Embedded newform 287.2.z.a.4.7

• Label: 287.2.z.a.4.7
• 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			4.7
Character	\(\chi\)	\(=\)	287.4
Dual form			287.2.z.a.72.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44506 - 0.643384i) q^{2} +(1.73343 + 1.00080i) q^{3} +(0.336004 + 0.373171i) q^{4} +(-3.23302 + 0.687199i) q^{5} +(-1.86102 - 2.56148i) q^{6} +(0.537448 - 2.59059i) q^{7} +(0.732162 + 2.25336i) q^{8} +(0.503196 + 0.871562i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} + 21 q^{4} + q^{5} - 40 q^{6} - 10 q^{7} - 8 q^{8} + 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44506	−	0.643384i	−1.02181	−	0.454941i	−0.173725	−	0.984794i	\(-0.555580\pi\)
							−0.848089	+	0.529853i	\(0.822247\pi\)
\(3\)	1.73343	+	1.00080i	1.00080	+	0.577811i	0.908485	−	0.417918i	\(-0.137240\pi\)
							0.0923143	+	0.995730i	\(0.470574\pi\)
\(5\)	−3.23302	+	0.687199i	−1.44585	+	0.307325i	−0.862979	−	0.505241i	\(-0.831404\pi\)
							−0.582870	+	0.812565i	\(0.698070\pi\)
\(7\)	0.537448	−	2.59059i	0.203136	−	0.979151i
							\(11\)	0.717142	−	3.37389i	0.216226	−	1.01727i	−0.727390	−	0.686224i	\(-0.759267\pi\)
0.943616	−	0.331041i	\(-0.107400\pi\)
\(13\)	−3.38964	−	4.66544i	−0.940116	−	1.29396i	−0.955780	−	0.294082i	\(-0.904986\pi\)
							0.0156639	−	0.999877i	\(-0.495014\pi\)
\(17\)	0.350179	−	1.64746i	0.0849309	−	0.399568i	−0.915062	−	0.403313i	\(-0.867859\pi\)
							0.999993	+	0.00374463i	\(0.00119195\pi\)
\(19\)	4.43294	+	0.465921i	1.01699	+	0.106890i	0.598325	−	0.801253i	\(-0.295833\pi\)
							0.418661	+	0.908143i	\(0.362500\pi\)
\(23\)	2.56222	+	1.14077i	0.534260	+	0.237868i	0.656089	−	0.754683i	\(-0.272209\pi\)
							−0.121829	+	0.992551i	\(0.538876\pi\)
\(29\)	−6.89772	−	2.24121i	−1.28087	−	0.416181i	−0.411987	−	0.911190i	\(-0.635165\pi\)
							−0.868888	+	0.495008i	\(0.835165\pi\)
\(31\)	0.966237	+	0.205380i	0.173541	+	0.0368873i	0.293863	−	0.955848i	\(-0.405059\pi\)
							−0.120321	+	0.992735i	\(0.538393\pi\)
\(37\)	−9.96799	+	2.11876i	−1.63873	+	0.348322i	−0.932921	−	0.360082i	\(-0.882749\pi\)
							−0.705807	+	0.708404i	\(0.749415\pi\)
\(41\)	−4.88581	+	4.13870i	−0.763035	+	0.646357i
							\(43\)	10.0389	−	7.29366i	1.53091	−	1.11227i	0.575184	−	0.818024i	\(-0.304930\pi\)
0.955729	−	0.294249i	\(-0.0950696\pi\)
\(47\)	−2.15477	+	4.83970i	−0.314306	+	0.705943i	−0.999754	−	0.0221788i	\(-0.992940\pi\)
							0.685448	+	0.728122i	\(0.259606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.z.a.4.7	✓	208
7.2	even	3	inner	287.2.z.a.86.20	yes	208
41.31	even	10	inner	287.2.z.a.277.20	yes	208
287.72	even	30	inner	287.2.z.a.72.7	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.z.a.4.7	✓	208	1.1	even	1	trivial
287.2.z.a.72.7	yes	208	287.72	even	30	inner
287.2.z.a.86.20	yes	208	7.2	even	3	inner
287.2.z.a.277.20	yes	208	41.31	even	10	inner