Embedded newform 483.2.y.a.163.16

• Label: 483.2.y.a.163.16
• 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			163.16
Character	\(\chi\)	\(=\)	483.163
Dual form			483.2.y.a.403.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.17293 - 1.12022i) q^{2} +(0.786053 - 0.618159i) q^{3} +(2.30661 - 3.23917i) q^{4} +(-1.06640 + 1.01681i) q^{5} +(1.01556 - 2.22377i) q^{6} +(2.64412 + 0.0929463i) q^{7} +(0.687662 - 4.78279i) q^{8} +(0.235759 - 0.971812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.17293	−	1.12022i	1.53649	−	0.792117i	0.537792	−	0.843078i	\(-0.319259\pi\)
							0.998701	+	0.0509607i	\(0.0162283\pi\)
\(3\)	0.786053	−	0.618159i	0.453828	−	0.356894i
							\(5\)	−1.06640	+	1.01681i	−0.476909	+	0.454732i	−0.890002	−	0.455957i	\(-0.849297\pi\)
0.413093	+	0.910689i	\(0.364449\pi\)
\(7\)	2.64412	+	0.0929463i	0.999383	+	0.0351304i
							\(11\)	−0.871776	−	0.449432i	−0.262850	−	0.135509i	0.321764	−	0.946820i	\(-0.395724\pi\)
−0.584615	+	0.811311i	\(0.698754\pi\)
\(13\)	2.23420	+	2.57841i	0.619656	+	0.715121i	0.975642	−	0.219370i	\(-0.0704003\pi\)
							−0.355986	+	0.934491i	\(0.615855\pi\)
\(17\)	−8.20040	+	0.783043i	−1.98889	+	0.189916i	−0.990126	+	0.140182i	\(0.955231\pi\)
							−0.998762	+	0.0497341i	\(0.984163\pi\)
\(19\)	−3.37506	−	0.322279i	−0.774292	−	0.0739359i	−0.299573	−	0.954073i	\(-0.596844\pi\)
							−0.474719	+	0.880137i	\(0.657450\pi\)
\(23\)	−2.78900	−	3.90147i	−0.581546	−	0.813513i
							\(29\)	2.87035	−	6.28519i	0.533010	−	1.16713i	−0.431265	−	0.902226i	\(-0.641932\pi\)
0.964275	−	0.264904i	\(-0.0853403\pi\)
\(31\)	−0.855818	+	0.342618i	−0.153709	+	0.0615360i	−0.447241	−	0.894413i	\(-0.647594\pi\)
							0.293532	+	0.955949i	\(0.405169\pi\)
\(37\)	−1.45474	+	5.99653i	−0.239158	+	0.985823i	0.717951	+	0.696094i	\(0.245080\pi\)
							−0.957109	+	0.289729i	\(0.906435\pi\)
\(41\)	10.2853	+	3.02003i	1.60629	+	0.471650i	0.957288	−	0.289136i	\(-0.0933681\pi\)
							0.649003	+	0.760786i	\(0.275186\pi\)
\(43\)	−0.461845	−	3.21220i	−0.0704307	−	0.489857i	−0.994255	−	0.107039i	\(-0.965863\pi\)
							0.923824	−	0.382817i	\(-0.125046\pi\)
\(47\)	−1.25535	+	2.17432i	−0.183111	+	0.317158i	−0.942938	−	0.332967i	\(-0.891950\pi\)
							0.759827	+	0.650125i	\(0.225283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.163.16	yes	320
7.4	even	3	inner	483.2.y.a.25.1	✓	320
23.12	even	11	inner	483.2.y.a.58.1	yes	320
161.81	even	33	inner	483.2.y.a.403.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.25.1	✓	320	7.4	even	3	inner
483.2.y.a.58.1	yes	320	23.12	even	11	inner
483.2.y.a.163.16	yes	320	1.1	even	1	trivial
483.2.y.a.403.16	yes	320	161.81	even	33	inner