Embedded newform 162.4.g.b.7.2

• Label: 162.4.g.b.7.2
• Level: $162$
• Weight: $4$
• Character: 162.7
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $252$
• 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:	\(252\)
Relative dimension:	\(14\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			7.2
Character	\(\chi\)	\(=\)	162.7
Dual form			162.4.g.b.139.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19432 - 1.60425i) q^{2} +(-4.83160 + 1.91197i) q^{3} +(-1.14721 + 3.83196i) q^{4} +(1.51049 - 0.993463i) q^{5} +(8.83774 + 5.46758i) q^{6} +(8.41336 - 8.91764i) q^{7} +(7.51754 - 2.73616i) q^{8} +(19.6887 - 18.4758i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 36 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{8}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19432	−	1.60425i	−0.422255	−	0.567187i
							\(3\)	−4.83160	+	1.91197i	−0.929842	+	0.367959i
\(5\)	1.51049	−	0.993463i	0.135102	−	0.0888580i								−0.480159	−	0.877182i	\(-0.659421\pi\)
							0.615261	+	0.788324i	\(0.289051\pi\)
\(7\)	8.41336	−	8.91764i	0.454279	−	0.481507i	−0.459235	−	0.888315i	\(-0.651877\pi\)
							0.913514	+	0.406808i	\(0.133358\pi\)
\(11\)	−10.4308	+	5.23854i	−0.285909	+	0.143589i	−0.585974	−	0.810330i	\(-0.699288\pi\)
							0.300065	+	0.953919i	\(0.402992\pi\)
\(13\)	0.415271	+	0.962706i	0.00885965	+	0.0205390i	0.922589	−	0.385786i	\(-0.126070\pi\)
							−0.913729	+	0.406324i	\(0.866810\pi\)
\(17\)	−88.2150	+	74.0212i	−1.25855	+	1.05605i	−0.262711	+	0.964875i	\(0.584617\pi\)
							−0.995835	+	0.0911710i	\(0.970939\pi\)
\(19\)	−38.8900	−	32.6326i	−0.469578	−	0.394023i	0.377063	−	0.926188i	\(-0.376934\pi\)
							−0.846641	+	0.532165i	\(0.821379\pi\)
\(23\)	4.96962	+	5.26749i	0.0450538	+	0.0477542i	0.749509	−	0.661995i	\(-0.230290\pi\)
							−0.704455	+	0.709749i	\(0.748808\pi\)
\(29\)	−283.786	+	33.1698i	−1.81716	+	0.212396i	−0.955481	−	0.295054i	\(-0.904662\pi\)
							−0.861682	+	0.507449i	\(0.830588\pi\)
\(31\)	87.4015	−	20.7145i	0.506380	−	0.120014i	0.0305178	−	0.999534i	\(-0.490284\pi\)
							0.475862	+	0.879520i	\(0.342136\pi\)
\(37\)	−18.4076	−	104.395i	−0.0817889	−	0.463848i	−0.998003	−	0.0631599i	\(-0.979882\pi\)
							0.916215	−	0.400688i	\(-0.131229\pi\)
\(41\)	−134.029	+	180.033i	−0.510534	+	0.685766i	−0.980410	−	0.196966i	\(-0.936891\pi\)
							0.469877	+	0.882732i	\(0.344298\pi\)
\(43\)	−4.67721	+	80.3047i	−0.0165876	+	0.284799i	0.979806	+	0.199950i	\(0.0640779\pi\)
							−0.996394	+	0.0848491i	\(0.972959\pi\)
\(47\)	−288.518	−	68.3800i	−0.895418	−	0.212218i	−0.242954	−	0.970038i	\(-0.578116\pi\)
							−0.652464	+	0.757820i	\(0.726265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.b.7.2	✓	252
81.58	even	27	inner	162.4.g.b.139.2	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.b.7.2	✓	252	1.1	even	1	trivial
162.4.g.b.139.2	yes	252	81.58	even	27	inner