Embedded newform 287.2.s.a.100.18

• Label: 287.2.s.a.100.18
• Level: $287$
• Weight: $2$
• Character: 287.100
• 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.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			100.18
Character	\(\chi\)	\(=\)	287.100
Dual form			287.2.s.a.221.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.01481 + 0.215704i) q^{2} +(-0.378549 - 0.655666i) q^{3} +(-0.843789 - 0.375679i) q^{4} +(-0.127823 - 1.21615i) q^{5} +(-0.242724 - 0.747028i) q^{6} +(-2.53216 - 0.766920i) q^{7} +(-2.45392 - 1.78288i) q^{8} +(1.21340 - 2.10167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.01481	+	0.215704i	0.717576	+	0.152525i	0.552208	−	0.833707i	\(-0.313786\pi\)
							0.165368	+	0.986232i	\(0.447119\pi\)
\(3\)	−0.378549	−	0.655666i	−0.218555	−	0.378549i	0.735811	−	0.677187i	\(-0.236801\pi\)
							−0.954367	+	0.298638i	\(0.903468\pi\)
\(5\)	−0.127823	−	1.21615i	−0.0571642	−	0.543881i	−0.985202	−	0.171396i	\(-0.945172\pi\)
							0.928038	−	0.372485i	\(-0.121494\pi\)
\(7\)	−2.53216	−	0.766920i	−0.957067	−	0.289868i
							\(11\)	−0.0788751	+	0.750446i	−0.0237817	+	0.226268i	0.976175	+	0.216984i	\(0.0696219\pi\)
−0.999957	+	0.00928410i	\(0.997045\pi\)
\(13\)	−0.194596	−	0.598906i	−0.0539713	−	0.166107i	0.920437	−	0.390890i	\(-0.127833\pi\)
							−0.974409	+	0.224783i	\(0.927833\pi\)
\(17\)	0.206112	−	1.96102i	0.0499894	−	0.475617i	−0.940676	−	0.339305i	\(-0.889808\pi\)
							0.990666	−	0.136313i	\(-0.0435251\pi\)
\(19\)	−1.25669	−	1.39570i	−0.288304	−	0.320195i	0.581543	−	0.813516i	\(-0.302449\pi\)
							−0.869847	+	0.493321i	\(0.835783\pi\)
\(23\)	6.42128	+	1.36488i	1.33893	+	0.284598i	0.821050	−	0.570856i	\(-0.193388\pi\)
							0.517879	+	0.855454i	\(0.326722\pi\)
\(29\)	−1.71322	+	1.24473i	−0.318136	+	0.231140i	−0.735380	−	0.677655i	\(-0.762996\pi\)
							0.417243	+	0.908795i	\(0.362996\pi\)
\(31\)	−0.0966405	+	0.919473i	−0.0173571	+	0.165142i	−0.999767	−	0.0215965i	\(-0.993125\pi\)
							0.982410	+	0.186739i	\(0.0597918\pi\)
\(37\)	−0.493935	−	4.69948i	−0.0812025	−	0.772590i	−0.957035	−	0.289974i	\(-0.906353\pi\)
							0.875832	−	0.482616i	\(-0.160313\pi\)
\(41\)	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	\(0.750423\pi\)
\(47\)	−3.20854	−	0.681996i	−0.468013	−	0.0994793i	−0.0321324	−	0.999484i	\(-0.510230\pi\)
							−0.435881	+	0.900004i	\(0.643563\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.9	✓	208	41.16	even	5	inner
287.2.s.a.18.9	yes	208	7.4	even	3	inner
287.2.s.a.100.18	yes	208	1.1	even	1	trivial
287.2.s.a.221.18	yes	208	287.221	even	15	inner