Embedded newform 483.2.y.a.193.16

• Label: 483.2.y.a.193.16
• 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			193.16
Character	\(\chi\)	\(=\)	483.193
Dual form			483.2.y.a.478.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.94254 + 1.85221i) q^{2} +(-0.981929 + 0.189251i) q^{3} +(0.247623 + 5.19824i) q^{4} +(-1.65917 + 1.30478i) q^{5} +(-2.25797 - 1.45111i) q^{6} +(1.45918 - 2.20699i) q^{7} +(-5.63184 + 6.49949i) q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.94254	+	1.85221i	1.37358	+	1.30971i	0.904449	+	0.426582i	\(0.140282\pi\)
							0.469134	+	0.883127i	\(0.344566\pi\)
\(3\)	−0.981929	+	0.189251i	−0.566917	+	0.109264i
							\(5\)	−1.65917	+	1.30478i	−0.742001	+	0.583516i	−0.915976	−	0.401234i	\(-0.868581\pi\)
0.173974	+	0.984750i	\(0.444339\pi\)
\(7\)	1.45918	−	2.20699i	0.551517	−	0.834163i
							\(11\)	−3.99423	+	3.80849i	−1.20430	+	1.14830i	−0.219321	+	0.975653i	\(0.570384\pi\)
−0.984983	+	0.172649i	\(0.944767\pi\)
\(13\)	0.558523	+	1.22299i	0.154906	+	0.339198i	0.971135	−	0.238532i	\(-0.0766661\pi\)
							−0.816228	+	0.577730i	\(0.803939\pi\)
\(17\)	5.44325	−	2.80619i	1.32018	−	0.680602i	0.353225	−	0.935538i	\(-0.385085\pi\)
							0.966958	+	0.254937i	\(0.0820546\pi\)
\(19\)	−3.52365	−	1.81657i	−0.808380	−	0.416749i	0.00391746	−	0.999992i	\(-0.498753\pi\)
							−0.812297	+	0.583244i	\(0.801783\pi\)
\(23\)	3.22323	+	3.55117i	0.672090	+	0.740469i
							\(29\)	2.23130	+	1.43397i	0.414342	+	0.266282i	0.731162	−	0.682204i	\(-0.238979\pi\)
−0.316819	+	0.948486i	\(0.602615\pi\)
\(31\)	−1.74881	−	5.05286i	−0.314096	−	0.907520i	−0.985980	−	0.166864i	\(-0.946636\pi\)
							0.671884	−	0.740656i	\(-0.265485\pi\)
\(37\)	7.89148	−	3.15927i	1.29735	−	0.519381i	0.382792	−	0.923834i	\(-0.374962\pi\)
							0.914559	+	0.404454i	\(0.132538\pi\)
\(41\)	0.829006	+	5.76586i	0.129469	+	0.900476i	0.946229	+	0.323498i	\(0.104859\pi\)
							−0.816760	+	0.576978i	\(0.804232\pi\)
\(43\)	5.84189	+	6.74190i	0.890880	+	1.02813i	0.999421	+	0.0340382i	\(0.0108368\pi\)
							−0.108541	+	0.994092i	\(0.534618\pi\)
\(47\)	5.81748	+	10.0762i	0.848567	+	1.46976i	0.882487	+	0.470337i	\(0.155868\pi\)
							−0.0339196	+	0.999425i	\(0.510799\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.193.16	✓	320
7.2	even	3	inner	483.2.y.a.331.16	yes	320
23.18	even	11	inner	483.2.y.a.340.16	yes	320
161.156	even	33	inner	483.2.y.a.478.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.193.16	✓	320	1.1	even	1	trivial
483.2.y.a.331.16	yes	320	7.2	even	3	inner
483.2.y.a.340.16	yes	320	23.18	even	11	inner
483.2.y.a.478.16	yes	320	161.156	even	33	inner