Embedded newform 483.2.y.b.4.15

• Label: 483.2.y.b.4.15
• 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			4.15
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38638 - 1.94690i) q^{2} +(0.235759 - 0.971812i) q^{3} +(-1.21423 - 3.50828i) q^{4} +(0.129724 - 2.72324i) q^{5} +(-1.56517 - 1.80630i) q^{6} +(-0.222669 + 2.63636i) q^{7} +(-3.92712 - 1.15311i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38638	−	1.94690i	0.980319	−	1.37667i	0.0545766	−	0.998510i	\(-0.482619\pi\)
							0.925742	−	0.378155i	\(-0.123442\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	0.129724	−	2.72324i	0.0580143	−	1.21787i	−0.762785	−	0.646652i	\(-0.776168\pi\)
0.820799	−	0.571217i	\(-0.193529\pi\)
\(7\)	−0.222669	+	2.63636i	−0.0841608	+	0.996452i
							\(11\)	−0.659545	−	0.926201i	−0.198860	−	0.279260i	0.703089	−	0.711102i	\(-0.251803\pi\)
−0.901950	+	0.431841i	\(0.857864\pi\)
\(13\)	−0.207775	+	1.44511i	−0.0576264	+	0.400800i	0.940509	+	0.339768i	\(0.110348\pi\)
							−0.998136	+	0.0610326i	\(0.980561\pi\)
\(17\)	1.95608	−	0.377003i	0.474418	−	0.0914366i	0.0535631	−	0.998564i	\(-0.482942\pi\)
							0.420855	+	0.907128i	\(0.361730\pi\)
\(19\)	−1.21893	−	0.234930i	−0.279642	−	0.0538966i	0.0475018	−	0.998871i	\(-0.484874\pi\)
							−0.327144	+	0.944975i	\(0.606086\pi\)
\(23\)	4.69404	−	0.982842i	0.978775	−	0.204937i
							\(29\)	1.79620	+	2.07293i	0.333546	+	0.384933i	0.897604	−	0.440802i	\(-0.145306\pi\)
−0.564058	+	0.825735i	\(0.690761\pi\)
\(31\)	−1.50849	+	1.43834i	−0.270932	+	0.258333i	−0.813349	−	0.581776i	\(-0.802358\pi\)
							0.542417	+	0.840110i	\(0.317509\pi\)
\(37\)	8.09956	+	4.17561i	1.33156	+	0.686466i	0.969362	−	0.245635i	\(-0.0789966\pi\)
							0.362197	+	0.932102i	\(0.382027\pi\)
\(41\)	0.00782816	+	0.00503085i	0.00122255	+	0.000785687i	0.541252	−	0.840860i	\(-0.317951\pi\)
							−0.540029	+	0.841646i	\(0.681587\pi\)
\(43\)	−10.5782	+	3.10603i	−1.61316	+	0.473666i	−0.959168	−	0.282837i	\(-0.908724\pi\)
							−0.653990	+	0.756503i	\(0.726906\pi\)
\(47\)	−4.79449	−	8.30429i	−0.699348	−	1.21131i	−0.968693	−	0.248262i	\(-0.920141\pi\)
							0.269345	−	0.963044i	\(-0.413193\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.15	✓	320
7.2	even	3	inner	483.2.y.b.142.2	yes	320
23.6	even	11	inner	483.2.y.b.466.2	yes	320
161.121	even	33	inner	483.2.y.b.121.15	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.15	✓	320	1.1	even	1	trivial
483.2.y.b.121.15	yes	320	161.121	even	33	inner
483.2.y.b.142.2	yes	320	7.2	even	3	inner
483.2.y.b.466.2	yes	320	23.6	even	11	inner