Embedded newform 287.3.bd.a.5.17

• Label: 287.3.bd.a.5.17
• 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			5.17
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84676 + 0.194102i) q^{2} +(-4.62713 - 1.23984i) q^{3} +(-0.539743 + 0.114726i) q^{4} +(-4.15770 - 4.61759i) q^{5} +(8.78585 + 1.39154i) q^{6} +(5.25002 + 4.63004i) q^{7} +(8.03870 - 2.61193i) q^{8} +(12.0789 + 6.97376i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84676	+	0.194102i	−0.923380	+	0.0970512i	−0.554272	−	0.832335i	\(-0.687003\pi\)
							−0.369108	+	0.929387i	\(0.620337\pi\)
\(3\)	−4.62713	−	1.23984i	−1.54238	−	0.413278i	−0.615344	−	0.788259i	\(-0.710983\pi\)
							−0.927033	+	0.374981i	\(0.877649\pi\)
\(5\)	−4.15770	−	4.61759i	−0.831540	−	0.923519i	0.166504	−	0.986041i	\(-0.446752\pi\)
							−0.998044	+	0.0625222i	\(0.980086\pi\)
\(7\)	5.25002	+	4.63004i	0.750003	+	0.661434i
							\(11\)	12.3187	−	0.645598i	1.11989	−	0.0586907i	0.516608	−	0.856222i	\(-0.327194\pi\)
0.603277	+	0.797531i	\(0.293861\pi\)
\(13\)	3.24478	−	20.4868i	0.249599	−	1.57590i	−0.470733	−	0.882276i	\(-0.656011\pi\)
							0.720332	−	0.693629i	\(-0.243989\pi\)
\(17\)	−3.89522	+	0.204140i	−0.229131	+	0.0120082i	−0.166556	−	0.986032i	\(-0.553265\pi\)
							−0.0625750	+	0.998040i	\(0.519931\pi\)
\(19\)	−19.7605	−	7.58533i	−1.04002	−	0.399228i	−0.222385	−	0.974959i	\(-0.571384\pi\)
							−0.817639	+	0.575731i	\(0.804718\pi\)
\(23\)	2.08950	+	19.8803i	0.0908478	+	0.864360i	0.941133	+	0.338037i	\(0.109763\pi\)
							−0.850285	+	0.526323i	\(0.823570\pi\)
\(29\)	35.8250	−	18.2537i	1.23534	−	0.629440i	0.290474	−	0.956883i	\(-0.406187\pi\)
							0.944871	+	0.327443i	\(0.106187\pi\)
\(31\)	−1.91200	−	1.72158i	−0.0616776	−	0.0555347i	0.637715	−	0.770272i	\(-0.279880\pi\)
							−0.699393	+	0.714737i	\(0.746546\pi\)
\(37\)	−1.81239	−	2.01286i	−0.0489835	−	0.0544017i	0.718155	−	0.695883i	\(-0.244987\pi\)
							−0.767139	+	0.641481i	\(0.778320\pi\)
\(41\)	39.2678	−	11.7915i	0.957751	−	0.287598i
							\(43\)	−21.5427	−	29.6509i	−0.500992	−	0.689557i	0.481376	−	0.876514i	\(-0.340137\pi\)
−0.982368	+	0.186958i	\(0.940137\pi\)
\(47\)	29.8183	−	36.8225i	0.634432	−	0.783458i	−0.353980	−	0.935253i	\(-0.615172\pi\)
							0.988412	+	0.151795i	\(0.0485053\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.17	✓	864
7.3	odd	6	inner	287.3.bd.a.87.38	yes	864
41.33	even	20	inner	287.3.bd.a.33.38	yes	864
287.115	odd	60	inner	287.3.bd.a.115.17	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.17	✓	864	1.1	even	1	trivial
287.3.bd.a.33.38	yes	864	41.33	even	20	inner
287.3.bd.a.87.38	yes	864	7.3	odd	6	inner
287.3.bd.a.115.17	yes	864	287.115	odd	60	inner