Embedded newform 287.3.v.a.44.9

• Label: 287.3.v.a.44.9
• Level: $287$
• Weight: $3$
• Character: 287.44
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.v (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			44.9
Character	\(\chi\)	\(=\)	287.44
Dual form			287.3.v.a.137.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743556 - 2.77499i) q^{2} +(-4.02310 + 0.529651i) q^{3} +(-3.68360 + 2.12672i) q^{4} +(5.12912 - 1.37434i) q^{5} +(4.46118 + 10.7702i) q^{6} +(-1.22522 - 6.89194i) q^{7} +(0.514865 + 0.514865i) q^{8} +(7.21145 - 1.93230i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 16 q^{6} - 8 q^{7} - 48 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.743556	−	2.77499i	−0.371778	−	1.38750i	−0.857995	−	0.513657i	\(-0.828290\pi\)
							0.486217	−	0.873838i	\(-0.338376\pi\)
\(3\)	−4.02310	+	0.529651i	−1.34103	+	0.176550i	−0.766665	−	0.642047i	\(-0.778085\pi\)
							−0.574367	+	0.818598i	\(0.694752\pi\)
\(5\)	5.12912	−	1.37434i	1.02582	−	0.274869i	0.293597	−	0.955929i	\(-0.405148\pi\)
							0.732227	+	0.681061i	\(0.238481\pi\)
\(7\)	−1.22522	−	6.89194i	−0.175031	−	0.984563i
							\(11\)	−1.05092	−	7.98255i	−0.0955384	−	0.725686i	−0.970433	−	0.241372i	\(-0.922403\pi\)
0.874894	−	0.484314i	\(-0.160931\pi\)
\(13\)	3.33337	−	8.04747i	0.256413	−	0.619036i	−0.742283	−	0.670086i	\(-0.766257\pi\)
							0.998696	+	0.0510507i	\(0.0162570\pi\)
\(17\)	0.533428	−	0.695177i	0.0313781	−	0.0408928i	−0.777401	−	0.629006i	\(-0.783462\pi\)
							0.808779	+	0.588113i	\(0.200129\pi\)
\(19\)	−20.4277	−	2.68935i	−1.07514	−	0.141545i	−0.427896	−	0.903828i	\(-0.640745\pi\)
							−0.647244	+	0.762283i	\(0.724078\pi\)
\(23\)	4.96620	+	2.86724i	0.215922	+	0.124663i	0.604061	−	0.796938i	\(-0.293549\pi\)
							−0.388139	+	0.921601i	\(0.626882\pi\)
\(29\)	−12.3388	+	29.7885i	−0.425475	+	1.02719i	0.555230	+	0.831697i	\(0.312630\pi\)
							−0.980705	+	0.195491i	\(0.937370\pi\)
\(31\)	12.7835	−	7.38054i	0.412370	−	0.238082i	−0.279437	−	0.960164i	\(-0.590148\pi\)
							0.691808	+	0.722082i	\(0.256815\pi\)
\(37\)	0.992686	−	1.71938i	0.0268294	−	0.0464698i	−0.852299	−	0.523055i	\(-0.824792\pi\)
							0.879128	+	0.476585i	\(0.158126\pi\)
\(41\)	−5.63931	+	40.6103i	−0.137544	+	0.990496i
							\(43\)	−4.18377	+	4.18377i	−0.0972970	+	0.0972970i	−0.754080	−	0.656783i	\(-0.771917\pi\)
0.656783	+	0.754080i	\(0.271917\pi\)
\(47\)	−82.0874	−	10.8070i	−1.74654	−	0.229936i	−0.811469	−	0.584396i	\(-0.801331\pi\)
							−0.935071	+	0.354460i	\(0.884665\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.v.a.44.9	✓	432
7.4	even	3	inner	287.3.v.a.249.46	yes	432
41.14	odd	8	inner	287.3.v.a.219.46	yes	432
287.137	odd	24	inner	287.3.v.a.137.9	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.v.a.44.9	✓	432	1.1	even	1	trivial
287.3.v.a.137.9	yes	432	287.137	odd	24	inner
287.3.v.a.219.46	yes	432	41.14	odd	8	inner
287.3.v.a.249.46	yes	432	7.4	even	3	inner