Embedded newform 287.3.bd.a.5.10

• Label: 287.3.bd.a.5.10
• 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			5.10
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.89312 + 0.304080i) q^{2} +(-0.510002 - 0.136655i) q^{3} +(4.36511 - 0.927833i) q^{4} +(3.45934 + 3.84199i) q^{5} +(1.51705 + 0.240278i) q^{6} +(6.80668 + 1.63372i) q^{7} +(-1.27995 + 0.415880i) q^{8} +(-7.55280 - 4.36061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.89312	+	0.304080i	−1.44656	+	0.152040i	−0.795025	−	0.606577i	\(-0.792542\pi\)
							−0.651537	+	0.758617i	\(0.725876\pi\)
\(3\)	−0.510002	−	0.136655i	−0.170001	−	0.0455516i	0.172815	−	0.984954i	\(-0.444714\pi\)
							−0.342815	+	0.939403i	\(0.611381\pi\)
\(5\)	3.45934	+	3.84199i	0.691869	+	0.768398i	0.982060	−	0.188571i	\(-0.0603855\pi\)
							−0.290191	+	0.956969i	\(0.593719\pi\)
\(7\)	6.80668	+	1.63372i	0.972383	+	0.233389i
							\(11\)	18.2278	−	0.955280i	1.65708	−	0.0868436i	0.799321	−	0.600904i	\(-0.205193\pi\)
0.857754	+	0.514060i	\(0.171859\pi\)
\(13\)	0.0487800	−	0.307985i	0.00375231	−	0.0236912i	−0.985741	−	0.168269i	\(-0.946182\pi\)
							0.989493	+	0.144578i	\(0.0461823\pi\)
\(17\)	−12.4992	+	0.655057i	−0.735249	+	0.0385328i	−0.416285	−	0.909234i	\(-0.636668\pi\)
							−0.318964	+	0.947767i	\(0.603335\pi\)
\(19\)	0.605136	+	0.232290i	0.0318493	+	0.0122258i	0.374240	−	0.927332i	\(-0.377903\pi\)
							−0.342391	+	0.939557i	\(0.611237\pi\)
\(23\)	−3.09008	−	29.4001i	−0.134351	−	1.27827i	−0.829135	−	0.559048i	\(-0.811167\pi\)
							0.694784	−	0.719219i	\(-0.255500\pi\)
\(29\)	28.5831	−	14.5638i	0.985626	−	0.502201i	0.114586	−	0.993413i	\(-0.463446\pi\)
							0.871040	+	0.491212i	\(0.163446\pi\)
\(31\)	39.6158	+	35.6703i	1.27793	+	1.15065i	0.980630	+	0.195869i	\(0.0627528\pi\)
							0.297300	+	0.954784i	\(0.403914\pi\)
\(37\)	26.9717	+	29.9551i	0.728965	+	0.809598i	0.987702	−	0.156350i	\(-0.0499728\pi\)
							−0.258736	+	0.965948i	\(0.583306\pi\)
\(41\)	−40.9986	−	0.335276i	−0.999967	−	0.00817747i
							\(43\)	29.0399	+	39.9699i	0.675346	+	0.929533i	0.999866	−	0.0163451i	\(-0.00520303\pi\)
−0.324521	+	0.945879i	\(0.605203\pi\)
\(47\)	−22.1330	+	27.3319i	−0.470914	+	0.581530i	−0.956159	−	0.292848i	\(-0.905397\pi\)
							0.485245	+	0.874378i	\(0.338730\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.10	✓	864
7.3	odd	6	inner	287.3.bd.a.87.45	yes	864
41.33	even	20	inner	287.3.bd.a.33.45	yes	864
287.115	odd	60	inner	287.3.bd.a.115.10	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.10	✓	864	1.1	even	1	trivial
287.3.bd.a.33.45	yes	864	41.33	even	20	inner
287.3.bd.a.87.45	yes	864	7.3	odd	6	inner
287.3.bd.a.115.10	yes	864	287.115	odd	60	inner