Embedded newform 483.2.y.a.163.4

• Label: 483.2.y.a.163.4
• Level: $483$
• Weight: $2$
• Character: 483.163
• 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			163.4
Character	\(\chi\)	\(=\)	483.163
Dual form			483.2.y.a.403.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59365 + 0.821585i) q^{2} +(0.786053 - 0.618159i) q^{3} +(0.704613 - 0.989491i) q^{4} +(-2.75786 + 2.62961i) q^{5} +(-0.744826 + 1.63094i) q^{6} +(1.28151 - 2.31468i) q^{7} +(0.200374 - 1.39363i) q^{8} +(0.235759 - 0.971812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59365	+	0.821585i	−1.12688	+	0.580948i	−0.917837	−	0.396957i	\(-0.870066\pi\)
							−0.209046	+	0.977906i	\(0.567036\pi\)
\(3\)	0.786053	−	0.618159i	0.453828	−	0.356894i
							\(5\)	−2.75786	+	2.62961i	−1.23335	+	1.17600i	−0.255474	+	0.966816i	\(0.582232\pi\)
−0.977877	+	0.209182i	\(0.932920\pi\)
\(7\)	1.28151	−	2.31468i	0.484365	−	0.874866i
							\(11\)	2.74064	+	1.41290i	0.826333	+	0.426004i	0.818878	−	0.573968i	\(-0.194597\pi\)
0.00745518	+	0.999972i	\(0.497627\pi\)
\(13\)	3.50975	+	4.05047i	0.973430	+	1.12340i	0.992335	+	0.123577i	\(0.0394365\pi\)
							−0.0189052	+	0.999821i	\(0.506018\pi\)
\(17\)	−4.57622	+	0.436976i	−1.10990	+	0.105982i	−0.633863	−	0.773445i	\(-0.718532\pi\)
							−0.476033	+	0.879428i	\(0.657926\pi\)
\(19\)	−2.44272	−	0.233251i	−0.560398	−	0.0535115i	−0.188986	−	0.981980i	\(-0.560520\pi\)
							−0.371412	+	0.928468i	\(0.621126\pi\)
\(23\)	0.0120486	+	4.79582i	0.00251231	+	0.999997i
							\(29\)	−2.52742	+	5.53428i	−0.469330	+	1.02769i	0.515931	+	0.856630i	\(0.327446\pi\)
−0.985261	+	0.171059i	\(0.945281\pi\)
\(31\)	−7.11480	+	2.84834i	−1.27786	+	0.511576i	−0.908567	−	0.417738i	\(-0.862823\pi\)
							−0.369289	+	0.929315i	\(0.620399\pi\)
\(37\)	−1.38229	+	5.69789i	−0.227248	+	0.936728i	0.737877	+	0.674936i	\(0.235829\pi\)
							−0.965124	+	0.261792i	\(0.915686\pi\)
\(41\)	−1.60565	−	0.471460i	−0.250760	−	0.0736297i	0.153937	−	0.988081i	\(-0.450805\pi\)
							−0.404697	+	0.914451i	\(0.632623\pi\)
\(43\)	0.888338	+	6.17852i	0.135470	+	0.942216i	0.938254	+	0.345946i	\(0.112442\pi\)
							−0.802784	+	0.596270i	\(0.796649\pi\)
\(47\)	4.89353	−	8.47585i	0.713795	−	1.23633i	−0.249627	−	0.968342i	\(-0.580308\pi\)
							0.963422	−	0.267988i	\(-0.0863586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.163.4	yes	320
7.4	even	3	inner	483.2.y.a.25.13	✓	320
23.12	even	11	inner	483.2.y.a.58.13	yes	320
161.81	even	33	inner	483.2.y.a.403.4	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.25.13	✓	320	7.4	even	3	inner
483.2.y.a.58.13	yes	320	23.12	even	11	inner
483.2.y.a.163.4	yes	320	1.1	even	1	trivial
483.2.y.a.403.4	yes	320	161.81	even	33	inner