Embedded newform 54.3.f.a.5.1

• Label: 54.3.f.a.5.1
• Level: $54$
• Weight: $3$
• Character: 54.5
• Analytic conductor: $1.471$
• Analytic rank: $0$
• Dimension: $36$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	54.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.47139342755\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	54.5
Dual form			54.3.f.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.909039 - 1.08335i) q^{2} +(-1.44503 + 2.62905i) q^{3} +(-0.347296 + 1.96962i) q^{4} +(0.696976 + 1.91493i) q^{5} +(4.16177 - 0.824439i) q^{6} +(2.23222 + 12.6596i) q^{7} +(2.44949 - 1.41421i) q^{8} +(-4.82380 - 7.59809i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 18 q^{5} + 12 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.909039	−	1.08335i	−0.454519	−	0.541675i
							\(3\)	−1.44503	+	2.62905i	−0.481675	+	0.876350i
\(5\)	0.696976	+	1.91493i	0.139395	+	0.382985i								0.989672	−	0.143351i	\(-0.0457877\pi\)
							−0.850277	+	0.526336i	\(0.823565\pi\)
\(7\)	2.23222	+	12.6596i	0.318889	+	1.80851i	0.549536	+	0.835470i	\(0.314805\pi\)
							−0.230647	+	0.973037i	\(0.574084\pi\)
\(11\)	1.28270	−	3.52418i	0.116609	−	0.320380i	−0.867634	−	0.497204i	\(-0.834360\pi\)
							0.984242	+	0.176824i	\(0.0565824\pi\)
\(13\)	6.69534	+	5.61806i	0.515026	+	0.432158i	0.862894	−	0.505385i	\(-0.168650\pi\)
							−0.347867	+	0.937544i	\(0.613094\pi\)
\(17\)	−20.4462	−	11.8046i	−1.20272	−	0.694390i	−0.241560	−	0.970386i	\(-0.577659\pi\)
							−0.961159	+	0.275996i	\(0.910992\pi\)
\(19\)	−2.58391	−	4.47546i	−0.135995	−	0.235551i	0.789982	−	0.613130i	\(-0.210090\pi\)
							−0.925977	+	0.377579i	\(0.876757\pi\)
\(23\)	34.3001	+	6.04803i	1.49131	+	0.262958i	0.859087	−	0.511829i	\(-0.171032\pi\)
							0.632222	+	0.774787i	\(0.282143\pi\)
\(29\)	18.1485	+	21.6285i	0.625809	+	0.745810i	0.982058	−	0.188581i	\(-0.0603889\pi\)
							−0.356249	+	0.934391i	\(0.615944\pi\)
\(31\)	−0.600583	+	3.40608i	−0.0193736	+	0.109873i	−0.992961	−	0.118441i	\(-0.962210\pi\)
							0.973587	+	0.228314i	\(0.0733214\pi\)
\(37\)	27.7928	−	48.1386i	0.751158	−	1.30104i	−0.196104	−	0.980583i	\(-0.562829\pi\)
							0.947262	−	0.320460i	\(-0.103838\pi\)
\(41\)	−10.1636	+	12.1124i	−0.247891	+	0.295426i	−0.875614	−	0.483012i	\(-0.839543\pi\)
							0.627722	+	0.778437i	\(0.283987\pi\)
\(43\)	49.2717	+	17.9334i	1.14585	+	0.417057i	0.844024	−	0.536305i	\(-0.180180\pi\)
							0.301830	+	0.953362i	\(0.402403\pi\)
\(47\)	−53.2114	+	9.38261i	−1.13216	+	0.199630i	−0.708173	−	0.706039i	\(-0.750480\pi\)
							−0.423985	+	0.905669i	\(0.639369\pi\)