Embedded newform 483.2.bf.a.10.2

• Label: 483.2.bf.a.10.2
• 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			10.2
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.850716 + 2.45798i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(-3.74585 - 2.94577i) q^{4} +(1.41466 - 0.135083i) q^{5} +(2.57456 - 0.370166i) q^{6} +(-1.58058 + 2.12174i) q^{7} +(6.05103 - 3.88876i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.850716	+	2.45798i	−0.601547	+	1.73806i	0.0704647	+	0.997514i	\(0.477552\pi\)
							−0.672011	+	0.740541i	\(0.734569\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	1.41466	−	0.135083i	0.632653	−	0.0604110i	0.226197	−	0.974082i	\(-0.427371\pi\)
0.406456	+	0.913670i	\(0.366765\pi\)
\(7\)	−1.58058	+	2.12174i	−0.597403	+	0.801941i
							\(11\)	−1.78014	+	0.616114i	−0.536733	+	0.185765i	−0.581969	−	0.813211i	\(-0.697717\pi\)
0.0452354	+	0.998976i	\(0.485596\pi\)
\(13\)	0.406488	+	1.38437i	0.112739	+	0.383955i	0.996461	−	0.0840579i	\(-0.0267881\pi\)
							−0.883721	+	0.468013i	\(0.844970\pi\)
\(17\)	−2.52072	−	1.00915i	−0.611365	−	0.244754i	0.0452547	−	0.998975i	\(-0.485590\pi\)
							−0.656620	+	0.754222i	\(0.728014\pi\)
\(19\)	−3.47681	+	1.39190i	−0.797635	+	0.319325i	−0.734438	−	0.678676i	\(-0.762554\pi\)
							−0.0631973	+	0.998001i	\(0.520130\pi\)
\(23\)	−3.90072	−	2.79005i	−0.813356	−	0.581767i
							\(29\)	−1.23921	−	8.61887i	−0.230115	−	1.60048i	−0.697603	−	0.716485i	\(-0.745750\pi\)
0.467488	−	0.883999i	\(-0.345159\pi\)
\(31\)	0.102376	−	0.00487677i	0.0183873	−	0.000875894i	−0.0383863	−	0.999263i	\(-0.512222\pi\)
							0.0567735	+	0.998387i	\(0.481919\pi\)
\(37\)	0.511929	+	0.364543i	0.0841606	+	0.0599305i	0.621363	−	0.783523i	\(-0.286579\pi\)
							−0.537202	+	0.843454i	\(0.680519\pi\)
\(41\)	−6.66902	−	3.04564i	−1.04153	−	0.475649i	−0.180162	−	0.983637i	\(-0.557662\pi\)
							−0.861364	+	0.507988i	\(0.830389\pi\)
\(43\)	−1.58344	+	2.46388i	−0.241472	+	0.375738i	−0.940744	−	0.339118i	\(-0.889871\pi\)
							0.699272	+	0.714856i	\(0.253508\pi\)
\(47\)	−8.83301	+	5.09974i	−1.28843	+	0.743873i	−0.978374	−	0.206846i	\(-0.933680\pi\)
							−0.310053	+	0.950719i	\(0.600347\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.2	✓	640
7.5	odd	6	inner	483.2.bf.a.355.31	yes	640
23.7	odd	22	inner	483.2.bf.a.283.31	yes	640
161.145	even	66	inner	483.2.bf.a.145.2	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.2	✓	640	1.1	even	1	trivial
483.2.bf.a.145.2	yes	640	161.145	even	66	inner
483.2.bf.a.283.31	yes	640	23.7	odd	22	inner
483.2.bf.a.355.31	yes	640	7.5	odd	6	inner