Embedded newform 287.3.m.a.85.15

• Label: 287.3.m.a.85.15
• 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			85.15
Character	\(\chi\)	\(=\)	287.85
Dual form			287.3.m.a.260.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04509 + 1.04509i) q^{2} +(1.67777 - 4.05049i) q^{3} +1.81559i q^{4} +(0.878824 + 0.878824i) q^{5} +(2.47970 + 5.98653i) q^{6} +(1.01249 - 2.44436i) q^{7} +(-6.07779 - 6.07779i) q^{8} +(-7.22761 - 7.22761i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 8 q^{2} - 16 q^{3} + 24 q^{6} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04509	+	1.04509i	−0.522544	+	0.522544i	−0.918339	−	0.395795i	\(-0.870469\pi\)
							0.395795	+	0.918339i	\(0.370469\pi\)
\(3\)	1.67777	−	4.05049i	0.559256	−	1.35016i	−0.351100	−	0.936338i	\(-0.614192\pi\)
							0.910356	−	0.413826i	\(-0.135808\pi\)
\(5\)	0.878824	+	0.878824i	0.175765	+	0.175765i	0.789507	−	0.613742i	\(-0.210336\pi\)
							−0.613742	+	0.789507i	\(0.710336\pi\)
\(7\)	1.01249	−	2.44436i	0.144641	−	0.349194i
							\(11\)	12.0408	+	4.98746i	1.09462	+	0.453405i	0.855615	−	0.517614i	\(-0.173180\pi\)
0.239003	+	0.971019i	\(0.423180\pi\)
\(13\)	6.13446	−	14.8099i	0.471881	−	1.13922i	−0.491450	−	0.870906i	\(-0.663533\pi\)
							0.963331	−	0.268316i	\(-0.0864672\pi\)
\(17\)	−1.59707	−	3.85567i	−0.0939454	−	0.226804i	0.869921	−	0.493192i	\(-0.164170\pi\)
							−0.963866	+	0.266388i	\(0.914170\pi\)
\(19\)	3.78124	+	9.12873i	0.199013	+	0.480460i	0.991607	−	0.129290i	\(-0.0412699\pi\)
							−0.792594	+	0.609750i	\(0.791270\pi\)
\(23\)			7.84121i			0.340922i	0.985364	+	0.170461i	\(0.0545257\pi\)
							−0.985364	+	0.170461i	\(0.945474\pi\)
\(29\)	21.1051	−	50.9523i	0.727763	−	1.75698i	0.0778524	−	0.996965i	\(-0.475194\pi\)
							0.649911	−	0.760011i	\(-0.274806\pi\)
\(31\)			3.76088i			0.121319i	0.998159	+	0.0606594i	\(0.0193203\pi\)
							−0.998159	+	0.0606594i	\(0.980680\pi\)
\(37\)	−41.7744			−1.12904			−0.564519	−	0.825420i	\(-0.690939\pi\)
							−0.564519	+	0.825420i	\(0.690939\pi\)
\(41\)	20.9236	+	35.2591i	0.510332	+	0.859977i
							\(43\)	40.7855	−	40.7855i	0.948500	−	0.948500i	−0.0502371	−	0.998737i	\(-0.515998\pi\)
0.998737	+	0.0502371i	\(0.0159977\pi\)
\(47\)	−1.46704	−	3.54175i	−0.0312136	−	0.0753563i	0.907504	−	0.420043i	\(-0.137985\pi\)
							−0.938718	+	0.344687i	\(0.887985\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.m.a.85.15	✓	168
41.14	odd	8	inner	287.3.m.a.260.15	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.m.a.85.15	✓	168	1.1	even	1	trivial
287.3.m.a.260.15	yes	168	41.14	odd	8	inner