Embedded newform 287.2.bb.a.6.4

• Label: 287.2.bb.a.6.4
• 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			6.4
Character	\(\chi\)	\(=\)	287.6
Dual form			287.2.bb.a.48.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01388 + 1.98985i) q^{2} +(1.23689 - 0.512338i) q^{3} +(-1.75598 - 2.41690i) q^{4} +(-2.31334 - 0.366396i) q^{5} +(-0.234585 + 2.98068i) q^{6} +(-0.970800 - 2.46121i) q^{7} +(2.17809 - 0.344976i) q^{8} +(-0.853904 + 0.853904i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 32 q^{2} - 40 q^{4} - 16 q^{7} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01388	+	1.98985i	−0.716921	+	1.40704i	0.188307	+	0.982110i	\(0.439700\pi\)
							−0.905228	+	0.424926i	\(0.860300\pi\)
\(3\)	1.23689	−	0.512338i	0.714121	−	0.295799i	0.00411243	−	0.999992i	\(-0.498691\pi\)
							0.710009	+	0.704193i	\(0.248691\pi\)
\(5\)	−2.31334	−	0.366396i	−1.03456	−	0.163857i	−0.384013	−	0.923328i	\(-0.625458\pi\)
							−0.650542	+	0.759470i	\(0.725458\pi\)
\(7\)	−0.970800	−	2.46121i	−0.366928	−	0.930249i
							\(11\)	−4.72774	−	1.13503i	−1.42547	−	0.342224i	−0.553933	−	0.832561i	\(-0.686874\pi\)
−0.871533	+	0.490337i	\(0.836874\pi\)
\(13\)	0.440811	+	0.516123i	0.122259	+	0.143147i	0.818160	−	0.574991i	\(-0.194995\pi\)
							−0.695901	+	0.718138i	\(0.744995\pi\)
\(17\)	−1.76730	−	2.88398i	−0.428634	−	0.699467i	0.563296	−	0.826255i	\(-0.309533\pi\)
							−0.991930	+	0.126788i	\(0.959533\pi\)
\(19\)	0.0379304	−	0.0444108i	0.00870183	−	0.0101885i	−0.756038	−	0.654528i	\(-0.772867\pi\)
							0.764740	+	0.644339i	\(0.222867\pi\)
\(23\)	−3.63127	+	1.17987i	−0.757172	+	0.246020i	−0.662064	−	0.749447i	\(-0.730320\pi\)
							−0.0951076	+	0.995467i	\(0.530320\pi\)
\(29\)	0.755714	+	0.463102i	0.140332	+	0.0859959i	0.590893	−	0.806750i	\(-0.298776\pi\)
							−0.450560	+	0.892746i	\(0.648776\pi\)
\(31\)	3.42626	+	2.48932i	0.615375	+	0.447096i	0.851303	−	0.524675i	\(-0.175813\pi\)
							−0.235928	+	0.971770i	\(0.575813\pi\)
\(37\)	6.90399	−	5.01604i	1.13501	−	0.824632i	0.148593	−	0.988898i	\(-0.452526\pi\)
							0.986416	+	0.164266i	\(0.0525256\pi\)
\(41\)	4.57332	−	4.48161i	0.714232	−	0.699909i
							\(43\)	−2.12031	+	4.16134i	−0.323344	+	0.634598i	−0.994267	−	0.106929i	\(-0.965898\pi\)
0.670923	+	0.741527i	\(0.265898\pi\)
\(47\)	0.341545	−	4.33974i	0.0498195	−	0.633017i	−0.920430	−	0.390908i	\(-0.872161\pi\)
							0.970249	−	0.242109i	\(-0.0778390\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bb.a.6.4	yes	416
7.6	odd	2	inner	287.2.bb.a.6.3	✓	416
41.7	odd	40	inner	287.2.bb.a.48.3	yes	416
287.48	even	40	inner	287.2.bb.a.48.4	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bb.a.6.3	✓	416	7.6	odd	2	inner
287.2.bb.a.6.4	yes	416	1.1	even	1	trivial
287.2.bb.a.48.3	yes	416	41.7	odd	40	inner
287.2.bb.a.48.4	yes	416	287.48	even	40	inner