Embedded newform 287.3.t.a.20.7

• Label: 287.3.t.a.20.7
• 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			20.7
Character	\(\chi\)	\(=\)	287.20
Dual form			287.3.t.a.244.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.91771 - 0.948022i) q^{2} +(-1.59164 - 1.59164i) q^{3} +(4.37823 + 3.18097i) q^{4} +(2.45492 + 1.78360i) q^{5} +(3.13503 + 6.15284i) q^{6} +(-4.48908 + 5.37104i) q^{7} +(-2.54581 - 3.50401i) q^{8} -3.93339i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 20 q^{2} + 196 q^{4} - 8 q^{7} - 20 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.91771	−	0.948022i	−1.45886	−	0.474011i	−0.531136	−	0.847287i	\(-0.678235\pi\)
							−0.927721	+	0.373275i	\(0.878235\pi\)
\(3\)	−1.59164	−	1.59164i	−0.530545	−	0.530545i	0.390189	−	0.920735i	\(-0.372410\pi\)
							−0.920735	+	0.390189i	\(0.872410\pi\)
\(5\)	2.45492	+	1.78360i	0.490984	+	0.356721i	0.805563	−	0.592511i	\(-0.201863\pi\)
							−0.314579	+	0.949231i	\(0.601863\pi\)
\(7\)	−4.48908	+	5.37104i	−0.641298	+	0.767292i
							\(11\)	−1.18707	+	7.49484i	−0.107915	+	0.681349i	0.873118	+	0.487509i	\(0.162094\pi\)
−0.981033	+	0.193840i	\(0.937906\pi\)
\(13\)	3.22377	−	1.64259i	0.247983	−	0.126353i	−0.325582	−	0.945514i	\(-0.605560\pi\)
							0.573565	+	0.819160i	\(0.305560\pi\)
\(17\)	2.75604	−	17.4010i	0.162120	−	1.02359i	−0.763686	−	0.645588i	\(-0.776612\pi\)
							0.925806	−	0.377998i	\(-0.123388\pi\)
\(19\)	8.82036	+	4.49420i	0.464229	+	0.236537i	0.670430	−	0.741973i	\(-0.266110\pi\)
							−0.206200	+	0.978510i	\(0.566110\pi\)
\(23\)	4.95582	−	15.2525i	0.215471	−	0.663150i	−0.783649	−	0.621204i	\(-0.786644\pi\)
							0.999120	−	0.0419468i	\(-0.0133560\pi\)
\(29\)	−18.4567	+	2.92325i	−0.636438	+	0.100802i	−0.466312	−	0.884621i	\(-0.654417\pi\)
							−0.170126	+	0.985422i	\(0.554417\pi\)
\(31\)	3.43178	+	4.72345i	0.110703	+	0.152369i	0.860773	−	0.508989i	\(-0.169980\pi\)
							−0.750071	+	0.661358i	\(0.769980\pi\)
\(37\)	−46.5367	−	33.8109i	−1.25775	−	0.913808i	−0.259104	−	0.965849i	\(-0.583427\pi\)
							−0.998645	+	0.0520409i	\(0.983427\pi\)
\(41\)	−9.46713	+	39.8920i	−0.230906	+	0.972976i
							\(43\)	−61.3821	−	19.9443i	−1.42749	−	0.463820i	−0.509518	−	0.860460i	\(-0.670176\pi\)
−0.917974	+	0.396640i	\(0.870176\pi\)
\(47\)	−0.453080	−	0.889219i	−0.00964000	−	0.0189196i	0.886136	−	0.463425i	\(-0.153380\pi\)
							−0.895776	+	0.444505i	\(0.853380\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.t.a.20.7	✓	432
7.6	odd	2	inner	287.3.t.a.20.8	yes	432
41.39	even	20	inner	287.3.t.a.244.8	yes	432
287.244	odd	20	inner	287.3.t.a.244.7	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.t.a.20.7	✓	432	1.1	even	1	trivial
287.3.t.a.20.8	yes	432	7.6	odd	2	inner
287.3.t.a.244.7	yes	432	287.244	odd	20	inner
287.3.t.a.244.8	yes	432	41.39	even	20	inner