Embedded newform 287.2.z.a.4.5

• Label: 287.2.z.a.4.5
• Level: $287$
• Weight: $2$
• Character: 287.4
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.z (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.5
Character	\(\chi\)	\(=\)	287.4
Dual form			287.2.z.a.72.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85683 - 0.826713i) q^{2} +(-2.83125 - 1.63462i) q^{3} +(1.42609 + 1.58384i) q^{4} +(2.48571 - 0.528355i) q^{5} +(3.90578 + 5.37584i) q^{6} +(-2.56260 - 0.658072i) q^{7} +(-0.0824439 - 0.253736i) q^{8} +(3.84399 + 6.65798i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} + 21 q^{4} + q^{5} - 40 q^{6} - 10 q^{7} - 8 q^{8} + 84 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85683	−	0.826713i	−1.31298	−	0.584574i	−0.373640	−	0.927574i	\(-0.621890\pi\)
							−0.939335	+	0.343000i	\(0.888557\pi\)
\(3\)	−2.83125	−	1.63462i	−1.63462	−	0.943750i	−0.982641	−	0.185516i	\(-0.940604\pi\)
							−0.651982	−	0.758234i	\(-0.726062\pi\)
\(5\)	2.48571	−	0.528355i	1.11165	−	0.236287i	0.384740	−	0.923025i	\(-0.374291\pi\)
							0.726905	+	0.686738i	\(0.240958\pi\)
\(7\)	−2.56260	−	0.658072i	−0.968573	−	0.248728i
							\(11\)	−0.686029	+	3.22751i	−0.206845	+	0.973131i	0.745121	+	0.666929i	\(0.232392\pi\)
−0.951967	+	0.306202i	\(0.900942\pi\)
\(13\)	2.83181	+	3.89766i	0.785404	+	1.08102i	0.994665	+	0.103156i	\(0.0328939\pi\)
							−0.209262	+	0.977860i	\(0.567106\pi\)
\(17\)	0.328530	−	1.54561i	0.0796803	−	0.374867i	−0.920181	−	0.391492i	\(-0.871959\pi\)
							0.999862	+	0.0166256i	\(0.00529235\pi\)
\(19\)	2.60501	+	0.273797i	0.597630	+	0.0628134i	0.398514	−	0.917162i	\(-0.369526\pi\)
							0.199116	+	0.979976i	\(0.436193\pi\)
\(23\)	−1.21928	−	0.542859i	−0.254238	−	0.113194i	0.275661	−	0.961255i	\(-0.411103\pi\)
							−0.529899	+	0.848061i	\(0.677770\pi\)
\(29\)	−3.11027	−	1.01059i	−0.577563	−	0.187662i	0.00564564	−	0.999984i	\(-0.498203\pi\)
							−0.583208	+	0.812323i	\(0.698203\pi\)
\(31\)	9.96329	+	2.11776i	1.78946	+	0.380362i	0.978745	−	0.205079i	\(-0.0657452\pi\)
							0.810715	+	0.585441i	\(0.199079\pi\)
\(37\)	4.74119	−	1.00777i	0.779446	−	0.165676i	0.199027	−	0.979994i	\(-0.436222\pi\)
							0.580420	+	0.814318i	\(0.302889\pi\)
\(41\)	−5.70208	+	2.91311i	−0.890516	+	0.454952i
							\(43\)	0.611500	−	0.444281i	0.0932529	−	0.0677522i	−0.540182	−	0.841548i	\(-0.681645\pi\)
0.633435	+	0.773796i	\(0.281645\pi\)
\(47\)	−1.08373	+	2.43410i	−0.158078	+	0.355050i	−0.975170	−	0.221458i	\(-0.928918\pi\)
							0.817091	+	0.576508i	\(0.195585\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.z.a.4.5	✓	208
7.2	even	3	inner	287.2.z.a.86.22	yes	208
41.31	even	10	inner	287.2.z.a.277.22	yes	208
287.72	even	30	inner	287.2.z.a.72.5	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.z.a.4.5	✓	208	1.1	even	1	trivial
287.2.z.a.72.5	yes	208	287.72	even	30	inner
287.2.z.a.86.22	yes	208	7.2	even	3	inner
287.2.z.a.277.22	yes	208	41.31	even	10	inner