Embedded newform 287.3.bd.a.5.1

• Label: 287.3.bd.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.86332 + 0.406051i) q^{2} +(0.385359 + 0.103257i) q^{3} +(10.8478 - 2.30576i) q^{4} +(5.36436 + 5.95772i) q^{5} +(-1.53069 - 0.242438i) q^{6} +(0.114923 - 6.99906i) q^{7} +(-26.1942 + 8.51102i) q^{8} +(-7.65639 - 4.42042i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.86332	+	0.406051i	−1.93166	+	0.203026i	−0.990757	−	0.135651i	\(-0.956687\pi\)
							−0.940903	+	0.338676i	\(0.890021\pi\)
\(3\)	0.385359	+	0.103257i	0.128453	+	0.0344189i	0.322473	−	0.946579i	\(-0.395486\pi\)
							−0.194020	+	0.980998i	\(0.562153\pi\)
\(5\)	5.36436	+	5.95772i	1.07287	+	1.19154i	0.980643	+	0.195806i	\(0.0627324\pi\)
							0.0922286	+	0.995738i	\(0.470601\pi\)
\(7\)	0.114923	−	6.99906i	0.0164176	−	0.999865i
							\(11\)	−13.1957	+	0.691556i	−1.19961	+	0.0628687i	−0.641647	−	0.767000i	\(-0.721749\pi\)
−0.557959	+	0.829868i	\(0.688415\pi\)
\(13\)	−0.481691	+	3.04128i	−0.0370532	+	0.233944i	−0.999264	−	0.0383613i	\(-0.987786\pi\)
							0.962211	+	0.272306i	\(0.0877862\pi\)
\(17\)	−4.80160	+	0.251641i	−0.282447	+	0.0148024i	−0.193033	−	0.981192i	\(-0.561833\pi\)
							−0.0894134	+	0.995995i	\(0.528499\pi\)
\(19\)	−23.1636	−	8.89169i	−1.21914	−	0.467983i	−0.338137	−	0.941097i	\(-0.609797\pi\)
							−0.881002	+	0.473113i	\(0.843130\pi\)
\(23\)	−0.711203	−	6.76664i	−0.0309219	−	0.294202i	−0.999044	−	0.0437222i	\(-0.986078\pi\)
							0.968122	−	0.250480i	\(-0.0805883\pi\)
\(29\)	−39.6365	+	20.1958i	−1.36678	+	0.696407i	−0.974698	−	0.223526i	\(-0.928243\pi\)
							−0.392077	+	0.919932i	\(0.628243\pi\)
\(31\)	19.9837	+	17.9934i	0.644636	+	0.580433i	0.925236	−	0.379392i	\(-0.123867\pi\)
							−0.280600	+	0.959825i	\(0.590533\pi\)
\(37\)	−43.6155	−	48.4399i	−1.17880	−	1.30919i	−0.941211	−	0.337819i	\(-0.890311\pi\)
							−0.237585	−	0.971367i	\(-0.576356\pi\)
\(41\)	−12.7648	+	38.9623i	−0.311337	+	0.950300i
							\(43\)	18.3758	+	25.2921i	0.427345	+	0.588189i	0.967341	−	0.253478i	\(-0.0815745\pi\)
−0.539997	+	0.841667i	\(0.681575\pi\)
\(47\)	26.6056	−	32.8552i	0.566077	−	0.699047i	−0.411072	−	0.911603i	\(-0.634845\pi\)
							0.977149	+	0.212556i	\(0.0681787\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.1	✓	864
7.3	odd	6	inner	287.3.bd.a.87.54	yes	864
41.33	even	20	inner	287.3.bd.a.33.54	yes	864
287.115	odd	60	inner	287.3.bd.a.115.1	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.1	✓	864	1.1	even	1	trivial
287.3.bd.a.33.54	yes	864	41.33	even	20	inner
287.3.bd.a.87.54	yes	864	7.3	odd	6	inner
287.3.bd.a.115.1	yes	864	287.115	odd	60	inner