Embedded newform 287.2.w.a.3.11

• Label: 287.2.w.a.3.11
• Level: $287$
• Weight: $2$
• Character: 287.3
• 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.w (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			3.11
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.626707 - 0.167926i) q^{2} +(-2.18400 + 1.67584i) q^{3} +(-1.36749 - 0.789519i) q^{4} +(-2.76165 - 0.739982i) q^{5} +(1.65015 - 0.683514i) q^{6} +(-2.20483 + 1.46244i) q^{7} +(1.64200 + 1.64200i) q^{8} +(1.18496 - 4.42232i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.626707	−	0.167926i	−0.443149	−	0.118741i	0.0303429	−	0.999540i	\(-0.490340\pi\)
							−0.473492	+	0.880798i	\(0.657007\pi\)
\(3\)	−2.18400	+	1.67584i	−1.26093	+	0.967549i	−0.260937	+	0.965356i	\(0.584031\pi\)
							−0.999998	+	0.00219303i	\(0.999302\pi\)
\(5\)	−2.76165	−	0.739982i	−1.23505	−	0.330930i	−0.418504	−	0.908215i	\(-0.637446\pi\)
							−0.816543	+	0.577285i	\(0.804112\pi\)
\(7\)	−2.20483	+	1.46244i	−0.833346	+	0.552752i
							\(11\)	3.63275	−	2.78751i	1.09532	−	0.840466i	0.107210	−	0.994236i	\(-0.465808\pi\)
0.988107	+	0.153770i	\(0.0491416\pi\)
\(13\)	2.37253	+	0.982736i	0.658022	+	0.272562i	0.686606	−	0.727030i	\(-0.259100\pi\)
							−0.0285837	+	0.999591i	\(0.509100\pi\)
\(17\)	−0.669075	+	5.08213i	−0.162275	+	1.23260i	0.695483	+	0.718543i	\(0.255191\pi\)
							−0.857757	+	0.514055i	\(0.828143\pi\)
\(19\)	−4.62097	−	3.54579i	−1.06012	−	0.813461i	−0.0771147	−	0.997022i	\(-0.524571\pi\)
							−0.983008	+	0.183561i	\(0.941237\pi\)
\(23\)	2.47005	−	1.42608i	0.515040	−	0.297359i	−0.219863	−	0.975531i	\(-0.570561\pi\)
							0.734903	+	0.678172i	\(0.237228\pi\)
\(29\)	−1.53703	+	3.71071i	−0.285419	+	0.689062i	−0.999944	−	0.0105476i	\(-0.996643\pi\)
							0.714526	+	0.699609i	\(0.246643\pi\)
\(31\)	5.35804	−	9.28041i	0.962333	−	1.66681i	0.245718	−	0.969341i	\(-0.420976\pi\)
							0.716615	−	0.697469i	\(-0.245690\pi\)
\(37\)	0.669284	+	1.15923i	0.110030	+	0.190577i	0.915782	−	0.401676i	\(-0.131572\pi\)
							−0.805752	+	0.592253i	\(0.798239\pi\)
\(41\)	2.26240	+	5.99012i	0.353328	+	0.935500i
							\(43\)	6.42954	−	6.42954i	0.980496	−	0.980496i	−0.0193171	−	0.999813i	\(-0.506149\pi\)
0.999813	+	0.0193171i	\(0.00614920\pi\)
\(47\)	−1.87039	−	1.43520i	−0.272824	−	0.209345i	0.463283	−	0.886210i	\(-0.346671\pi\)
							−0.736107	+	0.676865i	\(0.763338\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.11	✓	208
7.5	odd	6	inner	287.2.w.a.208.16	yes	208
41.14	odd	8	inner	287.2.w.a.178.16	yes	208
287.96	even	24	inner	287.2.w.a.96.11	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.11	✓	208	1.1	even	1	trivial
287.2.w.a.96.11	yes	208	287.96	even	24	inner
287.2.w.a.178.16	yes	208	41.14	odd	8	inner
287.2.w.a.208.16	yes	208	7.5	odd	6	inner