Embedded newform 483.2.y.b.193.1

• Label: 483.2.y.b.193.1
• Level: $483$
• Weight: $2$
• Character: 483.193
• 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			193.1
Character	\(\chi\)	\(=\)	483.193
Dual form			483.2.y.b.478.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95956 - 1.86843i) q^{2} +(0.981929 - 0.189251i) q^{3} +(0.253654 + 5.32486i) q^{4} +(0.0774480 - 0.0609058i) q^{5} +(-2.27775 - 1.46382i) q^{6} +(2.62353 - 0.342151i) q^{7} +(5.90594 - 6.81582i) q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95956	−	1.86843i	−1.38562	−	1.32118i	−0.892379	−	0.451286i	\(-0.850965\pi\)
							−0.493236	−	0.869896i	\(-0.664186\pi\)
\(3\)	0.981929	−	0.189251i	0.566917	−	0.109264i
							\(5\)	0.0774480	−	0.0609058i	0.0346358	−	0.0272379i	−0.600691	−	0.799481i	\(-0.705108\pi\)
0.635327	+	0.772243i	\(0.280865\pi\)
\(7\)	2.62353	−	0.342151i	0.991603	−	0.129321i
							\(11\)	−3.10855	+	2.96399i	−0.937262	+	0.893678i	−0.994506	−	0.104683i	\(-0.966617\pi\)
0.0572435	+	0.998360i	\(0.481769\pi\)
\(13\)	2.68262	+	5.87412i	0.744025	+	1.62919i	0.776818	+	0.629725i	\(0.216832\pi\)
							−0.0327937	+	0.999462i	\(0.510440\pi\)
\(17\)	−3.07460	+	1.58507i	−0.745700	+	0.384435i	−0.788811	−	0.614635i	\(-0.789303\pi\)
							0.0431115	+	0.999070i	\(0.486273\pi\)
\(19\)	−0.0534747	−	0.0275681i	−0.0122679	−	0.00632457i	0.452082	−	0.891977i	\(-0.350682\pi\)
							−0.464350	+	0.885652i	\(0.653712\pi\)
\(23\)	4.25404	+	2.21430i	0.887029	+	0.461714i
							\(29\)	4.91447	+	3.15834i	0.912594	+	0.586489i	0.910500	−	0.413509i	\(-0.135697\pi\)
0.00209412	+	0.999998i	\(0.499333\pi\)
\(31\)	−0.671067	−	1.93892i	−0.120527	−	0.348240i	0.868558	−	0.495588i	\(-0.165048\pi\)
							−0.989085	+	0.147348i	\(0.952926\pi\)
\(37\)	1.03559	−	0.414588i	0.170250	−	0.0681579i	−0.284972	−	0.958536i	\(-0.591984\pi\)
							0.455222	+	0.890378i	\(0.349560\pi\)
\(41\)	−1.14797	−	7.98429i	−0.179283	−	1.24694i	−0.858428	−	0.512934i	\(-0.828559\pi\)
							0.679146	−	0.734004i	\(-0.262350\pi\)
\(43\)	−7.14909	−	8.25048i	−1.09023	−	1.25819i	−0.963919	−	0.266196i	\(-0.914233\pi\)
							−0.126306	−	0.991991i	\(-0.540312\pi\)
\(47\)	5.45332	+	9.44542i	0.795448	+	1.37776i	0.922554	+	0.385867i	\(0.126098\pi\)
							−0.127106	+	0.991889i	\(0.540569\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.193.1	✓	320
7.2	even	3	inner	483.2.y.b.331.1	yes	320
23.18	even	11	inner	483.2.y.b.340.1	yes	320
161.156	even	33	inner	483.2.y.b.478.1	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.1	✓	320	1.1	even	1	trivial
483.2.y.b.331.1	yes	320	7.2	even	3	inner
483.2.y.b.340.1	yes	320	23.18	even	11	inner
483.2.y.b.478.1	yes	320	161.156	even	33	inner