Embedded newform 483.2.y.a.121.9

• Label: 483.2.y.a.121.9
• 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			121.9
Character	\(\chi\)	\(=\)	483.121
Dual form			483.2.y.a.4.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.282147 + 0.396220i) q^{2} +(-0.235759 - 0.971812i) q^{3} +(0.576752 - 1.66642i) q^{4} +(-0.0698936 - 1.46725i) q^{5} +(0.318533 - 0.367606i) q^{6} +(-2.64518 - 0.0550715i) q^{7} +(1.75642 - 0.515730i) q^{8} +(-0.888835 + 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.282147	+	0.396220i	0.199508	+	0.280170i	0.902194	−	0.431330i	\(-0.141956\pi\)
							−0.702686	+	0.711500i	\(0.748016\pi\)
\(3\)	−0.235759	−	0.971812i	−0.136115	−	0.561076i
							\(5\)	−0.0698936	−	1.46725i	−0.0312574	−	0.656173i	−0.959077	−	0.283145i	\(-0.908622\pi\)
0.927820	−	0.373029i	\(-0.121681\pi\)
\(7\)	−2.64518	−	0.0550715i	−0.999783	−	0.0208151i
							\(11\)	1.12389	−	1.57828i	0.338865	−	0.475869i	−0.609619	−	0.792694i	\(-0.708678\pi\)
0.948484	+	0.316826i	\(0.102617\pi\)
\(13\)	0.165604	+	1.15180i	0.0459303	+	0.319453i	0.999814	+	0.0192766i	\(0.00613631\pi\)
							−0.953884	+	0.300176i	\(0.902955\pi\)
\(17\)	−3.37403	−	0.650290i	−0.818321	−	0.157719i	−0.237122	−	0.971480i	\(-0.576204\pi\)
							−0.581199	+	0.813761i	\(0.697416\pi\)
\(19\)	2.74673	−	0.529389i	0.630143	−	0.121450i	0.135827	−	0.990733i	\(-0.456631\pi\)
							0.494316	+	0.869282i	\(0.335419\pi\)
\(23\)	−3.97880	−	2.67752i	−0.829638	−	0.558302i
							\(29\)	−5.33839	+	6.16083i	−0.991315	+	1.14404i	−0.00174286	+	0.999998i	\(0.500555\pi\)
−0.989572	+	0.144040i	\(0.953991\pi\)
\(31\)	−7.25316	−	6.91588i	−1.30271	−	1.24213i	−0.952166	−	0.305581i	\(-0.901149\pi\)
							−0.350541	−	0.936547i	\(-0.614002\pi\)
\(37\)	10.3186	−	5.31963i	1.69637	−	0.874541i	0.711198	−	0.702992i	\(-0.248153\pi\)
							0.985175	−	0.171550i	\(-0.0548774\pi\)
\(41\)	−1.53403	+	0.985859i	−0.239575	+	0.153965i	−0.654919	−	0.755699i	\(-0.727297\pi\)
							0.415345	+	0.909664i	\(0.363661\pi\)
\(43\)	8.54847	+	2.51006i	1.30363	+	0.382780i	0.858558	−	0.512716i	\(-0.171360\pi\)
							0.445070	+	0.895496i	\(0.353179\pi\)
\(47\)	−4.47886	+	7.75762i	−0.653309	+	1.13156i	0.329006	+	0.944328i	\(0.393286\pi\)
							−0.982315	+	0.187237i	\(0.940047\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.121.9	yes	320
7.4	even	3	inner	483.2.y.a.466.8	yes	320
23.4	even	11	inner	483.2.y.a.142.8	yes	320
161.4	even	33	inner	483.2.y.a.4.9	✓	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.9	✓	320	161.4	even	33	inner
483.2.y.a.121.9	yes	320	1.1	even	1	trivial
483.2.y.a.142.8	yes	320	23.4	even	11	inner
483.2.y.a.466.8	yes	320	7.4	even	3	inner