Embedded newform 483.2.bf.a.10.10

• Label: 483.2.bf.a.10.10
• Level: $483$
• Weight: $2$
• Character: 483.10
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			10.10
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.340926 + 0.985040i) q^{2} +(0.458227 + 0.888835i) q^{3} +(0.718033 + 0.564667i) q^{4} +(3.21073 - 0.306587i) q^{5} +(-1.03176 + 0.148345i) q^{6} +(-1.39165 - 2.25018i) q^{7} +(-2.55481 + 1.64188i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.340926	+	0.985040i	−0.241071	+	0.696528i	0.757951	+	0.652311i	\(0.226201\pi\)
							−0.999022	+	0.0442170i	\(0.985921\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	3.21073	−	0.306587i	1.43588	−	0.137110i	0.652093	−	0.758139i	\(-0.273891\pi\)
0.783787	+	0.621029i	\(0.213285\pi\)
\(7\)	−1.39165	−	2.25018i	−0.525996	−	0.850487i
							\(11\)	3.42423	−	1.18514i	1.03245	−	0.357333i	0.242267	−	0.970210i	\(-0.422109\pi\)
0.790178	+	0.612877i	\(0.209988\pi\)
\(13\)	1.57190	+	5.35341i	0.435967	+	1.48477i	0.825843	+	0.563901i	\(0.190700\pi\)
							−0.389875	+	0.920868i	\(0.627482\pi\)
\(17\)	−5.14547	−	2.05994i	−1.24796	−	0.499608i	−0.348787	−	0.937202i	\(-0.613406\pi\)
							−0.899173	+	0.437594i	\(0.855831\pi\)
\(19\)	3.70419	−	1.48293i	0.849799	−	0.340208i	0.0944488	−	0.995530i	\(-0.469891\pi\)
							0.755350	+	0.655322i	\(0.227467\pi\)
\(23\)	−0.334412	+	4.78416i	−0.0697297	+	0.997566i
							\(29\)	−0.591381	−	4.11315i	−0.109817	−	0.763792i	−0.968091	−	0.250600i	\(-0.919372\pi\)
0.858274	−	0.513192i	\(-0.171537\pi\)
\(31\)	6.14466	−	0.292706i	1.10361	−	0.0525715i	0.512144	−	0.858900i	\(-0.328851\pi\)
							0.591469	+	0.806328i	\(0.298548\pi\)
\(37\)	−3.45893	−	2.46310i	−0.568645	−	0.404931i	0.259252	−	0.965810i	\(-0.416524\pi\)
							−0.827898	+	0.560879i	\(0.810463\pi\)
\(41\)	−5.01471	−	2.29014i	−0.783166	−	0.357660i	−0.0166306	−	0.999862i	\(-0.505294\pi\)
							−0.766536	+	0.642202i	\(0.778021\pi\)
\(43\)	3.31403	−	5.15673i	0.505384	−	0.786393i	−0.491017	−	0.871150i	\(-0.663375\pi\)
							0.996402	+	0.0847563i	\(0.0270112\pi\)
\(47\)	−9.04873	+	5.22429i	−1.31989	+	0.762040i	−0.983711	−	0.179757i	\(-0.942469\pi\)
							−0.336182	+	0.941797i	\(0.609136\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.10	✓	640
7.5	odd	6	inner	483.2.bf.a.355.23	yes	640
23.7	odd	22	inner	483.2.bf.a.283.23	yes	640
161.145	even	66	inner	483.2.bf.a.145.10	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.10	✓	640	1.1	even	1	trivial
483.2.bf.a.145.10	yes	640	161.145	even	66	inner
483.2.bf.a.283.23	yes	640	23.7	odd	22	inner
483.2.bf.a.355.23	yes	640	7.5	odd	6	inner