Embedded newform 287.3.x.a.31.19

• Label: 287.3.x.a.31.19
• Level: $287$
• Weight: $3$
• Character: 287.31
• 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.x (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			31.19
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.139031 + 1.32279i) q^{2} +(0.972780 - 1.68490i) q^{3} +(2.18215 + 0.463831i) q^{4} +(-0.709834 - 0.639137i) q^{5} +(2.09352 + 1.52103i) q^{6} +(6.41681 + 2.79723i) q^{7} +(-2.56100 + 7.88194i) q^{8} +(2.60740 + 4.51615i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.139031	+	1.32279i	−0.0695153	+	0.661394i	0.903173	+	0.429277i	\(0.141232\pi\)
							−0.972688	+	0.232116i	\(0.925435\pi\)
\(3\)	0.972780	−	1.68490i	0.324260	−	0.561635i	−0.657102	−	0.753801i	\(-0.728218\pi\)
							0.981362	+	0.192167i	\(0.0615514\pi\)
\(5\)	−0.709834	−	0.639137i	−0.141967	−	0.127827i	0.595088	−	0.803661i	\(-0.297117\pi\)
							−0.737055	+	0.675833i	\(0.763784\pi\)
\(7\)	6.41681	+	2.79723i	0.916688	+	0.399605i
							\(11\)	1.99125	−	1.79293i	0.181022	−	0.162993i	−0.573638	−	0.819109i	\(-0.694468\pi\)
0.754660	+	0.656116i	\(0.227802\pi\)
\(13\)	−12.8605	−	9.34367i	−0.989266	−	0.718744i	−0.0295062	−	0.999565i	\(-0.509393\pi\)
							−0.959760	+	0.280820i	\(0.909393\pi\)
\(17\)	−0.779465	−	0.865684i	−0.0458509	−	0.0509226i	0.719783	−	0.694199i	\(-0.244241\pi\)
							−0.765634	+	0.643276i	\(0.777575\pi\)
\(19\)	32.5571	+	14.4954i	1.71353	+	0.762914i	0.997923	+	0.0644141i	\(0.0205179\pi\)
							0.715610	+	0.698500i	\(0.246149\pi\)
\(23\)	−1.83579	+	17.4664i	−0.0798170	+	0.759408i	0.879275	+	0.476314i	\(0.158028\pi\)
							−0.959092	+	0.283094i	\(0.908639\pi\)
\(29\)	4.99741	−	1.62376i	0.172325	−	0.0559917i	−0.221584	−	0.975141i	\(-0.571123\pi\)
							0.393909	+	0.919150i	\(0.371123\pi\)
\(31\)	10.8335	−	9.75452i	0.349467	−	0.314662i	−0.475638	−	0.879641i	\(-0.657783\pi\)
							0.825105	+	0.564979i	\(0.191116\pi\)
\(37\)	−14.7528	+	16.3847i	−0.398724	+	0.442828i	−0.908756	−	0.417327i	\(-0.862967\pi\)
							0.510032	+	0.860156i	\(0.329634\pi\)
\(41\)	−14.2001	−	38.4624i	−0.346343	−	0.938108i
							\(43\)	−7.50828	−	5.45508i	−0.174611	−	0.126862i	0.497047	−	0.867724i	\(-0.334418\pi\)
−0.671658	+	0.740861i	\(0.734418\pi\)
\(47\)	−4.31292	+	41.0347i	−0.0917642	+	0.873079i	0.847710	+	0.530460i	\(0.177981\pi\)
							−0.939474	+	0.342619i	\(0.888686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.19	✓	432
7.5	odd	6	inner	287.3.x.a.236.36	yes	432
41.4	even	10	inner	287.3.x.a.45.36	yes	432
287.250	odd	30	inner	287.3.x.a.250.19	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.19	✓	432	1.1	even	1	trivial
287.3.x.a.45.36	yes	432	41.4	even	10	inner
287.3.x.a.236.36	yes	432	7.5	odd	6	inner
287.3.x.a.250.19	yes	432	287.250	odd	30	inner