Embedded newform 287.3.k.a.124.7

• Label: 287.3.k.a.124.7
• 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			124.7
Character	\(\chi\)	\(=\)	287.124
Dual form			287.3.k.a.206.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62412 + 2.81306i) q^{2} +(-4.24432 + 2.45046i) q^{3} +(-3.27555 - 5.67343i) q^{4} +(2.25447 + 1.30162i) q^{5} -15.9194i q^{6} +(0.386365 - 6.98933i) q^{7} +8.28663 q^{8} +(7.50952 - 13.0069i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 6 q^{3} - 106 q^{4} - 6 q^{5} - 4 q^{7} - 4 q^{8} + 164 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62412	+	2.81306i	−0.812062	+	1.40653i	0.0993570	+	0.995052i	\(0.468321\pi\)
							−0.911419	+	0.411480i	\(0.865012\pi\)
\(3\)	−4.24432	+	2.45046i	−1.41477	+	0.816820i	−0.995833	−	0.0911929i	\(-0.970932\pi\)
							−0.418941	+	0.908013i	\(0.637599\pi\)
\(5\)	2.25447	+	1.30162i	0.450894	+	0.260324i	0.708208	−	0.706004i	\(-0.249504\pi\)
							−0.257314	+	0.966328i	\(0.582837\pi\)
\(7\)	0.386365	−	6.98933i	0.0551949	−	0.998476i
							\(11\)	8.36688	+	14.4919i	0.760625	+	1.31744i	0.942529	+	0.334125i	\(0.108441\pi\)
−0.181903	+	0.983316i	\(0.558226\pi\)
\(13\)		−	4.50686i		−	0.346682i	−0.984862	−	0.173341i	\(-0.944544\pi\)
							0.984862	−	0.173341i	\(-0.0554562\pi\)
\(17\)	27.7517	−	16.0225i	1.63246	−	0.942498i	0.649122	−	0.760684i	\(-0.275136\pi\)
							0.983333	−	0.181814i	\(-0.0581969\pi\)
\(19\)	5.36469	+	3.09731i	0.282352	+	0.163016i	0.634488	−	0.772933i	\(-0.281211\pi\)
							−0.352136	+	0.935949i	\(0.614544\pi\)
\(23\)	5.00720	−	8.67273i	0.217705	−	0.377075i	−0.736401	−	0.676545i	\(-0.763476\pi\)
							0.954106	+	0.299470i	\(0.0968098\pi\)
\(29\)	−41.8247			−1.44223			−0.721115	−	0.692815i	\(-0.756370\pi\)
							−0.721115	+	0.692815i	\(0.756370\pi\)
\(31\)	−19.1003	+	11.0276i	−0.616139	+	0.355728i	−0.775364	−	0.631514i	\(-0.782434\pi\)
							0.159225	+	0.987242i	\(0.449100\pi\)
\(37\)	17.3497	−	30.0506i	0.468912	−	0.812179i	−0.530456	−	0.847712i	\(-0.677979\pi\)
							0.999369	+	0.0355327i	\(0.0113128\pi\)
\(41\)			6.40312i			0.156174i
							\(43\)	63.4324			1.47517			0.737586	−	0.675253i	\(-0.235966\pi\)
0.737586	+	0.675253i	\(0.235966\pi\)
\(47\)	55.0351	+	31.7745i	1.17096	+	0.676054i	0.953906	−	0.300106i	\(-0.0970221\pi\)
							0.217053	+	0.976160i	\(0.430355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.k.a.124.7	✓	108
7.3	odd	6	inner	287.3.k.a.206.7	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.k.a.124.7	✓	108	1.1	even	1	trivial
287.3.k.a.206.7	yes	108	7.3	odd	6	inner