Embedded newform 483.2.bf.a.388.12

• Label: 483.2.bf.a.388.12
• Level: $483$
• Weight: $2$
• Character: 483.388
• 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			388.12
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.929110 + 0.730660i) q^{2} +(-0.814576 + 0.580057i) q^{3} +(-0.142137 + 0.585896i) q^{4} +(-3.37356 - 0.650200i) q^{5} +(0.333006 - 1.13411i) q^{6} +(2.36979 - 1.17647i) q^{7} +(-1.27807 - 2.79857i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.929110	+	0.730660i	−0.656980	+	0.516655i	−0.890084	−	0.455797i	\(-0.849354\pi\)
							0.233104	+	0.972452i	\(0.425112\pi\)
\(3\)	−0.814576	+	0.580057i	−0.470296	+	0.334896i
							\(5\)	−3.37356	−	0.650200i	−1.50870	−	0.290778i	−0.633198	−	0.773990i	\(-0.718258\pi\)
−0.875504	+	0.483212i	\(0.839470\pi\)
\(7\)	2.36979	−	1.17647i	0.895698	−	0.444664i
							\(11\)	−2.39072	+	3.04005i	−0.720829	+	0.916608i	−0.999035	−	0.0439128i	\(-0.986018\pi\)
0.278207	+	0.960521i	\(0.410260\pi\)
\(13\)	1.88551	−	2.93391i	0.522946	−	0.813719i	−0.474851	−	0.880066i	\(-0.657498\pi\)
							0.997797	+	0.0663469i	\(0.0211344\pi\)
\(17\)	1.19842	−	1.14269i	0.290660	−	0.277144i	−0.530693	−	0.847564i	\(-0.678068\pi\)
							0.821353	+	0.570421i	\(0.193220\pi\)
\(19\)	−1.99478	−	1.90202i	−0.457635	−	0.436354i	0.425864	−	0.904787i	\(-0.359970\pi\)
							−0.883499	+	0.468433i	\(0.844819\pi\)
\(23\)	4.73168	+	0.781824i	0.986622	+	0.163022i
							\(29\)	4.78405	+	1.40472i	0.888376	+	0.260851i	0.693913	−	0.720059i	\(-0.255885\pi\)
0.194463	+	0.980910i	\(0.437703\pi\)
\(31\)	−0.359307	+	3.76283i	−0.0645334	+	0.675825i	0.904098	+	0.427324i	\(0.140544\pi\)
							−0.968632	+	0.248500i	\(0.920062\pi\)
\(37\)	2.75219	+	0.952544i	0.452458	+	0.156597i	0.543786	−	0.839224i	\(-0.316990\pi\)
							−0.0913286	+	0.995821i	\(0.529111\pi\)
\(41\)	−0.0864582	−	0.0749165i	−0.0135025	−	0.0117000i	0.648083	−	0.761569i	\(-0.275571\pi\)
							−0.661586	+	0.749869i	\(0.730116\pi\)
\(43\)	−3.62491	−	1.65544i	−0.552794	−	0.252453i	0.119376	−	0.992849i	\(-0.461911\pi\)
							−0.672170	+	0.740396i	\(0.734638\pi\)
\(47\)	7.77973	−	4.49163i	1.13479	−	0.655172i	0.189655	−	0.981851i	\(-0.439263\pi\)
							0.945135	+	0.326679i	\(0.105930\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.12	yes	640
7.5	odd	6	inner	483.2.bf.a.250.21	yes	640
23.15	odd	22	inner	483.2.bf.a.199.21	yes	640
161.61	even	66	inner	483.2.bf.a.61.12	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.12	✓	640	161.61	even	66	inner
483.2.bf.a.199.21	yes	640	23.15	odd	22	inner
483.2.bf.a.250.21	yes	640	7.5	odd	6	inner
483.2.bf.a.388.12	yes	640	1.1	even	1	trivial