Embedded newform 216.3.x.a.101.41

• Label: 216.3.x.a.101.41
• 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			101.41
Character	\(\chi\)	\(=\)	216.101
Dual form			216.3.x.a.77.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.611247 + 1.90430i) q^{2} +(-0.704021 + 2.91622i) q^{3} +(-3.25275 + 2.32800i) q^{4} +(-1.59180 - 9.02752i) q^{5} +(-5.98371 + 0.441861i) q^{6} +(-0.928358 - 0.778985i) q^{7} +(-6.42146 - 4.77125i) q^{8} +(-8.00871 - 4.10616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.611247	+	1.90430i	0.305623	+	0.952152i
							\(3\)	−0.704021	+	2.91622i	−0.234674	+	0.972074i
\(5\)	−1.59180	−	9.02752i	−0.318359	−	1.80550i								−0.552734	−	0.833357i	\(-0.686416\pi\)
							0.234375	−	0.972146i	\(-0.424696\pi\)
\(7\)	−0.928358	−	0.778985i	−0.132623	−	0.111284i	0.574064	−	0.818811i	\(-0.305366\pi\)
							−0.706686	+	0.707527i	\(0.749811\pi\)
\(11\)	3.17314	−	17.9957i	0.288467	−	1.63598i	−0.404165	−	0.914686i	\(-0.632438\pi\)
							0.692632	−	0.721291i	\(-0.256451\pi\)
\(13\)	−4.20812	+	11.5617i	−0.323702	+	0.889363i	0.665966	+	0.745982i	\(0.268020\pi\)
							−0.989668	+	0.143381i	\(0.954203\pi\)
\(17\)	−16.5685	+	9.56585i	−0.974620	+	0.562697i	−0.900642	−	0.434563i	\(-0.856903\pi\)
							−0.0739784	+	0.997260i	\(0.523570\pi\)
\(19\)	−21.0219	−	12.1370i	−1.10642	−	0.638789i	−0.168517	−	0.985699i	\(-0.553898\pi\)
							−0.937899	+	0.346910i	\(0.887231\pi\)
\(23\)	15.5738	+	18.5602i	0.677123	+	0.806964i	0.989735	−	0.142917i	\(-0.0456482\pi\)
							−0.312611	+	0.949881i	\(0.601204\pi\)
\(29\)	−3.71127	+	1.35079i	−0.127975	+	0.0465790i	−0.405214	−	0.914222i	\(-0.632803\pi\)
							0.277239	+	0.960801i	\(0.410581\pi\)
\(31\)	17.2123	−	14.4428i	0.555236	−	0.465898i	−0.321473	−	0.946919i	\(-0.604178\pi\)
							0.876710	+	0.481020i	\(0.159734\pi\)
\(37\)	5.34337	−	3.08500i	0.144416	−	0.0833783i	−0.426051	−	0.904699i	\(-0.640096\pi\)
							0.570467	+	0.821321i	\(0.306762\pi\)
\(41\)	10.7994	−	29.6712i	0.263401	−	0.723687i	−0.735532	−	0.677490i	\(-0.763068\pi\)
							0.998932	−	0.0461968i	\(-0.0147101\pi\)
\(43\)	−55.2938	−	9.74979i	−1.28590	−	0.226739i	−0.511418	−	0.859332i	\(-0.670880\pi\)
							−0.774485	+	0.632593i	\(0.781991\pi\)
\(47\)	10.6764	−	12.7237i	0.227158	−	0.270716i	−0.640412	−	0.768032i	\(-0.721236\pi\)
							0.867570	+	0.497315i	\(0.165681\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.101.41	yes	420
8.5	even	2	inner	216.3.x.a.101.44	yes	420
27.23	odd	18	inner	216.3.x.a.77.44	yes	420
216.77	odd	18	inner	216.3.x.a.77.41	✓	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.77.41	✓	420	216.77	odd	18	inner
216.3.x.a.77.44	yes	420	27.23	odd	18	inner
216.3.x.a.101.41	yes	420	1.1	even	1	trivial
216.3.x.a.101.44	yes	420	8.5	even	2	inner