Embedded newform 216.3.x.a.101.23

• Label: 216.3.x.a.101.23
• Level: $216$
• Weight: $3$
• Character: 216.101
• Analytic conductor: $5.886$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.88557371018\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(70\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.23
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13871 - 1.64418i) q^{2} +(-2.88684 - 0.816183i) q^{3} +(-1.40666 + 3.74450i) q^{4} +(-0.293578 - 1.66496i) q^{5} +(1.94533 + 5.67589i) q^{6} +(5.24454 + 4.40069i) q^{7} +(7.75843 - 1.95111i) q^{8} +(7.66769 + 4.71238i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13871	−	1.64418i	−0.569357	−	0.822090i
							\(3\)	−2.88684	−	0.816183i	−0.962280	−	0.272061i
\(5\)	−0.293578	−	1.66496i	−0.0587156	−	0.332993i								0.941274	−	0.337645i	\(-0.109630\pi\)
							−0.999989	+	0.00465200i	\(0.998519\pi\)
\(7\)	5.24454	+	4.40069i	0.749221	+	0.628671i	0.935297	−	0.353865i	\(-0.115133\pi\)
							−0.186076	+	0.982535i	\(0.559577\pi\)
\(11\)	1.23412	−	6.99902i	0.112192	−	0.636274i	−0.875910	−	0.482475i	\(-0.839738\pi\)
							0.988102	−	0.153799i	\(-0.0491510\pi\)
\(13\)	−0.620016	+	1.70348i	−0.0476935	+	0.131037i	−0.961253	−	0.275669i	\(-0.911101\pi\)
							0.913559	+	0.406706i	\(0.133323\pi\)
\(17\)	7.80861	−	4.50830i	0.459330	−	0.265194i	−0.252432	−	0.967615i	\(-0.581231\pi\)
							0.711763	+	0.702420i	\(0.247897\pi\)
\(19\)	−10.6734	−	6.16231i	−0.561760	−	0.324332i	0.192092	−	0.981377i	\(-0.438473\pi\)
							−0.753852	+	0.657045i	\(0.771806\pi\)
\(23\)	−17.6120	−	20.9891i	−0.765738	−	0.912571i	0.232459	−	0.972606i	\(-0.425323\pi\)
							−0.998196	+	0.0600357i	\(0.980879\pi\)
\(29\)	32.2040	−	11.7213i	1.11048	−	0.404183i	0.279312	−	0.960200i	\(-0.409894\pi\)
							0.831170	+	0.556018i	\(0.187671\pi\)
\(31\)	−0.0746834	+	0.0626668i	−0.00240914	+	0.00202151i	−0.643991	−	0.765033i	\(-0.722723\pi\)
							0.641582	+	0.767054i	\(0.278278\pi\)
\(37\)	11.2551	−	6.49812i	0.304191	−	0.175625i	−0.340133	−	0.940377i	\(-0.610472\pi\)
							0.644324	+	0.764752i	\(0.277139\pi\)
\(41\)	19.7915	−	54.3768i	0.482720	−	1.32626i	−0.424432	−	0.905460i	\(-0.639526\pi\)
							0.907152	−	0.420803i	\(-0.138252\pi\)
\(43\)	−31.0791	−	5.48008i	−0.722769	−	0.127444i	−0.199851	−	0.979826i	\(-0.564046\pi\)
							−0.522919	+	0.852383i	\(0.675157\pi\)
\(47\)	10.5262	−	12.5446i	0.223961	−	0.266906i	−0.642350	−	0.766412i	\(-0.722040\pi\)
							0.866311	+	0.499505i	\(0.166485\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.23	yes	420
8.5	even	2	inner	216.3.x.a.101.34	yes	420
27.23	odd	18	inner	216.3.x.a.77.34	yes	420
216.77	odd	18	inner	216.3.x.a.77.23	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.23	✓	420	216.77	odd	18	inner
216.3.x.a.77.34	yes	420	27.23	odd	18	inner
216.3.x.a.101.23	yes	420	1.1	even	1	trivial
216.3.x.a.101.34	yes	420	8.5	even	2	inner