Embedded newform 287.2.r.c.114.8

• Label: 287.2.r.c.114.8
• Level: $287$
• Weight: $2$
• Character: 287.114
• 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			114.8
Character	\(\chi\)	\(=\)	287.114
Dual form			287.2.r.c.214.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12292 - 0.648320i) q^{2} +(-2.80858 - 0.752557i) q^{3} +(-0.159362 - 0.276022i) q^{4} +(1.50273 + 0.867599i) q^{5} +(2.66592 + 2.66592i) q^{6} +(2.55106 + 0.701479i) q^{7} +3.00655i q^{8} +(4.72371 + 2.72723i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 4 q^{3} + 48 q^{4} - 28 q^{6} - 14 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12292	−	0.648320i	−0.794027	−	0.458432i	0.0473514	−	0.998878i	\(-0.484922\pi\)
							−0.841378	+	0.540447i	\(0.818255\pi\)
\(3\)	−2.80858	−	0.752557i	−1.62153	−	0.434489i	−0.670083	−	0.742286i	\(-0.733741\pi\)
							−0.951452	+	0.307798i	\(0.900408\pi\)
\(5\)	1.50273	+	0.867599i	0.672040	+	0.388002i	0.796849	−	0.604179i	\(-0.206499\pi\)
							−0.124809	+	0.992181i	\(0.539832\pi\)
\(7\)	2.55106	+	0.701479i	0.964212	+	0.265134i
							\(11\)	−2.25167	−	0.603332i	−0.678903	−	0.181911i	−0.0971409	−	0.995271i	\(-0.530970\pi\)
−0.581762	+	0.813359i	\(0.697636\pi\)
\(13\)	3.55754	−	3.55754i	0.986685	−	0.986685i	−0.0132276	−	0.999913i	\(-0.504211\pi\)
							0.999913	+	0.0132276i	\(0.00421058\pi\)
\(17\)	−0.348914	+	1.30217i	−0.0846242	+	0.315822i	−0.995243	−	0.0974261i	\(-0.968939\pi\)
							0.910619	+	0.413248i	\(0.135606\pi\)
\(19\)	−2.18728	+	0.586080i	−0.501796	+	0.134456i	−0.500834	−	0.865544i	\(-0.666973\pi\)
							−0.000962780		1.00000i	\(0.500306\pi\)
\(23\)	3.47726	−	6.02278i	0.725058	−	1.25584i	−0.233892	−	0.972262i	\(-0.575146\pi\)
							0.958950	−	0.283574i	\(-0.0915203\pi\)
\(29\)	0.651898	−	0.651898i	0.121054	−	0.121054i	−0.643984	−	0.765039i	\(-0.722720\pi\)
							0.765039	+	0.643984i	\(0.222720\pi\)
\(31\)	2.55565	+	4.42652i	0.459009	+	0.795026i	0.998909	−	0.0467027i	\(-0.0148713\pi\)
							−0.539900	+	0.841729i	\(0.681538\pi\)
\(37\)	3.37484	−	5.84540i	0.554821	−	0.960977i	−0.443097	−	0.896474i	\(-0.646120\pi\)
							0.997917	−	0.0645037i	\(-0.0205464\pi\)
\(41\)	5.82327	+	2.66262i	0.909442	+	0.415831i
							\(43\)		−	11.9603i		−	1.82392i	−0.410276	−	0.911961i	\(-0.634568\pi\)
0.410276	−	0.911961i	\(-0.365432\pi\)
\(47\)	−10.6918	+	2.86486i	−1.55956	+	0.417883i	−0.932525	−	0.361107i	\(-0.882399\pi\)
							−0.627037	+	0.778990i	\(0.715732\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.r.c.114.8	yes	96
7.4	even	3	inner	287.2.r.c.32.17	yes	96
41.9	even	4	inner	287.2.r.c.9.17	✓	96
287.214	even	12	inner	287.2.r.c.214.8	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.r.c.9.17	✓	96	41.9	even	4	inner
287.2.r.c.32.17	yes	96	7.4	even	3	inner
287.2.r.c.114.8	yes	96	1.1	even	1	trivial
287.2.r.c.214.8	yes	96	287.214	even	12	inner