Embedded newform 483.2.bf.a.178.27

• Label: 483.2.bf.a.178.27
• Level: $483$
• Weight: $2$
• Character: 483.178
• 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			178.27
Character	\(\chi\)	\(=\)	483.178
Dual form			483.2.bf.a.19.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.04094 - 0.194886i) q^{2} +(-0.690079 - 0.723734i) q^{3} +(2.16361 - 0.417001i) q^{4} +(1.09672 + 0.565401i) q^{5} +(-1.54946 - 1.34261i) q^{6} +(2.20457 + 1.46284i) q^{7} +(0.400177 - 0.117503i) q^{8} +(-0.0475819 + 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.04094	−	0.194886i	1.44316	−	0.137805i	0.656099	−	0.754675i	\(-0.272205\pi\)
							0.787065	+	0.616869i	\(0.211599\pi\)
\(3\)	−0.690079	−	0.723734i	−0.398417	−	0.417848i
							\(5\)	1.09672	+	0.565401i	0.490470	+	0.252855i	0.685675	−	0.727908i	\(-0.259507\pi\)
−0.195205	+	0.980762i	\(0.562537\pi\)
\(7\)	2.20457	+	1.46284i	0.833247	+	0.552900i
							\(11\)	−0.0680321	+	0.712464i	−0.0205125	+	0.214816i	0.979404	+	0.201909i	\(0.0647146\pi\)
−0.999917	+	0.0129069i	\(0.995891\pi\)
\(13\)	6.47361	−	0.930764i	1.79546	−	0.258148i	0.837783	−	0.546004i	\(-0.183852\pi\)
							0.957674	+	0.287856i	\(0.0929426\pi\)
\(17\)	−1.65630	−	4.78557i	−0.401712	−	1.16067i	−0.946669	−	0.322207i	\(-0.895575\pi\)
							0.544957	−	0.838464i	\(-0.316546\pi\)
\(19\)	−0.163786	+	0.473228i	−0.0375750	+	0.108566i	−0.962228	−	0.272245i	\(-0.912234\pi\)
							0.924653	+	0.380811i	\(0.124355\pi\)
\(23\)	−1.28231	−	4.62122i	−0.267380	−	0.963591i
							\(29\)	−4.92177	+	5.68003i	−0.913950	+	1.05475i	0.0843483	+	0.996436i	\(0.473119\pi\)
−0.998298	+	0.0583179i	\(0.981426\pi\)
\(31\)	0.129056	+	0.0313087i	0.0231792	+	0.00562320i	0.247332	−	0.968931i	\(-0.420446\pi\)
							−0.224153	+	0.974554i	\(0.571961\pi\)
\(37\)	−4.66083	−	0.222023i	−0.766235	−	0.0365003i	−0.339175	−	0.940723i	\(-0.610148\pi\)
							−0.427061	+	0.904223i	\(0.640451\pi\)
\(41\)	3.08787	+	4.80482i	0.482245	+	0.750387i	0.994075	−	0.108699i	\(-0.0346686\pi\)
							−0.511830	+	0.859087i	\(0.671032\pi\)
\(43\)	−2.20926	+	7.52406i	−0.336909	+	1.14741i	0.600629	+	0.799528i	\(0.294917\pi\)
							−0.937539	+	0.347881i	\(0.886901\pi\)
\(47\)	−6.00923	+	3.46943i	−0.876536	+	0.506068i	−0.869515	−	0.493907i	\(-0.835568\pi\)
							−0.00702112	+	0.999975i	\(0.502235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.178.27	yes	640
7.5	odd	6	inner	483.2.bf.a.40.6	yes	640
23.19	odd	22	inner	483.2.bf.a.157.6	yes	640
161.19	even	66	inner	483.2.bf.a.19.27	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.27	✓	640	161.19	even	66	inner
483.2.bf.a.40.6	yes	640	7.5	odd	6	inner
483.2.bf.a.157.6	yes	640	23.19	odd	22	inner
483.2.bf.a.178.27	yes	640	1.1	even	1	trivial