Embedded newform 287.3.q.a.73.16

• Label: 287.3.q.a.73.16
• 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			73.16
Character	\(\chi\)	\(=\)	287.73
Dual form			287.3.q.a.173.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65656 + 0.956418i) q^{2} +(1.46388 - 0.392246i) q^{3} +(-0.170530 + 0.295367i) q^{4} +(2.30350 + 3.98978i) q^{5} +(-2.04986 + 2.04986i) q^{6} +(-6.96672 - 0.681733i) q^{7} -8.30373i q^{8} +(-5.80513 + 3.35159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 6 q^{3} + 204 q^{4} - 4 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65656	+	0.956418i	−0.828282	+	0.478209i	−0.853264	−	0.521479i	\(-0.825380\pi\)
							0.0249821	+	0.999688i	\(0.492047\pi\)
\(3\)	1.46388	−	0.392246i	0.487961	−	0.130749i	−0.00644609	−	0.999979i	\(-0.502052\pi\)
							0.494407	+	0.869230i	\(0.335385\pi\)
\(5\)	2.30350	+	3.98978i	0.460700	+	0.797956i	0.998996	−	0.0447999i	\(-0.0142650\pi\)
							−0.538296	+	0.842756i	\(0.680932\pi\)
\(7\)	−6.96672	−	0.681733i	−0.995246	−	0.0973904i
							\(11\)	−4.49229	−	16.7655i	−0.408390	−	1.52413i	−0.797716	−	0.603033i	\(-0.793959\pi\)
0.389326	−	0.921100i	\(-0.372708\pi\)
\(13\)	−3.12281	−	3.12281i	−0.240216	−	0.240216i	0.576723	−	0.816939i	\(-0.304331\pi\)
							−0.816939	+	0.576723i	\(0.804331\pi\)
\(17\)	0.0223954	+	0.0835809i	0.00131738	+	0.00491653i	0.966581	−	0.256360i	\(-0.0825232\pi\)
							−0.965264	+	0.261276i	\(0.915857\pi\)
\(19\)	5.11312	+	1.37006i	0.269112	+	0.0721082i	0.390851	−	0.920454i	\(-0.372181\pi\)
							−0.121740	+	0.992562i	\(0.538847\pi\)
\(23\)	−3.53407	−	6.12118i	−0.153655	−	0.266138i	0.778913	−	0.627132i	\(-0.215771\pi\)
							−0.932569	+	0.360993i	\(0.882438\pi\)
\(29\)	12.3423	−	12.3423i	0.425595	−	0.425595i	−0.461530	−	0.887125i	\(-0.652699\pi\)
							0.887125	+	0.461530i	\(0.152699\pi\)
\(31\)	−31.5574	−	18.2197i	−1.01798	−	0.587731i	−0.104462	−	0.994529i	\(-0.533312\pi\)
							−0.913518	+	0.406798i	\(0.866645\pi\)
\(37\)	−5.85087	−	10.1340i	−0.158132	−	0.273892i	0.776063	−	0.630655i	\(-0.217214\pi\)
							−0.934195	+	0.356763i	\(0.883880\pi\)
\(41\)	−31.9216	+	25.7296i	−0.778575	+	0.627552i
							\(43\)		−	65.1338i		−	1.51474i	−0.652986	−	0.757370i	\(-0.726484\pi\)
0.652986	−	0.757370i	\(-0.273516\pi\)
\(47\)	−6.37196	−	1.70736i	−0.135574	−	0.0363268i	0.190394	−	0.981708i	\(-0.439023\pi\)
							−0.325968	+	0.945381i	\(0.605690\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.q.a.73.16	✓	216
7.5	odd	6	inner	287.3.q.a.278.39	yes	216
41.9	even	4	inner	287.3.q.a.255.39	yes	216
287.173	odd	12	inner	287.3.q.a.173.16	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.q.a.73.16	✓	216	1.1	even	1	trivial
287.3.q.a.173.16	yes	216	287.173	odd	12	inner
287.3.q.a.255.39	yes	216	41.9	even	4	inner
287.3.q.a.278.39	yes	216	7.5	odd	6	inner