Embedded newform 287.3.m.a.85.11

• Label: 287.3.m.a.85.11
• 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			85.11
Character	\(\chi\)	\(=\)	287.85
Dual form			287.3.m.a.260.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.56178 + 1.56178i) q^{2} +(-1.33869 + 3.23188i) q^{3} -0.878306i q^{4} +(-2.93781 - 2.93781i) q^{5} +(-2.95674 - 7.13821i) q^{6} +(1.01249 - 2.44436i) q^{7} +(-4.87540 - 4.87540i) q^{8} +(-2.28898 - 2.28898i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 8 q^{2} - 16 q^{3} + 24 q^{6} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.56178	+	1.56178i	−0.780889	+	0.780889i	−0.979981	−	0.199092i	\(-0.936201\pi\)
							0.199092	+	0.979981i	\(0.436201\pi\)
\(3\)	−1.33869	+	3.23188i	−0.446229	+	1.07729i	0.527495	+	0.849558i	\(0.323131\pi\)
							−0.973724	+	0.227733i	\(0.926869\pi\)
\(5\)	−2.93781	−	2.93781i	−0.587563	−	0.587563i	0.349408	−	0.936971i	\(-0.386383\pi\)
							−0.936971	+	0.349408i	\(0.886383\pi\)
\(7\)	1.01249	−	2.44436i	0.144641	−	0.349194i
							\(11\)	4.46626	+	1.84999i	0.406024	+	0.168181i	0.576342	−	0.817208i	\(-0.304480\pi\)
−0.170318	+	0.985389i	\(0.554480\pi\)
\(13\)	6.92683	−	16.7229i	0.532833	−	1.28637i	−0.396806	−	0.917902i	\(-0.629881\pi\)
							0.929639	−	0.368471i	\(-0.120119\pi\)
\(17\)	2.72159	+	6.57050i	0.160094	+	0.386500i	0.983489	−	0.180968i	\(-0.0579230\pi\)
							−0.823395	+	0.567468i	\(0.807923\pi\)
\(19\)	−12.9076	−	31.1616i	−0.679346	−	1.64009i	−0.765211	−	0.643780i	\(-0.777365\pi\)
							0.0858650	−	0.996307i	\(-0.472635\pi\)
\(23\)			24.5998i			1.06956i	0.844992	+	0.534779i	\(0.179605\pi\)
							−0.844992	+	0.534779i	\(0.820395\pi\)
\(29\)	4.84009	−	11.6850i	0.166900	−	0.402931i	−0.818196	−	0.574940i	\(-0.805026\pi\)
							0.985096	+	0.172008i	\(0.0550256\pi\)
\(31\)			28.5476i			0.920890i	0.887688	+	0.460445i	\(0.152310\pi\)
							−0.887688	+	0.460445i	\(0.847690\pi\)
\(37\)	9.64523			0.260682			0.130341	−	0.991469i	\(-0.458393\pi\)
							0.130341	+	0.991469i	\(0.458393\pi\)
\(41\)	40.7438	−	4.57609i	0.993752	−	0.111612i
							\(43\)	7.52380	−	7.52380i	0.174972	−	0.174972i	−0.614188	−	0.789160i	\(-0.710516\pi\)
0.789160	+	0.614188i	\(0.210516\pi\)
\(47\)	−19.8219	−	47.8542i	−0.421742	−	1.01817i	−0.981834	−	0.189744i	\(-0.939234\pi\)
							0.560092	−	0.828430i	\(-0.310766\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.m.a.85.11	✓	168
41.14	odd	8	inner	287.3.m.a.260.11	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.m.a.85.11	✓	168	1.1	even	1	trivial
287.3.m.a.260.11	yes	168	41.14	odd	8	inner