.10308i q^{4} +(0.770141 + 0.770141i) q^{5} +(3.62959 + 8.76261i) q^{6} +(-1.01249 + 2.44436i) q^{7} +(-6.14166 - 6.14166i) q^{8} +(-15.5936 - 15.5936i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 8 q^{2} - 16 q^{3} + 24 q^{6} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20352	+	1.20352i	−0.601760	+	0.601760i	−0.940779	−	0.339020i	\(-0.889905\pi\)
							0.339020	+	0.940779i	\(0.389905\pi\)
\(3\)	2.13250	−	5.14832i	0.710834	−	1.71611i	0.0129287	−	0.999916i	\(-0.495885\pi\)
							0.697906	−	0.716190i	\(-0.254115\pi\)
\(5\)	0.770141	+	0.770141i	0.154028	+	0.154028i	0.779914	−	0.625886i	\(-0.215263\pi\)
							−0.625886	+	0.779914i	\(0.715263\pi\)
\(7\)	−1.01249	+	2.44436i	−0.144641	+	0.349194i
							\(11\)	−19.3428	−	8.01206i	−1.75844	−	0.728369i	−0.996762	−	0.0804111i	\(-0.974377\pi\)
−0.761676	−	0.647958i	\(-0.775623\pi\)
\(13\)	−2.60975	+	6.30050i	−0.200750	+	0.484654i	−0.991908	−	0.126958i	\(-0.959479\pi\)
							0.791158	+	0.611612i	\(0.209479\pi\)
\(17\)	−5.45200	−	13.1623i	−0.320706	−	0.774252i	−0.999213	−	0.0396586i	\(-0.987373\pi\)
							0.678508	−	0.734593i	\(-0.262627\pi\)
\(19\)	10.0778	+	24.3299i	0.530409	+	1.28052i	0.931253	+	0.364374i	\(0.118717\pi\)
							−0.400843	+	0.916147i	\(0.631283\pi\)
\(23\)		−	19.2152i		−	0.835442i	−0.908575	−	0.417721i	\(-0.862829\pi\)
							0.908575	−	0.417721i	\(-0.137171\pi\)
\(29\)	−10.3641	+	25.0212i	−0.357383	+	0.862799i	0.638283	+	0.769802i	\(0.279645\pi\)
							−0.995666	+	0.0929974i	\(0.970355\pi\)
\(31\)		−	32.1258i		−	1.03632i	−0.855285	−	0.518158i	\(-0.826618\pi\)
							0.855285	−	0.518158i	\(-0.173382\pi\)
\(37\)	−64.0832			−1.73198			−0.865989	−	0.500063i	\(-0.833310\pi\)
							−0.865989	+	0.500063i	\(0.833310\pi\)
\(41\)	40.3681	−	7.17063i	0.984587	−	0.174893i
							\(43\)	−21.6512	+	21.6512i	−0.503517	+	0.503517i	−0.912529	−	0.409012i	\(-0.865873\pi\)
0.409012	+	0.912529i	\(0.365873\pi\)
\(47\)	1.26676	+	3.05823i	0.0269524	+	0.0650688i	0.936783	−	0.349912i	\(-0.113788\pi\)
							−0.909830	+	0.414980i	\(0.863788\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.m.a.85.14	✓	168
41.14	odd	8	inner	287.3.m.a.260.14	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.m.a.85.14	✓	168	1.1	even	1	trivial
287.3.m.a.260.14	yes	168	41.14	odd	8	inner