Embedded newform 216.3.r.b.43.8

• Label: 216.3.r.b.43.8
• 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			43.8
Character	\(\chi\)	\(=\)	216.43
Dual form			216.3.r.b.211.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87801 - 0.687799i) q^{2} +(2.64392 - 1.41764i) q^{3} +(3.05387 + 2.58339i) q^{4} +(-1.12060 + 3.07881i) q^{5} +(-5.94036 + 0.843855i) q^{6} +(1.91560 + 0.337772i) q^{7} +(-3.95835 - 6.95208i) q^{8} +(4.98062 - 7.49623i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87801	−	0.687799i	−0.939007	−	0.343899i
							\(3\)	2.64392	−	1.41764i	0.881306	−	0.472545i
\(5\)	−1.12060	+	3.07881i	−0.224119	+	0.615763i								−0.999884	−	0.0152600i	\(-0.995142\pi\)
							0.775764	+	0.631023i	\(0.217365\pi\)
\(7\)	1.91560	+	0.337772i	0.273657	+	0.0482532i	0.308793	−	0.951129i	\(-0.400075\pi\)
							−0.0351352	+	0.999383i	\(0.511186\pi\)
\(11\)	3.67380	−	1.33715i	0.333982	−	0.121559i	−0.169586	−	0.985515i	\(-0.554243\pi\)
							0.503567	+	0.863956i	\(0.332021\pi\)
\(13\)	12.9154	+	15.3920i	0.993495	+	1.18400i	0.982916	+	0.184057i	\(0.0589230\pi\)
							0.0105790	+	0.999944i	\(0.496633\pi\)
\(17\)	1.04473	+	1.80953i	0.0614547	+	0.106443i	0.895116	−	0.445834i	\(-0.147093\pi\)
							−0.833661	+	0.552276i	\(0.813759\pi\)
\(19\)	7.94506	−	13.7613i	0.418161	−	0.724276i	−0.577593	−	0.816325i	\(-0.696008\pi\)
							0.995755	+	0.0920482i	\(0.0293414\pi\)
\(23\)	−0.409701	+	0.0722414i	−0.0178131	+	0.00314093i	−0.182548	−	0.983197i	\(-0.558434\pi\)
							0.164735	+	0.986338i	\(0.447323\pi\)
\(29\)	0.0479760	−	0.0571755i	0.00165434	−	0.00197157i	−0.765217	−	0.643773i	\(-0.777368\pi\)
							0.766871	+	0.641801i	\(0.221813\pi\)
\(31\)	30.3737	−	5.35570i	0.979796	−	0.172764i	0.339260	−	0.940693i	\(-0.389823\pi\)
							0.640536	+	0.767928i	\(0.278712\pi\)
\(37\)	13.0050	−	7.50846i	0.351487	−	0.202931i	−0.313853	−	0.949472i	\(-0.601620\pi\)
							0.665340	+	0.746540i	\(0.268287\pi\)
\(41\)	−7.58446	+	6.36412i	−0.184987	+	0.155222i	−0.730578	−	0.682829i	\(-0.760749\pi\)
							0.545592	+	0.838051i	\(0.316305\pi\)
\(43\)	24.4263	−	8.89045i	0.568054	−	0.206755i	−0.0419959	−	0.999118i	\(-0.513372\pi\)
							0.610050	+	0.792363i	\(0.291149\pi\)
\(47\)	−40.7768	−	7.19005i	−0.867591	−	0.152980i	−0.277902	−	0.960610i	\(-0.589639\pi\)
							−0.589690	+	0.807630i	\(0.700750\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.b.43.8	✓	408
8.3	odd	2	inner	216.3.r.b.43.22	yes	408
27.22	even	9	inner	216.3.r.b.211.22	yes	408
216.211	odd	18	inner	216.3.r.b.211.8	yes	408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.b.43.8	✓	408	1.1	even	1	trivial
216.3.r.b.43.22	yes	408	8.3	odd	2	inner
216.3.r.b.211.8	yes	408	216.211	odd	18	inner
216.3.r.b.211.22	yes	408	27.22	even	9	inner