Embedded newform 483.2.bf.a.178.8

• Label: 483.2.bf.a.178.8
• 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			178.8
Character	\(\chi\)	\(=\)	483.178
Dual form			483.2.bf.a.19.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80344 + 0.172208i) q^{2} +(0.690079 + 0.723734i) q^{3} +(1.25889 - 0.242631i) q^{4} +(2.30542 + 1.18853i) q^{5} +(-1.36915 - 1.18638i) q^{6} +(2.63811 + 0.200976i) q^{7} +(1.24797 - 0.366437i) q^{8} +(-0.0475819 + 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80344	+	0.172208i	−1.27523	+	0.121769i	−0.710611	−	0.703585i	\(-0.751581\pi\)
							−0.564615	+	0.825354i	\(0.690975\pi\)
\(3\)	0.690079	+	0.723734i	0.398417	+	0.417848i
							\(5\)	2.30542	+	1.18853i	1.03102	+	0.531526i	0.888793	−	0.458309i	\(-0.151545\pi\)
0.142223	+	0.989835i	\(0.454575\pi\)
\(7\)	2.63811	+	0.200976i	0.997111	+	0.0759617i
							\(11\)	−0.336232	+	3.52118i	−0.101378	+	1.06167i	0.791437	+	0.611251i	\(0.209333\pi\)
−0.892815	+	0.450424i	\(0.851273\pi\)
\(13\)	3.67557	−	0.528467i	1.01942	−	0.146570i	0.387715	−	0.921779i	\(-0.373264\pi\)
							0.631705	+	0.775209i	\(0.282355\pi\)
\(17\)	−1.78237	−	5.14982i	−0.432288	−	1.24901i	−0.925381	−	0.379038i	\(-0.876255\pi\)
							0.493093	−	0.869977i	\(-0.335866\pi\)
\(19\)	2.68797	−	7.76638i	0.616663	−	1.78173i	−0.00692320	−	0.999976i	\(-0.502204\pi\)
							0.623586	−	0.781755i	\(-0.285675\pi\)
\(23\)	3.96084	+	2.70402i	0.825892	+	0.563828i
							\(29\)	−5.10492	+	5.89139i	−0.947960	+	1.09400i	0.0475050	+	0.998871i	\(0.484873\pi\)
−0.995465	+	0.0951327i	\(0.969672\pi\)
\(31\)	−6.35914	−	1.54271i	−1.14214	−	0.277079i	−0.380309	−	0.924860i	\(-0.624182\pi\)
							−0.761827	+	0.647781i	\(0.775697\pi\)
\(37\)	1.18866	+	0.0566228i	0.195414	+	0.00930873i	0.145061	−	0.989423i	\(-0.453662\pi\)
							0.0503534	+	0.998731i	\(0.483965\pi\)
\(41\)	2.75749	+	4.29074i	0.430648	+	0.670101i	0.986977	−	0.160861i	\(-0.0514270\pi\)
							−0.556329	+	0.830962i	\(0.687791\pi\)
\(43\)	1.88201	−	6.40953i	0.287003	−	0.977443i	−0.682197	−	0.731169i	\(-0.738975\pi\)
							0.969200	−	0.246275i	\(-0.0792066\pi\)
\(47\)	−0.107463	+	0.0620438i	−0.0156751	+	0.00905002i	−0.507817	−	0.861465i	\(-0.669547\pi\)
							0.492142	+	0.870515i	\(0.336214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.178.8	yes	640
7.5	odd	6	inner	483.2.bf.a.40.25	yes	640
23.19	odd	22	inner	483.2.bf.a.157.25	yes	640
161.19	even	66	inner	483.2.bf.a.19.8	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.8	✓	640	161.19	even	66	inner
483.2.bf.a.40.25	yes	640	7.5	odd	6	inner
483.2.bf.a.157.25	yes	640	23.19	odd	22	inner
483.2.bf.a.178.8	yes	640	1.1	even	1	trivial