Embedded newform 287.3.q.a.73.6

• Label: 287.3.q.a.73.6
• Level: $287$
• Weight: $3$
• Character: 287.73
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.q (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			73.6
Character	\(\chi\)	\(=\)	287.73
Dual form			287.3.q.a.173.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.86475 + 1.65396i) q^{2} +(-1.20386 + 0.322573i) q^{3} +(3.47118 - 6.01226i) q^{4} +(-0.352767 - 0.611010i) q^{5} +(2.91523 - 2.91523i) q^{6} +(-6.65179 + 2.18029i) q^{7} +9.73311i q^{8} +(-6.44901 + 3.72334i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{3} + 204 q^{4} - 4 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.86475	+	1.65396i	−1.43237	+	0.826981i	−0.997301	−	0.0734147i	\(-0.976610\pi\)
							−0.435072	+	0.900396i	\(0.643277\pi\)
\(3\)	−1.20386	+	0.322573i	−0.401286	+	0.107524i	−0.453817	−	0.891095i	\(-0.649938\pi\)
							0.0525305	+	0.998619i	\(0.483271\pi\)
\(5\)	−0.352767	−	0.611010i	−0.0705534	−	0.122202i	0.828591	−	0.559855i	\(-0.189143\pi\)
							−0.899144	+	0.437653i	\(0.855810\pi\)
\(7\)	−6.65179	+	2.18029i	−0.950256	+	0.311470i
							\(11\)	0.202439	+	0.755513i	0.0184035	+	0.0686830i	0.974517	−	0.224314i	\(-0.0720141\pi\)
−0.956113	+	0.292997i	\(0.905347\pi\)
\(13\)	−0.0873358	−	0.0873358i	−0.00671814	−	0.00671814i	0.703740	−	0.710458i	\(-0.251512\pi\)
							−0.710458	+	0.703740i	\(0.751512\pi\)
\(17\)	−3.47814	−	12.9806i	−0.204596	−	0.763564i	−0.989572	−	0.144038i	\(-0.953991\pi\)
							0.784976	−	0.619526i	\(-0.212675\pi\)
\(19\)	−5.43220	−	1.45555i	−0.285905	−	0.0766080i	0.113016	−	0.993593i	\(-0.463949\pi\)
							−0.398921	+	0.916985i	\(0.630615\pi\)
\(23\)	−6.83907	−	11.8456i	−0.297351	−	0.515027i	0.678178	−	0.734897i	\(-0.262770\pi\)
							−0.975529	+	0.219871i	\(0.929436\pi\)
\(29\)	8.41194	−	8.41194i	0.290067	−	0.290067i	−0.547040	−	0.837107i	\(-0.684245\pi\)
							0.837107	+	0.547040i	\(0.184245\pi\)
\(31\)	44.9488	+	25.9512i	1.44996	+	0.837136i	0.998478	−	0.0551440i	\(-0.0175618\pi\)
							0.451483	+	0.892280i	\(0.350895\pi\)
\(37\)	28.4135	+	49.2136i	0.767933	+	1.33010i	0.938682	+	0.344784i	\(0.112048\pi\)
							−0.170749	+	0.985314i	\(0.554619\pi\)
\(41\)	28.2369	+	29.7268i	0.688704	+	0.725043i
							\(43\)			14.4575i			0.336220i	0.985768	+	0.168110i	\(0.0537664\pi\)
−0.985768	+	0.168110i	\(0.946234\pi\)
\(47\)	−10.0172	−	2.68410i	−0.213132	−	0.0571085i	0.150673	−	0.988584i	\(-0.451856\pi\)
							−0.363805	+	0.931475i	\(0.618523\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.q.a.73.6	✓	216
7.5	odd	6	inner	287.3.q.a.278.49	yes	216
41.9	even	4	inner	287.3.q.a.255.49	yes	216
287.173	odd	12	inner	287.3.q.a.173.6	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.q.a.73.6	✓	216	1.1	even	1	trivial
287.3.q.a.173.6	yes	216	287.173	odd	12	inner
287.3.q.a.255.49	yes	216	41.9	even	4	inner
287.3.q.a.278.49	yes	216	7.5	odd	6	inner