Embedded newform 287.3.bd.a.5.14

• Label: 287.3.bd.a.5.14
• 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			5.14
Character	\(\chi\)	\(=\)	287.5
Dual form			287.3.bd.a.115.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22184 + 0.233525i) q^{2} +(-2.69284 - 0.721545i) q^{3} +(0.969471 - 0.206067i) q^{4} +(2.30180 + 2.55641i) q^{5} +(6.15158 + 0.974314i) q^{6} +(-6.38644 - 2.86590i) q^{7} +(6.39307 - 2.07723i) q^{8} +(-1.06346 - 0.613988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22184	+	0.233525i	−1.11092	+	0.116763i	−0.642167	−	0.766565i	\(-0.721964\pi\)
							−0.468756	+	0.883328i	\(0.655298\pi\)
\(3\)	−2.69284	−	0.721545i	−0.897614	−	0.240515i	−0.219623	−	0.975585i	\(-0.570483\pi\)
							−0.677991	+	0.735070i	\(0.737149\pi\)
\(5\)	2.30180	+	2.55641i	0.460361	+	0.511283i	0.927971	−	0.372652i	\(-0.121552\pi\)
							−0.467610	+	0.883935i	\(0.654885\pi\)
\(7\)	−6.38644	−	2.86590i	−0.912349	−	0.409414i
							\(11\)	−9.39099	+	0.492161i	−0.853726	+	0.0447419i	−0.474177	−	0.880430i	\(-0.657254\pi\)
−0.379549	+	0.925172i	\(0.623921\pi\)
\(13\)	0.506815	−	3.19990i	0.0389858	−	0.246146i	−0.960498	−	0.278288i	\(-0.910233\pi\)
							0.999483	+	0.0321420i	\(0.0102329\pi\)
\(17\)	2.30733	−	0.120922i	0.135725	−	0.00711306i	0.0156477	−	0.999878i	\(-0.495019\pi\)
							0.120078	+	0.992765i	\(0.461686\pi\)
\(19\)	−19.9604	−	7.66207i	−1.05055	−	0.403267i	−0.229011	−	0.973424i	\(-0.573549\pi\)
							−0.821536	+	0.570157i	\(0.806882\pi\)
\(23\)	1.45837	+	13.8755i	0.0634074	+	0.603282i	0.979378	+	0.202036i	\(0.0647559\pi\)
							−0.915971	+	0.401245i	\(0.868577\pi\)
\(29\)	28.3357	−	14.4378i	0.977093	−	0.497854i	0.108885	−	0.994054i	\(-0.465272\pi\)
							0.868208	+	0.496200i	\(0.165272\pi\)
\(31\)	35.2666	+	31.7542i	1.13763	+	1.02433i	0.999424	+	0.0339263i	\(0.0108012\pi\)
							0.138209	+	0.990403i	\(0.455866\pi\)
\(37\)	27.2227	+	30.2339i	0.735749	+	0.817132i	0.988630	−	0.150366i	\(-0.0480451\pi\)
							−0.252882	+	0.967497i	\(0.581378\pi\)
\(41\)	40.4605	−	6.62941i	0.986841	−	0.161693i
							\(43\)	13.2891	+	18.2909i	0.309050	+	0.425370i	0.935085	−	0.354425i	\(-0.115323\pi\)
−0.626035	+	0.779795i	\(0.715323\pi\)
\(47\)	−18.0205	+	22.2534i	−0.383414	+	0.473477i	−0.931917	−	0.362671i	\(-0.881865\pi\)
							0.548503	+	0.836149i	\(0.315198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.5.14	✓	864
7.3	odd	6	inner	287.3.bd.a.87.41	yes	864
41.33	even	20	inner	287.3.bd.a.33.41	yes	864
287.115	odd	60	inner	287.3.bd.a.115.14	yes	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.14	✓	864	1.1	even	1	trivial
287.3.bd.a.33.41	yes	864	41.33	even	20	inner
287.3.bd.a.87.41	yes	864	7.3	odd	6	inner
287.3.bd.a.115.14	yes	864	287.115	odd	60	inner