Embedded newform 287.2.bb.a.6.7

• Label: 287.2.bb.a.6.7
• Level: $287$
• Weight: $2$
• Character: 287.6
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $416$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.bb (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(26\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			6.7
Character	\(\chi\)	\(=\)	287.6
Dual form			287.2.bb.a.48.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.645342 + 1.26655i) q^{2} +(-2.64325 + 1.09487i) q^{3} +(-0.0121231 - 0.0166860i) q^{4} +(-1.74535 - 0.276436i) q^{5} +(0.319088 - 4.05439i) q^{6} +(1.63135 - 2.08295i) q^{7} +(-2.77901 + 0.440153i) q^{8} +(3.66673 - 3.66673i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 32 q^{2} - 40 q^{4} - 16 q^{7} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.645342	+	1.26655i	−0.456325	+	0.895589i	0.542144	+	0.840285i	\(0.317613\pi\)
							−0.998470	+	0.0553035i	\(0.982387\pi\)
\(3\)	−2.64325	+	1.09487i	−1.52608	+	0.632125i	−0.978799	−	0.204822i	\(-0.934338\pi\)
							−0.547285	+	0.836947i	\(0.684338\pi\)
\(5\)	−1.74535	−	0.276436i	−0.780544	−	0.123626i	−0.246568	−	0.969126i	\(-0.579303\pi\)
							−0.533976	+	0.845500i	\(0.679303\pi\)
\(7\)	1.63135	−	2.08295i	0.616593	−	0.787282i
							\(11\)	−5.14360	−	1.23487i	−1.55086	−	0.372327i	−0.634574	−	0.772862i	\(-0.718824\pi\)
−0.916281	+	0.400535i	\(0.868824\pi\)
\(13\)	0.722238	+	0.845632i	0.200313	+	0.234536i	0.851465	−	0.524411i	\(-0.175714\pi\)
							−0.651153	+	0.758947i	\(0.725714\pi\)
\(17\)	2.91529	+	4.75732i	0.707062	+	1.15382i	0.981361	+	0.192175i	\(0.0615542\pi\)
							−0.274299	+	0.961644i	\(0.588446\pi\)
\(19\)	3.34644	−	3.91818i	0.767727	−	0.898893i	−0.229500	−	0.973309i	\(-0.573709\pi\)
							0.997227	+	0.0744159i	\(0.0237092\pi\)
\(23\)	7.30599	−	2.37386i	1.52340	−	0.494984i	0.576664	−	0.816981i	\(-0.304354\pi\)
							0.946741	+	0.321997i	\(0.104354\pi\)
\(29\)	−4.08442	−	2.50294i	−0.758458	−	0.464784i	0.0887442	−	0.996054i	\(-0.471715\pi\)
							−0.847202	+	0.531271i	\(0.821715\pi\)
\(31\)	−4.17434	−	3.03283i	−0.749733	−	0.544713i	0.146011	−	0.989283i	\(-0.453356\pi\)
							−0.895744	+	0.444570i	\(0.853356\pi\)
\(37\)	−4.65941	+	3.38526i	−0.766002	+	0.556533i	−0.900745	−	0.434348i	\(-0.856979\pi\)
							0.134744	+	0.990880i	\(0.456979\pi\)
\(41\)	−6.06053	−	2.06638i	−0.946496	−	0.322714i
							\(43\)	1.46572	−	2.87665i	0.223521	−	0.438684i	−0.751826	−	0.659361i	\(-0.770827\pi\)
0.975347	+	0.220677i	\(0.0708267\pi\)
\(47\)	0.666717	−	8.47145i	0.0972507	−	1.23569i	−0.733751	−	0.679418i	\(-0.762232\pi\)
							0.831002	−	0.556269i	\(-0.187768\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bb.a.6.7	✓	416
7.6	odd	2	inner	287.2.bb.a.6.8	yes	416
41.7	odd	40	inner	287.2.bb.a.48.8	yes	416
287.48	even	40	inner	287.2.bb.a.48.7	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bb.a.6.7	✓	416	1.1	even	1	trivial
287.2.bb.a.6.8	yes	416	7.6	odd	2	inner
287.2.bb.a.48.7	yes	416	287.48	even	40	inner
287.2.bb.a.48.8	yes	416	41.7	odd	40	inner