Embedded newform 483.2.y.a.478.15

• Label: 483.2.y.a.478.15
• Level: $483$
• Weight: $2$
• Character: 483.478
• 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			478.15
Character	\(\chi\)	\(=\)	483.478
Dual form			483.2.y.a.193.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.71111 - 1.63154i) q^{2} +(-0.981929 - 0.189251i) q^{3} +(0.170812 - 3.58578i) q^{4} +(2.82540 + 2.22192i) q^{5} +(-1.98896 + 1.27823i) q^{6} +(-1.76234 - 1.97336i) q^{7} +(-2.46152 - 2.84074i) q^{8} +(0.928368 + 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.71111	−	1.63154i	1.20994	−	1.15368i	0.226199	−	0.974081i	\(-0.427370\pi\)
							0.983741	−	0.179594i	\(-0.0574785\pi\)
\(3\)	−0.981929	−	0.189251i	−0.566917	−	0.109264i
							\(5\)	2.82540	+	2.22192i	1.26356	+	0.993673i	0.999575	+	0.0291463i	\(0.00927887\pi\)
0.263983	+	0.964527i	\(0.414964\pi\)
\(7\)	−1.76234	−	1.97336i	−0.666101	−	0.745862i
							\(11\)	−3.32800	−	3.17324i	−1.00343	−	0.956767i	−0.00436318	−	0.999990i	\(-0.501389\pi\)
−0.999065	+	0.0432232i	\(0.986237\pi\)
\(13\)	2.20273	−	4.82330i	0.610927	−	1.33774i	−0.311011	−	0.950406i	\(-0.600668\pi\)
							0.921938	−	0.387337i	\(-0.126605\pi\)
\(17\)	1.26200	+	0.650608i	0.306081	+	0.157796i	0.604424	−	0.796663i	\(-0.293403\pi\)
							−0.298343	+	0.954459i	\(0.596434\pi\)
\(19\)	4.64926	−	2.39686i	1.06661	−	0.549878i	0.166728	−	0.986003i	\(-0.446680\pi\)
							0.899886	+	0.436125i	\(0.143650\pi\)
\(23\)	1.87717	+	4.41319i	0.391417	+	0.920214i
							\(29\)	−4.42458	+	2.84350i	−0.821624	+	0.528026i	−0.882606	−	0.470113i	\(-0.844213\pi\)
0.0609823	+	0.998139i	\(0.480577\pi\)
\(31\)	−1.56243	+	4.51434i	−0.280620	+	0.810799i	0.713316	+	0.700843i	\(0.247192\pi\)
							−0.993936	+	0.109957i	\(0.964929\pi\)
\(37\)	0.228280	+	0.0913894i	0.0375289	+	0.0150243i	0.390351	−	0.920666i	\(-0.372354\pi\)
							−0.352822	+	0.935690i	\(0.614778\pi\)
\(41\)	−0.958375	+	6.66564i	−0.149673	+	1.04100i	0.767082	+	0.641549i	\(0.221708\pi\)
							−0.916755	+	0.399450i	\(0.869201\pi\)
\(43\)	−7.20513	+	8.31516i	−1.09877	+	1.26805i	−0.138084	+	0.990421i	\(0.544094\pi\)
							−0.960688	+	0.277630i	\(0.910451\pi\)
\(47\)	−3.98690	+	6.90551i	−0.581549	+	1.00727i	0.413747	+	0.910392i	\(0.364220\pi\)
							−0.995296	+	0.0968804i	\(0.969114\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.478.15	yes	320
7.4	even	3	inner	483.2.y.a.340.15	yes	320
23.9	even	11	inner	483.2.y.a.331.15	yes	320
161.32	even	33	inner	483.2.y.a.193.15	✓	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.193.15	✓	320	161.32	even	33	inner
483.2.y.a.331.15	yes	320	23.9	even	11	inner
483.2.y.a.340.15	yes	320	7.4	even	3	inner
483.2.y.a.478.15	yes	320	1.1	even	1	trivial