Embedded newform 287.2.s.a.16.1

• Label: 287.2.s.a.16.1
• Level: $287$
• Weight: $2$
• Character: 287.16
• 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			16.1
Character	\(\chi\)	\(=\)	287.16
Dual form			287.2.s.a.18.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79193 - 1.99014i) q^{2} +(-0.888984 - 1.53977i) q^{3} +(-0.540586 + 5.14333i) q^{4} +(2.48601 - 1.10684i) q^{5} +(-1.47135 + 4.52835i) q^{6} +(2.12797 - 1.57217i) q^{7} +(6.87154 - 4.99247i) q^{8} +(-0.0805851 + 0.139577i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79193	−	1.99014i	−1.26708	−	1.40724i	−0.872672	−	0.488307i	\(-0.837615\pi\)
							−0.394413	−	0.918933i	\(-0.629052\pi\)
\(3\)	−0.888984	−	1.53977i	−0.513255	−	0.888984i	−0.999882	−	0.0153740i	\(-0.995106\pi\)
							0.486627	−	0.873610i	\(-0.338227\pi\)
\(5\)	2.48601	−	1.10684i	1.11178	−	0.494995i	0.233119	−	0.972448i	\(-0.425107\pi\)
							0.878657	+	0.477453i	\(0.158440\pi\)
\(7\)	2.12797	−	1.57217i	0.804298	−	0.594226i
							\(11\)	1.10161	+	0.490467i	0.332147	+	0.147881i	0.566033	−	0.824383i	\(-0.308478\pi\)
−0.233886	+	0.972264i	\(0.575144\pi\)
\(13\)	1.17519	−	3.61687i	0.325939	−	1.00314i	−0.645075	−	0.764119i	\(-0.723174\pi\)
							0.971015	−	0.239019i	\(-0.0768259\pi\)
\(17\)	4.80605	+	2.13979i	1.16564	+	0.518976i	0.896030	−	0.443994i	\(-0.146439\pi\)
							0.269609	+	0.962970i	\(0.413106\pi\)
\(19\)	−2.89298	−	0.614921i	−0.663694	−	0.141073i	−0.136266	−	0.990672i	\(-0.543510\pi\)
							−0.527428	+	0.849600i	\(0.676844\pi\)
\(23\)	3.56304	+	3.95716i	0.742945	+	0.825124i	0.989580	−	0.143984i	\(-0.0459915\pi\)
							−0.246635	+	0.969108i	\(0.579325\pi\)
\(29\)	4.82133	+	3.50290i	0.895298	+	0.650472i	0.937254	−	0.348648i	\(-0.113359\pi\)
							−0.0419561	+	0.999119i	\(0.513359\pi\)
\(31\)	−3.24997	−	1.44698i	−0.583712	−	0.259885i	0.0935652	−	0.995613i	\(-0.470174\pi\)
							−0.677277	+	0.735728i	\(0.736840\pi\)
\(37\)	−6.52226	+	2.90390i	−1.07225	+	0.477397i	−0.865455	−	0.500986i	\(-0.832971\pi\)
							−0.206797	+	0.978384i	\(0.566304\pi\)
\(41\)	−4.76709	+	4.27491i	−0.744495	+	0.667628i
							\(43\)	−2.86830	+	8.82771i	−0.437411	+	1.34621i	0.453184	+	0.891417i	\(0.350288\pi\)
−0.890595	+	0.454797i	\(0.849712\pi\)
\(47\)	−4.28201	−	4.75566i	−0.624596	−	0.693684i	0.344942	−	0.938624i	\(-0.387898\pi\)
							−0.969538	+	0.244940i	\(0.921232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.s.a.16.1	✓	208
7.4	even	3	inner	287.2.s.a.221.26	yes	208
41.18	even	5	inner	287.2.s.a.100.26	yes	208
287.18	even	15	inner	287.2.s.a.18.1	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.1	✓	208	1.1	even	1	trivial
287.2.s.a.18.1	yes	208	287.18	even	15	inner
287.2.s.a.100.26	yes	208	41.18	even	5	inner
287.2.s.a.221.26	yes	208	7.4	even	3	inner