Embedded newform 287.2.s.a.18.9

• Label: 287.2.s.a.18.9
• Level: $287$
• Weight: $2$
• Character: 287.18
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.s (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			18.9
Character	\(\chi\)	\(=\)	287.18
Dual form			287.2.s.a.16.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.694207 + 0.770995i) q^{2} +(-0.378549 + 0.655666i) q^{3} +(0.0965469 + 0.918582i) q^{4} +(1.11713 + 0.497379i) q^{5} +(-0.242724 - 0.747028i) q^{6} +(2.49934 + 0.867915i) q^{7} +(-2.45392 - 1.78288i) q^{8} +(1.21340 + 2.10167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.694207	+	0.770995i	−0.490879	+	0.545176i	−0.936786	−	0.349903i	\(-0.886214\pi\)
							0.445907	+	0.895079i	\(0.352881\pi\)
\(3\)	−0.378549	+	0.655666i	−0.218555	+	0.378549i	−0.954367	−	0.298638i	\(-0.903468\pi\)
							0.735811	+	0.677187i	\(0.236801\pi\)
\(5\)	1.11713	+	0.497379i	0.499597	+	0.222435i	0.641034	−	0.767512i	\(-0.278506\pi\)
							−0.141437	+	0.989947i	\(0.545172\pi\)
\(7\)	2.49934	+	0.867915i	0.944663	+	0.328041i
							\(11\)	0.689343	−	0.306915i	0.207845	−	0.0925384i	−0.300174	−	0.953884i	\(-0.597045\pi\)
0.508019	+	0.861346i	\(0.330378\pi\)
\(13\)	−0.194596	−	0.598906i	−0.0539713	−	0.166107i	0.920437	−	0.390890i	\(-0.127833\pi\)
							−0.974409	+	0.224783i	\(0.927833\pi\)
\(17\)	−1.80135	+	0.802012i	−0.436891	+	0.194517i	−0.613383	−	0.789786i	\(-0.710192\pi\)
							0.176492	+	0.984302i	\(0.443525\pi\)
\(19\)	1.83705	−	0.390478i	0.421449	−	0.0895817i	0.00769473	−	0.999970i	\(-0.497551\pi\)
							0.413754	+	0.910389i	\(0.364217\pi\)
\(23\)	−4.39266	+	4.87855i	−0.915934	+	1.01725i	0.0838504	+	0.996478i	\(0.473278\pi\)
							−0.999784	+	0.0207693i	\(0.993388\pi\)
\(29\)	−1.71322	+	1.24473i	−0.318136	+	0.231140i	−0.735380	−	0.677655i	\(-0.762996\pi\)
							0.417243	+	0.908795i	\(0.362996\pi\)
\(31\)	0.844607	−	0.376043i	0.151696	−	0.0675394i	−0.329484	−	0.944161i	\(-0.606875\pi\)
							0.481180	+	0.876622i	\(0.340208\pi\)
\(37\)	4.31684	+	1.92198i	0.709684	+	0.315972i	0.729642	−	0.683829i	\(-0.239687\pi\)
							−0.0199583	+	0.999801i	\(0.506353\pi\)
\(41\)	1.79462	−	6.14649i	0.280273	−	0.959920i
							\(43\)	1.83509	+	5.64783i	0.279849	+	0.861286i	0.987896	+	0.155121i	\(0.0495766\pi\)
−0.708047	+	0.706166i	\(0.750423\pi\)
\(47\)	2.19490	−	2.43768i	0.320158	−	0.355572i	−0.561486	−	0.827486i	\(-0.689770\pi\)
							0.881644	+	0.471914i	\(0.156437\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.s.a.18.9	yes	208
7.2	even	3	inner	287.2.s.a.100.18	yes	208
41.16	even	5	inner	287.2.s.a.221.18	yes	208
287.16	even	15	inner	287.2.s.a.16.9	✓	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.9	✓	208	287.16	even	15	inner
287.2.s.a.18.9	yes	208	1.1	even	1	trivial
287.2.s.a.100.18	yes	208	7.2	even	3	inner
287.2.s.a.221.18	yes	208	41.16	even	5	inner