Embedded newform 287.4.e.a.247.32

• Label: 287.4.e.a.247.32
• Level: $287$
• Weight: $4$
• Character: 287.247
• 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			247.32
Character	\(\chi\)	\(=\)	287.247
Dual form			287.4.e.a.165.32

$q$-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 \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.32895	−	4.03386i	0.823407	−	1.42618i	−0.0797231	−	0.996817i	\(-0.525404\pi\)
							0.903130	−	0.429366i	\(-0.141263\pi\)
\(3\)	3.17737	+	5.50337i	0.611485	+	1.05912i	0.990990	+	0.133934i	\(0.0427610\pi\)
							−0.379505	+	0.925190i	\(0.623906\pi\)
\(5\)	5.62194	−	9.73749i	0.502842	−	0.870947i	−0.497153	−	0.867663i	\(-0.665621\pi\)
							0.999995	−	0.00328437i	\(-0.00104545\pi\)
\(7\)	2.97547	−	18.2797i	0.160660	−	0.987010i
							\(11\)	14.6373	+	25.3525i	0.401209	+	0.694914i	0.993872	−	0.110536i	\(-0.0352568\pi\)
−0.592663	+	0.805450i	\(0.701923\pi\)
\(13\)	−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	0.649856	−	0.760058i	\(-0.274829\pi\)
							−0.983157	+	0.182763i	\(0.941496\pi\)
\(19\)	−0.881768	+	1.52727i	−0.0106469	+	0.0184410i	−0.871300	−	0.490751i	\(-0.836722\pi\)
							0.860653	+	0.509192i	\(0.170056\pi\)
\(23\)	34.2195	−	59.2699i	0.310228	−	0.537331i	−0.668183	−	0.743997i	\(-0.732928\pi\)
							0.978412	+	0.206665i	\(0.0662611\pi\)
\(29\)	−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	0.972129	−	0.234447i	\(-0.0753280\pi\)
							−0.689102	+	0.724665i	\(0.741995\pi\)
\(37\)	44.1231	−	76.4234i	0.196048	−	0.339566i	−0.751195	−	0.660080i	\(-0.770522\pi\)
							0.947244	+	0.320514i	\(0.103856\pi\)
\(41\)	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	−0.616465	−	0.787382i	\(-0.711436\pi\)
							0.990126	+	0.140183i	\(0.0447692\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.4.e.a.165.32	✓	72	7.4	even	3	inner
287.4.e.a.247.32	yes	72	1.1	even	1	trivial
2009.4.a.j.1.5		36	7.2	even	3
2009.4.a.k.1.5		36	7.5	odd	6