Embedded newform 483.2.y.b.25.2

• Label: 483.2.y.b.25.2
• Level: $483$
• Weight: $2$
• Character: 483.25
• 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			25.2
Character	\(\chi\)	\(=\)	483.25
Dual form			483.2.y.b.58.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.108052 + 2.26829i) q^{2} +(0.928368 + 0.371662i) q^{3} +(-3.14254 - 0.300076i) q^{4} +(0.756875 + 3.11988i) q^{5} +(-0.943352 + 2.06565i) q^{6} +(-2.46743 - 0.954876i) q^{7} +(0.373863 - 2.60027i) q^{8} +(0.723734 + 0.690079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.108052	+	2.26829i	−0.0764044	+	1.60393i	0.557273	+	0.830329i	\(0.311848\pi\)
							−0.633678	+	0.773597i	\(0.718456\pi\)
\(3\)	0.928368	+	0.371662i	0.535993	+	0.214579i
							\(5\)	0.756875	+	3.11988i	0.338485	+	1.39525i	0.845804	+	0.533495i	\(0.179122\pi\)
−0.507319	+	0.861758i	\(0.669363\pi\)
\(7\)	−2.46743	−	0.954876i	−0.932601	−	0.360909i
							\(11\)	0.0489591	+	1.02778i	0.0147617	+	0.309887i	0.994158	+	0.107937i	\(0.0344246\pi\)
−0.979396	+	0.201949i	\(0.935272\pi\)
\(13\)	−0.882063	−	1.01795i	−0.244640	−	0.282330i	0.620128	−	0.784500i	\(-0.287080\pi\)
							−0.864769	+	0.502170i	\(0.832535\pi\)
\(17\)	1.07086	+	1.50381i	0.259722	+	0.364729i	0.923761	−	0.382969i	\(-0.125099\pi\)
							−0.664039	+	0.747698i	\(0.731159\pi\)
\(19\)	1.50930	−	2.11951i	0.346257	−	0.486250i	−0.604319	−	0.796743i	\(-0.706555\pi\)
							0.950576	+	0.310493i	\(0.100494\pi\)
\(23\)	−0.750041	+	4.73682i	−0.156394	+	0.987695i
							\(29\)	1.20181	−	2.63159i	0.223170	−	0.488675i	−0.764617	−	0.644485i	\(-0.777072\pi\)
0.987787	+	0.155811i	\(0.0497990\pi\)
\(31\)	5.55612	+	4.36938i	0.997908	+	0.784763i	0.976612	−	0.215010i	\(-0.0689783\pi\)
							0.0212960	+	0.999773i	\(0.493221\pi\)
\(37\)	−1.57628	−	1.50298i	−0.259139	−	0.247088i	0.549360	−	0.835585i	\(-0.314871\pi\)
							−0.808499	+	0.588497i	\(0.799720\pi\)
\(41\)	−9.25775	−	2.71832i	−1.44582	−	0.424531i	−0.537661	−	0.843161i	\(-0.680692\pi\)
							−0.908157	+	0.418630i	\(0.862510\pi\)
\(43\)	0.528860	+	3.67830i	0.0806505	+	0.560936i	0.989580	+	0.143986i	\(0.0459920\pi\)
							−0.908929	+	0.416950i	\(0.863099\pi\)
\(47\)	2.39235	+	4.14367i	0.348960	+	0.604417i	0.986065	−	0.166360i	\(-0.0532015\pi\)
							−0.637105	+	0.770777i	\(0.719868\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.25.2	✓	320
7.2	even	3	inner	483.2.y.b.163.15	yes	320
23.12	even	11	inner	483.2.y.b.403.15	yes	320
161.58	even	33	inner	483.2.y.b.58.2	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.25.2	✓	320	1.1	even	1	trivial
483.2.y.b.58.2	yes	320	161.58	even	33	inner
483.2.y.b.163.15	yes	320	7.2	even	3	inner
483.2.y.b.403.15	yes	320	23.12	even	11	inner