Embedded newform 287.3.t.a.20.4

• Label: 287.3.t.a.20.4
• Level: $287$
• Weight: $3$
• Character: 287.20
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.t (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			20.4
Character	\(\chi\)	\(=\)	287.20
Dual form			287.3.t.a.244.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.56829 - 1.15941i) q^{2} +(0.388163 + 0.388163i) q^{3} +(8.15243 + 5.92309i) q^{4} +(4.84900 + 3.52300i) q^{5} +(-0.935040 - 1.83512i) q^{6} +(6.42417 - 2.78029i) q^{7} +(-13.4017 - 18.4458i) q^{8} -8.69866i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 20 q^{2} + 196 q^{4} - 8 q^{7} - 20 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.56829	−	1.15941i	−1.78415	−	0.579705i	−0.784942	−	0.619569i	\(-0.787307\pi\)
							−0.999205	+	0.0398641i	\(0.987307\pi\)
\(3\)	0.388163	+	0.388163i	0.129388	+	0.129388i	0.768835	−	0.639447i	\(-0.220837\pi\)
							−0.639447	+	0.768835i	\(0.720837\pi\)
\(5\)	4.84900	+	3.52300i	0.969799	+	0.704600i	0.955406	−	0.295296i	\(-0.0954183\pi\)
							0.0143934	+	0.999896i	\(0.495418\pi\)
\(7\)	6.42417	−	2.78029i	0.917739	−	0.397184i
							\(11\)	0.279541	−	1.76495i	0.0254128	−	0.160450i	−0.971718	−	0.236143i	\(-0.924117\pi\)
0.997131	+	0.0756924i	\(0.0241167\pi\)
\(13\)	2.57109	−	1.31003i	0.197776	−	0.100772i	−0.352298	−	0.935888i	\(-0.614600\pi\)
							0.550074	+	0.835116i	\(0.314600\pi\)
\(17\)	2.06678	−	13.0492i	0.121576	−	0.767598i	−0.849282	−	0.527940i	\(-0.822965\pi\)
							0.970857	−	0.239658i	\(-0.0770353\pi\)
\(19\)	20.8929	+	10.6454i	1.09962	+	0.560287i	0.907062	−	0.420998i	\(-0.138320\pi\)
							0.192563	+	0.981285i	\(0.438320\pi\)
\(23\)	−0.396288	+	1.21965i	−0.0172299	+	0.0530283i	−0.959302	−	0.282383i	\(-0.908875\pi\)
							0.942072	+	0.335411i	\(0.108875\pi\)
\(29\)	−26.3333	+	4.17079i	−0.908045	+	0.143820i	−0.592950	−	0.805240i	\(-0.702037\pi\)
							−0.315096	+	0.949060i	\(0.602037\pi\)
\(31\)	−30.0767	−	41.3970i	−0.970215	−	1.33539i	−0.941938	−	0.335787i	\(-0.890998\pi\)
							−0.0282774	−	0.999600i	\(-0.509002\pi\)
\(37\)	25.6294	+	18.6208i	0.692686	+	0.503266i	0.877542	−	0.479500i	\(-0.159182\pi\)
							−0.184856	+	0.982766i	\(0.559182\pi\)
\(41\)	−22.9113	+	34.0011i	−0.558812	+	0.829295i
							\(43\)	67.8652	+	22.0507i	1.57826	+	0.512808i	0.961608	−	0.274428i	\(-0.0884885\pi\)
0.616652	+	0.787236i	\(0.288489\pi\)
\(47\)	−39.3831	−	77.2936i	−0.837937	−	1.64454i	−0.762132	−	0.647422i	\(-0.775847\pi\)
							−0.0758055	−	0.997123i	\(-0.524153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.t.a.20.4	yes	432
7.6	odd	2	inner	287.3.t.a.20.3	✓	432
41.39	even	20	inner	287.3.t.a.244.3	yes	432
287.244	odd	20	inner	287.3.t.a.244.4	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.t.a.20.3	✓	432	7.6	odd	2	inner
287.3.t.a.20.4	yes	432	1.1	even	1	trivial
287.3.t.a.244.3	yes	432	41.39	even	20	inner
287.3.t.a.244.4	yes	432	287.244	odd	20	inner