Embedded newform 483.2.bf.a.10.11

• Label: 483.2.bf.a.10.11
• 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			10.11
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.319328 + 0.922636i) q^{2} +(0.458227 + 0.888835i) q^{3} +(0.822819 + 0.647072i) q^{4} +(2.77405 - 0.264890i) q^{5} +(-0.966396 + 0.138947i) q^{6} +(1.16682 + 2.37456i) q^{7} +(-2.50245 + 1.60823i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.319328	+	0.922636i	−0.225799	+	0.652402i	0.774016	+	0.633166i	\(0.218245\pi\)
							−0.999815	+	0.0192367i	\(0.993876\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	2.77405	−	0.264890i	1.24059	−	0.118462i	0.545951	−	0.837817i	\(-0.316168\pi\)
0.694643	+	0.719354i	\(0.255562\pi\)
\(7\)	1.16682	+	2.37456i	0.441018	+	0.897498i
							\(11\)	−1.78836	+	0.618957i	−0.539211	+	0.186623i	−0.583080	−	0.812415i	\(-0.698153\pi\)
0.0438695	+	0.999037i	\(0.486031\pi\)
\(13\)	−1.20021	−	4.08754i	−0.332879	−	1.13368i	−0.940596	−	0.339527i	\(-0.889733\pi\)
							0.607718	−	0.794153i	\(-0.292085\pi\)
\(17\)	1.57904	+	0.632151i	0.382973	+	0.153319i	0.555157	−	0.831746i	\(-0.312658\pi\)
							−0.172184	+	0.985065i	\(0.555082\pi\)
\(19\)	3.37986	−	1.35309i	0.775393	−	0.310420i	0.0499942	−	0.998750i	\(-0.484080\pi\)
							0.725399	+	0.688329i	\(0.241655\pi\)
\(23\)	−1.67869	−	4.49244i	−0.350030	−	0.936738i
							\(29\)	−0.571461	−	3.97460i	−0.106118	−	0.738064i	−0.971515	−	0.236978i	\(-0.923843\pi\)
0.865397	−	0.501086i	\(-0.167066\pi\)
\(31\)	3.99327	−	0.190223i	0.717212	−	0.0341650i	0.314196	−	0.949358i	\(-0.398265\pi\)
							0.403016	+	0.915193i	\(0.367962\pi\)
\(37\)	5.27961	+	3.75959i	0.867962	+	0.618073i	0.924860	−	0.380307i	\(-0.124182\pi\)
							−0.0568982	+	0.998380i	\(0.518121\pi\)
\(41\)	−6.98182	−	3.18849i	−1.09038	−	0.497959i	−0.212655	−	0.977127i	\(-0.568211\pi\)
							−0.877723	+	0.479169i	\(0.840938\pi\)
\(43\)	−1.81961	+	2.83137i	−0.277488	+	0.431780i	−0.951824	−	0.306645i	\(-0.900794\pi\)
							0.674336	+	0.738425i	\(0.264430\pi\)
\(47\)	−2.22689	+	1.28570i	−0.324825	+	0.187538i	−0.653541	−	0.756891i	\(-0.726717\pi\)
							0.328716	+	0.944429i	\(0.393384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.11	✓	640
7.5	odd	6	inner	483.2.bf.a.355.22	yes	640
23.7	odd	22	inner	483.2.bf.a.283.22	yes	640
161.145	even	66	inner	483.2.bf.a.145.11	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.11	✓	640	1.1	even	1	trivial
483.2.bf.a.145.11	yes	640	161.145	even	66	inner
483.2.bf.a.283.22	yes	640	23.7	odd	22	inner
483.2.bf.a.355.22	yes	640	7.5	odd	6	inner