Embedded newform 287.3.q.a.73.9

• Label: 287.3.q.a.73.9
• 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			73.9
Character	\(\chi\)	\(=\)	287.73
Dual form			287.3.q.a.173.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60389 + 1.50336i) q^{2} +(5.11320 - 1.37008i) q^{3} +(2.52016 - 4.36505i) q^{4} +(1.80353 + 3.12380i) q^{5} +(-11.2545 + 11.2545i) q^{6} +(1.87307 - 6.74475i) q^{7} +3.12796i q^{8} +(16.4735 - 9.51099i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{3} + 204 q^{4} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60389	+	1.50336i	−1.30195	+	0.751678i	−0.980737	−	0.195330i	\(-0.937422\pi\)
							−0.321208	+	0.947009i	\(0.604089\pi\)
\(3\)	5.11320	−	1.37008i	1.70440	−	0.456693i	0.730360	−	0.683062i	\(-0.239352\pi\)
							0.974042	+	0.226369i	\(0.0726856\pi\)
\(5\)	1.80353	+	3.12380i	0.360706	+	0.624761i	0.988077	−	0.153959i	\(-0.0492025\pi\)
							−0.627371	+	0.778720i	\(0.715869\pi\)
\(7\)	1.87307	−	6.74475i	0.267581	−	0.963535i
							\(11\)	−4.57793	−	17.0851i	−0.416175	−	1.55319i	−0.782470	−	0.622688i	\(-0.786040\pi\)
0.366295	−	0.930499i	\(-0.380626\pi\)
\(13\)	6.54551	+	6.54551i	0.503501	+	0.503501i	0.912524	−	0.409023i	\(-0.134131\pi\)
							−0.409023	+	0.912524i	\(0.634131\pi\)
\(17\)	−2.08404	−	7.77774i	−0.122590	−	0.457514i	0.877152	−	0.480213i	\(-0.159441\pi\)
							−0.999742	+	0.0226992i	\(0.992774\pi\)
\(19\)	−9.22609	−	2.47212i	−0.485584	−	0.130112i	0.00771761	−	0.999970i	\(-0.497543\pi\)
							−0.493301	+	0.869858i	\(0.664210\pi\)
\(23\)	−14.8883	−	25.7873i	−0.647317	−	1.12119i	−0.983761	−	0.179483i	\(-0.942558\pi\)
							0.336444	−	0.941703i	\(-0.390776\pi\)
\(29\)	−34.6240	+	34.6240i	−1.19393	+	1.19393i	−0.217977	+	0.975954i	\(0.569946\pi\)
							−0.975954	+	0.217977i	\(0.930054\pi\)
\(31\)	21.4339	+	12.3748i	0.691415	+	0.399189i	0.804142	−	0.594437i	\(-0.202625\pi\)
							−0.112727	+	0.993626i	\(0.535959\pi\)
\(37\)	20.8834	+	36.1711i	0.564416	+	0.977598i	0.997104	+	0.0760536i	\(0.0242320\pi\)
							−0.432688	+	0.901544i	\(0.642435\pi\)
\(41\)	28.6589	−	29.3201i	0.698996	−	0.715125i
							\(43\)			61.6669i			1.43411i	0.697015	+	0.717057i	\(0.254511\pi\)
−0.697015	+	0.717057i	\(0.745489\pi\)
\(47\)	31.6703	+	8.48604i	0.673837	+	0.180554i	0.579482	−	0.814985i	\(-0.303255\pi\)
							0.0943542	+	0.995539i	\(0.469921\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.q.a.73.9	✓	216
7.5	odd	6	inner	287.3.q.a.278.46	yes	216
41.9	even	4	inner	287.3.q.a.255.46	yes	216
287.173	odd	12	inner	287.3.q.a.173.9	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.q.a.73.9	✓	216	1.1	even	1	trivial
287.3.q.a.173.9	yes	216	287.173	odd	12	inner
287.3.q.a.255.46	yes	216	41.9	even	4	inner
287.3.q.a.278.46	yes	216	7.5	odd	6	inner