Embedded newform 287.2.s.a.16.20

• Label: 287.2.s.a.16.20
• 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			16.20
Character	\(\chi\)	\(=\)	287.16
Dual form			287.2.s.a.18.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.07005 + 1.18841i) q^{2} +(1.15675 + 2.00356i) q^{3} +(-0.0582569 + 0.554278i) q^{4} +(2.56833 - 1.14349i) q^{5} +(-1.14327 + 3.51861i) q^{6} +(-2.64569 + 0.0176917i) q^{7} +(1.86646 - 1.35606i) q^{8} +(-1.17616 + 2.03717i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.07005	+	1.18841i	0.756640	+	0.840334i	0.991283	−	0.131748i	\(-0.0420588\pi\)
							−0.234643	+	0.972082i	\(0.575392\pi\)
\(3\)	1.15675	+	2.00356i	0.667852	+	1.15675i	0.978503	+	0.206230i	\(0.0661196\pi\)
							−0.310651	+	0.950524i	\(0.600547\pi\)
\(5\)	2.56833	−	1.14349i	1.14859	−	0.511385i	0.257979	−	0.966151i	\(-0.416943\pi\)
							0.890611	+	0.454765i	\(0.150277\pi\)
\(7\)	−2.64569	+	0.0176917i	−0.999978	+	0.00668685i
							\(11\)	−1.45652	−	0.648483i	−0.439157	−	0.195525i	0.175235	−	0.984527i	\(-0.443932\pi\)
−0.614391	+	0.789002i	\(0.710598\pi\)
\(13\)	−0.717956	+	2.20964i	−0.199125	+	0.612845i	0.800778	+	0.598961i	\(0.204419\pi\)
							−0.999904	+	0.0138838i	\(0.995581\pi\)
\(17\)	−2.37671	−	1.05818i	−0.576437	−	0.256646i	0.0977453	−	0.995211i	\(-0.468837\pi\)
							−0.674182	+	0.738565i	\(0.735504\pi\)
\(19\)	−7.71584	−	1.64005i	−1.77013	−	0.376254i	−0.796549	−	0.604575i	\(-0.793343\pi\)
							−0.973586	+	0.228321i	\(0.926676\pi\)
\(23\)	−4.46400	−	4.95777i	−0.930808	−	1.03377i	−0.999347	−	0.0361190i	\(-0.988500\pi\)
							0.0685392	−	0.997648i	\(-0.478166\pi\)
\(29\)	−0.901260	−	0.654804i	−0.167360	−	0.121594i	0.500953	−	0.865475i	\(-0.332983\pi\)
							−0.668312	+	0.743881i	\(0.732983\pi\)
\(31\)	6.19736	+	2.75924i	1.11308	+	0.495575i	0.879086	−	0.476664i	\(-0.158154\pi\)
							0.233993	+	0.972238i	\(0.424821\pi\)
\(37\)	−3.69112	+	1.64339i	−0.606817	+	0.270172i	−0.687057	−	0.726603i	\(-0.741098\pi\)
							0.0802402	+	0.996776i	\(0.474431\pi\)
\(41\)	5.68028	+	2.95540i	0.887111	+	0.461557i
							\(43\)	−1.94278	+	5.97927i	−0.296271	+	0.911830i	0.686520	+	0.727111i	\(0.259137\pi\)
−0.982791	+	0.184719i	\(0.940863\pi\)
\(47\)	6.54361	+	7.26742i	0.954484	+	1.06006i	0.998137	+	0.0610147i	\(0.0194337\pi\)
							−0.0436533	+	0.999047i	\(0.513900\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.s.a.16.20	✓	208
7.4	even	3	inner	287.2.s.a.221.7	yes	208
41.18	even	5	inner	287.2.s.a.100.7	yes	208
287.18	even	15	inner	287.2.s.a.18.20	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.20	✓	208	1.1	even	1	trivial
287.2.s.a.18.20	yes	208	287.18	even	15	inner
287.2.s.a.100.7	yes	208	41.18	even	5	inner
287.2.s.a.221.7	yes	208	7.4	even	3	inner