Embedded newform 287.4.e.a.165.32

• Label: 287.4.e.a.165.32
• Level: $287$
• Weight: $4$
• Character: 287.165
• Analytic conductor: $16.934$
• Analytic rank: $0$
• Dimension: $72$
• 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			165.32
Character	\(\chi\)	\(=\)	287.165
Dual form			287.4.e.a.247.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.32895 + 4.03386i) q^{2} +(3.17737 - 5.50337i) q^{3} +(-6.84799 + 11.8611i) q^{4} +(5.62194 + 9.73749i) q^{5} +29.5997 q^{6} +(2.97547 + 18.2797i) q^{7} -26.5313 q^{8} +(-6.69137 - 11.5898i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 5 q^{2} + 6 q^{3} - 117 q^{4} - 4 q^{5} - 24 q^{6} - 30 q^{7} - 78 q^{8} - 236 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.32895	+	4.03386i	0.823407	+	1.42618i	0.903130	+	0.429366i	\(0.141263\pi\)
							−0.0797231	+	0.996817i	\(0.525404\pi\)
\(3\)	3.17737	−	5.50337i	0.611485	−	1.05912i	−0.379505	−	0.925190i	\(-0.623906\pi\)
							0.990990	−	0.133934i	\(-0.0427610\pi\)
\(5\)	5.62194	+	9.73749i	0.502842	+	0.870947i	0.999995	+	0.00328437i	\(0.00104545\pi\)
							−0.497153	+	0.867663i	\(0.665621\pi\)
\(7\)	2.97547	+	18.2797i	0.160660	+	0.987010i
							\(11\)	14.6373	−	25.3525i	0.401209	−	0.694914i	−0.592663	−	0.805450i	\(-0.701923\pi\)
0.993872	+	0.110536i	\(0.0352568\pi\)
\(13\)	−35.0862			−0.748551			−0.374275	−	0.927318i	\(-0.622108\pi\)
							−0.374275	+	0.927318i	\(0.622108\pi\)
\(17\)	−23.3620	+	40.4642i	−0.333301	+	0.577295i	−0.983157	−	0.182763i	\(-0.941496\pi\)
							0.649856	+	0.760058i	\(0.274829\pi\)
\(19\)	−0.881768	−	1.52727i	−0.0106469	−	0.0184410i	0.860653	−	0.509192i	\(-0.170056\pi\)
							−0.871300	+	0.490751i	\(0.836722\pi\)
\(23\)	34.2195	+	59.2699i	0.310228	+	0.537331i	0.978412	−	0.206665i	\(-0.0662611\pi\)
							−0.668183	+	0.743997i	\(0.732928\pi\)
\(29\)	−33.0184			−0.211426			−0.105713	−	0.994397i	\(-0.533713\pi\)
							−0.105713	+	0.994397i	\(0.533713\pi\)
\(31\)	48.8507	−	84.6119i	0.283027	−	0.490218i	−0.689102	−	0.724665i	\(-0.741995\pi\)
							0.972129	+	0.234447i	\(0.0753280\pi\)
\(37\)	44.1231	+	76.4234i	0.196048	+	0.339566i	0.947244	−	0.320514i	\(-0.103856\pi\)
							−0.751195	+	0.660080i	\(0.770522\pi\)
\(41\)	41.0000			0.156174
							\(43\)	−128.780			−0.456714			−0.228357	−	0.973577i	\(-0.573335\pi\)
−0.228357	+	0.973577i	\(0.573335\pi\)
\(47\)	120.399	+	208.538i	0.373661	+	0.647199i	0.990126	−	0.140183i	\(-0.0447692\pi\)
							−0.616465	+	0.787382i	\(0.711436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.4.e.a.165.32	✓	72
7.2	even	3	inner	287.4.e.a.247.32	yes	72
7.3	odd	6		2009.4.a.k.1.5		36
7.4	even	3		2009.4.a.j.1.5		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.4.e.a.165.32	✓	72	1.1	even	1	trivial
287.4.e.a.247.32	yes	72	7.2	even	3	inner
2009.4.a.j.1.5		36	7.4	even	3
2009.4.a.k.1.5		36	7.3	odd	6