Embedded newform 216.2.t.a.13.3

• Label: 216.2.t.a.13.3
• Level: $216$
• Weight: $2$
• Character: 216.13
• Analytic conductor: $1.725$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.72476868366\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(34\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	216.13
Dual form			216.2.t.a.133.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36295 - 0.377313i) q^{2} +(0.411299 - 1.68251i) q^{3} +(1.71527 + 1.02852i) q^{4} +(-2.15975 + 2.57389i) q^{5} +(-1.19541 + 2.13799i) q^{6} +(-1.78453 - 0.649515i) q^{7} +(-1.94976 - 2.04901i) q^{8} +(-2.66167 - 1.38403i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36295	−	0.377313i	−0.963752	−	0.266801i
							\(3\)	0.411299	−	1.68251i	0.237463	−	0.971396i
\(5\)	−2.15975	+	2.57389i	−0.965871	+	1.15108i								0.0226118	+	0.999744i	\(0.492802\pi\)
							−0.988482	+	0.151335i	\(0.951643\pi\)
\(7\)	−1.78453	−	0.649515i	−0.674489	−	0.245494i	−0.0180093	−	0.999838i	\(-0.505733\pi\)
							−0.656479	+	0.754344i	\(0.727955\pi\)
\(11\)	−3.21944	−	3.83678i	−0.970699	−	1.15683i	−0.987602	−	0.156977i	\(-0.949825\pi\)
							0.0169036	−	0.999857i	\(-0.494619\pi\)
\(13\)	−3.61424	−	0.637289i	−1.00241	−	0.176752i	−0.351728	−	0.936102i	\(-0.614406\pi\)
							−0.650682	+	0.759350i	\(0.725517\pi\)
\(17\)	−1.74380	+	3.02034i	−0.422932	+	0.732541i	−0.996225	−	0.0868103i	\(-0.972333\pi\)
							0.573292	+	0.819351i	\(0.305666\pi\)
\(19\)	2.73085	−	1.57666i	0.626501	−	0.361710i	−0.152895	−	0.988242i	\(-0.548860\pi\)
							0.779396	+	0.626532i	\(0.215526\pi\)
\(23\)	−2.93428	+	1.06799i	−0.611839	+	0.222691i	−0.629308	−	0.777156i	\(-0.716662\pi\)
							0.0174685	+	0.999847i	\(0.494439\pi\)
\(29\)	6.61611	−	1.16660i	1.22858	−	0.216632i	0.478565	−	0.878052i	\(-0.341157\pi\)
							0.750015	+	0.661420i	\(0.230046\pi\)
\(31\)	0.842639	−	0.306695i	0.151342	−	0.0550841i	−0.265238	−	0.964183i	\(-0.585451\pi\)
							0.416581	+	0.909099i	\(0.363228\pi\)
\(37\)	−6.21485	−	3.58815i	−1.02172	−	0.589887i	−0.107115	−	0.994247i	\(-0.534161\pi\)
							−0.914600	+	0.404359i	\(0.867495\pi\)
\(41\)	−0.341649	+	1.93759i	−0.0533567	+	0.302601i	−0.999794	−	0.0202862i	\(-0.993542\pi\)
							0.946438	+	0.322887i	\(0.104653\pi\)
\(43\)	5.36405	+	6.39263i	0.818010	+	0.974867i	0.999964	−	0.00844018i	\(-0.00268662\pi\)
							−0.181954	+	0.983307i	\(0.558242\pi\)
\(47\)	2.63733	+	0.959909i	0.384694	+	0.140017i	0.527125	−	0.849788i	\(-0.323270\pi\)
							−0.142432	+	0.989805i	\(0.545492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.t.a.13.3	✓	204
3.2	odd	2		648.2.t.a.253.32		204
4.3	odd	2		864.2.bf.a.337.15		204
8.3	odd	2		864.2.bf.a.337.20		204
8.5	even	2	inner	216.2.t.a.13.29	yes	204
24.5	odd	2		648.2.t.a.253.6		204
27.2	odd	18		648.2.t.a.397.6		204
27.25	even	9	inner	216.2.t.a.133.29	yes	204
108.79	odd	18		864.2.bf.a.241.20		204
216.29	odd	18		648.2.t.a.397.32		204
216.133	even	18	inner	216.2.t.a.133.3	yes	204
216.187	odd	18		864.2.bf.a.241.15		204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.t.a.13.3	✓	204	1.1	even	1	trivial
216.2.t.a.13.29	yes	204	8.5	even	2	inner
216.2.t.a.133.3	yes	204	216.133	even	18	inner
216.2.t.a.133.29	yes	204	27.25	even	9	inner
648.2.t.a.253.6		204	24.5	odd	2
648.2.t.a.253.32		204	3.2	odd	2
648.2.t.a.397.6		204	27.2	odd	18
648.2.t.a.397.32		204	216.29	odd	18
864.2.bf.a.241.15		204	216.187	odd	18
864.2.bf.a.241.20		204	108.79	odd	18
864.2.bf.a.337.15		204	4.3	odd	2
864.2.bf.a.337.20		204	8.3	odd	2