Embedded newform 162.3.h.a.59.1

• Label: 162.3.h.a.59.1
• Level: $162$
• Weight: $3$
• Character: 162.59
• 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			59.1
Character	\(\chi\)	\(=\)	162.59
Dual form			162.3.h.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02866 + 0.970492i) q^{2} +(-2.92008 + 0.687859i) q^{3} +(0.116290 - 1.99662i) q^{4} +(0.533779 + 4.56677i) q^{5} +(2.33621 - 3.54149i) q^{6} +(7.85421 + 3.94453i) q^{7} +(1.81808 + 2.16670i) q^{8} +(8.05370 - 4.01720i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

Character values

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

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{41}{54}\right)\)

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
							\(3\)	−2.92008	+	0.687859i	−0.973359	+	0.229286i
\(5\)	0.533779	+	4.56677i	0.106756	+	0.913355i								0.935403	+	0.353583i	\(0.115037\pi\)
							−0.828647	+	0.559771i	\(0.810889\pi\)
\(7\)	7.85421	+	3.94453i	1.12203	+	0.563505i	0.910352	−	0.413835i	\(-0.135811\pi\)
							0.211678	+	0.977339i	\(0.432107\pi\)
\(11\)	−13.7218	+	5.91902i	−1.24744	+	0.538093i	−0.914397	−	0.404819i	\(-0.867334\pi\)
							−0.333042	+	0.942912i	\(0.608075\pi\)
\(13\)	−17.5398	−	4.15701i	−1.34921	−	0.319770i	−0.508413	−	0.861113i	\(-0.669768\pi\)
							−0.840802	+	0.541344i	\(0.817916\pi\)
\(17\)	−20.6502	−	3.64120i	−1.21472	−	0.214188i	−0.470669	−	0.882310i	\(-0.655987\pi\)
							−0.744052	+	0.668122i	\(0.767098\pi\)
\(19\)	−0.538862	−	3.05604i	−0.0283611	−	0.160844i	0.967338	−	0.253490i	\(-0.0815786\pi\)
							−0.995699	+	0.0926462i	\(0.970467\pi\)
\(23\)	13.2936	+	26.4697i	0.577982	+	1.15086i	0.972702	+	0.232056i	\(0.0745453\pi\)
							−0.394720	+	0.918801i	\(0.629158\pi\)
\(29\)	−15.4329	−	4.62031i	−0.532169	−	0.159321i	0.00943371	−	0.999956i	\(-0.496997\pi\)
							−0.541603	+	0.840635i	\(0.682182\pi\)
\(31\)	−14.0339	−	9.23025i	−0.452707	−	0.297750i	0.302604	−	0.953116i	\(-0.402144\pi\)
							−0.755311	+	0.655366i	\(0.772514\pi\)
\(37\)	−48.4177	+	17.6226i	−1.30859	+	0.476287i	−0.899784	−	0.436335i	\(-0.856276\pi\)
							−0.408804	+	0.912622i	\(0.634054\pi\)
\(41\)	−8.66263	−	8.17277i	−0.211284	−	0.199336i	0.573323	−	0.819329i	\(-0.305654\pi\)
							−0.784607	+	0.619993i	\(0.787135\pi\)
\(43\)	19.9544	−	26.8034i	0.464055	−	0.623335i	−0.507070	−	0.861905i	\(-0.669271\pi\)
							0.971125	+	0.238570i	\(0.0766788\pi\)
\(47\)	40.0181	+	60.8446i	0.851449	+	1.29456i	0.953991	+	0.299836i	\(0.0969321\pi\)
							−0.102542	+	0.994729i	\(0.532698\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.59.1	yes	324
81.11	odd	54	inner	162.3.h.a.11.1	✓	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.11.1	✓	324	81.11	odd	54	inner
162.3.h.a.59.1	yes	324	1.1	even	1	trivial