Embedded newform 216.3.r.b.43.17

• Label: 216.3.r.b.43.17
• 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			43.17
Character	\(\chi\)	\(=\)	216.43
Dual form			216.3.r.b.211.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39225 + 1.43584i) q^{2} +(-2.77266 - 1.14558i) q^{3} +(-0.123276 - 3.99810i) q^{4} +(2.62649 - 7.21622i) q^{5} +(5.50511 - 2.38616i) q^{6} +(-0.683553 - 0.120529i) q^{7} +(5.91227 + 5.38935i) q^{8} +(6.37529 + 6.35261i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 51 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39225	+	1.43584i	−0.696125	+	0.717920i
							\(3\)	−2.77266	−	1.14558i	−0.924220	−	0.381860i
\(5\)	2.62649	−	7.21622i	0.525298	−	1.44324i								−0.339251	−	0.940696i	\(-0.610173\pi\)
							0.864549	−	0.502549i	\(-0.167604\pi\)
\(7\)	−0.683553	−	0.120529i	−0.0976504	−	0.0172184i	0.124609	−	0.992206i	\(-0.460232\pi\)
							−0.222260	+	0.974987i	\(0.571343\pi\)
\(11\)	−4.80977	+	1.75061i	−0.437252	+	0.159147i	−0.551261	−	0.834333i	\(-0.685853\pi\)
							0.114009	+	0.993480i	\(0.463631\pi\)
\(13\)	−12.4318	−	14.8156i	−0.956291	−	1.13966i	−0.990118	−	0.140236i	\(-0.955214\pi\)
							0.0338270	−	0.999428i	\(-0.489230\pi\)
\(17\)	0.869041	+	1.50522i	0.0511201	+	0.0885426i	0.890453	−	0.455075i	\(-0.150388\pi\)
							−0.839333	+	0.543618i	\(0.817054\pi\)
\(19\)	−8.19524	+	14.1946i	−0.431328	+	0.747082i	−0.996988	−	0.0775564i	\(-0.975288\pi\)
							0.565660	+	0.824639i	\(0.308622\pi\)
\(23\)	13.9921	−	2.46718i	0.608352	−	0.107269i	0.139018	−	0.990290i	\(-0.455605\pi\)
							0.469333	+	0.883021i	\(0.344494\pi\)
\(29\)	17.3059	−	20.6244i	0.596755	−	0.711185i	−0.380134	−	0.924932i	\(-0.624122\pi\)
							0.976889	+	0.213746i	\(0.0685665\pi\)
\(31\)	−50.2232	+	8.85571i	−1.62010	+	0.285668i	−0.908805	−	0.417222i	\(-0.863004\pi\)
							−0.711299	+	0.702890i	\(0.751893\pi\)
\(37\)	−17.2572	+	9.96346i	−0.466411	+	0.269283i	−0.714736	−	0.699394i	\(-0.753453\pi\)
							0.248325	+	0.968677i	\(0.420120\pi\)
\(41\)	−58.7527	+	49.2994i	−1.43299	+	1.20242i	−0.489077	+	0.872241i	\(0.662666\pi\)
							−0.943917	+	0.330184i	\(0.892889\pi\)
\(43\)	−58.9197	+	21.4450i	−1.37022	+	0.498721i	−0.919200	−	0.393790i	\(-0.871164\pi\)
							−0.451025	+	0.892511i	\(0.648941\pi\)
\(47\)	17.6548	+	3.11301i	0.375633	+	0.0662342i	0.358278	−	0.933615i	\(-0.383364\pi\)
							0.0173555	+	0.999849i	\(0.494475\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.b.43.17	✓	408
8.3	odd	2	inner	216.3.r.b.43.48	yes	408
27.22	even	9	inner	216.3.r.b.211.48	yes	408
216.211	odd	18	inner	216.3.r.b.211.17	yes	408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.b.43.17	✓	408	1.1	even	1	trivial
216.3.r.b.43.48	yes	408	8.3	odd	2	inner
216.3.r.b.211.17	yes	408	216.211	odd	18	inner
216.3.r.b.211.48	yes	408	27.22	even	9	inner