Embedded newform 287.4.e.a.165.33

• Label: 287.4.e.a.165.33
• 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			165.33
Character	\(\chi\)	\(=\)	287.165
Dual form			287.4.e.a.247.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.33615 + 4.04633i) q^{2} +(2.28901 - 3.96468i) q^{3} +(-6.91517 + 11.9774i) q^{4} +(-4.39511 - 7.61255i) q^{5} +21.3898 q^{6} +(-9.59497 - 15.8410i) q^{7} -27.2411 q^{8} +(3.02088 + 5.23232i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 5 q^{2} + 6 q^{3} - 117 q^{4} - 4 q^{5} - 24 q^{6} - 30 q^{7} - 78 q^{8} - 236 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.33615	+	4.04633i	0.825953	+	1.43059i	0.901189	+	0.433427i	\(0.142696\pi\)
							−0.0752361	+	0.997166i	\(0.523971\pi\)
\(3\)	2.28901	−	3.96468i	0.440520	−	0.763003i	−0.557208	−	0.830373i	\(-0.688127\pi\)
							0.997728	+	0.0673701i	\(0.0214608\pi\)
\(5\)	−4.39511	−	7.61255i	−0.393110	−	0.680887i	0.599748	−	0.800189i	\(-0.295268\pi\)
							−0.992858	+	0.119302i	\(0.961934\pi\)
\(7\)	−9.59497	−	15.8410i	−0.518080	−	0.855332i
							\(11\)	21.0655	−	36.4865i	0.577408	−	1.00010i	−0.418368	−	0.908278i	\(-0.637398\pi\)
0.995775	−	0.0918216i	\(-0.0292690\pi\)
\(13\)	−83.3095			−1.77738			−0.888688	−	0.458512i	\(-0.848383\pi\)
							−0.888688	+	0.458512i	\(0.848383\pi\)
\(17\)	59.7504	−	103.491i	0.852447	−	1.47648i	−0.0265468	−	0.999648i	\(-0.508451\pi\)
							0.878994	−	0.476834i	\(-0.158216\pi\)
\(19\)	−42.6188	−	73.8178i	−0.514601	−	0.891314i	−0.999856	−	0.0169422i	\(-0.994607\pi\)
							0.485256	−	0.874372i	\(-0.338726\pi\)
\(23\)	−17.1399	−	29.6871i	−0.155388	−	0.269139i	0.777813	−	0.628496i	\(-0.216329\pi\)
							−0.933200	+	0.359357i	\(0.882996\pi\)
\(29\)	127.196			0.814476			0.407238	−	0.913322i	\(-0.366492\pi\)
							0.407238	+	0.913322i	\(0.366492\pi\)
\(31\)	−70.4310	+	121.990i	−0.408057	+	0.706776i	−0.994672	−	0.103090i	\(-0.967127\pi\)
							0.586615	+	0.809866i	\(0.300460\pi\)
\(37\)	29.6323	+	51.3247i	0.131663	+	0.228047i	0.924318	−	0.381624i	\(-0.124635\pi\)
							−0.792655	+	0.609671i	\(0.791302\pi\)
\(41\)	41.0000			0.156174
							\(43\)	79.8833			0.283304			0.141652	−	0.989916i	\(-0.454759\pi\)
0.141652	+	0.989916i	\(0.454759\pi\)
\(47\)	−277.854	−	481.258i	−0.862324	−	1.49359i	−0.869680	−	0.493616i	\(-0.835675\pi\)
							0.00735621	−	0.999973i	\(-0.497658\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.4.e.a.165.33	✓	72
7.2	even	3	inner	287.4.e.a.247.33	yes	72
7.3	odd	6		2009.4.a.k.1.4		36
7.4	even	3		2009.4.a.j.1.4		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.4.e.a.165.33	✓	72	1.1	even	1	trivial
287.4.e.a.247.33	yes	72	7.2	even	3	inner
2009.4.a.j.1.4		36	7.4	even	3
2009.4.a.k.1.4		36	7.3	odd	6