Embedded newform 216.3.x.a.101.8

• Label: 216.3.x.a.101.8
• 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			101.8
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91161 + 0.588017i) q^{2} +(-1.68752 - 2.48038i) q^{3} +(3.30847 - 2.24811i) q^{4} +(-0.0158330 - 0.0897932i) q^{5} +(4.68438 + 3.74921i) q^{6} +(4.80570 + 4.03246i) q^{7} +(-5.00256 + 6.24294i) q^{8} +(-3.30455 + 8.37138i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91161	+	0.588017i	−0.955803	+	0.294009i
							\(3\)	−1.68752	−	2.48038i	−0.562507	−	0.826793i
\(5\)	−0.0158330	−	0.0897932i	−0.00316659	−	0.0179586i								0.983184	−	0.182620i	\(-0.0584579\pi\)
							−0.986350	+	0.164662i	\(0.947347\pi\)
\(7\)	4.80570	+	4.03246i	0.686528	+	0.576066i	0.917906	−	0.396798i	\(-0.129879\pi\)
							−0.231378	+	0.972864i	\(0.574323\pi\)
\(11\)	−1.16979	+	6.63419i	−0.106344	+	0.603108i	0.884331	+	0.466861i	\(0.154615\pi\)
							−0.990675	+	0.136247i	\(0.956496\pi\)
\(13\)	3.44076	−	9.45340i	0.264674	−	0.727184i	−0.734164	−	0.678973i	\(-0.762426\pi\)
							0.998837	−	0.0482119i	\(-0.0153523\pi\)
\(17\)	8.06265	−	4.65498i	0.474274	−	0.273822i	−0.243753	−	0.969837i	\(-0.578379\pi\)
							0.718027	+	0.696015i	\(0.245045\pi\)
\(19\)	16.7972	+	9.69785i	0.884062	+	0.510413i	0.871996	−	0.489514i	\(-0.162826\pi\)
							0.0120664	+	0.999927i	\(0.496159\pi\)
\(23\)	0.477626	+	0.569213i	0.0207664	+	0.0247484i	0.776328	−	0.630330i	\(-0.217080\pi\)
							−0.755561	+	0.655078i	\(0.772636\pi\)
\(29\)	5.36461	−	1.95256i	0.184987	−	0.0673296i	−0.247866	−	0.968794i	\(-0.579729\pi\)
							0.432853	+	0.901465i	\(0.357507\pi\)
\(31\)	−8.84587	+	7.42257i	−0.285351	+	0.239438i	−0.774216	−	0.632922i	\(-0.781855\pi\)
							0.488865	+	0.872359i	\(0.337411\pi\)
\(37\)	45.0507	−	26.0100i	1.21759	−	0.702973i	0.253184	−	0.967418i	\(-0.418522\pi\)
							0.964401	+	0.264445i	\(0.0851887\pi\)
\(41\)	16.1245	−	44.3018i	0.393281	−	1.08053i	−0.572213	−	0.820105i	\(-0.693915\pi\)
							0.965494	−	0.260426i	\(-0.0838630\pi\)
\(43\)	33.2309	+	5.85950i	0.772811	+	0.136268i	0.546129	−	0.837701i	\(-0.316101\pi\)
							0.226683	+	0.973969i	\(0.427212\pi\)
\(47\)	−12.2731	+	14.6265i	−0.261130	+	0.311203i	−0.880640	−	0.473786i	\(-0.842887\pi\)
							0.619510	+	0.784989i	\(0.287331\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.8	yes	420
8.5	even	2	inner	216.3.x.a.101.64	yes	420
27.23	odd	18	inner	216.3.x.a.77.64	yes	420
216.77	odd	18	inner	216.3.x.a.77.8	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.8	✓	420	216.77	odd	18	inner
216.3.x.a.77.64	yes	420	27.23	odd	18	inner
216.3.x.a.101.8	yes	420	1.1	even	1	trivial
216.3.x.a.101.64	yes	420	8.5	even	2	inner