Embedded newform 287.2.bb.a.6.8

• Label: 287.2.bb.a.6.8
• 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			6.8
Character	\(\chi\)	\(=\)	287.6
Dual form			287.2.bb.a.48.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.645342 + 1.26655i) q^{2} +(2.64325 - 1.09487i) q^{3} +(-0.0121231 - 0.0166860i) q^{4} +(1.74535 + 0.276436i) q^{5} +(-0.319088 + 4.05439i) q^{6} +(1.93711 + 1.80211i) q^{7} +(-2.77901 + 0.440153i) q^{8} +(3.66673 - 3.66673i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 32 q^{2} - 40 q^{4} - 16 q^{7} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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	0.547285	−	0.836947i	\(-0.315662\pi\)
							0.978799	+	0.204822i	\(0.0656615\pi\)
\(5\)	1.74535	+	0.276436i	0.780544	+	0.123626i	0.533976	−	0.845500i	\(-0.320697\pi\)
							0.246568	+	0.969126i	\(0.420697\pi\)
\(7\)	1.93711	+	1.80211i	0.732160	+	0.681133i
							\(11\)	−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	0.651153	−	0.758947i	\(-0.274286\pi\)
							−0.851465	+	0.524411i	\(0.824286\pi\)
\(17\)	−2.91529	−	4.75732i	−0.707062	−	1.15382i	−0.981361	−	0.192175i	\(-0.938446\pi\)
							0.274299	−	0.961644i	\(-0.411554\pi\)
\(19\)	−3.34644	+	3.91818i	−0.767727	+	0.898893i	−0.997227	−	0.0744159i	\(-0.976291\pi\)
							0.229500	+	0.973309i	\(0.426291\pi\)
\(23\)	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	0.895744	−	0.444570i	\(-0.146644\pi\)
							−0.146011	+	0.989283i	\(0.546644\pi\)
\(37\)	−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	\(0.237768\pi\)
							−0.831002	+	0.556269i	\(0.812232\pi\)

(See \(a_n\) instead)

Twists

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

    

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