Embedded newform 54.3.f.a.11.1

• Label: 54.3.f.a.11.1
• Level: $54$
• Weight: $3$
• Character: 54.11
• Analytic conductor: $1.471$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• 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			11.1
Character	\(\chi\)	\(=\)	54.11
Dual form			54.3.f.a.5.1

$q$-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 \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.3.f.a.11.1	yes	36
3.2	odd	2		162.3.f.a.35.5		36
4.3	odd	2		432.3.bc.c.65.5		36
27.5	odd	18	inner	54.3.f.a.5.1	✓	36
27.7	even	9		1458.3.b.c.1457.29		36
27.20	odd	18		1458.3.b.c.1457.8		36
27.22	even	9		162.3.f.a.125.5		36
108.59	even	18		432.3.bc.c.113.5		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.5.1	✓	36	27.5	odd	18	inner
54.3.f.a.11.1	yes	36	1.1	even	1	trivial
162.3.f.a.35.5		36	3.2	odd	2
162.3.f.a.125.5		36	27.22	even	9
432.3.bc.c.65.5		36	4.3	odd	2
432.3.bc.c.113.5		36	108.59	even	18
1458.3.b.c.1457.8		36	27.20	odd	18
1458.3.b.c.1457.29		36	27.7	even	9