Embedded newform 483.2.y.b.16.4

• Label: 483.2.y.b.16.4
• Level: $483$
• Weight: $2$
• Character: 483.16
• 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			16.4
Character	\(\chi\)	\(=\)	483.16
Dual form			483.2.y.b.151.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.620909 - 1.79400i) q^{2} +(-0.888835 - 0.458227i) q^{3} +(-1.26080 + 0.991503i) q^{4} +(-3.16803 - 0.302510i) q^{5} +(-0.270172 + 1.87909i) q^{6} +(1.40046 - 2.24471i) q^{7} +(-0.632491 - 0.406478i) q^{8} +(0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.620909	−	1.79400i	−0.439049	−	1.26855i	−0.920040	−	0.391824i	\(-0.871844\pi\)
							0.480991	−	0.876725i	\(-0.340277\pi\)
\(3\)	−0.888835	−	0.458227i	−0.513169	−	0.264557i
							\(5\)	−3.16803	−	0.302510i	−1.41679	−	0.135287i	−0.641610	−	0.767031i	\(-0.721733\pi\)
−0.775177	+	0.631744i	\(0.782339\pi\)
\(7\)	1.40046	−	2.24471i	0.529325	−	0.848419i
							\(11\)	−0.842932	+	2.43549i	−0.254154	+	0.734328i	0.743680	+	0.668536i	\(0.233079\pi\)
−0.997833	+	0.0657925i	\(0.979042\pi\)
\(13\)	−3.50062	−	1.02787i	−0.970897	−	0.285081i	−0.242436	−	0.970168i	\(-0.577946\pi\)
							−0.728462	+	0.685086i	\(0.759764\pi\)
\(17\)	−1.43410	+	0.574128i	−0.347821	+	0.139247i	−0.538993	−	0.842310i	\(-0.681195\pi\)
							0.191172	+	0.981556i	\(0.438771\pi\)
\(19\)	2.71924	+	1.08862i	0.623837	+	0.249747i	0.661975	−	0.749526i	\(-0.269718\pi\)
							−0.0381382	+	0.999272i	\(0.512143\pi\)
\(23\)	2.95904	−	3.77413i	0.617003	−	0.786961i
							\(29\)	−1.35096	+	9.39616i	−0.250868	+	1.74482i	0.342161	+	0.939641i	\(0.388841\pi\)
−0.593028	+	0.805182i	\(0.702068\pi\)
\(31\)	−0.0236884	+	0.497280i	−0.00425456	+	0.0893141i	−0.999996	−	0.00287728i	\(-0.999084\pi\)
							0.995741	+	0.0921914i	\(0.0293872\pi\)
\(37\)	4.33020	+	6.08092i	0.711881	+	0.999697i	0.999124	+	0.0418396i	\(0.0133218\pi\)
							−0.287243	+	0.957858i	\(0.592739\pi\)
\(41\)	−3.04181	−	6.66063i	−0.475050	−	1.04022i	−0.983795	−	0.179298i	\(-0.942618\pi\)
							0.508745	−	0.860917i	\(-0.330110\pi\)
\(43\)	2.95900	−	1.90163i	0.451243	−	0.289996i	−0.295213	−	0.955432i	\(-0.595390\pi\)
							0.746455	+	0.665435i	\(0.231754\pi\)
\(47\)	−4.58771	+	7.94615i	−0.669186	+	1.15906i	0.308946	+	0.951080i	\(0.400024\pi\)
							−0.978132	+	0.207985i	\(0.933309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.16.4	✓	320
7.4	even	3	inner	483.2.y.b.361.13	yes	320
23.13	even	11	inner	483.2.y.b.289.13	yes	320
161.151	even	33	inner	483.2.y.b.151.4	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.16.4	✓	320	1.1	even	1	trivial
483.2.y.b.151.4	yes	320	161.151	even	33	inner
483.2.y.b.289.13	yes	320	23.13	even	11	inner
483.2.y.b.361.13	yes	320	7.4	even	3	inner