Embedded newform 287.3.ba.a.15.14

• Label: 287.3.ba.a.15.14
• Level: $287$
• Weight: $3$
• Character: 287.15
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.ba (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(42\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			15.14
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.83089 - 0.289985i) q^{2} +(-4.82616 + 1.99906i) q^{3} +(-0.536147 - 0.174205i) q^{4} +(-2.39266 - 4.69587i) q^{5} +(9.41587 - 2.26055i) q^{6} +(2.57265 + 0.617638i) q^{7} +(7.53780 + 3.84070i) q^{8} +(12.9316 - 12.9316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{37}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.83089	−	0.289985i	−0.915447	−	0.144993i	−0.319101	−	0.947721i	\(-0.603381\pi\)
							−0.596345	+	0.802728i	\(0.703381\pi\)
\(3\)	−4.82616	+	1.99906i	−1.60872	+	0.666353i	−0.992614	−	0.121312i	\(-0.961290\pi\)
							−0.616104	+	0.787665i	\(0.711290\pi\)
\(5\)	−2.39266	−	4.69587i	−0.478533	−	0.939174i	−0.996486	−	0.0837644i	\(-0.973306\pi\)
							0.517953	−	0.855409i	\(-0.326694\pi\)
\(7\)	2.57265	+	0.617638i	0.367521	+	0.0882341i
							\(11\)	−0.540937	+	0.462004i	−0.0491761	+	0.0420003i	−0.673696	−	0.739008i	\(-0.735294\pi\)
0.624520	+	0.781008i	\(0.285294\pi\)
\(13\)	4.56184	−	7.44425i	0.350911	−	0.572634i	−0.627675	−	0.778476i	\(-0.715993\pi\)
							0.978585	+	0.205842i	\(0.0659931\pi\)
\(17\)	0.876008	−	11.1307i	0.0515299	−	0.654749i	−0.915867	−	0.401483i	\(-0.868495\pi\)
							0.967397	−	0.253267i	\(-0.0815050\pi\)
\(19\)	4.79401	+	7.82312i	0.252317	+	0.411743i	0.953571	−	0.301168i	\(-0.0973764\pi\)
							−0.701255	+	0.712911i	\(0.747376\pi\)
\(23\)	−6.05477	−	8.33368i	−0.263251	−	0.362334i	0.656846	−	0.754025i	\(-0.271890\pi\)
							−0.920097	+	0.391691i	\(0.871890\pi\)
\(29\)	−0.571871	−	7.26631i	−0.0197197	−	0.250562i	−0.998874	−	0.0474440i	\(-0.984892\pi\)
							0.979154	−	0.203118i	\(-0.0651076\pi\)
\(31\)	−42.7615	+	13.8941i	−1.37940	+	0.448195i	−0.902473	−	0.430746i	\(-0.858250\pi\)
							−0.476930	+	0.878941i	\(0.658250\pi\)
\(37\)	−5.43095	+	16.7147i	−0.146782	+	0.451750i	−0.997236	−	0.0743004i	\(-0.976328\pi\)
							0.850453	+	0.526050i	\(0.176328\pi\)
\(41\)	−26.3947	−	31.3739i	−0.643773	−	0.765217i
							\(43\)	28.0619	+	4.44456i	0.652602	+	0.103362i	0.473952	−	0.880551i	\(-0.342827\pi\)
0.178650	+	0.983913i	\(0.442827\pi\)
\(47\)	−12.8298	+	3.08016i	−0.272974	+	0.0655353i	−0.367621	−	0.929976i	\(-0.619828\pi\)
							0.0946471	+	0.995511i	\(0.469828\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.14	✓	672
41.11	odd	40	inner	287.3.ba.a.134.14	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.14	✓	672	1.1	even	1	trivial
287.3.ba.a.134.14	yes	672	41.11	odd	40	inner