Embedded newform 483.2.y.a.193.3

• Label: 483.2.y.a.193.3
• Level: $483$
• Weight: $2$
• Character: 483.193
• 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			193.3
Character	\(\chi\)	\(=\)	483.193
Dual form			483.2.y.a.478.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.49269 - 1.42328i) q^{2} +(-0.981929 + 0.189251i) q^{3} +(0.107243 + 2.25132i) q^{4} +(2.56270 - 2.01533i) q^{5} +(1.73508 + 1.11507i) q^{6} +(0.200482 - 2.63814i) q^{7} +(0.342886 - 0.395711i) q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.49269	−	1.42328i	−1.05549	−	1.00641i	−0.999968	−	0.00795484i	\(-0.997468\pi\)
							−0.0555262	−	0.998457i	\(-0.517684\pi\)
\(3\)	−0.981929	+	0.189251i	−0.566917	+	0.109264i
							\(5\)	2.56270	−	2.01533i	1.14607	−	0.901282i	0.149901	−	0.988701i	\(-0.452104\pi\)
0.996172	+	0.0874192i	\(0.0278620\pi\)
\(7\)	0.200482	−	2.63814i	0.0757750	−	0.997125i
							\(11\)	2.38321	−	2.27239i	0.718565	−	0.685150i	−0.239759	−	0.970832i	\(-0.577069\pi\)
0.958324	+	0.285682i	\(0.0922201\pi\)
\(13\)	2.65776	+	5.81968i	0.737130	+	1.61409i	0.788219	+	0.615395i	\(0.211003\pi\)
							−0.0510893	+	0.998694i	\(0.516269\pi\)
\(17\)	6.09068	−	3.13996i	1.47721	−	0.761553i	0.483689	−	0.875240i	\(-0.339296\pi\)
							0.993517	+	0.113687i	\(0.0362662\pi\)
\(19\)	−4.63740	−	2.39075i	−1.06389	−	0.548475i	−0.164843	−	0.986320i	\(-0.552712\pi\)
							−0.899051	+	0.437845i	\(0.855742\pi\)
\(23\)	4.03592	+	2.59063i	0.841547	+	0.540184i
							\(29\)	1.05851	+	0.680262i	0.196560	+	0.126322i	0.635219	−	0.772332i	\(-0.280910\pi\)
−0.438659	+	0.898654i	\(0.644546\pi\)
\(31\)	−1.03694	−	2.99604i	−0.186240	−	0.538104i	0.812771	−	0.582584i	\(-0.197958\pi\)
							−0.999010	+	0.0444796i	\(0.985837\pi\)
\(37\)	−5.16799	+	2.06895i	−0.849613	+	0.340134i	−0.755276	−	0.655407i	\(-0.772497\pi\)
							−0.0943366	+	0.995540i	\(0.530073\pi\)
\(41\)	0.196975	+	1.36999i	0.0307623	+	0.213956i	0.999405	−	0.0345031i	\(-0.0109849\pi\)
							−0.968642	+	0.248459i	\(0.920076\pi\)
\(43\)	−2.12285	−	2.44990i	−0.323731	−	0.373606i	0.570434	−	0.821344i	\(-0.306775\pi\)
							−0.894165	+	0.447738i	\(0.852230\pi\)
\(47\)	−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.193.3	✓	320
7.2	even	3	inner	483.2.y.a.331.3	yes	320
23.18	even	11	inner	483.2.y.a.340.3	yes	320
161.156	even	33	inner	483.2.y.a.478.3	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.193.3	✓	320	1.1	even	1	trivial
483.2.y.a.331.3	yes	320	7.2	even	3	inner
483.2.y.a.340.3	yes	320	23.18	even	11	inner
483.2.y.a.478.3	yes	320	161.156	even	33	inner