Embedded newform 483.2.y.a.4.3

• Label: 483.2.y.a.4.3
• Level: $483$
• Weight: $2$
• Character: 483.4
• 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			4.3
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.a.121.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09070 + 1.53167i) q^{2} +(-0.235759 + 0.971812i) q^{3} +(-0.502260 - 1.45118i) q^{4} +(-0.0927520 + 1.94710i) q^{5} +(-1.23135 - 1.42106i) q^{6} +(0.850242 - 2.50541i) q^{7} +(-0.837777 - 0.245993i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09070	+	1.53167i	−0.771240	+	1.08306i	0.222887	+	0.974844i	\(0.428452\pi\)
							−0.994128	+	0.108211i	\(0.965488\pi\)
\(3\)	−0.235759	+	0.971812i	−0.136115	+	0.561076i
							\(5\)	−0.0927520	+	1.94710i	−0.0414800	+	0.870771i	0.878246	+	0.478210i	\(0.158714\pi\)
−0.919726	+	0.392562i	\(0.871589\pi\)
\(7\)	0.850242	−	2.50541i	0.321361	−	0.946957i
							\(11\)	−2.62185	−	3.68187i	−0.790517	−	1.11013i	−0.991446	−	0.130520i	\(-0.958335\pi\)
0.200929	−	0.979606i	\(-0.435604\pi\)
\(13\)	0.355241	−	2.47076i	0.0985262	−	0.685265i	−0.879365	−	0.476149i	\(-0.842032\pi\)
							0.977891	−	0.209116i	\(-0.0670585\pi\)
\(17\)	−5.23770	+	1.00948i	−1.27033	+	0.244836i	−0.779490	−	0.626415i	\(-0.784522\pi\)
							−0.490838	+	0.871251i	\(0.663309\pi\)
\(19\)	−5.79587	−	1.11706i	−1.32966	−	0.256272i	−0.525558	−	0.850758i	\(-0.676143\pi\)
							−0.804106	+	0.594486i	\(0.797355\pi\)
\(23\)	−4.10511	−	2.47953i	−0.855974	−	0.517018i
							\(29\)	1.14804	+	1.32491i	0.213185	+	0.246029i	0.852263	−	0.523113i	\(-0.175229\pi\)
−0.639078	+	0.769142i	\(0.720684\pi\)
\(31\)	−4.23019	+	4.03348i	−0.759764	+	0.724434i	−0.967256	−	0.253803i	\(-0.918319\pi\)
							0.207492	+	0.978237i	\(0.433470\pi\)
\(37\)	−4.15434	−	2.14171i	−0.682970	−	0.352095i	0.0815738	−	0.996667i	\(-0.474005\pi\)
							−0.764544	+	0.644572i	\(0.777036\pi\)
\(41\)	6.12161	+	3.93412i	0.956035	+	0.614406i	0.922898	−	0.385044i	\(-0.125814\pi\)
							0.0331369	+	0.999451i	\(0.489450\pi\)
\(43\)	−2.62030	+	0.769389i	−0.399592	+	0.117331i	−0.475353	−	0.879795i	\(-0.657680\pi\)
							0.0757610	+	0.997126i	\(0.475861\pi\)
\(47\)	−2.16886	−	3.75657i	−0.316360	−	0.547952i	0.663365	−	0.748296i	\(-0.269128\pi\)
							−0.979726	+	0.200344i	\(0.935794\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.4.3	✓	320
7.2	even	3	inner	483.2.y.a.142.14	yes	320
23.6	even	11	inner	483.2.y.a.466.14	yes	320
161.121	even	33	inner	483.2.y.a.121.3	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.3	✓	320	1.1	even	1	trivial
483.2.y.a.121.3	yes	320	161.121	even	33	inner
483.2.y.a.142.14	yes	320	7.2	even	3	inner
483.2.y.a.466.14	yes	320	23.6	even	11	inner