Embedded newform 483.2.bf.a.199.15

• Label: 483.2.bf.a.199.15
• Level: $483$
• Weight: $2$
• Character: 483.199
• 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			199.15
Character	\(\chi\)	\(=\)	483.199
Dual form			483.2.bf.a.250.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0667300 + 0.0267147i) q^{2} +(0.0950560 - 0.995472i) q^{3} +(-1.44373 + 1.37659i) q^{4} +(1.45422 - 4.20169i) q^{5} +(0.0202506 + 0.0689672i) q^{6} +(1.33629 - 2.28349i) q^{7} +(0.119284 - 0.261195i) q^{8} +(-0.981929 - 0.189251i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{17}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0667300	+	0.0267147i	−0.0471852	+	0.0188901i	−0.395135	−	0.918623i	\(-0.629302\pi\)
							0.347950	+	0.937513i	\(0.386878\pi\)
\(3\)	0.0950560	−	0.995472i	0.0548806	−	0.574736i
							\(5\)	1.45422	−	4.20169i	0.650346	−	1.87905i	0.223593	−	0.974683i	\(-0.428221\pi\)
0.426753	−	0.904368i	\(-0.359657\pi\)
\(7\)	1.33629	−	2.28349i	0.505072	−	0.863078i
							\(11\)	−1.76969	+	4.42047i	−0.533581	+	1.33282i	0.379555	+	0.925169i	\(0.376077\pi\)
−0.913137	+	0.407653i	\(0.866347\pi\)
\(13\)	−2.81483	−	4.37996i	−0.780693	−	1.21478i	−0.972398	−	0.233330i	\(-0.925038\pi\)
							0.191705	−	0.981453i	\(-0.438598\pi\)
\(17\)	−0.773076	+	3.18666i	−0.187499	+	0.772879i	0.797925	+	0.602757i	\(0.205931\pi\)
							−0.985423	+	0.170122i	\(0.945584\pi\)
\(19\)	0.858640	+	3.53936i	0.196986	+	0.811985i	0.981435	+	0.191793i	\(0.0614303\pi\)
							−0.784450	+	0.620192i	\(0.787055\pi\)
\(23\)	−0.606841	−	4.75728i	−0.126535	−	0.991962i
							\(29\)	−1.25490	+	0.368473i	−0.233030	+	0.0684236i	−0.396162	−	0.918180i	\(-0.629658\pi\)
0.163133	+	0.986604i	\(0.447840\pi\)
\(31\)	3.91862	−	2.79043i	0.703804	−	0.501177i	−0.171234	−	0.985230i	\(-0.554775\pi\)
							0.875038	+	0.484054i	\(0.160836\pi\)
\(37\)	0.646531	−	3.35452i	0.106289	−	0.551480i	−0.889273	−	0.457376i	\(-0.848789\pi\)
							0.995562	−	0.0941040i	\(-0.0299986\pi\)
\(41\)	1.13853	−	0.986544i	0.177809	−	0.154072i	−0.561409	−	0.827538i	\(-0.689741\pi\)
							0.739218	+	0.673466i	\(0.235195\pi\)
\(43\)	6.41277	−	2.92862i	0.977939	−	0.446610i	0.138677	−	0.990338i	\(-0.455715\pi\)
							0.839262	+	0.543728i	\(0.182988\pi\)
\(47\)	3.38890	−	1.95658i	0.494322	−	0.285397i	−0.232044	−	0.972705i	\(-0.574541\pi\)
							0.726366	+	0.687308i	\(0.241208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.199.15	yes	640
7.5	odd	6	inner	483.2.bf.a.61.18	✓	640
23.20	odd	22	inner	483.2.bf.a.388.18	yes	640
161.89	even	66	inner	483.2.bf.a.250.15	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.18	✓	640	7.5	odd	6	inner
483.2.bf.a.199.15	yes	640	1.1	even	1	trivial
483.2.bf.a.250.15	yes	640	161.89	even	66	inner
483.2.bf.a.388.18	yes	640	23.20	odd	22	inner