Embedded newform 287.3.t.a.20.17

• Label: 287.3.t.a.20.17
• 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			20.17
Character	\(\chi\)	\(=\)	287.20
Dual form			287.3.t.a.244.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62424 - 0.527747i) q^{2} +(-2.43233 - 2.43233i) q^{3} +(-0.876434 - 0.636766i) q^{4} +(1.95820 + 1.42272i) q^{5} +(2.66703 + 5.23435i) q^{6} +(3.54135 - 6.03811i) q^{7} +(5.10282 + 7.02344i) q^{8} +2.83249i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 20 q^{2} + 196 q^{4} - 8 q^{7} - 20 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62424	−	0.527747i	−0.812119	−	0.263874i	−0.126624	−	0.991951i	\(-0.540414\pi\)
							−0.685495	+	0.728077i	\(0.740414\pi\)
\(3\)	−2.43233	−	2.43233i	−0.810778	−	0.810778i	0.173973	−	0.984750i	\(-0.444340\pi\)
							−0.984750	+	0.173973i	\(0.944340\pi\)
\(5\)	1.95820	+	1.42272i	0.391641	+	0.284544i	0.766127	−	0.642689i	\(-0.222181\pi\)
							−0.374487	+	0.927232i	\(0.622181\pi\)
\(7\)	3.54135	−	6.03811i	0.505908	−	0.862588i
							\(11\)	2.58268	−	16.3064i	0.234789	−	1.48240i	−0.535409	−	0.844593i	\(-0.679842\pi\)
0.770198	−	0.637805i	\(-0.220158\pi\)
\(13\)	16.2202	−	8.26459i	1.24771	−	0.635738i	0.299713	−	0.954029i	\(-0.403109\pi\)
							0.947993	+	0.318292i	\(0.103109\pi\)
\(17\)	0.652217	−	4.11794i	0.0383657	−	0.242232i	−0.961051	−	0.276370i	\(-0.910869\pi\)
							0.999417	+	0.0341380i	\(0.0108686\pi\)
\(19\)	15.2072	+	7.74843i	0.800377	+	0.407812i	0.805811	−	0.592173i	\(-0.201730\pi\)
							−0.00543424	+	0.999985i	\(0.501730\pi\)
\(23\)	−2.91227	+	8.96304i	−0.126620	+	0.389697i	−0.994193	−	0.107613i	\(-0.965679\pi\)
							0.867572	+	0.497311i	\(0.165679\pi\)
\(29\)	−42.0610	+	6.66181i	−1.45038	+	0.229717i	−0.831393	−	0.555685i	\(-0.812456\pi\)
							−0.618986	+	0.785402i	\(0.712456\pi\)
\(31\)	14.5331	+	20.0031i	0.468809	+	0.645260i	0.976306	−	0.216393i	\(-0.0694293\pi\)
							−0.507497	+	0.861653i	\(0.669429\pi\)
\(37\)	−24.4431	−	17.7589i	−0.660624	−	0.479971i	0.206250	−	0.978499i	\(-0.433874\pi\)
							−0.866874	+	0.498528i	\(0.833874\pi\)
\(41\)	9.16253	−	39.9631i	0.223476	−	0.974709i
							\(43\)	27.9604	+	9.08489i	0.650242	+	0.211276i	0.615521	−	0.788121i	\(-0.288946\pi\)
0.0347212	+	0.999397i	\(0.488946\pi\)
\(47\)	11.7888	+	23.1368i	0.250825	+	0.492272i	0.981748	−	0.190188i	\(-0.0609099\pi\)
							−0.730922	+	0.682461i	\(0.760910\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.t.a.20.17	✓	432
7.6	odd	2	inner	287.3.t.a.20.18	yes	432
41.39	even	20	inner	287.3.t.a.244.18	yes	432
287.244	odd	20	inner	287.3.t.a.244.17	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.t.a.20.17	✓	432	1.1	even	1	trivial
287.3.t.a.20.18	yes	432	7.6	odd	2	inner
287.3.t.a.244.17	yes	432	287.244	odd	20	inner
287.3.t.a.244.18	yes	432	41.39	even	20	inner