Embedded newform 483.2.bf.a.61.18

• Label: 483.2.bf.a.61.18
• Level: $483$
• Weight: $2$
• Character: 483.61
• 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			61.18
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0565006 + 0.0444325i) q^{2} +(-0.814576 - 0.580057i) q^{3} +(-0.470300 - 1.93860i) q^{4} +(4.36588 - 0.841453i) q^{5} +(-0.0202506 - 0.0689672i) q^{6} +(1.52202 - 2.16413i) q^{7} +(0.119284 - 0.261195i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0565006	+	0.0444325i	0.0399519	+	0.0314185i	0.637935	−	0.770090i	\(-0.279789\pi\)
							−0.597983	+	0.801509i	\(0.704031\pi\)
\(3\)	−0.814576	−	0.580057i	−0.470296	−	0.334896i
							\(5\)	4.36588	−	0.841453i	1.95248	−	0.376309i	0.955896	−	0.293704i	\(-0.0948880\pi\)
0.996582	−	0.0826055i	\(-0.0263241\pi\)
\(7\)	1.52202	−	2.16413i	0.575269	−	0.817965i
							\(11\)	−2.94340	−	3.74283i	−0.887467	−	1.12851i	−0.990998	−	0.133880i	\(-0.957256\pi\)
0.103530	−	0.994626i	\(-0.466986\pi\)
\(13\)	2.81483	+	4.37996i	0.780693	+	1.21478i	0.972398	+	0.233330i	\(0.0749621\pi\)
							−0.191705	+	0.981453i	\(0.561402\pi\)
\(17\)	2.37319	+	2.26284i	0.575584	+	0.548818i	0.920965	−	0.389644i	\(-0.127402\pi\)
							−0.345382	+	0.938462i	\(0.612250\pi\)
\(19\)	−2.63586	+	2.51329i	−0.604707	+	0.576587i	−0.929327	−	0.369257i	\(-0.879612\pi\)
							0.324620	+	0.945845i	\(0.394764\pi\)
\(23\)	−3.81651	+	2.90418i	−0.795797	+	0.605564i
							\(29\)	−1.25490	+	0.368473i	−0.233030	+	0.0684236i	−0.396162	−	0.918180i	\(-0.629658\pi\)
0.163133	+	0.986604i	\(0.447840\pi\)
\(31\)	−0.457278	−	4.78884i	−0.0821296	−	0.860101i	−0.938852	−	0.344322i	\(-0.888109\pi\)
							0.856722	−	0.515779i	\(-0.172497\pi\)
\(37\)	−3.22837	+	1.11735i	−0.530740	+	0.183691i	−0.579278	−	0.815130i	\(-0.696665\pi\)
							0.0485372	+	0.998821i	\(0.484544\pi\)
\(41\)	−1.13853	+	0.986544i	−0.177809	+	0.154072i	−0.739218	−	0.673466i	\(-0.764805\pi\)
							0.561409	+	0.827538i	\(0.310259\pi\)
\(43\)	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	0.726366	−	0.687308i	\(-0.241208\pi\)
							−0.232044	+	0.972705i	\(0.574541\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.18	✓	640	1.1	even	1	trivial
483.2.bf.a.199.15	yes	640	7.3	odd	6	inner
483.2.bf.a.250.15	yes	640	23.20	odd	22	inner
483.2.bf.a.388.18	yes	640	161.66	even	66	inner