Embedded newform 483.2.bf.a.10.28

• Label: 483.2.bf.a.10.28
• 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			10.28
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.679230 - 1.96250i) q^{2} +(0.458227 + 0.888835i) q^{3} +(-1.81797 - 1.42966i) q^{4} +(-3.97123 + 0.379207i) q^{5} +(2.05558 - 0.295548i) q^{6} +(0.921223 - 2.48019i) q^{7} +(-0.546439 + 0.351175i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.679230	−	1.96250i	0.480288	−	1.38770i	−0.401415	−	0.915896i	\(-0.631482\pi\)
							0.881703	−	0.471804i	\(-0.156397\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	−3.97123	+	0.379207i	−1.77599	+	0.169586i	−0.930957	−	0.365129i	\(-0.881025\pi\)
−0.845032	+	0.534716i	\(0.820419\pi\)
\(7\)	0.921223	−	2.48019i	0.348190	−	0.937424i
							\(11\)	−3.49404	+	1.20930i	−1.05349	+	0.364617i	−0.798292	−	0.602271i	\(-0.794263\pi\)
−0.255201	+	0.966888i	\(0.582142\pi\)
\(13\)	−1.42333	−	4.84742i	−0.394761	−	1.34443i	−0.882036	−	0.471183i	\(-0.843827\pi\)
							0.487275	−	0.873249i	\(-0.337991\pi\)
\(17\)	−3.63594	−	1.45561i	−0.881845	−	0.353037i	−0.113853	−	0.993498i	\(-0.536319\pi\)
							−0.767991	+	0.640460i	\(0.778744\pi\)
\(19\)	−6.01028	+	2.40615i	−1.37885	+	0.552010i	−0.938333	−	0.345733i	\(-0.887630\pi\)
							−0.440521	+	0.897743i	\(0.645206\pi\)
\(23\)	4.30517	−	2.11318i	0.897690	−	0.440628i
							\(29\)	0.765412	+	5.32356i	0.142133	+	0.988560i	0.928641	+	0.370979i	\(0.120978\pi\)
−0.786508	+	0.617580i	\(0.788113\pi\)
\(31\)	2.45132	−	0.116771i	0.440270	−	0.0209726i	0.173723	−	0.984795i	\(-0.444420\pi\)
							0.266547	+	0.963822i	\(0.414117\pi\)
\(37\)	0.999459	+	0.711711i	0.164310	+	0.117005i	0.659254	−	0.751920i	\(-0.270872\pi\)
							−0.494944	+	0.868925i	\(0.664812\pi\)
\(41\)	1.81963	+	0.830995i	0.284178	+	0.129780i	0.552405	−	0.833576i	\(-0.313710\pi\)
							−0.268227	+	0.963356i	\(0.586438\pi\)
\(43\)	5.70399	−	8.87559i	0.869851	−	1.35351i	−0.0647818	−	0.997899i	\(-0.520635\pi\)
							0.934633	−	0.355615i	\(-0.115729\pi\)
\(47\)	3.43100	−	1.98089i	0.500463	−	0.288942i	−0.228442	−	0.973558i	\(-0.573363\pi\)
							0.728905	+	0.684615i	\(0.240030\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.28	✓	640
7.5	odd	6	inner	483.2.bf.a.355.5	yes	640
23.7	odd	22	inner	483.2.bf.a.283.5	yes	640
161.145	even	66	inner	483.2.bf.a.145.28	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.28	✓	640	1.1	even	1	trivial
483.2.bf.a.145.28	yes	640	161.145	even	66	inner
483.2.bf.a.283.5	yes	640	23.7	odd	22	inner
483.2.bf.a.355.5	yes	640	7.5	odd	6	inner