Embedded newform 287.2.r.c.9.6

• Label: 287.2.r.c.9.6
• 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			9.6
Character	\(\chi\)	\(=\)	287.9
Dual form			287.2.r.c.32.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53151 - 0.884220i) q^{2} +(0.753885 - 2.81354i) q^{3} +(0.563689 + 0.976338i) q^{4} +(-1.29997 - 0.750535i) q^{5} +(-3.64237 + 3.64237i) q^{6} +(-1.57140 + 2.12855i) q^{7} +1.54318i q^{8} +(-4.74957 - 2.74217i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 4 q^{3} + 48 q^{4} - 28 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53151	−	0.884220i	−1.08294	−	0.625238i	−0.151255	−	0.988495i	\(-0.548331\pi\)
							−0.931689	+	0.363257i	\(0.881665\pi\)
\(3\)	0.753885	−	2.81354i	0.435256	−	1.62440i	−0.305198	−	0.952289i	\(-0.598723\pi\)
							0.740454	−	0.672107i	\(-0.234611\pi\)
\(5\)	−1.29997	−	0.750535i	−0.581362	−	0.335650i	0.180312	−	0.983609i	\(-0.442289\pi\)
							−0.761675	+	0.647960i	\(0.775622\pi\)
\(7\)	−1.57140	+	2.12855i	−0.593932	+	0.804515i
							\(11\)	−1.29741	+	4.84199i	−0.391183	+	1.45991i	0.437002	+	0.899460i	\(0.356040\pi\)
−0.828185	+	0.560454i	\(0.810627\pi\)
\(13\)	−2.78851	−	2.78851i	−0.773394	−	0.773394i	0.205304	−	0.978698i	\(-0.434182\pi\)
							−0.978698	+	0.205304i	\(0.934182\pi\)
\(17\)	2.16513	+	0.580146i	0.525122	+	0.140706i	0.511635	−	0.859203i	\(-0.329040\pi\)
							0.0134872	+	0.999909i	\(0.495707\pi\)
\(19\)	−1.49179	−	5.56742i	−0.342239	−	1.27725i	−0.895804	−	0.444448i	\(-0.853400\pi\)
							0.553565	−	0.832806i	\(-0.313267\pi\)
\(23\)	−0.655493	+	1.13535i	−0.136680	+	0.236736i	−0.926238	−	0.376939i	\(-0.876976\pi\)
							0.789558	+	0.613676i	\(0.210310\pi\)
\(29\)	−2.57275	−	2.57275i	−0.477749	−	0.477749i	0.426662	−	0.904411i	\(-0.359689\pi\)
							−0.904411	+	0.426662i	\(0.859689\pi\)
\(31\)	−3.14709	−	5.45091i	−0.565233	−	0.979013i	−0.997028	−	0.0770410i	\(-0.975453\pi\)
							0.431795	−	0.901972i	\(-0.357881\pi\)
\(37\)	−2.49055	+	4.31377i	−0.409445	+	0.709179i	−0.994828	−	0.101578i	\(-0.967611\pi\)
							0.585383	+	0.810757i	\(0.300944\pi\)
\(41\)	−3.63338	+	5.27244i	−0.567438	+	0.823416i
							\(43\)			7.45895i			1.13748i	0.822517	+	0.568740i	\(0.192569\pi\)
−0.822517	+	0.568740i	\(0.807431\pi\)
\(47\)	0.0209040	+	0.0780149i	0.00304917	+	0.0113796i	0.967434	−	0.253125i	\(-0.0814583\pi\)
							−0.964384	+	0.264504i	\(0.914792\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.r.c.9.6	✓	96
7.4	even	3	inner	287.2.r.c.214.19	yes	96
41.32	even	4	inner	287.2.r.c.114.19	yes	96
287.32	even	12	inner	287.2.r.c.32.6	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.r.c.9.6	✓	96	1.1	even	1	trivial
287.2.r.c.32.6	yes	96	287.32	even	12	inner
287.2.r.c.114.19	yes	96	41.32	even	4	inner
287.2.r.c.214.19	yes	96	7.4	even	3	inner