Embedded newform 483.2.y.a.4.12

• Label: 483.2.y.a.4.12
• 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			4.12
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.a.121.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.618045 - 0.867923i) q^{2} +(-0.235759 + 0.971812i) q^{3} +(0.282825 + 0.817169i) q^{4} +(0.184708 - 3.87751i) q^{5} +(0.697748 + 0.805244i) q^{6} +(0.447567 + 2.60762i) q^{7} +(2.92870 + 0.859944i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.618045	−	0.867923i	0.437024	−	0.613715i	−0.536056	−	0.844183i	\(-0.680086\pi\)
							0.973080	+	0.230468i	\(0.0740257\pi\)
\(3\)	−0.235759	+	0.971812i	−0.136115	+	0.561076i
							\(5\)	0.184708	−	3.87751i	0.0826041	−	1.73407i	−0.454665	−	0.890662i	\(-0.650241\pi\)
0.537269	−	0.843411i	\(-0.319456\pi\)
\(7\)	0.447567	+	2.60762i	0.169165	+	0.985588i
							\(11\)	1.22931	+	1.72632i	0.370651	+	0.520506i	0.957208	−	0.289402i	\(-0.0934564\pi\)
−0.586557	+	0.809908i	\(0.699517\pi\)
\(13\)	0.563532	−	3.91945i	0.156296	−	1.08706i	−0.749089	−	0.662469i	\(-0.769509\pi\)
							0.905385	−	0.424592i	\(-0.139582\pi\)
\(17\)	4.35346	−	0.839060i	1.05587	−	0.203502i	0.368364	−	0.929682i	\(-0.379918\pi\)
							0.687505	+	0.726180i	\(0.258706\pi\)
\(19\)	0.791694	+	0.152586i	0.181627	+	0.0350057i	0.279254	−	0.960217i	\(-0.409913\pi\)
							−0.0976266	+	0.995223i	\(0.531125\pi\)
\(23\)	4.18453	+	2.34301i	0.872535	+	0.488552i
							\(29\)	−1.96210	−	2.26439i	−0.364353	−	0.420486i	0.543740	−	0.839253i	\(-0.317008\pi\)
−0.908093	+	0.418768i	\(0.862462\pi\)
\(31\)	−2.04075	+	1.94585i	−0.366530	+	0.349485i	−0.850853	−	0.525404i	\(-0.823914\pi\)
							0.484323	+	0.874889i	\(0.339066\pi\)
\(37\)	−0.703388	−	0.362622i	−0.115636	−	0.0596147i	0.399438	−	0.916760i	\(-0.369205\pi\)
							−0.515074	+	0.857146i	\(0.672236\pi\)
\(41\)	1.18935	+	0.764350i	0.185746	+	0.119371i	0.630209	−	0.776426i	\(-0.282969\pi\)
							−0.444463	+	0.895797i	\(0.646606\pi\)
\(43\)	8.48447	−	2.49126i	1.29387	−	0.379914i	0.438873	−	0.898549i	\(-0.355378\pi\)
							0.854996	+	0.518635i	\(0.173560\pi\)
\(47\)	−2.08992	−	3.61984i	−0.304846	−	0.528009i	0.672381	−	0.740205i	\(-0.265272\pi\)
							−0.977227	+	0.212197i	\(0.931938\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.4.12	✓	320
7.2	even	3	inner	483.2.y.a.142.5	yes	320
23.6	even	11	inner	483.2.y.a.466.5	yes	320
161.121	even	33	inner	483.2.y.a.121.12	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.12	✓	320	1.1	even	1	trivial
483.2.y.a.121.12	yes	320	161.121	even	33	inner
483.2.y.a.142.5	yes	320	7.2	even	3	inner
483.2.y.a.466.5	yes	320	23.6	even	11	inner