Embedded newform 216.3.x.a.101.1

• Label: 216.3.x.a.101.1
• 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			101.1
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99910 - 0.0598947i) q^{2} +(-2.63546 + 1.43329i) q^{3} +(3.99283 + 0.239471i) q^{4} +(-1.71984 - 9.75368i) q^{5} +(5.35441 - 2.70745i) q^{6} +(4.26266 + 3.57680i) q^{7} +(-7.96773 - 0.717877i) q^{8} +(4.89134 - 7.55479i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99910	−	0.0598947i	−0.999551	−	0.0299473i
							\(3\)	−2.63546	+	1.43329i	−0.878488	+	0.477764i
\(5\)	−1.71984	−	9.75368i	−0.343967	−	1.95074i								−0.308014	−	0.951382i	\(-0.599664\pi\)
							−0.0359534	−	0.999353i	\(-0.511447\pi\)
\(7\)	4.26266	+	3.57680i	0.608952	+	0.510971i	0.894309	−	0.447450i	\(-0.147668\pi\)
							−0.285357	+	0.958421i	\(0.592112\pi\)
\(11\)	−2.25849	+	12.8085i	−0.205317	+	1.16441i	0.691623	+	0.722259i	\(0.256896\pi\)
							−0.896940	+	0.442152i	\(0.854215\pi\)
\(13\)	2.65254	−	7.28778i	0.204041	−	0.560599i	−0.794893	−	0.606749i	\(-0.792473\pi\)
							0.998934	+	0.0461507i	\(0.0146954\pi\)
\(17\)	−6.20949	+	3.58505i	−0.365264	+	0.210885i	−0.671387	−	0.741107i	\(-0.734301\pi\)
							0.306123	+	0.951992i	\(0.400968\pi\)
\(19\)	−12.6300	−	7.29191i	−0.664735	−	0.383785i	0.129344	−	0.991600i	\(-0.458713\pi\)
							−0.794079	+	0.607815i	\(0.792046\pi\)
\(23\)	−28.7434	−	34.2550i	−1.24971	−	1.48935i	−0.804535	−	0.593905i	\(-0.797585\pi\)
							−0.445177	−	0.895443i	\(-0.646859\pi\)
\(29\)	20.5013	−	7.46185i	0.706941	−	0.257305i	0.0365693	−	0.999331i	\(-0.488357\pi\)
							0.670371	+	0.742026i	\(0.266135\pi\)
\(31\)	−4.24682	+	3.56350i	−0.136994	+	0.114952i	−0.708710	−	0.705500i	\(-0.750722\pi\)
							0.571716	+	0.820452i	\(0.306278\pi\)
\(37\)	−38.3496	+	22.1412i	−1.03648	+	0.598410i	−0.918833	−	0.394646i	\(-0.870867\pi\)
							−0.117643	+	0.993056i	\(0.537534\pi\)
\(41\)	−10.1553	+	27.9014i	−0.247690	+	0.680522i	0.752080	+	0.659072i	\(0.229051\pi\)
							−0.999770	+	0.0214502i	\(0.993172\pi\)
\(43\)	−35.2830	−	6.22135i	−0.820536	−	0.144683i	−0.252406	−	0.967621i	\(-0.581222\pi\)
							−0.568130	+	0.822939i	\(0.692333\pi\)
\(47\)	−35.4512	+	42.2491i	−0.754281	+	0.898917i	−0.997472	−	0.0710618i	\(-0.977361\pi\)
							0.243191	+	0.969978i	\(0.421806\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.1	yes	420
8.5	even	2	inner	216.3.x.a.101.55	yes	420
27.23	odd	18	inner	216.3.x.a.77.55	yes	420
216.77	odd	18	inner	216.3.x.a.77.1	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.1	✓	420	216.77	odd	18	inner
216.3.x.a.77.55	yes	420	27.23	odd	18	inner
216.3.x.a.101.1	yes	420	1.1	even	1	trivial
216.3.x.a.101.55	yes	420	8.5	even	2	inner