Embedded newform 287.3.bd.a.5.11

• Label: 287.3.bd.a.5.11
• 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			5.11
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.46927 + 0.259531i) q^{2} +(0.688721 + 0.184542i) q^{3} +(2.11735 - 0.450057i) q^{4} +(-5.13570 - 5.70377i) q^{5} +(-1.74853 - 0.276940i) q^{6} +(2.59590 + 6.50087i) q^{7} +(4.33389 - 1.40817i) q^{8} +(-7.35395 - 4.24580i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.46927	+	0.259531i	−1.23464	+	0.129765i	−0.699272	−	0.714856i	\(-0.746492\pi\)
							−0.535364	+	0.844622i	\(0.679826\pi\)
\(3\)	0.688721	+	0.184542i	0.229574	+	0.0615141i	0.371772	−	0.928324i	\(-0.378750\pi\)
							−0.142198	+	0.989838i	\(0.545417\pi\)
\(5\)	−5.13570	−	5.70377i	−1.02714	−	1.14075i	−0.989946	−	0.141446i	\(-0.954825\pi\)
							−0.0371936	−	0.999308i	\(-0.511842\pi\)
\(7\)	2.59590	+	6.50087i	0.370843	+	0.928696i
							\(11\)	−13.5239	+	0.708757i	−1.22944	+	0.0644324i	−0.655967	−	0.754789i	\(-0.727739\pi\)
−0.573477	+	0.819222i	\(0.694406\pi\)
\(13\)	0.303245	−	1.91461i	0.0233265	−	0.147278i	−0.973276	−	0.229638i	\(-0.926246\pi\)
							0.996603	+	0.0823601i	\(0.0262458\pi\)
\(17\)	25.1302	−	1.31702i	1.47825	−	0.0774717i	0.703931	−	0.710269i	\(-0.251427\pi\)
							0.774318	+	0.632797i	\(0.218093\pi\)
\(19\)	14.5790	+	5.59635i	0.767316	+	0.294545i	0.710366	−	0.703832i	\(-0.248529\pi\)
							0.0569493	+	0.998377i	\(0.481863\pi\)
\(23\)	−0.234921	−	2.23513i	−0.0102140	−	0.0971795i	0.988227	−	0.152993i	\(-0.0488913\pi\)
							−0.998441	+	0.0558139i	\(0.982225\pi\)
\(29\)	−31.3160	+	15.9563i	−1.07986	+	0.550217i	−0.901072	−	0.433670i	\(-0.857218\pi\)
							−0.178790	+	0.983887i	\(0.557218\pi\)
\(31\)	38.9350	+	35.0573i	1.25597	+	1.13088i	0.985774	+	0.168074i	\(0.0537547\pi\)
							0.270194	+	0.962806i	\(0.412912\pi\)
\(37\)	8.20129	+	9.10845i	0.221656	+	0.246174i	0.843709	−	0.536801i	\(-0.180368\pi\)
							−0.622052	+	0.782976i	\(0.713701\pi\)
\(41\)	0.984025	+	40.9882i	0.0240006	+	0.999712i
							\(43\)	29.8505	+	41.0857i	0.694197	+	0.955481i	0.999994	+	0.00338400i	\(0.00107716\pi\)
−0.305797	+	0.952097i	\(0.598923\pi\)
\(47\)	−0.411507	+	0.508169i	−0.00875546	+	0.0108121i	−0.781505	−	0.623899i	\(-0.785548\pi\)
							0.772749	+	0.634711i	\(0.218881\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.11	✓	864
7.3	odd	6	inner	287.3.bd.a.87.44	yes	864
41.33	even	20	inner	287.3.bd.a.33.44	yes	864
287.115	odd	60	inner	287.3.bd.a.115.11	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.11	✓	864	1.1	even	1	trivial
287.3.bd.a.33.44	yes	864	41.33	even	20	inner
287.3.bd.a.87.44	yes	864	7.3	odd	6	inner
287.3.bd.a.115.11	yes	864	287.115	odd	60	inner