Embedded newform 162.3.h.a.11.9

• Label: 162.3.h.a.11.9
• Level: $162$
• Weight: $3$
• Character: 162.11
• Analytic conductor: $4.414$
• Analytic rank: $0$
• Dimension: $324$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	162.h (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.41418028264\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(18\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			11.9
Character	\(\chi\)	\(=\)	162.11
Dual form			162.3.h.a.59.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02866 - 0.970492i) q^{2} +(2.79752 - 1.08347i) q^{3} +(0.116290 + 1.99662i) q^{4} +(-0.238043 + 2.03659i) q^{5} +(-3.92920 - 1.60044i) q^{6} +(2.85624 - 1.43446i) q^{7} +(1.81808 - 2.16670i) q^{8} +(6.65219 - 6.06204i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{13}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02866	−	0.970492i	−0.514331	−	0.485246i
							\(3\)	2.79752	−	1.08347i	0.932505	−	0.361156i
\(5\)	−0.238043	+	2.03659i	−0.0476087	+	0.407318i								0.948330	+	0.317285i	\(0.102771\pi\)
							−0.995939	+	0.0900328i	\(0.971303\pi\)
\(7\)	2.85624	−	1.43446i	0.408034	−	0.204922i	−0.232930	−	0.972494i	\(-0.574831\pi\)
							0.640964	+	0.767571i	\(0.278535\pi\)
\(11\)	4.29462	+	1.85252i	0.390420	+	0.168411i	0.582235	−	0.813020i	\(-0.302178\pi\)
							−0.191815	+	0.981431i	\(0.561438\pi\)
\(13\)	18.7345	−	4.44015i	1.44111	−	0.341550i	0.565575	−	0.824697i	\(-0.308654\pi\)
							0.875539	+	0.483147i	\(0.160506\pi\)
\(17\)	−16.9540	+	2.98944i	−0.997292	+	0.175849i	−0.648389	−	0.761309i	\(-0.724557\pi\)
							−0.348903	+	0.937159i	\(0.613446\pi\)
\(19\)	0.592020	−	3.35751i	0.0311589	−	0.176711i	−0.965257	−	0.261303i	\(-0.915848\pi\)
							0.996416	+	0.0845923i	\(0.0269588\pi\)
\(23\)	9.86814	−	19.6491i	0.429050	−	0.854308i	−0.570407	−	0.821362i	\(-0.693215\pi\)
							0.999457	−	0.0329464i	\(-0.0104891\pi\)
\(29\)	−40.2372	+	12.0462i	−1.38749	+	0.415387i	−0.891531	−	0.452959i	\(-0.850368\pi\)
							−0.495958	+	0.868346i	\(0.665183\pi\)
\(31\)	4.45093	−	2.92742i	0.143578	−	0.0944329i	−0.475688	−	0.879614i	\(-0.657801\pi\)
							0.619266	+	0.785181i	\(0.287430\pi\)
\(37\)	−27.1410	−	9.87853i	−0.733542	−	0.266987i	−0.0518785	−	0.998653i	\(-0.516521\pi\)
							−0.681663	+	0.731666i	\(0.738743\pi\)
\(41\)	1.32566	−	1.25070i	0.0323332	−	0.0305048i	−0.669906	−	0.742446i	\(-0.733665\pi\)
							0.702239	+	0.711942i	\(0.252184\pi\)
\(43\)	19.8396	+	26.6492i	0.461387	+	0.619750i	0.970544	−	0.240923i	\(-0.0774503\pi\)
							−0.509157	+	0.860673i	\(0.670043\pi\)
\(47\)	−28.5491	+	43.4067i	−0.607427	+	0.923548i	0.392566	+	0.919724i	\(0.371587\pi\)
							−0.999993	+	0.00382365i	\(0.998783\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.11.9	✓	324
81.59	odd	54	inner	162.3.h.a.59.9	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.11.9	✓	324	1.1	even	1	trivial
162.3.h.a.59.9	yes	324	81.59	odd	54	inner