Embedded newform 162.4.g.a.13.7

• Label: 162.4.g.a.13.7
• Level: $162$
• Weight: $4$
• Character: 162.13
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $234$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.55830942093\)
Analytic rank:	\(0\)
Dimension:	\(234\)
Relative dimension:	\(13\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			13.7
Character	\(\chi\)	\(=\)	162.13
Dual form			162.4.g.a.25.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.78727 + 0.897598i) q^{2} +(-1.30843 - 5.02872i) q^{3} +(2.38863 + 3.20849i) q^{4} +(0.879965 + 2.93929i) q^{5} +(2.17526 - 10.1621i) q^{6} +(10.8789 + 25.2202i) q^{7} +(1.38919 + 7.87846i) q^{8} +(-23.5760 + 13.1595i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 234 q + 36 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{4}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.78727	+	0.897598i	0.631894	+	0.317349i
							\(3\)	−1.30843	−	5.02872i	−0.251807	−	0.967777i
\(5\)	0.879965	+	2.93929i	0.0787065	+	0.262898i								0.988282	−	0.152641i	\(-0.0487780\pi\)
							−0.909575	+	0.415540i	\(0.863593\pi\)
\(7\)	10.8789	+	25.2202i	0.587408	+	1.36176i	0.909756	+	0.415144i	\(0.136269\pi\)
							−0.322348	+	0.946621i	\(0.604472\pi\)
\(11\)	40.0058	−	9.48155i	1.09656	−	0.259890i	0.357762	−	0.933813i	\(-0.383540\pi\)
							0.738802	+	0.673923i	\(0.235392\pi\)
\(13\)	14.9100	+	9.80648i	0.318100	+	0.209218i	0.698510	−	0.715600i	\(-0.253847\pi\)
							−0.380410	+	0.924818i	\(0.624217\pi\)
\(17\)	92.5128	−	33.6719i	1.31986	−	0.480390i	0.416447	−	0.909160i	\(-0.363275\pi\)
							0.903414	+	0.428770i	\(0.141053\pi\)
\(19\)	−17.7820	−	6.47213i	−0.214709	−	0.0781478i	0.232426	−	0.972614i	\(-0.425334\pi\)
							−0.447135	+	0.894466i	\(0.647556\pi\)
\(23\)	−17.6442	+	40.9039i	−0.159960	+	0.370828i	−0.979281	−	0.202506i	\(-0.935091\pi\)
							0.819321	+	0.573335i	\(0.194351\pi\)
\(29\)	7.24089	+	124.321i	0.0463655	+	0.796065i	0.938710	+	0.344709i	\(0.112022\pi\)
							−0.892344	+	0.451356i	\(0.850941\pi\)
\(31\)	−112.091	+	13.1015i	−0.649422	+	0.0759066i	−0.434425	−	0.900708i	\(-0.643048\pi\)
							−0.214997	+	0.976615i	\(0.568974\pi\)
\(37\)	−9.37317	−	7.86502i	−0.0416470	−	0.0349460i	0.621727	−	0.783234i	\(-0.286432\pi\)
							−0.663374	+	0.748288i	\(0.730876\pi\)
\(41\)	−238.759	+	119.909i	−0.909459	+	0.456748i	−0.841079	−	0.540912i	\(-0.818079\pi\)
							−0.0683799	+	0.997659i	\(0.521783\pi\)
\(43\)	−227.574	−	241.214i	−0.807085	−	0.855460i	0.184776	−	0.982781i	\(-0.440844\pi\)
							−0.991862	+	0.127320i	\(0.959362\pi\)
\(47\)	79.7236	+	9.31836i	0.247423	+	0.0289196i	0.238900	−	0.971044i	\(-0.423213\pi\)
							0.00852268	+	0.999964i	\(0.497287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.a.13.7	✓	234
81.25	even	27	inner	162.4.g.a.25.7	yes	234

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.a.13.7	✓	234	1.1	even	1	trivial
162.4.g.a.25.7	yes	234	81.25	even	27	inner