Embedded newform 483.2.y.a.340.3

• Label: 483.2.y.a.340.3
• Level: $483$
• Weight: $2$
• Character: 483.340
• 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			340.3
Character	\(\chi\)	\(=\)	483.340
Dual form			483.2.y.a.331.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.486251 - 2.00435i) q^{2} +(0.327068 + 0.945001i) q^{3} +(-2.00332 + 1.03278i) q^{4} +(-3.02667 + 1.21170i) q^{5} +(1.73508 - 1.11507i) q^{6} +(1.86249 + 1.87913i) q^{7} +(0.342886 + 0.395711i) q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.486251	−	2.00435i	−0.343831	−	1.41729i	−0.836926	−	0.547316i	\(-0.815650\pi\)
							0.493095	−	0.869975i	\(-0.335865\pi\)
\(3\)	0.327068	+	0.945001i	0.188833	+	0.545596i
							\(5\)	−3.02667	+	1.21170i	−1.35357	+	0.541887i	−0.931191	−	0.364532i	\(-0.881229\pi\)
−0.422379	+	0.906420i	\(0.638805\pi\)
\(7\)	1.86249	+	1.87913i	0.703955	+	0.710245i
							\(11\)	0.776339	−	3.20011i	0.234075	−	0.964871i	−0.726570	−	0.687092i	\(-0.758887\pi\)
0.960645	−	0.277778i	\(-0.0895981\pi\)
\(13\)	2.65776	−	5.81968i	0.737130	−	1.61409i	−0.0510893	−	0.998694i	\(-0.516269\pi\)
							0.788219	−	0.615395i	\(-0.211003\pi\)
\(17\)	−0.326051	−	6.84466i	−0.0790791	−	1.66007i	−0.595214	−	0.803567i	\(-0.702933\pi\)
							0.516135	−	0.856507i	\(-0.327370\pi\)
\(19\)	0.248254	−	5.21148i	0.0569533	−	1.19560i	−0.771757	−	0.635918i	\(-0.780622\pi\)
							0.828710	−	0.559678i	\(-0.189075\pi\)
\(23\)	0.225596	+	4.79052i	0.0470399	+	0.998893i
							\(29\)	1.05851	−	0.680262i	0.196560	−	0.126322i	−0.438659	−	0.898654i	\(-0.644546\pi\)
0.635219	+	0.772332i	\(0.280910\pi\)
\(31\)	3.11311	−	0.600004i	0.559132	−	0.107764i	0.0981471	−	0.995172i	\(-0.468708\pi\)
							0.460985	+	0.887408i	\(0.347496\pi\)
\(37\)	4.37576	−	3.44114i	0.719371	−	0.565719i	−0.189961	−	0.981792i	\(-0.560836\pi\)
							0.909332	+	0.416072i	\(0.136594\pi\)
\(41\)	0.196975	−	1.36999i	0.0307623	−	0.213956i	−0.968642	−	0.248459i	\(-0.920076\pi\)
							0.999405	+	0.0345031i	\(0.0109849\pi\)
\(43\)	−2.12285	+	2.44990i	−0.323731	+	0.373606i	−0.894165	−	0.447738i	\(-0.852230\pi\)
							0.570434	+	0.821344i	\(0.306775\pi\)
\(47\)	−1.95592	−	3.38776i	−0.285301	−	0.494155i	0.687381	−	0.726297i	\(-0.258760\pi\)
							−0.972682	+	0.232141i	\(0.925427\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.340.3	yes	320
7.2	even	3	inner	483.2.y.a.478.3	yes	320
23.9	even	11	inner	483.2.y.a.193.3	✓	320
161.9	even	33	inner	483.2.y.a.331.3	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.193.3	✓	320	23.9	even	11	inner
483.2.y.a.331.3	yes	320	161.9	even	33	inner
483.2.y.a.340.3	yes	320	1.1	even	1	trivial
483.2.y.a.478.3	yes	320	7.2	even	3	inner