Embedded newform 216.3.r.b.43.10

• Label: 216.3.r.b.43.10
• Level: $216$
• Weight: $3$
• Character: 216.43
• Analytic conductor: $5.886$
• Analytic rank: $0$
• Dimension: $408$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.r (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.88557371018\)
Analytic rank:	\(0\)
Dimension:	\(408\)
Relative dimension:	\(68\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			43.10
Character	\(\chi\)	\(=\)	216.43
Dual form			216.3.r.b.211.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84655 + 0.768275i) q^{2} +(-1.75085 + 2.43608i) q^{3} +(2.81951 - 2.83732i) q^{4} +(0.197813 - 0.543487i) q^{5} +(1.36146 - 5.84349i) q^{6} +(-0.709060 - 0.125026i) q^{7} +(-3.02652 + 7.40541i) q^{8} +(-2.86902 - 8.53046i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84655	+	0.768275i	−0.923276	+	0.384138i
							\(3\)	−1.75085	+	2.43608i	−0.583618	+	0.812028i
\(5\)	0.197813	−	0.543487i	0.0395626	−	0.108697i								−0.918338	−	0.395796i	\(-0.870469\pi\)
							0.957901	+	0.287099i	\(0.0926909\pi\)
\(7\)	−0.709060	−	0.125026i	−0.101294	−	0.0178609i	0.122771	−	0.992435i	\(-0.460822\pi\)
							−0.224066	+	0.974574i	\(0.571933\pi\)
\(11\)	13.9780	−	5.08758i	1.27073	−	0.462507i	0.383373	−	0.923594i	\(-0.374762\pi\)
							0.887355	+	0.461086i	\(0.152540\pi\)
\(13\)	−16.3836	−	19.5252i	−1.26027	−	1.50194i	−0.781368	−	0.624071i	\(-0.785478\pi\)
							−0.478906	−	0.877866i	\(-0.658967\pi\)
\(17\)	10.9657	+	18.9931i	0.645041	+	1.11724i	0.984292	+	0.176548i	\(0.0564930\pi\)
							−0.339251	+	0.940696i	\(0.610174\pi\)
\(19\)	8.64700	−	14.9770i	0.455105	−	0.788266i	−0.543589	−	0.839352i	\(-0.682935\pi\)
							0.998694	+	0.0510859i	\(0.0162682\pi\)
\(23\)	9.82761	−	1.73287i	0.427287	−	0.0753423i	0.0441309	−	0.999026i	\(-0.485948\pi\)
							0.383156	+	0.923683i	\(0.374837\pi\)
\(29\)	−21.1792	+	25.2404i	−0.730317	+	0.870358i	−0.995590	−	0.0938151i	\(-0.970094\pi\)
							0.265272	+	0.964174i	\(0.414538\pi\)
\(31\)	57.1681	−	10.0803i	1.84413	−	0.325170i	0.861076	−	0.508477i	\(-0.169791\pi\)
							0.983056	+	0.183307i	\(0.0586802\pi\)
\(37\)	24.9942	−	14.4304i	0.675519	−	0.390011i	−0.122646	−	0.992451i	\(-0.539138\pi\)
							0.798165	+	0.602440i	\(0.205805\pi\)
\(41\)	43.9783	−	36.9022i	1.07264	−	0.900053i	0.0773521	−	0.997004i	\(-0.475353\pi\)
							0.995289	+	0.0969510i	\(0.0309090\pi\)
\(43\)	−22.3127	+	8.12116i	−0.518900	+	0.188864i	−0.588175	−	0.808733i	\(-0.700154\pi\)
							0.0692752	+	0.997598i	\(0.477931\pi\)
\(47\)	−25.8287	−	4.55430i	−0.549547	−	0.0968999i	−0.108021	−	0.994149i	\(-0.534451\pi\)
							−0.441526	+	0.897249i	\(0.645563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.b.43.10	✓	408
8.3	odd	2	inner	216.3.r.b.43.38	yes	408
27.22	even	9	inner	216.3.r.b.211.38	yes	408
216.211	odd	18	inner	216.3.r.b.211.10	yes	408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.b.43.10	✓	408	1.1	even	1	trivial
216.3.r.b.43.38	yes	408	8.3	odd	2	inner
216.3.r.b.211.10	yes	408	216.211	odd	18	inner
216.3.r.b.211.38	yes	408	27.22	even	9	inner