Embedded newform 216.3.x.a.5.11

• Label: 216.3.x.a.5.11
• Level: $216$
• Weight: $3$
• Character: 216.5
• 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			5.11
Character	\(\chi\)	\(=\)	216.5
Dual form			216.3.x.a.173.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71853 - 1.02307i) q^{2} +(-2.36755 - 1.84247i) q^{3} +(1.90667 + 3.51633i) q^{4} +(-7.66878 + 2.79121i) q^{5} +(2.18374 + 5.58849i) q^{6} +(0.943246 + 5.34941i) q^{7} +(0.320773 - 7.99357i) q^{8} +(2.21063 + 8.72428i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(55\)	\(109\)	\(137\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71853	−	1.02307i	−0.859264	−	0.511533i
							\(3\)	−2.36755	−	1.84247i	−0.789185	−	0.614156i
\(5\)	−7.66878	+	2.79121i	−1.53376	+	0.558241i								−0.964538	−	0.263945i	\(-0.914976\pi\)
							−0.569218	+	0.822187i	\(0.692754\pi\)
\(7\)	0.943246	+	5.34941i	0.134749	+	0.764202i	0.975034	+	0.222056i	\(0.0712768\pi\)
							−0.840284	+	0.542146i	\(0.817612\pi\)
\(11\)	−1.17032	−	0.425962i	−0.106393	−	0.0387238i	0.288275	−	0.957548i	\(-0.406918\pi\)
							−0.394668	+	0.918824i	\(0.629140\pi\)
\(13\)	15.6218	−	18.6174i	1.20168	−	1.43211i	0.328643	−	0.944454i	\(-0.393409\pi\)
							0.873037	−	0.487653i	\(-0.162147\pi\)
\(17\)	−11.6996	−	6.75475i	−0.688211	−	0.397339i	0.114731	−	0.993397i	\(-0.463399\pi\)
							−0.802941	+	0.596058i	\(0.796733\pi\)
\(19\)	−21.7975	+	12.5848i	−1.14724	+	0.662358i	−0.948213	−	0.317636i	\(-0.897111\pi\)
							−0.199025	+	0.979994i	\(0.563778\pi\)
\(23\)	22.2487	+	3.92305i	0.967336	+	0.170567i	0.634931	−	0.772569i	\(-0.281029\pi\)
							0.332405	+	0.943137i	\(0.392140\pi\)
\(29\)	3.04849	−	2.55799i	0.105120	−	0.0882065i	−0.588713	−	0.808342i	\(-0.700365\pi\)
							0.693833	+	0.720136i	\(0.255921\pi\)
\(31\)	7.93971	−	45.0283i	0.256120	−	1.45253i	−0.537063	−	0.843542i	\(-0.680466\pi\)
							0.793183	−	0.608984i	\(-0.208423\pi\)
\(37\)	28.2567	+	16.3140i	0.763696	+	0.440920i	0.830621	−	0.556838i	\(-0.187986\pi\)
							−0.0669254	+	0.997758i	\(0.521319\pi\)
\(41\)	32.0507	−	38.1965i	0.781724	−	0.931622i	−0.217286	−	0.976108i	\(-0.569721\pi\)
							0.999010	+	0.0444860i	\(0.0141650\pi\)
\(43\)	−3.58732	+	9.85609i	−0.0834261	+	0.229211i	−0.974391	−	0.224861i	\(-0.927807\pi\)
							0.890965	+	0.454073i	\(0.150029\pi\)
\(47\)	−6.65648	+	1.17372i	−0.141627	+	0.0249727i	−0.244012	−	0.969772i	\(-0.578464\pi\)
							0.102385	+	0.994745i	\(0.467353\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.x.a.5.11	✓	420
8.5	even	2	inner	216.3.x.a.5.29	yes	420
27.11	odd	18	inner	216.3.x.a.173.29	yes	420
216.173	odd	18	inner	216.3.x.a.173.11	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.x.a.5.11	✓	420	1.1	even	1	trivial
216.3.x.a.5.29	yes	420	8.5	even	2	inner
216.3.x.a.173.11	yes	420	216.173	odd	18	inner
216.3.x.a.173.29	yes	420	27.11	odd	18	inner