Embedded newform 287.2.s.a.16.16

• Label: 287.2.s.a.16.16
• 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			16.16
Character	\(\chi\)	\(=\)	287.16
Dual form			287.2.s.a.18.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.472483 + 0.524745i) q^{2} +(-1.44621 - 2.50492i) q^{3} +(0.156939 - 1.49318i) q^{4} +(1.37084 - 0.610336i) q^{5} +(0.631132 - 1.94242i) q^{6} +(2.23920 + 1.40925i) q^{7} +(2.00021 - 1.45323i) q^{8} +(-2.68307 + 4.64721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.472483	+	0.524745i	0.334096	+	0.371051i	0.886662	−	0.462418i	\(-0.153018\pi\)
							−0.552566	+	0.833469i	\(0.686351\pi\)
\(3\)	−1.44621	−	2.50492i	−0.834972	−	1.44621i	−0.894053	−	0.447961i	\(-0.852150\pi\)
							0.0590811	−	0.998253i	\(-0.481183\pi\)
\(5\)	1.37084	−	0.610336i	0.613057	−	0.272951i	−0.0766284	−	0.997060i	\(-0.524416\pi\)
							0.689685	+	0.724109i	\(0.257749\pi\)
\(7\)	2.23920	+	1.40925i	0.846337	+	0.532648i
							\(11\)	−5.87235	−	2.61454i	−1.77058	−	0.788314i	−0.985666	−	0.168711i	\(-0.946040\pi\)
−0.784916	−	0.619603i	\(-0.787294\pi\)
\(13\)	0.461948	−	1.42173i	0.128121	−	0.394317i	−0.866336	−	0.499462i	\(-0.833531\pi\)
							0.994457	+	0.105146i	\(0.0335309\pi\)
\(17\)	4.98280	+	2.21849i	1.20851	+	0.538062i	0.909306	−	0.416128i	\(-0.136613\pi\)
							0.299201	+	0.954190i	\(0.403280\pi\)
\(19\)	−0.303816	−	0.0645780i	−0.0697001	−	0.0148152i	0.172930	−	0.984934i	\(-0.444677\pi\)
							−0.242630	+	0.970119i	\(0.578010\pi\)
\(23\)	−0.678200	−	0.753218i	−0.141415	−	0.157057i	0.668277	−	0.743913i	\(-0.267032\pi\)
							−0.809691	+	0.586856i	\(0.800365\pi\)
\(29\)	−2.12494	−	1.54386i	−0.394591	−	0.286687i	0.372743	−	0.927934i	\(-0.378417\pi\)
							−0.767334	+	0.641248i	\(0.778417\pi\)
\(31\)	3.62884	+	1.61566i	0.651759	+	0.290182i	0.705858	−	0.708354i	\(-0.250562\pi\)
							−0.0540986	+	0.998536i	\(0.517229\pi\)
\(37\)	9.89823	−	4.40698i	1.62726	−	0.724503i	0.628674	−	0.777669i	\(-0.283598\pi\)
							0.998586	+	0.0531664i	\(0.0169314\pi\)
\(41\)	6.37947	+	0.549865i	0.996306	+	0.0858745i
							\(43\)	−3.90676	+	12.0238i	−0.595775	+	1.83361i	−0.0449437	+	0.998990i	\(0.514311\pi\)
−0.550831	+	0.834617i	\(0.685689\pi\)
\(47\)	−3.11105	−	3.45518i	−0.453794	−	0.503989i	0.472219	−	0.881481i	\(-0.343453\pi\)
							−0.926013	+	0.377492i	\(0.876787\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.s.a.16.16	✓	208
7.4	even	3	inner	287.2.s.a.221.11	yes	208
41.18	even	5	inner	287.2.s.a.100.11	yes	208
287.18	even	15	inner	287.2.s.a.18.16	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.16	✓	208	1.1	even	1	trivial
287.2.s.a.18.16	yes	208	287.18	even	15	inner
287.2.s.a.100.11	yes	208	41.18	even	5	inner
287.2.s.a.221.11	yes	208	7.4	even	3	inner