Embedded newform 287.3.t.a.20.20

• Label: 287.3.t.a.20.20
• 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			20.20
Character	\(\chi\)	\(=\)	287.20
Dual form			287.3.t.a.244.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.56087 - 0.507158i) q^{2} +(3.90043 + 3.90043i) q^{3} +(-1.05696 - 0.767924i) q^{4} +(1.95861 + 1.42301i) q^{5} +(-4.10994 - 8.06621i) q^{6} +(-3.77633 - 5.89401i) q^{7} +(5.11900 + 7.04570i) q^{8} +21.4267i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 20 q^{2} + 196 q^{4} - 8 q^{7} - 20 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.56087	−	0.507158i	−0.780436	−	0.253579i	−0.108410	−	0.994106i	\(-0.534576\pi\)
							−0.672026	+	0.740527i	\(0.734576\pi\)
\(3\)	3.90043	+	3.90043i	1.30014	+	1.30014i	0.928292	+	0.371851i	\(0.121277\pi\)
							0.371851	+	0.928292i	\(0.378723\pi\)
\(5\)	1.95861	+	1.42301i	0.391721	+	0.284602i	0.766161	−	0.642649i	\(-0.222165\pi\)
							−0.374439	+	0.927251i	\(0.622165\pi\)
\(7\)	−3.77633	−	5.89401i	−0.539476	−	0.842001i
							\(11\)	−0.808733	+	5.10614i	−0.0735212	+	0.464194i	0.923270	+	0.384151i	\(0.125506\pi\)
−0.996791	+	0.0800433i	\(0.974494\pi\)
\(13\)	1.15259	−	0.587273i	0.0886606	−	0.0451748i	−0.409098	−	0.912490i	\(-0.634157\pi\)
							0.497759	+	0.867316i	\(0.334157\pi\)
\(17\)	−3.45609	+	21.8209i	−0.203300	+	1.28358i	0.649105	+	0.760699i	\(0.275144\pi\)
							−0.852404	+	0.522884i	\(0.824856\pi\)
\(19\)	29.7128	+	15.1395i	1.56383	+	0.796813i	0.999585	−	0.0287915i	\(-0.00916590\pi\)
							0.564249	+	0.825605i	\(0.309166\pi\)
\(23\)	−5.77563	+	17.7755i	−0.251114	+	0.772850i	0.743456	+	0.668785i	\(0.233185\pi\)
							−0.994570	+	0.104065i	\(0.966815\pi\)
\(29\)	−41.7039	+	6.60525i	−1.43807	+	0.227767i	−0.826282	−	0.563257i	\(-0.809548\pi\)
							−0.611785	+	0.791024i	\(0.709548\pi\)
\(31\)	−18.6919	−	25.7273i	−0.602966	−	0.829912i	0.393010	−	0.919534i	\(-0.371434\pi\)
							−0.995976	+	0.0896227i	\(0.971434\pi\)
\(37\)	45.4009	+	32.9857i	1.22705	+	0.891505i	0.996666	−	0.0815945i	\(-0.0260012\pi\)
							0.230386	+	0.973099i	\(0.426001\pi\)
\(41\)	5.18160	+	40.6713i	0.126380	+	0.991982i
							\(43\)	−32.3495	−	10.5110i	−0.752315	−	0.244442i	−0.0923380	−	0.995728i	\(-0.529434\pi\)
−0.659977	+	0.751286i	\(0.729434\pi\)
\(47\)	−0.912603	−	1.79109i	−0.0194171	−	0.0381082i	0.881094	−	0.472941i	\(-0.156808\pi\)
							−0.900511	+	0.434833i	\(0.856808\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.t.a.20.20	yes	432
7.6	odd	2	inner	287.3.t.a.20.19	✓	432
41.39	even	20	inner	287.3.t.a.244.19	yes	432
287.244	odd	20	inner	287.3.t.a.244.20	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.t.a.20.19	✓	432	7.6	odd	2	inner
287.3.t.a.20.20	yes	432	1.1	even	1	trivial
287.3.t.a.244.19	yes	432	41.39	even	20	inner
287.3.t.a.244.20	yes	432	287.244	odd	20	inner