Embedded newform 287.2.r.c.9.4

• Label: 287.2.r.c.9.4
• Level: $287$
• Weight: $2$
• Character: 287.9
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.r (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			9.4
Character	\(\chi\)	\(=\)	287.9
Dual form			287.2.r.c.32.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85131 - 1.06886i) q^{2} +(0.785917 - 2.93308i) q^{3} +(1.28491 + 2.22553i) q^{4} +(1.72481 + 0.995818i) q^{5} +(-4.59003 + 4.59003i) q^{6} +(2.23519 - 1.41560i) q^{7} -1.21811i q^{8} +(-5.38724 - 3.11032i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 4 q^{3} + 48 q^{4} - 28 q^{6} - 14 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85131	−	1.06886i	−1.30908	−	0.755796i	−0.327135	−	0.944978i	\(-0.606083\pi\)
							−0.981942	+	0.189182i	\(0.939416\pi\)
\(3\)	0.785917	−	2.93308i	0.453750	−	1.69342i	−0.237989	−	0.971268i	\(-0.576488\pi\)
							0.691738	−	0.722148i	\(-0.256845\pi\)
\(5\)	1.72481	+	0.995818i	0.771357	+	0.445343i	0.833358	−	0.552733i	\(-0.186415\pi\)
							−0.0620015	+	0.998076i	\(0.519748\pi\)
\(7\)	2.23519	−	1.41560i	0.844822	−	0.535048i
							\(11\)	1.37087	−	5.11615i	0.413332	−	1.54258i	−0.374821	−	0.927097i	\(-0.622296\pi\)
0.788153	−	0.615479i	\(-0.211037\pi\)
\(13\)	1.35544	+	1.35544i	0.375931	+	0.375931i	0.869632	−	0.493701i	\(-0.164356\pi\)
							−0.493701	+	0.869632i	\(0.664356\pi\)
\(17\)	−1.95467	−	0.523753i	−0.474078	−	0.127029i	0.0138657	−	0.999904i	\(-0.495586\pi\)
							−0.487944	+	0.872875i	\(0.662253\pi\)
\(19\)	0.638018	+	2.38111i	0.146371	+	0.546265i	0.999691	+	0.0248763i	\(0.00791920\pi\)
							−0.853319	+	0.521389i	\(0.825414\pi\)
\(23\)	−3.53533	+	6.12337i	−0.737168	+	1.27681i	0.216598	+	0.976261i	\(0.430504\pi\)
							−0.953766	+	0.300551i	\(0.902829\pi\)
\(29\)	0.435030	+	0.435030i	0.0807830	+	0.0807830i	0.746344	−	0.665561i	\(-0.231808\pi\)
							−0.665561	+	0.746344i	\(0.731808\pi\)
\(31\)	4.48223	+	7.76345i	0.805032	+	1.39436i	0.916269	+	0.400563i	\(0.131185\pi\)
							−0.111237	+	0.993794i	\(0.535481\pi\)
\(37\)	3.09304	−	5.35731i	0.508493	−	0.880736i	−0.491459	−	0.870901i	\(-0.663536\pi\)
							0.999952	−	0.00983485i	\(-0.00313058\pi\)
\(41\)	−6.40114	−	0.159260i	−0.999691	−	0.0248722i
							\(43\)			9.09511i			1.38699i	0.720461	+	0.693496i	\(0.243930\pi\)
−0.720461	+	0.693496i	\(0.756070\pi\)
\(47\)	−0.511886	−	1.91038i	−0.0746662	−	0.278658i	0.918491	−	0.395441i	\(-0.129408\pi\)
							−0.993157	+	0.116783i	\(0.962742\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.r.c.9.4	✓	96
7.4	even	3	inner	287.2.r.c.214.21	yes	96
41.32	even	4	inner	287.2.r.c.114.21	yes	96
287.32	even	12	inner	287.2.r.c.32.4	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.r.c.9.4	✓	96	1.1	even	1	trivial
287.2.r.c.32.4	yes	96	287.32	even	12	inner
287.2.r.c.114.21	yes	96	41.32	even	4	inner
287.2.r.c.214.21	yes	96	7.4	even	3	inner