Embedded newform 216.3.r.b.43.2

• Label: 216.3.r.b.43.2
• Level: $216$
• Weight: $3$
• Character: 216.43
• Analytic conductor: $5.886$
• Analytic rank: $0$
• Dimension: $408$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.88557371018\)
Analytic rank:	\(0\)
Dimension:	\(408\)
Relative dimension:	\(68\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			43.2
Character	\(\chi\)	\(=\)	216.43
Dual form			216.3.r.b.211.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99900 + 0.0631437i) q^{2} +(2.99997 + 0.0137424i) q^{3} +(3.99203 - 0.252449i) q^{4} +(2.50897 - 6.89334i) q^{5} +(-5.99781 + 0.161958i) q^{6} +(12.4678 + 2.19840i) q^{7} +(-7.96413 + 0.756718i) q^{8} +(8.99962 + 0.0824533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99900	+	0.0631437i	−0.999501	+	0.0315719i
							\(3\)	2.99997	+	0.0137424i	0.999990	+	0.00458079i
\(5\)	2.50897	−	6.89334i	0.501794	−	1.37867i								−0.387727	−	0.921774i	\(-0.626740\pi\)
							0.889521	−	0.456894i	\(-0.151038\pi\)
\(7\)	12.4678	+	2.19840i	1.78111	+	0.314058i	0.964687	−	0.263398i	\(-0.0848433\pi\)
							0.816422	+	0.577456i	\(0.195954\pi\)
\(11\)	−6.03825	+	2.19774i	−0.548932	+	0.199795i	−0.601572	−	0.798819i	\(-0.705459\pi\)
							0.0526398	+	0.998614i	\(0.483236\pi\)
\(13\)	−11.7133	−	13.9594i	−0.901027	−	1.07380i	−0.996921	−	0.0784103i	\(-0.975016\pi\)
							0.0958945	−	0.995392i	\(-0.469429\pi\)
\(17\)	−2.36876	−	4.10281i	−0.139339	−	0.241342i	0.787908	−	0.615793i	\(-0.211164\pi\)
							−0.927247	+	0.374451i	\(0.877831\pi\)
\(19\)	−18.1448	+	31.4277i	−0.954990	+	1.65409i	−0.220597	+	0.975365i	\(0.570801\pi\)
							−0.734392	+	0.678725i	\(0.762533\pi\)
\(23\)	−2.90811	+	0.512778i	−0.126439	+	0.0222947i	−0.236510	−	0.971629i	\(-0.576004\pi\)
							0.110070	+	0.993924i	\(0.464892\pi\)
\(29\)	−4.94275	+	5.89054i	−0.170440	+	0.203122i	−0.844502	−	0.535552i	\(-0.820103\pi\)
							0.674062	+	0.738674i	\(0.264548\pi\)
\(31\)	34.1835	−	6.02747i	1.10269	−	0.194435i	0.407463	−	0.913222i	\(-0.366413\pi\)
							0.695230	+	0.718787i	\(0.255302\pi\)
\(37\)	−0.348508	+	0.201211i	−0.00941913	+	0.00543814i	−0.504702	−	0.863294i	\(-0.668398\pi\)
							0.495283	+	0.868732i	\(0.335064\pi\)
\(41\)	27.1043	−	22.7432i	0.661081	−	0.554713i	−0.249329	−	0.968419i	\(-0.580210\pi\)
							0.910411	+	0.413706i	\(0.135766\pi\)
\(43\)	−20.8576	+	7.59156i	−0.485061	+	0.176548i	−0.572963	−	0.819581i	\(-0.694206\pi\)
							0.0879016	+	0.996129i	\(0.471984\pi\)
\(47\)	6.69583	+	1.18066i	0.142465	+	0.0251203i	0.244426	−	0.969668i	\(-0.421401\pi\)
							−0.101961	+	0.994788i	\(0.532512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.b.43.2	✓	408
8.3	odd	2	inner	216.3.r.b.43.31	yes	408
27.22	even	9	inner	216.3.r.b.211.31	yes	408
216.211	odd	18	inner	216.3.r.b.211.2	yes	408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.b.43.2	✓	408	1.1	even	1	trivial
216.3.r.b.43.31	yes	408	8.3	odd	2	inner
216.3.r.b.211.2	yes	408	216.211	odd	18	inner
216.3.r.b.211.31	yes	408	27.22	even	9	inner