Embedded newform 483.2.y.b.403.7

• Label: 483.2.y.b.403.7
• Level: $483$
• Weight: $2$
• Character: 483.403
• 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			403.7
Character	\(\chi\)	\(=\)	483.403
Dual form			483.2.y.b.163.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.818913 - 0.422179i) q^{2} +(-0.786053 - 0.618159i) q^{3} +(-0.667730 - 0.937696i) q^{4} +(2.57491 + 2.45517i) q^{5} +(0.382736 + 0.838074i) q^{6} +(-2.33165 - 1.25037i) q^{7} +(0.413176 + 2.87370i) q^{8} +(0.235759 + 0.971812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{10}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.818913	−	0.422179i	−0.579059	−	0.298526i	0.143705	−	0.989621i	\(-0.454098\pi\)
							−0.722764	+	0.691095i	\(0.757129\pi\)
\(3\)	−0.786053	−	0.618159i	−0.453828	−	0.356894i
							\(5\)	2.57491	+	2.45517i	1.15153	+	1.09799i	0.993877	+	0.110492i	\(0.0352426\pi\)
0.157657	+	0.987494i	\(0.449606\pi\)
\(7\)	−2.33165	−	1.25037i	−0.881280	−	0.472595i
							\(11\)	1.66746	−	0.859638i	0.502759	−	0.259190i	−0.188139	−	0.982142i	\(-0.560246\pi\)
0.690898	+	0.722952i	\(0.257215\pi\)
\(13\)	−4.13780	+	4.77527i	−1.14762	+	1.32442i	−0.209621	+	0.977783i	\(0.567223\pi\)
							−0.937998	+	0.346640i	\(0.887323\pi\)
\(17\)	−6.90505	−	0.659352i	−1.67472	−	0.159916i	−0.785956	−	0.618282i	\(-0.787829\pi\)
							−0.888764	+	0.458366i	\(0.848435\pi\)
\(19\)	3.32955	−	0.317933i	0.763851	−	0.0729389i	0.294144	−	0.955761i	\(-0.404966\pi\)
							0.469707	+	0.882822i	\(0.344360\pi\)
\(23\)	−0.621743	+	4.75536i	−0.129642	+	0.991561i
							\(29\)	2.46382	+	5.39502i	0.457521	+	1.00183i	0.988046	+	0.154161i	\(0.0492675\pi\)
−0.530525	+	0.847669i	\(0.678005\pi\)
\(31\)	0.474880	+	0.190113i	0.0852910	+	0.0341454i	0.413916	−	0.910315i	\(-0.364161\pi\)
							−0.328625	+	0.944461i	\(0.606585\pi\)
\(37\)	0.772056	+	3.18246i	0.126925	+	0.523193i	0.999403	+	0.0345396i	\(0.0109965\pi\)
							−0.872478	+	0.488653i	\(0.837488\pi\)
\(41\)	1.00259	−	0.294386i	0.156578	−	0.0459753i	−0.202504	−	0.979281i	\(-0.564908\pi\)
							0.359082	+	0.933306i	\(0.383090\pi\)
\(43\)	−1.52897	+	10.6342i	−0.233165	+	1.62170i	0.451104	+	0.892471i	\(0.351030\pi\)
							−0.684270	+	0.729229i	\(0.739879\pi\)
\(47\)	−1.36200	−	2.35906i	−0.198668	−	0.344104i	0.749429	−	0.662085i	\(-0.230328\pi\)
							−0.948097	+	0.317981i	\(0.896995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.403.7	yes	320
7.2	even	3	inner	483.2.y.b.58.10	yes	320
23.2	even	11	inner	483.2.y.b.25.10	✓	320
161.2	even	33	inner	483.2.y.b.163.7	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.25.10	✓	320	23.2	even	11	inner
483.2.y.b.58.10	yes	320	7.2	even	3	inner
483.2.y.b.163.7	yes	320	161.2	even	33	inner
483.2.y.b.403.7	yes	320	1.1	even	1	trivial