Embedded newform 483.2.y.b.163.9

• Label: 483.2.y.b.163.9
• Level: $483$
• Weight: $2$
• Character: 483.163
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			163.9
Character	\(\chi\)	\(=\)	483.163
Dual form			483.2.y.b.403.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.189594 - 0.0977427i) q^{2} +(-0.786053 + 0.618159i) q^{3} +(-1.13372 + 1.59209i) q^{4} +(2.30911 - 2.20173i) q^{5} +(-0.0886107 + 0.194030i) q^{6} +(2.00840 - 1.72231i) q^{7} +(-0.120045 + 0.834935i) q^{8} +(0.235759 - 0.971812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.189594	−	0.0977427i	0.134063	−	0.0691145i	−0.389890	−	0.920861i	\(-0.627487\pi\)
							0.523953	+	0.851747i	\(0.324457\pi\)
\(3\)	−0.786053	+	0.618159i	−0.453828	+	0.356894i
							\(5\)	2.30911	−	2.20173i	1.03267	−	0.984644i	0.0327749	−	0.999463i	\(-0.489566\pi\)
0.999890	+	0.0148186i	\(0.00471708\pi\)
\(7\)	2.00840	−	1.72231i	0.759103	−	0.650971i
							\(11\)	−1.11039	−	0.572444i	−0.334794	−	0.172598i	0.282630	−	0.959229i	\(-0.408793\pi\)
−0.617424	+	0.786631i	\(0.711824\pi\)
\(13\)	−1.60717	−	1.85477i	−0.445748	−	0.514420i	0.487760	−	0.872978i	\(-0.337814\pi\)
							−0.933508	+	0.358558i	\(0.883269\pi\)
\(17\)	5.71037	−	0.545274i	1.38497	−	0.132248i	0.624196	−	0.781268i	\(-0.285427\pi\)
							0.760772	+	0.649019i	\(0.224820\pi\)
\(19\)	−0.160437	−	0.0153199i	−0.0368068	−	0.00351462i	0.0766364	−	0.997059i	\(-0.475582\pi\)
							−0.113443	+	0.993544i	\(0.536188\pi\)
\(23\)	3.65661	+	3.10310i	0.762455	+	0.647041i
							\(29\)	0.470414	−	1.03006i	0.0873537	−	0.191278i	−0.860913	−	0.508752i	\(-0.830107\pi\)
0.948267	+	0.317474i	\(0.102834\pi\)
\(31\)	8.66104	−	3.46736i	1.55557	−	0.622756i	0.574864	−	0.818249i	\(-0.305055\pi\)
							0.980705	+	0.195493i	\(0.0626308\pi\)
\(37\)	0.170041	−	0.700919i	0.0279546	−	0.115230i	−0.956125	−	0.292959i	\(-0.905360\pi\)
							0.984079	+	0.177729i	\(0.0568751\pi\)
\(41\)	4.70839	+	1.38251i	0.735326	+	0.215911i	0.627892	−	0.778301i	\(-0.283918\pi\)
							0.107435	+	0.994212i	\(0.465736\pi\)
\(43\)	1.06050	+	7.37596i	0.161725	+	1.12482i	0.895380	+	0.445302i	\(0.146904\pi\)
							−0.733655	+	0.679522i	\(0.762187\pi\)
\(47\)	−2.15646	+	3.73510i	−0.314552	+	0.544821i	−0.979342	−	0.202210i	\(-0.935188\pi\)
							0.664790	+	0.747030i	\(0.268521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.163.9	yes	320
7.4	even	3	inner	483.2.y.b.25.8	✓	320
23.12	even	11	inner	483.2.y.b.58.8	yes	320
161.81	even	33	inner	483.2.y.b.403.9	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.25.8	✓	320	7.4	even	3	inner
483.2.y.b.58.8	yes	320	23.12	even	11	inner
483.2.y.b.163.9	yes	320	1.1	even	1	trivial
483.2.y.b.403.9	yes	320	161.81	even	33	inner