Embedded newform 287.3.i.a.40.9

• Label: 287.3.i.a.40.9
• Level: $287$
• Weight: $3$
• Character: 287.40
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(54\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			40.9
Character	\(\chi\)	\(=\)	287.40
Dual form			287.3.i.a.122.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50161 + 2.60086i) q^{2} +(-0.735502 - 1.27393i) q^{3} +(-2.50965 - 4.34684i) q^{4} +(0.0228039 + 0.0131658i) q^{5} +4.41774 q^{6} +(-1.47843 + 6.84209i) q^{7} +3.06116 q^{8} +(3.41807 - 5.92028i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 106 q^{4} - 6 q^{5} + 20 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50161	+	2.60086i	−0.750803	+	1.30043i	0.196630	+	0.980478i	\(0.437000\pi\)
							−0.947434	+	0.319952i	\(0.896333\pi\)
\(3\)	−0.735502	−	1.27393i	−0.245167	−	0.424642i	0.717011	−	0.697062i	\(-0.245510\pi\)
							−0.962179	+	0.272419i	\(0.912176\pi\)
\(5\)	0.0228039	+	0.0131658i	0.00456077	+	0.00263316i	0.502279	−	0.864706i	\(-0.332495\pi\)
							−0.497718	+	0.867339i	\(0.665829\pi\)
\(7\)	−1.47843	+	6.84209i	−0.211204	+	0.977442i
							\(11\)	−2.09053	+	1.20697i	−0.190048	+	0.109724i	−0.592005	−	0.805934i	\(-0.701663\pi\)
0.401957	+	0.915658i	\(0.368330\pi\)
\(13\)	−16.1064			−1.23895			−0.619475	−	0.785016i	\(-0.712655\pi\)
							−0.619475	+	0.785016i	\(0.712655\pi\)
\(17\)	0.400975	+	0.694508i	0.0235867	+	0.0408534i	0.877578	−	0.479434i	\(-0.159158\pi\)
							−0.853991	+	0.520288i	\(0.825825\pi\)
\(19\)	5.22571	−	9.05120i	0.275038	−	0.476379i	−0.695107	−	0.718906i	\(-0.744643\pi\)
							0.970145	+	0.242527i	\(0.0779764\pi\)
\(23\)	12.2711	−	21.2541i	0.533524	−	0.924091i	−0.465709	−	0.884938i	\(-0.654201\pi\)
							0.999233	−	0.0391530i	\(-0.0124660\pi\)
\(29\)		−	19.7196i		−	0.679985i	−0.940428	−	0.339993i	\(-0.889575\pi\)
							0.940428	−	0.339993i	\(-0.110425\pi\)
\(31\)	41.7855	−	24.1249i	1.34792	−	0.778222i	0.359965	−	0.932966i	\(-0.382789\pi\)
							0.987955	+	0.154744i	\(0.0494552\pi\)
\(37\)	−1.49673	+	2.59242i	−0.0404522	+	0.0700653i	−0.885543	−	0.464558i	\(-0.846213\pi\)
							0.845090	+	0.534623i	\(0.179547\pi\)
\(41\)	29.8032	+	28.1562i	0.726907	+	0.686736i
							\(43\)	−35.7793			−0.832077			−0.416039	−	0.909347i	\(-0.636582\pi\)
−0.416039	+	0.909347i	\(0.636582\pi\)
\(47\)	34.6585	−	60.0303i	0.737415	−	1.27724i	−0.216241	−	0.976340i	\(-0.569380\pi\)
							0.953656	−	0.300900i	\(-0.0972870\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.i.a.40.9	✓	108
7.3	odd	6	inner	287.3.i.a.122.10	yes	108
41.40	even	2	inner	287.3.i.a.40.10	yes	108
287.122	odd	6	inner	287.3.i.a.122.9	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.i.a.40.9	✓	108	1.1	even	1	trivial
287.3.i.a.40.10	yes	108	41.40	even	2	inner
287.3.i.a.122.9	yes	108	287.122	odd	6	inner
287.3.i.a.122.10	yes	108	7.3	odd	6	inner