Embedded newform 483.2.y.a.121.12

• Label: 483.2.y.a.121.12
• Level: $483$
• Weight: $2$
• Character: 483.121
• 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			121.12
Character	\(\chi\)	\(=\)	483.121
Dual form			483.2.y.a.4.12

$q$-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 \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{9}{11}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.12	✓	320	161.4	even	33	inner
483.2.y.a.121.12	yes	320	1.1	even	1	trivial
483.2.y.a.142.5	yes	320	23.4	even	11	inner
483.2.y.a.466.5	yes	320	7.4	even	3	inner