Embedded newform 216.3.x.a.101.24

• Label: 216.3.x.a.101.24
• 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			101.24
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.994649 - 1.73513i) q^{2} +(1.87416 - 2.34255i) q^{3} +(-2.02135 + 3.45169i) q^{4} +(0.836712 + 4.74523i) q^{5} +(-5.92875 - 0.921898i) q^{6} +(1.29808 + 1.08922i) q^{7} +(7.99966 + 0.0740797i) q^{8} +(-1.97505 - 8.78061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.994649	−	1.73513i	−0.497324	−	0.867565i
							\(3\)	1.87416	−	2.34255i	0.624720	−	0.780849i
\(5\)	0.836712	+	4.74523i	0.167342	+	0.949046i								0.946616	+	0.322363i	\(0.104477\pi\)
							−0.779274	+	0.626684i	\(0.784412\pi\)
\(7\)	1.29808	+	1.08922i	0.185441	+	0.155603i	0.730782	−	0.682611i	\(-0.239156\pi\)
							−0.545341	+	0.838214i	\(0.683600\pi\)
\(11\)	1.26824	−	7.19255i	0.115295	−	0.653868i	−0.871309	−	0.490734i	\(-0.836729\pi\)
							0.986604	−	0.163134i	\(-0.0521603\pi\)
\(13\)	7.76988	−	21.3476i	0.597683	−	1.64212i	−0.158195	−	0.987408i	\(-0.550567\pi\)
							0.755878	−	0.654713i	\(-0.227210\pi\)
\(17\)	20.0696	−	11.5872i	1.18057	−	0.681600i	0.224421	−	0.974492i	\(-0.427951\pi\)
							0.956145	+	0.292892i	\(0.0946177\pi\)
\(19\)	−13.5648	−	7.83165i	−0.713937	−	0.412192i	0.0985799	−	0.995129i	\(-0.468570\pi\)
							−0.812517	+	0.582937i	\(0.801903\pi\)
\(23\)	26.9749	+	32.1474i	1.17282	+	1.39771i	0.900135	+	0.435611i	\(0.143468\pi\)
							0.272686	+	0.962103i	\(0.412088\pi\)
\(29\)	−11.6027	+	4.22305i	−0.400094	+	0.145622i	−0.534227	−	0.845341i	\(-0.679397\pi\)
							0.134133	+	0.990963i	\(0.457175\pi\)
\(31\)	−15.1005	+	12.6709i	−0.487114	+	0.408738i	−0.852991	−	0.521926i	\(-0.825214\pi\)
							0.365876	+	0.930663i	\(0.380769\pi\)
\(37\)	55.7962	−	32.2139i	1.50800	−	0.870647i	0.508048	−	0.861329i	\(-0.330367\pi\)
							0.999957	−	0.00931756i	\(-0.00296591\pi\)
\(41\)	−24.1199	+	66.2688i	−0.588290	+	1.61631i	0.185339	+	0.982675i	\(0.440662\pi\)
							−0.773629	+	0.633639i	\(0.781561\pi\)
\(43\)	−13.9386	−	2.45775i	−0.324154	−	0.0571570i	0.00920353	−	0.999958i	\(-0.497070\pi\)
							−0.333357	+	0.942801i	\(0.608181\pi\)
\(47\)	15.3825	−	18.3322i	0.327288	−	0.390046i	−0.577160	−	0.816631i	\(-0.695839\pi\)
							0.904448	+	0.426585i	\(0.140283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.24	yes	420
8.5	even	2	inner	216.3.x.a.101.31	yes	420
27.23	odd	18	inner	216.3.x.a.77.31	yes	420
216.77	odd	18	inner	216.3.x.a.77.24	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.24	✓	420	216.77	odd	18	inner
216.3.x.a.77.31	yes	420	27.23	odd	18	inner
216.3.x.a.101.24	yes	420	1.1	even	1	trivial
216.3.x.a.101.31	yes	420	8.5	even	2	inner