Embedded newform 287.3.bd.a.5.15

• Label: 287.3.bd.a.5.15
• 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			5.15
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19003 + 0.230182i) q^{2} +(5.08333 + 1.36208i) q^{3} +(0.830673 - 0.176565i) q^{4} +(-3.40707 - 3.78394i) q^{5} +(-11.4462 - 1.81290i) q^{6} +(-6.83271 - 1.52119i) q^{7} +(6.59871 - 2.14405i) q^{8} +(16.1908 + 9.34777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19003	+	0.230182i	−1.09502	+	0.115091i	−0.634771	−	0.772701i	\(-0.718905\pi\)
							−0.460246	+	0.887791i	\(0.652239\pi\)
\(3\)	5.08333	+	1.36208i	1.69444	+	0.454025i	0.971531	−	0.236913i	\(-0.0761356\pi\)
							0.722914	+	0.690938i	\(0.242802\pi\)
\(5\)	−3.40707	−	3.78394i	−0.681415	−	0.756788i	0.298888	−	0.954288i	\(-0.403384\pi\)
							−0.980303	+	0.197500i	\(0.936718\pi\)
\(7\)	−6.83271	−	1.52119i	−0.976102	−	0.217313i
							\(11\)	−12.0751	+	0.632830i	−1.09774	+	0.0575300i	−0.592591	−	0.805504i	\(-0.701895\pi\)
−0.505146	+	0.863034i	\(0.668561\pi\)
\(13\)	2.46285	−	15.5498i	0.189450	−	1.19614i	−0.691304	−	0.722564i	\(-0.742964\pi\)
							0.880754	−	0.473574i	\(-0.157036\pi\)
\(17\)	−12.3063	+	0.644944i	−0.723897	+	0.0379379i	−0.410726	−	0.911759i	\(-0.634725\pi\)
							−0.313171	+	0.949697i	\(0.601391\pi\)
\(19\)	−23.3716	−	8.97153i	−1.23009	−	0.472186i	−0.345436	−	0.938442i	\(-0.612269\pi\)
							−0.884650	+	0.466257i	\(0.845602\pi\)
\(23\)	−0.869792	−	8.27552i	−0.0378171	−	0.359805i	−0.997023	−	0.0770986i	\(-0.975434\pi\)
							0.959206	−	0.282707i	\(-0.0912323\pi\)
\(29\)	19.1744	−	9.76984i	0.661186	−	0.336891i	−0.0909917	−	0.995852i	\(-0.529004\pi\)
							0.752177	+	0.658961i	\(0.229004\pi\)
\(31\)	23.5947	+	21.2447i	0.761118	+	0.685314i	0.955296	−	0.295652i	\(-0.0955370\pi\)
							−0.194177	+	0.980966i	\(0.562204\pi\)
\(37\)	−45.4749	−	50.5050i	−1.22905	−	1.36500i	−0.908563	−	0.417748i	\(-0.862820\pi\)
							−0.320489	−	0.947252i	\(-0.603847\pi\)
\(41\)	−35.4546	+	20.5905i	−0.864747	+	0.502208i
							\(43\)	−13.8626	−	19.0802i	−0.322385	−	0.443725i	0.616808	−	0.787113i	\(-0.288425\pi\)
−0.939194	+	0.343388i	\(0.888425\pi\)
\(47\)	15.1774	−	18.7425i	0.322924	−	0.398778i	−0.589671	−	0.807644i	\(-0.700742\pi\)
							0.912594	+	0.408866i	\(0.134076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.15	✓	864
7.3	odd	6	inner	287.3.bd.a.87.40	yes	864
41.33	even	20	inner	287.3.bd.a.33.40	yes	864
287.115	odd	60	inner	287.3.bd.a.115.15	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.15	✓	864	1.1	even	1	trivial
287.3.bd.a.33.40	yes	864	41.33	even	20	inner
287.3.bd.a.87.40	yes	864	7.3	odd	6	inner
287.3.bd.a.115.15	yes	864	287.115	odd	60	inner