Embedded newform 483.2.bf.a.157.6

• Label: 483.2.bf.a.157.6
• Level: $483$
• Weight: $2$
• Character: 483.157
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $640$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			157.6
Character	\(\chi\)	\(=\)	483.157
Dual form			483.2.bf.a.40.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18925 + 1.67006i) q^{2} +(-0.971812 - 0.235759i) q^{3} +(-0.720670 - 2.08224i) q^{4} +(0.0587108 - 1.23249i) q^{5} +(1.54946 - 1.34261i) q^{6} +(2.52739 + 0.782484i) q^{7} +(0.400177 + 0.117503i) q^{8} +(0.888835 + 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18925	+	1.67006i	−0.840925	+	1.18091i	0.140692	+	0.990053i	\(0.455067\pi\)
							−0.981617	+	0.190861i	\(0.938872\pi\)
\(3\)	−0.971812	−	0.235759i	−0.561076	−	0.136115i
							\(5\)	0.0587108	−	1.23249i	0.0262563	−	0.551187i	−0.946968	−	0.321328i	\(-0.895871\pi\)
0.973224	−	0.229858i	\(-0.0738263\pi\)
\(7\)	2.52739	+	0.782484i	0.955265	+	0.295751i
							\(11\)	−0.582996	+	0.415150i	−0.175780	+	0.125172i	−0.664561	−	0.747234i	\(-0.731381\pi\)
0.488781	+	0.872407i	\(0.337442\pi\)
\(13\)	−6.47361	−	0.930764i	−1.79546	−	0.258148i	−0.837783	−	0.546004i	\(-0.816148\pi\)
							−0.957674	+	0.287856i	\(0.907057\pi\)
\(17\)	−4.97258	+	0.958386i	−1.20603	+	0.232443i	−0.752374	−	0.658736i	\(-0.771092\pi\)
							−0.453653	+	0.891178i	\(0.649879\pi\)
\(19\)	−0.491720	−	0.0947713i	−0.112808	−	0.0217420i	0.132535	−	0.991178i	\(-0.457688\pi\)
							−0.245343	+	0.969436i	\(0.578901\pi\)
\(23\)	−3.36094	−	3.42112i	−0.700805	−	0.713353i
							\(29\)	−4.92177	−	5.68003i	−0.913950	−	1.05475i	−0.998298	−	0.0583179i	\(-0.981426\pi\)
0.0843483	−	0.996436i	\(-0.473119\pi\)
\(31\)	0.0916422	+	0.0961115i	0.0164594	+	0.0172621i	0.731912	−	0.681399i	\(-0.238628\pi\)
							−0.715453	+	0.698661i	\(0.753780\pi\)
\(37\)	2.13814	−	4.14741i	0.351508	−	0.681830i	−0.645103	−	0.764096i	\(-0.723185\pi\)
							0.996610	+	0.0822662i	\(0.0262158\pi\)
\(41\)	−3.08787	+	4.80482i	−0.482245	+	0.750387i	−0.994075	−	0.108699i	\(-0.965331\pi\)
							0.511830	+	0.859087i	\(0.328968\pi\)
\(43\)	−2.20926	−	7.52406i	−0.336909	−	1.14741i	−0.937539	−	0.347881i	\(-0.886901\pi\)
							0.600629	−	0.799528i	\(-0.294917\pi\)
\(47\)	−6.00923	+	3.46943i	−0.876536	+	0.506068i	−0.869515	−	0.493907i	\(-0.835568\pi\)
							−0.00702112	+	0.999975i	\(0.502235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.157.6	yes	640
7.5	odd	6	inner	483.2.bf.a.19.27	✓	640
23.17	odd	22	inner	483.2.bf.a.178.27	yes	640
161.40	even	66	inner	483.2.bf.a.40.6	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.27	✓	640	7.5	odd	6	inner
483.2.bf.a.40.6	yes	640	161.40	even	66	inner
483.2.bf.a.157.6	yes	640	1.1	even	1	trivial
483.2.bf.a.178.27	yes	640	23.17	odd	22	inner