Embedded newform 287.2.r.c.9.13

• Label: 287.2.r.c.9.13
• 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			9.13
Character	\(\chi\)	\(=\)	287.9
Dual form			287.2.r.c.32.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366500 + 0.211599i) q^{2} +(-0.581121 + 2.16877i) q^{3} +(-0.910452 - 1.57695i) q^{4} +(-3.23301 - 1.86658i) q^{5} +(-0.671891 + 0.671891i) q^{6} +(0.706651 + 2.54964i) q^{7} -1.61700i q^{8} +(-1.76780 - 1.02064i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 4 q^{3} + 48 q^{4} - 28 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366500	+	0.211599i	0.259155	+	0.149623i	0.623949	−	0.781465i	\(-0.285527\pi\)
							−0.364794	+	0.931088i	\(0.618861\pi\)
\(3\)	−0.581121	+	2.16877i	−0.335510	+	1.25214i	0.567805	+	0.823163i	\(0.307793\pi\)
							−0.903315	+	0.428978i	\(0.858874\pi\)
\(5\)	−3.23301	−	1.86658i	−1.44585	−	0.834760i	−0.447616	−	0.894226i	\(-0.647727\pi\)
							−0.998230	+	0.0594659i	\(0.981060\pi\)
\(7\)	0.706651	+	2.54964i	0.267089	+	0.963672i
							\(11\)	1.38989	−	5.18713i	0.419067	−	1.56398i	−0.357480	−	0.933921i	\(-0.616364\pi\)
0.776547	−	0.630059i	\(-0.216969\pi\)
\(13\)	−3.75552	−	3.75552i	−1.04159	−	1.04159i	−0.999097	−	0.0424975i	\(-0.986469\pi\)
							−0.0424975	−	0.999097i	\(-0.513531\pi\)
\(17\)	−4.84407	−	1.29797i	−1.17486	−	0.314803i	−0.381974	−	0.924173i	\(-0.624756\pi\)
							−0.792886	+	0.609370i	\(0.791422\pi\)
\(19\)	−0.310069	−	1.15719i	−0.0711346	−	0.265478i	0.921194	−	0.389103i	\(-0.127215\pi\)
							−0.992329	+	0.123625i	\(0.960548\pi\)
\(23\)	0.522827	−	0.905563i	0.109017	−	0.188823i	−0.806355	−	0.591431i	\(-0.798563\pi\)
							0.915372	+	0.402609i	\(0.131896\pi\)
\(29\)	−0.595726	−	0.595726i	−0.110624	−	0.110624i	0.649628	−	0.760252i	\(-0.274925\pi\)
							−0.760252	+	0.649628i	\(0.774925\pi\)
\(31\)	−0.694583	−	1.20305i	−0.124751	−	0.216075i	0.796885	−	0.604131i	\(-0.206480\pi\)
							−0.921636	+	0.388057i	\(0.873146\pi\)
\(37\)	−3.02610	+	5.24137i	−0.497488	+	0.861675i	−0.999996	−	0.00289768i	\(-0.999078\pi\)
							0.502507	+	0.864573i	\(0.332411\pi\)
\(41\)	5.68606	+	2.94428i	0.888013	+	0.459819i
							\(43\)			2.73513i			0.417104i	0.978011	+	0.208552i	\(0.0668750\pi\)
−0.978011	+	0.208552i	\(0.933125\pi\)
\(47\)	1.45309	+	5.42299i	0.211954	+	0.791024i	0.987217	+	0.159385i	\(0.0509510\pi\)
							−0.775262	+	0.631639i	\(0.782382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.r.c.9.13	✓	96
7.4	even	3	inner	287.2.r.c.214.12	yes	96
41.32	even	4	inner	287.2.r.c.114.12	yes	96
287.32	even	12	inner	287.2.r.c.32.13	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.r.c.9.13	✓	96	1.1	even	1	trivial
287.2.r.c.32.13	yes	96	287.32	even	12	inner
287.2.r.c.114.12	yes	96	41.32	even	4	inner
287.2.r.c.214.12	yes	96	7.4	even	3	inner