Embedded newform 287.3.k.a.124.9

• Label: 287.3.k.a.124.9
• Level: $287$
• Weight: $3$
• Character: 287.124
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(54\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			124.9
Character	\(\chi\)	\(=\)	287.124
Dual form			287.3.k.a.206.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.47349 + 2.55216i) q^{2} +(-1.81510 + 1.04795i) q^{3} +(-2.34234 - 4.05705i) q^{4} +(-2.56042 - 1.47826i) q^{5} -6.17655i q^{6} +(-6.99953 + 0.0810157i) q^{7} +2.01773 q^{8} +(-2.30362 + 3.98998i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 6 q^{3} - 106 q^{4} - 6 q^{5} - 4 q^{7} - 4 q^{8} + 164 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.47349	+	2.55216i	−0.736744	+	1.27608i	0.217209	+	0.976125i	\(0.430305\pi\)
							−0.953954	+	0.299954i	\(0.903029\pi\)
\(3\)	−1.81510	+	1.04795i	−0.605032	+	0.349315i	−0.771019	−	0.636813i	\(-0.780252\pi\)
							0.165987	+	0.986128i	\(0.446919\pi\)
\(5\)	−2.56042	−	1.47826i	−0.512084	−	0.295652i	0.221606	−	0.975136i	\(-0.428870\pi\)
							−0.733690	+	0.679485i	\(0.762203\pi\)
\(7\)	−6.99953	+	0.0810157i	−0.999933	+	0.0115737i
							\(11\)	−2.22976	−	3.86206i	−0.202705	−	0.351096i	0.746694	−	0.665168i	\(-0.231640\pi\)
−0.949399	+	0.314072i	\(0.898307\pi\)
\(13\)			1.06413i			0.0818562i	0.999162	+	0.0409281i	\(0.0130315\pi\)
							−0.999162	+	0.0409281i	\(0.986969\pi\)
\(17\)	0.599579	−	0.346167i	0.0352693	−	0.0203628i	−0.482262	−	0.876027i	\(-0.660185\pi\)
							0.517531	+	0.855664i	\(0.326851\pi\)
\(19\)	27.5853	+	15.9264i	1.45186	+	0.838232i	0.998587	−	0.0531402i	\(-0.0169230\pi\)
							0.453273	+	0.891372i	\(0.350256\pi\)
\(23\)	6.10011	−	10.5657i	0.265222	−	0.459378i	−0.702400	−	0.711783i	\(-0.747888\pi\)
							0.967622	+	0.252405i	\(0.0812214\pi\)
\(29\)	30.5257			1.05261			0.526304	−	0.850296i	\(-0.323577\pi\)
							0.526304	+	0.850296i	\(0.323577\pi\)
\(31\)	−25.8828	+	14.9434i	−0.834928	+	0.482046i	−0.855537	−	0.517742i	\(-0.826773\pi\)
							0.0206091	+	0.999788i	\(0.493439\pi\)
\(37\)	−23.9468	+	41.4772i	−0.647212	+	1.12100i	0.336574	+	0.941657i	\(0.390732\pi\)
							−0.983786	+	0.179347i	\(0.942601\pi\)
\(41\)		−	6.40312i		−	0.156174i
							\(43\)	77.0235			1.79125			0.895623	−	0.444815i	\(-0.146730\pi\)
0.895623	+	0.444815i	\(0.146730\pi\)
\(47\)	−73.7776	−	42.5955i	−1.56974	−	0.906288i	−0.996199	−	0.0871095i	\(-0.972237\pi\)
							−0.573538	−	0.819179i	\(-0.694430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.k.a.124.9	✓	108
7.3	odd	6	inner	287.3.k.a.206.9	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.k.a.124.9	✓	108	1.1	even	1	trivial
287.3.k.a.206.9	yes	108	7.3	odd	6	inner