Embedded newform 216.3.x.a.101.33

• Label: 216.3.x.a.101.33
• 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			101.33
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.236183 + 1.98601i) q^{2} +(2.87676 + 0.851044i) q^{3} +(-3.88844 - 0.938120i) q^{4} +(0.707891 + 4.01465i) q^{5} +(-2.36962 + 5.51225i) q^{6} +(7.48970 + 6.28460i) q^{7} +(2.78149 - 7.50089i) q^{8} +(7.55145 + 4.89649i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.236183	+	1.98601i	−0.118091	+	0.993003i
							\(3\)	2.87676	+	0.851044i	0.958919	+	0.283681i
\(5\)	0.707891	+	4.01465i	0.141578	+	0.802930i								0.970051	+	0.242901i	\(0.0780990\pi\)
							−0.828473	+	0.560029i	\(0.810790\pi\)
\(7\)	7.48970	+	6.28460i	1.06996	+	0.897800i	0.995049	−	0.0993901i	\(-0.0316892\pi\)
							0.0749082	+	0.997190i	\(0.476134\pi\)
\(11\)	0.996559	−	5.65177i	0.0905963	−	0.513797i	−0.905412	−	0.424534i	\(-0.860438\pi\)
							0.996008	−	0.0892626i	\(-0.0284510\pi\)
\(13\)	3.01620	−	8.28694i	0.232015	−	0.637457i	−0.767980	−	0.640474i	\(-0.778738\pi\)
							0.999995	+	0.00301678i	\(0.000960272\pi\)
\(17\)	−27.7410	+	16.0163i	−1.63182	+	0.942133i	−0.648291	+	0.761393i	\(0.724516\pi\)
							−0.983531	+	0.180740i	\(0.942151\pi\)
\(19\)	−17.5956	−	10.1588i	−0.926082	−	0.534674i	−0.0405115	−	0.999179i	\(-0.512899\pi\)
							−0.885570	+	0.464506i	\(0.846232\pi\)
\(23\)	11.5866	+	13.8084i	0.503767	+	0.600366i	0.956663	−	0.291197i	\(-0.0940535\pi\)
							−0.452896	+	0.891563i	\(0.649609\pi\)
\(29\)	−26.3004	+	9.57256i	−0.906910	+	0.330088i	−0.753019	−	0.657999i	\(-0.771403\pi\)
							−0.153892	+	0.988088i	\(0.549181\pi\)
\(31\)	8.39742	−	7.04627i	0.270884	−	0.227299i	−0.497219	−	0.867625i	\(-0.665645\pi\)
							0.768103	+	0.640326i	\(0.221201\pi\)
\(37\)	36.3004	−	20.9581i	0.981092	−	0.566434i	0.0784926	−	0.996915i	\(-0.474989\pi\)
							0.902600	+	0.430481i	\(0.141656\pi\)
\(41\)	13.6066	−	37.3837i	0.331867	−	0.911798i	−0.655759	−	0.754971i	\(-0.727651\pi\)
							0.987626	−	0.156828i	\(-0.0501267\pi\)
\(43\)	−54.3192	−	9.57794i	−1.26324	−	0.222743i	−0.498389	−	0.866953i	\(-0.666075\pi\)
							−0.764848	+	0.644211i	\(0.777186\pi\)
\(47\)	37.0704	−	44.1788i	0.788732	−	0.939974i	−0.210561	−	0.977581i	\(-0.567529\pi\)
							0.999293	+	0.0376069i	\(0.0119735\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.33	yes	420
8.5	even	2	inner	216.3.x.a.101.53	yes	420
27.23	odd	18	inner	216.3.x.a.77.53	yes	420
216.77	odd	18	inner	216.3.x.a.77.33	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.33	✓	420	216.77	odd	18	inner
216.3.x.a.77.53	yes	420	27.23	odd	18	inner
216.3.x.a.101.33	yes	420	1.1	even	1	trivial
216.3.x.a.101.53	yes	420	8.5	even	2	inner