Embedded newform 287.2.bb.a.6.16

• Label: 287.2.bb.a.6.16
• 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			6.16
Character	\(\chi\)	\(=\)	287.6
Dual form			287.2.bb.a.48.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.171065 - 0.335734i) q^{2} +(2.78559 - 1.15383i) q^{3} +(1.09212 + 1.50317i) q^{4} +(0.146833 + 0.0232561i) q^{5} +(0.0891375 - 1.13260i) q^{6} +(-2.58883 - 0.545871i) q^{7} +(1.43582 - 0.227411i) q^{8} +(4.30688 - 4.30688i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 32 q^{2} - 40 q^{4} - 16 q^{7} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.171065	−	0.335734i	0.120961	−	0.237400i	−0.822584	−	0.568644i	\(-0.807468\pi\)
							0.943545	+	0.331244i	\(0.107468\pi\)
\(3\)	2.78559	−	1.15383i	1.60826	−	0.666164i	0.615709	−	0.787974i	\(-0.288870\pi\)
							0.992554	+	0.121809i	\(0.0388697\pi\)
\(5\)	0.146833	+	0.0232561i	0.0656657	+	0.0104004i	0.189181	−	0.981942i	\(-0.439417\pi\)
							−0.123515	+	0.992343i	\(0.539417\pi\)
\(7\)	−2.58883	−	0.545871i	−0.978485	−	0.206320i
							\(11\)	−4.43477	−	1.06469i	−1.33713	−	0.321017i	−0.499057	−	0.866569i	\(-0.666320\pi\)
−0.838077	+	0.545552i	\(0.816320\pi\)
\(13\)	−0.857800	−	1.00436i	−0.237911	−	0.278558i	0.628586	−	0.777740i	\(-0.283634\pi\)
							−0.866497	+	0.499182i	\(0.833634\pi\)
\(17\)	4.12722	+	6.73501i	1.00100	+	1.63348i	0.745308	+	0.666720i	\(0.232302\pi\)
							0.255689	+	0.966759i	\(0.417698\pi\)
\(19\)	1.46364	−	1.71370i	0.335781	−	0.393150i	−0.566667	−	0.823947i	\(-0.691768\pi\)
							0.902449	+	0.430797i	\(0.141768\pi\)
\(23\)	−4.79564	+	1.55820i	−0.999960	+	0.324907i	−0.762850	−	0.646576i	\(-0.776200\pi\)
							−0.237110	+	0.971483i	\(0.576200\pi\)
\(29\)	5.36519	+	3.28780i	0.996292	+	0.610528i	0.922196	−	0.386723i	\(-0.126393\pi\)
							0.0740956	+	0.997251i	\(0.476393\pi\)
\(31\)	−3.48279	−	2.53040i	−0.625528	−	0.454472i	0.229320	−	0.973351i	\(-0.426350\pi\)
							−0.854848	+	0.518879i	\(0.826350\pi\)
\(37\)	−4.38096	+	3.18296i	−0.720226	+	0.523275i	−0.886456	−	0.462812i	\(-0.846840\pi\)
							0.166231	+	0.986087i	\(0.446840\pi\)
\(41\)	4.30676	−	4.73834i	0.672602	−	0.740004i
							\(43\)	3.05748	−	6.00064i	0.466261	−	0.915089i	−0.531426	−	0.847105i	\(-0.678343\pi\)
0.997686	−	0.0679835i	\(-0.0216565\pi\)
\(47\)	0.224728	−	2.85544i	0.0327799	−	0.416508i	−0.958533	−	0.284982i	\(-0.908012\pi\)
							0.991313	−	0.131526i	\(-0.0419877\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bb.a.6.16	yes	416
7.6	odd	2	inner	287.2.bb.a.6.15	✓	416
41.7	odd	40	inner	287.2.bb.a.48.15	yes	416
287.48	even	40	inner	287.2.bb.a.48.16	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bb.a.6.15	✓	416	7.6	odd	2	inner
287.2.bb.a.6.16	yes	416	1.1	even	1	trivial
287.2.bb.a.48.15	yes	416	41.7	odd	40	inner
287.2.bb.a.48.16	yes	416	287.48	even	40	inner