Embedded newform 483.2.y.a.340.7

• Label: 483.2.y.a.340.7
• Level: $483$
• Weight: $2$
• Character: 483.340
• 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			340.7
Character	\(\chi\)	\(=\)	483.340
Dual form			483.2.y.a.331.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0240647 - 0.0991961i) q^{2} +(0.327068 + 0.945001i) q^{3} +(1.76841 - 0.911679i) q^{4} +(-2.72932 + 1.09266i) q^{5} +(0.0858696 - 0.0551850i) q^{6} +(-0.0983016 + 2.64392i) q^{7} +(-0.266679 - 0.307764i) q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0240647	−	0.0991961i	−0.0170163	−	0.0701422i	0.962670	−	0.270676i	\(-0.0872474\pi\)
							−0.979687	+	0.200534i	\(0.935732\pi\)
\(3\)	0.327068	+	0.945001i	0.188833	+	0.545596i
							\(5\)	−2.72932	+	1.09266i	−1.22059	+	0.488651i	−0.890363	−	0.455251i	\(-0.849550\pi\)
−0.330227	+	0.943901i	\(0.607125\pi\)
\(7\)	−0.0983016	+	2.64392i	−0.0371545	+	0.999310i
							\(11\)	−0.169965	+	0.700603i	−0.0512462	+	0.211240i	−0.991678	−	0.128742i	\(-0.958906\pi\)
0.940432	+	0.339982i	\(0.110421\pi\)
\(13\)	−0.589314	+	1.29042i	−0.163446	+	0.357897i	−0.973579	−	0.228349i	\(-0.926667\pi\)
							0.810133	+	0.586246i	\(0.199395\pi\)
\(17\)	0.239911	+	5.03635i	0.0581870	+	1.22150i	0.819488	+	0.573096i	\(0.194258\pi\)
							−0.761301	+	0.648399i	\(0.775439\pi\)
\(19\)	−0.191297	+	4.01581i	−0.0438865	+	0.921290i	0.864365	+	0.502865i	\(0.167721\pi\)
							−0.908251	+	0.418425i	\(0.862582\pi\)
\(23\)	−4.70641	+	0.921816i	−0.981353	+	0.192212i
							\(29\)	1.87064	−	1.20219i	0.347369	−	0.223241i	−0.355308	−	0.934749i	\(-0.615624\pi\)
0.702678	+	0.711508i	\(0.251988\pi\)
\(31\)	9.14480	−	1.76252i	1.64246	−	0.316557i	0.717478	−	0.696581i	\(-0.245296\pi\)
							0.924977	+	0.380024i	\(0.124084\pi\)
\(37\)	0.290668	−	0.228584i	0.0477856	−	0.0375790i	−0.593981	−	0.804479i	\(-0.702444\pi\)
							0.641766	+	0.766900i	\(0.278202\pi\)
\(41\)	0.912464	−	6.34633i	0.142503	−	0.991130i	−0.785581	−	0.618759i	\(-0.787636\pi\)
							0.928084	−	0.372371i	\(-0.121455\pi\)
\(43\)	−2.97097	+	3.42868i	−0.453068	+	0.522868i	−0.935625	−	0.352996i	\(-0.885163\pi\)
							0.482557	+	0.875865i	\(0.339708\pi\)
\(47\)	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.340.7	yes	320
7.2	even	3	inner	483.2.y.a.478.7	yes	320
23.9	even	11	inner	483.2.y.a.193.7	✓	320
161.9	even	33	inner	483.2.y.a.331.7	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.193.7	✓	320	23.9	even	11	inner
483.2.y.a.331.7	yes	320	161.9	even	33	inner
483.2.y.a.340.7	yes	320	1.1	even	1	trivial
483.2.y.a.478.7	yes	320	7.2	even	3	inner