Embedded newform 287.3.bd.a.5.8

• Label: 287.3.bd.a.5.8
• Level: $287$
• Weight: $3$
• Character: 287.5
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $864$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.bd (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(864\)
Relative dimension:	\(54\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			5.8
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.15003 + 0.331082i) q^{2} +(5.01583 + 1.34399i) q^{3} +(5.90050 - 1.25419i) q^{4} +(3.23952 + 3.59786i) q^{5} +(-16.2450 - 2.57296i) q^{6} +(4.02106 + 5.72984i) q^{7} +(-6.12207 + 1.98918i) q^{8} +(15.5580 + 8.98244i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.15003	+	0.331082i	−1.57502	+	0.165541i	−0.851346	−	0.524605i	\(-0.824213\pi\)
							−0.723670	+	0.690146i	\(0.757546\pi\)
\(3\)	5.01583	+	1.34399i	1.67194	+	0.447996i	0.965634	−	0.259906i	\(-0.0836916\pi\)
							0.706310	+	0.707902i	\(0.250358\pi\)
\(5\)	3.23952	+	3.59786i	0.647905	+	0.719571i	0.974197	−	0.225697i	\(-0.0724661\pi\)
							−0.326292	+	0.945269i	\(0.605799\pi\)
\(7\)	4.02106	+	5.72984i	0.574437	+	0.818549i
							\(11\)	−10.4177	+	0.545970i	−0.947066	+	0.0496336i	−0.519611	−	0.854403i	\(-0.673923\pi\)
−0.427455	+	0.904037i	\(0.640590\pi\)
\(13\)	−2.91020	+	18.3743i	−0.223861	+	1.41340i	0.578069	+	0.815988i	\(0.303806\pi\)
							−0.801931	+	0.597417i	\(0.796194\pi\)
\(17\)	21.1675	−	1.10934i	1.24515	−	0.0652555i	0.581654	−	0.813436i	\(-0.302406\pi\)
							0.663495	+	0.748181i	\(0.269072\pi\)
\(19\)	−23.1402	−	8.88270i	−1.21791	−	0.467511i	−0.337318	−	0.941391i	\(-0.609520\pi\)
							−0.880589	+	0.473880i	\(0.842853\pi\)
\(23\)	−3.85094	−	36.6392i	−0.167432	−	1.59301i	−0.679244	−	0.733913i	\(-0.737692\pi\)
							0.511812	−	0.859098i	\(-0.328975\pi\)
\(29\)	22.5681	−	11.4990i	0.778212	−	0.396519i	−0.0193064	−	0.999814i	\(-0.506146\pi\)
							0.797518	+	0.603295i	\(0.206146\pi\)
\(31\)	−43.7741	−	39.4144i	−1.41207	−	1.27143i	−0.914844	−	0.403808i	\(-0.867686\pi\)
							−0.497223	−	0.867623i	\(-0.665647\pi\)
\(37\)	23.1634	+	25.7256i	0.626038	+	0.695286i	0.969836	−	0.243758i	\(-0.0783802\pi\)
							−0.343798	+	0.939044i	\(0.611714\pi\)
\(41\)	8.42543	+	40.1250i	0.205498	+	0.978657i
							\(43\)	−6.91986	−	9.52436i	−0.160927	−	0.221497i	0.720937	−	0.693000i	\(-0.243712\pi\)
−0.881864	+	0.471503i	\(0.843712\pi\)
\(47\)	15.7402	−	19.4375i	0.334898	−	0.413564i	−0.581682	−	0.813416i	\(-0.697605\pi\)
							0.916580	+	0.399852i	\(0.130939\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.8	✓	864
7.3	odd	6	inner	287.3.bd.a.87.47	yes	864
41.33	even	20	inner	287.3.bd.a.33.47	yes	864
287.115	odd	60	inner	287.3.bd.a.115.8	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.8	✓	864	1.1	even	1	trivial
287.3.bd.a.33.47	yes	864	41.33	even	20	inner
287.3.bd.a.87.47	yes	864	7.3	odd	6	inner
287.3.bd.a.115.8	yes	864	287.115	odd	60	inner