Embedded newform 287.3.ba.a.15.20

• Label: 287.3.ba.a.15.20
• Level: $287$
• Weight: $3$
• Character: 287.15
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.ba (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(42\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			15.20
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.140552 - 0.0222612i) q^{2} +(1.74628 - 0.723333i) q^{3} +(-3.78497 - 1.22981i) q^{4} +(-4.24244 - 8.32626i) q^{5} +(-0.261545 + 0.0627914i) q^{6} +(2.57265 + 0.617638i) q^{7} +(1.01178 + 0.515528i) q^{8} +(-3.83768 + 3.83768i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{37}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.140552	−	0.0222612i	−0.0702759	−	0.0111306i	0.121197	−	0.992628i	\(-0.461327\pi\)
							−0.191473	+	0.981498i	\(0.561327\pi\)
\(3\)	1.74628	−	0.723333i	0.582093	−	0.241111i	−0.0721516	−	0.997394i	\(-0.522987\pi\)
							0.654245	+	0.756283i	\(0.272987\pi\)
\(5\)	−4.24244	−	8.32626i	−0.848488	−	1.66525i	−0.741462	−	0.670995i	\(-0.765867\pi\)
							−0.107026	−	0.994256i	\(-0.534133\pi\)
\(7\)	2.57265	+	0.617638i	0.367521	+	0.0882341i
							\(11\)	−7.67679	+	6.55660i	−0.697890	+	0.596054i	−0.925974	−	0.377587i	\(-0.876754\pi\)
0.228084	+	0.973641i	\(0.426754\pi\)
\(13\)	−1.19921	+	1.95693i	−0.0922466	+	0.150533i	−0.895435	−	0.445192i	\(-0.853135\pi\)
							0.803188	+	0.595725i	\(0.203135\pi\)
\(17\)	−0.292981	+	3.72267i	−0.0172342	+	0.218981i	0.982269	+	0.187475i	\(0.0600303\pi\)
							−0.999504	+	0.0315059i	\(0.989970\pi\)
\(19\)	10.9275	+	17.8321i	0.575133	+	0.938532i	0.999455	+	0.0330203i	\(0.0105126\pi\)
							−0.424322	+	0.905512i	\(0.639487\pi\)
\(23\)	−16.1184	−	22.1851i	−0.700800	−	0.964568i	−0.999946	−	0.0103607i	\(-0.996702\pi\)
							0.299147	−	0.954207i	\(-0.403298\pi\)
\(29\)	−0.959319	−	12.1893i	−0.0330800	−	0.420321i	−0.991054	−	0.133464i	\(-0.957390\pi\)
							0.957974	−	0.286856i	\(-0.0926102\pi\)
\(31\)	−14.0662	+	4.57039i	−0.453749	+	0.147432i	−0.526970	−	0.849884i	\(-0.676672\pi\)
							0.0732215	+	0.997316i	\(0.476672\pi\)
\(37\)	21.8054	−	67.1101i	0.589335	−	1.81379i	0.00821821	−	0.999966i	\(-0.497384\pi\)
							0.581117	−	0.813820i	\(-0.302616\pi\)
\(41\)	−16.6912	−	37.4487i	−0.407103	−	0.913382i
							\(43\)	−42.6480	−	6.75477i	−0.991813	−	0.157088i	−0.360619	−	0.932713i	\(-0.617435\pi\)
−0.631194	+	0.775625i	\(0.717435\pi\)
\(47\)	−39.9576	+	9.59297i	−0.850161	+	0.204106i	−0.635038	−	0.772481i	\(-0.719015\pi\)
							−0.215124	+	0.976587i	\(0.569015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.20	✓	672
41.11	odd	40	inner	287.3.ba.a.134.20	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.20	✓	672	1.1	even	1	trivial
287.3.ba.a.134.20	yes	672	41.11	odd	40	inner