Embedded newform 287.4.e.a.247.23

• Label: 287.4.e.a.247.23
• Level: $287$
• Weight: $4$
• Character: 287.247
• Analytic conductor: $16.934$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	287.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.9335481716\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			247.23
Character	\(\chi\)	\(=\)	287.247
Dual form			287.4.e.a.165.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.844645 - 1.46297i) q^{2} +(1.21994 + 2.11299i) q^{3} +(2.57315 + 4.45683i) q^{4} +(-1.57454 + 2.72718i) q^{5} +4.12166 q^{6} +(-0.952937 + 18.4957i) q^{7} +22.2079 q^{8} +(10.5235 - 18.2272i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 5 q^{2} + 6 q^{3} - 117 q^{4} - 4 q^{5} - 24 q^{6} - 30 q^{7} - 78 q^{8} - 236 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.844645	−	1.46297i	0.298627	−	0.517237i	−0.677195	−	0.735804i	\(-0.736805\pi\)
							0.975822	+	0.218566i	\(0.0701380\pi\)
\(3\)	1.21994	+	2.11299i	0.234777	+	0.406646i	0.959208	−	0.282702i	\(-0.0912306\pi\)
							−0.724431	+	0.689348i	\(0.757897\pi\)
\(5\)	−1.57454	+	2.72718i	−0.140831	+	0.243926i	−0.927810	−	0.373054i	\(-0.878311\pi\)
							0.786979	+	0.616980i	\(0.211644\pi\)
\(7\)	−0.952937	+	18.4957i	−0.0514538	+	0.998675i
							\(11\)	−15.4596	−	26.7769i	−0.423750	−	0.733957i	0.572552	−	0.819868i	\(-0.305953\pi\)
−0.996303	+	0.0859109i	\(0.972620\pi\)
\(13\)	31.4823			0.671662			0.335831	−	0.941922i	\(-0.390983\pi\)
							0.335831	+	0.941922i	\(0.390983\pi\)
\(17\)	44.4353	+	76.9643i	0.633950	+	1.09803i	0.986737	+	0.162330i	\(0.0519009\pi\)
							−0.352786	+	0.935704i	\(0.614766\pi\)
\(19\)	−22.2350	+	38.5121i	−0.268476	+	0.465015i	−0.968469	−	0.249136i	\(-0.919853\pi\)
							0.699992	+	0.714151i	\(0.253187\pi\)
\(23\)	−46.4235	+	80.4079i	−0.420869	+	0.728966i	−0.996025	−	0.0890779i	\(-0.971608\pi\)
							0.575156	+	0.818044i	\(0.304941\pi\)
\(29\)	−267.929			−1.71563			−0.857814	−	0.513960i	\(-0.828178\pi\)
							−0.857814	+	0.513960i	\(0.828178\pi\)
\(31\)	47.8543	+	82.8860i	0.277254	+	0.480218i	0.970701	−	0.240289i	\(-0.0772423\pi\)
							−0.693447	+	0.720507i	\(0.743909\pi\)
\(37\)	−74.3982	+	128.861i	−0.330567	+	0.572559i	−0.982623	−	0.185612i	\(-0.940573\pi\)
							0.652056	+	0.758171i	\(0.273907\pi\)
\(41\)	41.0000			0.156174
							\(43\)	168.504			0.597596			0.298798	−	0.954316i	\(-0.403414\pi\)
0.298798	+	0.954316i	\(0.403414\pi\)
\(47\)	123.081	−	213.183i	0.381985	−	0.661617i	−0.609361	−	0.792893i	\(-0.708574\pi\)
							0.991346	+	0.131276i	\(0.0419074\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.4.e.a.247.23	yes	72
7.2	even	3		2009.4.a.j.1.14		36
7.4	even	3	inner	287.4.e.a.165.23	✓	72
7.5	odd	6		2009.4.a.k.1.14		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.4.e.a.165.23	✓	72	7.4	even	3	inner
287.4.e.a.247.23	yes	72	1.1	even	1	trivial
2009.4.a.j.1.14		36	7.2	even	3
2009.4.a.k.1.14		36	7.5	odd	6