Embedded newform 483.2.bf.a.178.23

• Label: 483.2.bf.a.178.23
• 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			178.23
Character	\(\chi\)	\(=\)	483.178
Dual form			483.2.bf.a.19.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.11888 - 0.106840i) q^{2} +(0.690079 + 0.723734i) q^{3} +(-0.723384 + 0.139421i) q^{4} +(-0.742288 - 0.382676i) q^{5} +(0.849438 + 0.736042i) q^{6} +(1.48873 + 2.18716i) q^{7} +(-2.95136 + 0.866598i) q^{8} +(-0.0475819 + 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.11888	−	0.106840i	0.791166	−	0.0755472i	0.308355	−	0.951271i	\(-0.400222\pi\)
							0.482812	+	0.875724i	\(0.339616\pi\)
\(3\)	0.690079	+	0.723734i	0.398417	+	0.417848i
							\(5\)	−0.742288	−	0.382676i	−0.331961	−	0.171138i	0.284185	−	0.958769i	\(-0.408277\pi\)
−0.616147	+	0.787631i	\(0.711307\pi\)
\(7\)	1.48873	+	2.18716i	0.562686	+	0.826670i
							\(11\)	−0.449946	+	4.71204i	−0.135664	+	1.42073i	0.629551	+	0.776959i	\(0.283239\pi\)
−0.765214	+	0.643776i	\(0.777367\pi\)
\(13\)	2.59781	−	0.373509i	0.720503	−	0.103593i	0.227697	−	0.973732i	\(-0.426880\pi\)
							0.492806	+	0.870139i	\(0.335971\pi\)
\(17\)	1.73328	+	5.00799i	0.420383	+	1.21462i	0.934210	+	0.356723i	\(0.116106\pi\)
							−0.513827	+	0.857894i	\(0.671773\pi\)
\(19\)	0.477861	−	1.38069i	0.109629	−	0.316752i	−0.876820	−	0.480818i	\(-0.840340\pi\)
							0.986449	+	0.164066i	\(0.0524610\pi\)
\(23\)	3.76869	−	2.96598i	0.785825	−	0.618449i
							\(29\)	−0.680193	+	0.784985i	−0.126309	+	0.145768i	−0.815381	−	0.578924i	\(-0.803473\pi\)
0.689073	+	0.724692i	\(0.258018\pi\)
\(31\)	−0.978400	−	0.237357i	−0.175726	−	0.0426306i	0.146930	−	0.989147i	\(-0.453061\pi\)
							−0.322656	+	0.946516i	\(0.604576\pi\)
\(37\)	−0.294563	−	0.0140318i	−0.0484258	−	0.00230681i	0.0233550	−	0.999727i	\(-0.492565\pi\)
							−0.0717809	+	0.997420i	\(0.522868\pi\)
\(41\)	−0.542058	−	0.843459i	−0.0846553	−	0.131726i	0.796344	−	0.604844i	\(-0.206765\pi\)
							−0.880999	+	0.473118i	\(0.843128\pi\)
\(43\)	1.86774	−	6.36095i	0.284828	−	0.970035i	−0.685465	−	0.728105i	\(-0.740401\pi\)
							0.970294	−	0.241930i	\(-0.0777806\pi\)
\(47\)	−2.80940	+	1.62201i	−0.409792	+	0.236594i	−0.690701	−	0.723141i	\(-0.742698\pi\)
							0.280908	+	0.959735i	\(0.409364\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.178.23	yes	640
7.5	odd	6	inner	483.2.bf.a.40.10	yes	640
23.19	odd	22	inner	483.2.bf.a.157.10	yes	640
161.19	even	66	inner	483.2.bf.a.19.23	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.23	✓	640	161.19	even	66	inner
483.2.bf.a.40.10	yes	640	7.5	odd	6	inner
483.2.bf.a.157.10	yes	640	23.19	odd	22	inner
483.2.bf.a.178.23	yes	640	1.1	even	1	trivial