Embedded newform 483.2.y.b.193.7

• Label: 483.2.y.b.193.7
• 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			193.7
Character	\(\chi\)	\(=\)	483.193
Dual form			483.2.y.b.478.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.222171 - 0.211840i) q^{2} +(0.981929 - 0.189251i) q^{3} +(-0.0906799 - 1.90361i) q^{4} +(2.08025 - 1.63593i) q^{5} +(-0.258247 - 0.165965i) q^{6} +(-1.45194 - 2.21176i) q^{7} +(-0.785169 + 0.906134i) q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.222171	−	0.211840i	−0.157099	−	0.149793i	0.607453	−	0.794355i	\(-0.292191\pi\)
							−0.764552	+	0.644562i	\(0.777040\pi\)
\(3\)	0.981929	−	0.189251i	0.566917	−	0.109264i
							\(5\)	2.08025	−	1.63593i	0.930315	−	0.731608i	−0.0331475	−	0.999450i	\(-0.510553\pi\)
0.963463	+	0.267843i	\(0.0863107\pi\)
\(7\)	−1.45194	−	2.21176i	−0.548780	−	0.835967i
							\(11\)	1.46723	−	1.39900i	0.442387	−	0.421815i	−0.435866	−	0.900012i	\(-0.643558\pi\)
0.878253	+	0.478197i	\(0.158709\pi\)
\(13\)	1.94718	+	4.26374i	0.540052	+	1.18255i	0.961275	+	0.275592i	\(0.0888737\pi\)
							−0.421223	+	0.906957i	\(0.638399\pi\)
\(17\)	−2.16594	+	1.11662i	−0.525318	+	0.270820i	−0.700421	−	0.713730i	\(-0.747004\pi\)
							0.175103	+	0.984550i	\(0.443974\pi\)
\(19\)	−2.43507	−	1.25537i	−0.558644	−	0.288001i	0.155690	−	0.987806i	\(-0.450240\pi\)
							−0.714334	+	0.699805i	\(0.753270\pi\)
\(23\)	−3.70893	−	3.04037i	−0.773365	−	0.633961i
							\(29\)	6.42397	+	4.12844i	1.19290	+	0.766632i	0.977714	−	0.209941i	\(-0.0673272\pi\)
0.215188	+	0.976573i	\(0.430964\pi\)
\(31\)	1.91210	+	5.52466i	0.343424	+	0.992258i	0.976002	+	0.217762i	\(0.0698755\pi\)
							−0.632578	+	0.774497i	\(0.718003\pi\)
\(37\)	7.91858	−	3.17012i	1.30181	−	0.521165i	0.385894	−	0.922543i	\(-0.373893\pi\)
							0.915912	+	0.401378i	\(0.131469\pi\)
\(41\)	−0.609770	−	4.24104i	−0.0952301	−	0.662340i	−0.980392	−	0.197057i	\(-0.936862\pi\)
							0.885162	−	0.465283i	\(-0.154047\pi\)
\(43\)	2.70855	+	3.12583i	0.413050	+	0.476685i	0.923707	−	0.383099i	\(-0.125143\pi\)
							−0.510657	+	0.859784i	\(0.670598\pi\)
\(47\)	−3.20703	−	5.55474i	−0.467793	−	0.810242i	0.531530	−	0.847040i	\(-0.321617\pi\)
							−0.999323	+	0.0367982i	\(0.988284\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.193.7	✓	320
7.2	even	3	inner	483.2.y.b.331.7	yes	320
23.18	even	11	inner	483.2.y.b.340.7	yes	320
161.156	even	33	inner	483.2.y.b.478.7	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.7	✓	320	1.1	even	1	trivial
483.2.y.b.331.7	yes	320	7.2	even	3	inner
483.2.y.b.340.7	yes	320	23.18	even	11	inner
483.2.y.b.478.7	yes	320	161.156	even	33	inner