Embedded newform 483.2.y.b.4.2

• Label: 483.2.y.b.4.2
• Level: $483$
• Weight: $2$
• Character: 483.4
• 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			4.2
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37000 + 1.92390i) q^{2} +(0.235759 - 0.971812i) q^{3} +(-1.17034 - 3.38148i) q^{4} +(0.00886507 - 0.186101i) q^{5} +(1.54668 + 1.78496i) q^{6} +(0.229059 + 2.63582i) q^{7} +(3.57665 + 1.05020i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37000	+	1.92390i	−0.968737	+	1.36040i	−0.0361926	+	0.999345i	\(0.511523\pi\)
							−0.932544	+	0.361056i	\(0.882416\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	0.00886507	−	0.186101i	0.00396458	−	0.0832268i	−0.996018	−	0.0891540i	\(-0.971584\pi\)
0.999982	+	0.00592722i	\(0.00188670\pi\)
\(7\)	0.229059	+	2.63582i	0.0865763	+	0.996245i
							\(11\)	1.99704	+	2.80445i	0.602131	+	0.845575i	0.997330	−	0.0730206i	\(-0.0232639\pi\)
−0.395199	+	0.918595i	\(0.629324\pi\)
\(13\)	1.00107	−	6.96260i	0.277647	−	1.93108i	−0.0790776	−	0.996868i	\(-0.525197\pi\)
							0.356725	−	0.934210i	\(-0.383893\pi\)
\(17\)	0.821886	−	0.158406i	0.199337	−	0.0384190i	−0.0886055	−	0.996067i	\(-0.528241\pi\)
							0.287942	+	0.957648i	\(0.407029\pi\)
\(19\)	1.86797	+	0.360021i	0.428541	+	0.0825945i	0.398961	−	0.916968i	\(-0.369371\pi\)
							0.0295801	+	0.999562i	\(0.490583\pi\)
\(23\)	−0.101065	+	4.79477i	−0.0210734	+	0.999778i
							\(29\)	4.93730	+	5.69795i	0.916834	+	1.05808i	0.998114	+	0.0613915i	\(0.0195538\pi\)
−0.0812797	+	0.996691i	\(0.525901\pi\)
\(31\)	−7.25015	+	6.91300i	−1.30217	+	1.24161i	−0.349738	+	0.936848i	\(0.613729\pi\)
							−0.952428	+	0.304765i	\(0.901422\pi\)
\(37\)	1.27855	+	0.659140i	0.210193	+	0.108362i	0.560108	−	0.828419i	\(-0.310759\pi\)
							−0.349916	+	0.936781i	\(0.613790\pi\)
\(41\)	1.71177	+	1.10009i	0.267333	+	0.171805i	0.667439	−	0.744664i	\(-0.267390\pi\)
							−0.400106	+	0.916469i	\(0.631027\pi\)
\(43\)	10.1515	−	2.98076i	1.54809	−	0.454561i	0.607563	−	0.794271i	\(-0.292147\pi\)
							0.940530	+	0.339710i	\(0.110329\pi\)
\(47\)	0.421245	+	0.729618i	0.0614449	+	0.106426i	0.895111	−	0.445842i	\(-0.147096\pi\)
							−0.833667	+	0.552268i	\(0.813762\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.2	✓	320
7.2	even	3	inner	483.2.y.b.142.15	yes	320
23.6	even	11	inner	483.2.y.b.466.15	yes	320
161.121	even	33	inner	483.2.y.b.121.2	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.2	✓	320	1.1	even	1	trivial
483.2.y.b.121.2	yes	320	161.121	even	33	inner
483.2.y.b.142.15	yes	320	7.2	even	3	inner
483.2.y.b.466.15	yes	320	23.6	even	11	inner