Embedded newform 483.2.y.a.193.7

• Label: 483.2.y.a.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.a.478.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0738740 - 0.0704387i) q^{2} +(-0.981929 + 0.189251i) q^{3} +(-0.0946681 - 1.98733i) q^{4} +(2.31093 - 1.81734i) q^{5} +(0.0858696 + 0.0551850i) q^{6} +(2.06252 - 1.65711i) q^{7} +(-0.266679 + 0.307764i) q^{8} +(0.928368 - 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0738740	−	0.0704387i	−0.0522368	−	0.0498077i	0.663511	−	0.748166i	\(-0.269066\pi\)
							−0.715748	+	0.698359i	\(0.753914\pi\)
\(3\)	−0.981929	+	0.189251i	−0.566917	+	0.109264i
							\(5\)	2.31093	−	1.81734i	1.03348	−	0.812737i	0.0509229	−	0.998703i	\(-0.483784\pi\)
0.982556	+	0.185966i	\(0.0595413\pi\)
\(7\)	2.06252	−	1.65711i	0.779559	−	0.626329i
							\(11\)	−0.521758	+	0.497495i	−0.157316	+	0.150000i	−0.764649	−	0.644447i	\(-0.777088\pi\)
0.607333	+	0.794447i	\(0.292239\pi\)
\(13\)	−0.589314	−	1.29042i	−0.163446	−	0.357897i	0.810133	−	0.586246i	\(-0.199395\pi\)
							−0.973579	+	0.228349i	\(0.926667\pi\)
\(17\)	−4.48157	+	2.31041i	−1.08694	+	0.560356i	−0.906060	−	0.423149i	\(-0.860925\pi\)
							−0.180879	+	0.983505i	\(0.557894\pi\)
\(19\)	3.57344	+	1.84224i	0.819804	+	0.422638i	0.816492	−	0.577356i	\(-0.195916\pi\)
							0.00331163	+	0.999995i	\(0.498946\pi\)
\(23\)	3.15152	−	3.61496i	0.657137	−	0.753771i
							\(29\)	1.87064	+	1.20219i	0.347369	+	0.223241i	0.702678	−	0.711508i	\(-0.251988\pi\)
−0.355308	+	0.934749i	\(0.615624\pi\)
\(31\)	−3.04602	−	8.80089i	−0.547081	−	1.58069i	−0.791598	−	0.611042i	\(-0.790751\pi\)
							0.244518	−	0.969645i	\(-0.421370\pi\)
\(37\)	−0.343294	+	0.137434i	−0.0564372	+	0.0225940i	−0.399709	−	0.916642i	\(-0.630889\pi\)
							0.343272	+	0.939236i	\(0.388465\pi\)
\(41\)	0.912464	+	6.34633i	0.142503	+	0.991130i	0.928084	+	0.372371i	\(0.121455\pi\)
							−0.785581	+	0.618759i	\(0.787636\pi\)
\(43\)	−2.97097	−	3.42868i	−0.453068	−	0.522868i	0.482557	−	0.875865i	\(-0.339708\pi\)
							−0.935625	+	0.352996i	\(0.885163\pi\)
\(47\)	3.35673	+	5.81403i	0.489630	+	0.848064i	0.999929	−	0.0119334i	\(-0.00379861\pi\)
							−0.510299	+	0.859997i	\(0.670465\pi\)

(See \(a_n\) instead)

Twists

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

    

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