Embedded newform 287.2.bb.a.6.5

• Label: 287.2.bb.a.6.5
• 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			6.5
Character	\(\chi\)	\(=\)	287.6
Dual form			287.2.bb.a.48.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.966243 + 1.89636i) q^{2} +(-1.91386 + 0.792745i) q^{3} +(-1.48698 - 2.04665i) q^{4} +(-3.95080 - 0.625746i) q^{5} +(0.345921 - 4.39534i) q^{6} +(-1.55491 + 2.14062i) q^{7} +(1.11370 - 0.176393i) q^{8} +(0.913082 - 0.913082i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 32 q^{2} - 40 q^{4} - 16 q^{7} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.966243	+	1.89636i	−0.683237	+	1.34093i	0.245213	+	0.969469i	\(0.421142\pi\)
							−0.928450	+	0.371458i	\(0.878858\pi\)
\(3\)	−1.91386	+	0.792745i	−1.10497	+	0.457692i	−0.859201	−	0.511638i	\(-0.829039\pi\)
							−0.245764	+	0.969330i	\(0.579039\pi\)
\(5\)	−3.95080	−	0.625746i	−1.76685	−	0.279842i	−0.813470	−	0.581606i	\(-0.802424\pi\)
							−0.953383	+	0.301764i	\(0.902424\pi\)
\(7\)	−1.55491	+	2.14062i	−0.587702	+	0.809077i
							\(11\)	4.18307	+	1.00427i	1.26124	+	0.302798i	0.808316	−	0.588749i	\(-0.200379\pi\)
0.452928	+	0.891547i	\(0.350379\pi\)
\(13\)	2.68543	+	3.14424i	0.744805	+	0.872055i	0.995377	−	0.0960430i	\(-0.0306186\pi\)
							−0.250572	+	0.968098i	\(0.580619\pi\)
\(17\)	−2.49674	−	4.07431i	−0.605549	−	0.988166i	−0.997668	−	0.0682517i	\(-0.978258\pi\)
							0.392119	−	0.919914i	\(-0.371742\pi\)
\(19\)	0.167442	−	0.196049i	0.0384138	−	0.0449768i	−0.740866	−	0.671652i	\(-0.765585\pi\)
							0.779280	+	0.626676i	\(0.215585\pi\)
\(23\)	−3.01997	+	0.981248i	−0.629707	+	0.204604i	−0.606445	−	0.795125i	\(-0.707405\pi\)
							−0.0232619	+	0.999729i	\(0.507405\pi\)
\(29\)	−8.00576	−	4.90594i	−1.48663	−	0.911009i	−0.999285	−	0.0378010i	\(-0.987965\pi\)
							−0.487347	−	0.873208i	\(-0.662035\pi\)
\(31\)	0.283495	+	0.205971i	0.0509171	+	0.0369935i	0.612953	−	0.790120i	\(-0.289982\pi\)
							−0.562036	+	0.827113i	\(0.689982\pi\)
\(37\)	−7.38859	+	5.36812i	−1.21468	+	0.882514i	−0.995647	−	0.0932044i	\(-0.970289\pi\)
							−0.219029	+	0.975718i	\(0.570289\pi\)
\(41\)	6.39958	+	0.212904i	0.999447	+	0.0332500i
							\(43\)	2.06275	−	4.04837i	0.314566	−	0.617371i	−0.678543	−	0.734560i	\(-0.737388\pi\)
0.993110	+	0.117189i	\(0.0373884\pi\)
\(47\)	0.0705291	−	0.896158i	0.0102877	−	0.130718i	−0.989629	−	0.143649i	\(-0.954116\pi\)
							0.999916	+	0.0129313i	\(0.00411627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bb.a.6.5	✓	416
7.6	odd	2	inner	287.2.bb.a.6.6	yes	416
41.7	odd	40	inner	287.2.bb.a.48.6	yes	416
287.48	even	40	inner	287.2.bb.a.48.5	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bb.a.6.5	✓	416	1.1	even	1	trivial
287.2.bb.a.6.6	yes	416	7.6	odd	2	inner
287.2.bb.a.48.5	yes	416	287.48	even	40	inner
287.2.bb.a.48.6	yes	416	41.7	odd	40	inner