Embedded newform 287.3.t.a.20.1

• Label: 287.3.t.a.20.1
• 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			20.1
Character	\(\chi\)	\(=\)	287.20
Dual form			287.3.t.a.244.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.60896 - 1.17262i) q^{2} +(-3.31582 - 3.31582i) q^{3} +(8.41351 + 6.11277i) q^{4} +(-1.82266 - 1.32424i) q^{5} +(8.07846 + 15.8549i) q^{6} +(-4.25521 - 5.55816i) q^{7} +(-14.2742 - 19.6468i) q^{8} +12.9893i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 20 q^{2} + 196 q^{4} - 8 q^{7} - 20 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.60896	−	1.17262i	−1.80448	−	0.586312i	−0.804511	−	0.593937i	\(-0.797573\pi\)
							−0.999971	+	0.00762565i	\(0.997573\pi\)
\(3\)	−3.31582	−	3.31582i	−1.10527	−	1.10527i	−0.993763	−	0.111508i	\(-0.964432\pi\)
							−0.111508	−	0.993763i	\(-0.535568\pi\)
\(5\)	−1.82266	−	1.32424i	−0.364531	−	0.264848i	0.390408	−	0.920642i	\(-0.372334\pi\)
							−0.754940	+	0.655794i	\(0.772334\pi\)
\(7\)	−4.25521	−	5.55816i	−0.607887	−	0.794023i
							\(11\)	−1.95136	+	12.3204i	−0.177397	+	1.12004i	0.724879	+	0.688877i	\(0.241896\pi\)
−0.902275	+	0.431161i	\(0.858104\pi\)
\(13\)	15.1744	−	7.73176i	1.16726	−	0.594751i	0.240594	−	0.970626i	\(-0.422658\pi\)
							0.926670	+	0.375875i	\(0.122658\pi\)
\(17\)	−2.97046	+	18.7548i	−0.174733	+	1.10322i	0.731934	+	0.681375i	\(0.238618\pi\)
							−0.906668	+	0.421846i	\(0.861382\pi\)
\(19\)	24.7682	+	12.6201i	1.30359	+	0.664213i	0.961332	−	0.275392i	\(-0.0888076\pi\)
							0.342260	+	0.939605i	\(0.388808\pi\)
\(23\)	−10.2029	+	31.4014i	−0.443605	+	1.36528i	0.440401	+	0.897801i	\(0.354836\pi\)
							−0.884006	+	0.467475i	\(0.845164\pi\)
\(29\)	37.9655	−	6.01314i	1.30916	−	0.207350i	0.537431	−	0.843308i	\(-0.319395\pi\)
							0.771724	+	0.635958i	\(0.219395\pi\)
\(31\)	−3.95544	−	5.44419i	−0.127595	−	0.175619i	0.740440	−	0.672122i	\(-0.234617\pi\)
							−0.868035	+	0.496503i	\(0.834617\pi\)
\(37\)	−10.3666	−	7.53177i	−0.280178	−	0.203561i	0.438817	−	0.898576i	\(-0.355398\pi\)
							−0.718995	+	0.695015i	\(0.755398\pi\)
\(41\)	−7.13038	−	40.3752i	−0.173912	−	0.984761i
							\(43\)	−14.6445	−	4.75828i	−0.340569	−	0.110658i	0.133739	−	0.991017i	\(-0.457302\pi\)
−0.474308	+	0.880359i	\(0.657302\pi\)
\(47\)	−4.20449	−	8.25178i	−0.0894573	−	0.175570i	0.841943	−	0.539567i	\(-0.181412\pi\)
							−0.931400	+	0.363997i	\(0.881412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.t.a.20.1	✓	432
7.6	odd	2	inner	287.3.t.a.20.2	yes	432
41.39	even	20	inner	287.3.t.a.244.2	yes	432
287.244	odd	20	inner	287.3.t.a.244.1	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.t.a.20.1	✓	432	1.1	even	1	trivial
287.3.t.a.20.2	yes	432	7.6	odd	2	inner
287.3.t.a.244.1	yes	432	287.244	odd	20	inner
287.3.t.a.244.2	yes	432	41.39	even	20	inner