Embedded newform 483.2.y.b.193.11

• Label: 483.2.y.b.193.11
• 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			193.11
Character	\(\chi\)	\(=\)	483.193
Dual form			483.2.y.b.478.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.821216 + 0.783028i) q^{2} +(0.981929 - 0.189251i) q^{3} +(-0.0339007 - 0.711664i) q^{4} +(-1.35459 + 1.06526i) q^{5} +(0.954565 + 0.613461i) q^{6} +(1.68871 - 2.03673i) q^{7} +(2.01555 - 2.32606i) q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.821216	+	0.783028i	0.580687	+	0.553684i	0.922457	−	0.386099i	\(-0.126178\pi\)
							−0.341770	+	0.939784i	\(0.611026\pi\)
\(3\)	0.981929	−	0.189251i	0.566917	−	0.109264i
							\(5\)	−1.35459	+	1.06526i	−0.605791	+	0.476399i	−0.873318	−	0.487151i	\(-0.838036\pi\)
0.267527	+	0.963550i	\(0.413794\pi\)
\(7\)	1.68871	−	2.03673i	0.638273	−	0.769810i
							\(11\)	1.67774	−	1.59972i	0.505858	−	0.482335i	−0.393637	−	0.919266i	\(-0.628783\pi\)
0.899495	+	0.436931i	\(0.143935\pi\)
\(13\)	−0.181622	−	0.397697i	−0.0503730	−	0.110301i	0.882772	−	0.469801i	\(-0.155675\pi\)
							−0.933145	+	0.359500i	\(0.882947\pi\)
\(17\)	−0.319548	+	0.164739i	−0.0775018	+	0.0399550i	−0.496542	−	0.868013i	\(-0.665397\pi\)
							0.419040	+	0.907968i	\(0.362367\pi\)
\(19\)	−3.90166	−	2.01145i	−0.895103	−	0.461458i	−0.0516527	−	0.998665i	\(-0.516449\pi\)
							−0.843450	+	0.537208i	\(0.819479\pi\)
\(23\)	2.83865	+	3.86550i	0.591899	+	0.806012i
							\(29\)	8.72573	+	5.60769i	1.62033	+	1.04132i	0.955783	+	0.294074i	\(0.0950111\pi\)
0.664545	+	0.747248i	\(0.268625\pi\)
\(31\)	0.951160	+	2.74820i	0.170833	+	0.493591i	0.997622	−	0.0689221i	\(-0.0219560\pi\)
							−0.826789	+	0.562513i	\(0.809835\pi\)
\(37\)	−0.0425635	+	0.0170399i	−0.00699740	+	0.00280134i	−0.375159	−	0.926961i	\(-0.622412\pi\)
							0.368161	+	0.929762i	\(0.379987\pi\)
\(41\)	0.876300	+	6.09480i	0.136855	+	0.951848i	0.936323	+	0.351140i	\(0.114206\pi\)
							−0.799468	+	0.600709i	\(0.794885\pi\)
\(43\)	−6.63969	−	7.66261i	−1.01254	−	1.16854i	−0.985633	−	0.168900i	\(-0.945979\pi\)
							−0.0269101	−	0.999638i	\(-0.508567\pi\)
\(47\)	−2.87323	−	4.97658i	−0.419103	−	0.725908i	0.576746	−	0.816923i	\(-0.304322\pi\)
							−0.995850	+	0.0910151i	\(0.970989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.193.11	✓	320
7.2	even	3	inner	483.2.y.b.331.11	yes	320
23.18	even	11	inner	483.2.y.b.340.11	yes	320
161.156	even	33	inner	483.2.y.b.478.11	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.11	✓	320	1.1	even	1	trivial
483.2.y.b.331.11	yes	320	7.2	even	3	inner
483.2.y.b.340.11	yes	320	23.18	even	11	inner
483.2.y.b.478.11	yes	320	161.156	even	33	inner