Embedded newform 287.2.r.c.9.18

• Label: 287.2.r.c.9.18
• 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			9.18
Character	\(\chi\)	\(=\)	287.9
Dual form			287.2.r.c.32.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.23416 + 0.712544i) q^{2} +(0.388603 - 1.45029i) q^{3} +(0.0154369 + 0.0267375i) q^{4} +(3.19102 + 1.84234i) q^{5} +(1.51299 - 1.51299i) q^{6} +(-2.63716 - 0.213102i) q^{7} -2.80618i q^{8} +(0.645761 + 0.372830i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 4 q^{3} + 48 q^{4} - 28 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.23416	+	0.712544i	0.872684	+	0.503844i	0.868239	−	0.496146i	\(-0.165252\pi\)
							0.00444489	+	0.999990i	\(0.498585\pi\)
\(3\)	0.388603	−	1.45029i	0.224360	−	0.837323i	−0.758300	−	0.651906i	\(-0.773970\pi\)
							0.982660	−	0.185417i	\(-0.0593636\pi\)
\(5\)	3.19102	+	1.84234i	1.42707	+	0.823917i	0.996888	−	0.0788267i	\(-0.0251174\pi\)
							0.430178	+	0.902744i	\(0.358451\pi\)
\(7\)	−2.63716	−	0.213102i	−0.996751	−	0.0805450i
							\(11\)	−0.0928437	+	0.346497i	−0.0279934	+	0.104473i	−0.978509	−	0.206204i	\(-0.933889\pi\)
0.950516	+	0.310677i	\(0.100556\pi\)
\(13\)	1.57654	+	1.57654i	0.437254	+	0.437254i	0.891087	−	0.453833i	\(-0.149944\pi\)
							−0.453833	+	0.891087i	\(0.649944\pi\)
\(17\)	−3.50475	−	0.939096i	−0.850028	−	0.227764i	−0.192596	−	0.981278i	\(-0.561691\pi\)
							−0.657432	+	0.753514i	\(0.728357\pi\)
\(19\)	−1.15505	−	4.31071i	−0.264987	−	0.988945i	−0.962258	−	0.272137i	\(-0.912269\pi\)
							0.697272	−	0.716807i	\(-0.254397\pi\)
\(23\)	−3.66252	+	6.34367i	−0.763688	+	1.32275i	0.177249	+	0.984166i	\(0.443280\pi\)
							−0.940937	+	0.338581i	\(0.890053\pi\)
\(29\)	−2.08655	−	2.08655i	−0.387463	−	0.387463i	0.486319	−	0.873781i	\(-0.338339\pi\)
							−0.873781	+	0.486319i	\(0.838339\pi\)
\(31\)	2.05478	+	3.55898i	0.369049	+	0.639212i	0.989417	−	0.145099i	\(-0.0463500\pi\)
							−0.620368	+	0.784311i	\(0.713017\pi\)
\(37\)	−3.63870	+	6.30242i	−0.598199	+	1.03611i	0.394888	+	0.918729i	\(0.370783\pi\)
							−0.993087	+	0.117382i	\(0.962550\pi\)
\(41\)	−6.33865	+	0.906387i	−0.989931	+	0.141554i
							\(43\)		−	0.831349i		−	0.126780i	−0.997989	−	0.0633898i	\(-0.979809\pi\)
0.997989	−	0.0633898i	\(-0.0201911\pi\)
\(47\)	−0.163171	−	0.608963i	−0.0238010	−	0.0888264i	0.953004	−	0.302959i	\(-0.0979744\pi\)
							−0.976805	+	0.214132i	\(0.931308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.r.c.9.18	✓	96
7.4	even	3	inner	287.2.r.c.214.7	yes	96
41.32	even	4	inner	287.2.r.c.114.7	yes	96
287.32	even	12	inner	287.2.r.c.32.18	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.r.c.9.18	✓	96	1.1	even	1	trivial
287.2.r.c.32.18	yes	96	287.32	even	12	inner
287.2.r.c.114.7	yes	96	41.32	even	4	inner
287.2.r.c.214.7	yes	96	7.4	even	3	inner