Embedded newform 483.2.y.b.193.6

• Label: 483.2.y.b.193.6
• 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			193.6
Character	\(\chi\)	\(=\)	483.193
Dual form			483.2.y.b.478.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.575609 - 0.548842i) q^{2} +(0.981929 - 0.189251i) q^{3} +(-0.0650657 - 1.36590i) q^{4} +(-1.82076 + 1.43186i) q^{5} +(-0.669076 - 0.429989i) q^{6} +(-2.13456 + 1.56322i) q^{7} +(-1.75387 + 2.02408i) q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.575609	−	0.548842i	−0.407017	−	0.388090i	0.458727	−	0.888577i	\(-0.348306\pi\)
							−0.865744	+	0.500487i	\(0.833154\pi\)
\(3\)	0.981929	−	0.189251i	0.566917	−	0.109264i
							\(5\)	−1.82076	+	1.43186i	−0.814270	+	0.640349i	−0.935915	−	0.352225i	\(-0.885425\pi\)
0.121646	+	0.992574i	\(0.461183\pi\)
\(7\)	−2.13456	+	1.56322i	−0.806789	+	0.590840i
							\(11\)	0.142334	−	0.135715i	0.0429153	−	0.0409196i	−0.668318	−	0.743876i	\(-0.732985\pi\)
0.711233	+	0.702956i	\(0.248137\pi\)
\(13\)	−0.365286	−	0.799865i	−0.101312	−	0.221843i	0.852188	−	0.523236i	\(-0.175275\pi\)
							−0.953500	+	0.301394i	\(0.902548\pi\)
\(17\)	−6.76196	+	3.48603i	−1.64002	+	0.845487i	−0.643204	+	0.765695i	\(0.722395\pi\)
							−0.996812	+	0.0797915i	\(0.974575\pi\)
\(19\)	−3.22261	−	1.66137i	−0.739318	−	0.381145i	0.0470585	−	0.998892i	\(-0.485015\pi\)
							−0.786377	+	0.617747i	\(0.788046\pi\)
\(23\)	−0.241665	+	4.78974i	−0.0503907	+	0.998730i
							\(29\)	−1.02138	−	0.656400i	−0.189665	−	0.121890i	0.442362	−	0.896837i	\(-0.354141\pi\)
−0.632027	+	0.774946i	\(0.717777\pi\)
\(31\)	−0.489952	−	1.41562i	−0.0879980	−	0.254254i	0.892412	−	0.451222i	\(-0.149012\pi\)
							−0.980410	+	0.196968i	\(0.936890\pi\)
\(37\)	2.55743	−	1.02384i	0.420439	−	0.168318i	−0.151789	−	0.988413i	\(-0.548503\pi\)
							0.572228	+	0.820095i	\(0.306079\pi\)
\(41\)	−0.172968	−	1.20302i	−0.0270131	−	0.187880i	0.971847	−	0.235612i	\(-0.0757096\pi\)
							−0.998860	+	0.0477325i	\(0.984801\pi\)
\(43\)	1.16802	+	1.34797i	0.178121	+	0.205563i	0.837788	−	0.545995i	\(-0.183848\pi\)
							−0.659667	+	0.751558i	\(0.729303\pi\)
\(47\)	0.751264	+	1.30123i	0.109583	+	0.189804i	0.915601	−	0.402087i	\(-0.131715\pi\)
							−0.806018	+	0.591891i	\(0.798382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.193.6	✓	320
7.2	even	3	inner	483.2.y.b.331.6	yes	320
23.18	even	11	inner	483.2.y.b.340.6	yes	320
161.156	even	33	inner	483.2.y.b.478.6	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.6	✓	320	1.1	even	1	trivial
483.2.y.b.331.6	yes	320	7.2	even	3	inner
483.2.y.b.340.6	yes	320	23.18	even	11	inner
483.2.y.b.478.6	yes	320	161.156	even	33	inner