Embedded newform 162.4.g.b.7.9

• Label: 162.4.g.b.7.9
• 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			7.9
Character	\(\chi\)	\(=\)	162.7
Dual form			162.4.g.b.139.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19432 - 1.60425i) q^{2} +(2.06068 + 4.77007i) q^{3} +(-1.14721 + 3.83196i) q^{4} +(6.94326 - 4.56666i) q^{5} +(5.19127 - 9.00282i) q^{6} +(-24.6415 + 26.1185i) q^{7} +(7.51754 - 2.73616i) q^{8} +(-18.5072 + 19.6592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 36 q^{6}+O(q^{10}) \)

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
							\(3\)	2.06068	+	4.77007i	0.396578	+	0.918001i
\(5\)	6.94326	−	4.56666i	0.621024	−	0.408454i								−0.199636	−	0.979870i	\(-0.563976\pi\)
							0.820661	+	0.571416i	\(0.193606\pi\)
\(7\)	−24.6415	+	26.1185i	−1.33052	+	1.41027i	−0.486768	+	0.873531i	\(0.661824\pi\)
							−0.843750	+	0.536736i	\(0.819657\pi\)
\(11\)	2.84620	−	1.42942i	0.0780148	−	0.0391805i	−0.409364	−	0.912371i	\(-0.634249\pi\)
							0.487379	+	0.873191i	\(0.337953\pi\)
\(13\)	−27.4355	−	63.6027i	−0.585327	−	1.35694i	−0.911371	−	0.411586i	\(-0.864975\pi\)
							0.326044	−	0.945355i	\(-0.394284\pi\)
\(17\)	−47.6955	+	40.0212i	−0.680462	+	0.570975i	−0.916141	−	0.400856i	\(-0.868713\pi\)
							0.235680	+	0.971831i	\(0.424268\pi\)
\(19\)	−11.3779	−	9.54720i	−0.137383	−	0.115278i	0.571507	−	0.820597i	\(-0.306359\pi\)
							−0.708890	+	0.705320i	\(0.750804\pi\)
\(23\)	−5.05421	−	5.35714i	−0.0458206	−	0.0485670i	0.704058	−	0.710143i	\(-0.251370\pi\)
							−0.749878	+	0.661576i	\(0.769888\pi\)
\(29\)	−211.681	+	24.7419i	−1.35545	+	0.158430i	−0.762594	−	0.646877i	\(-0.776075\pi\)
							−0.592858	+	0.805307i	\(0.702001\pi\)
\(31\)	106.746	−	25.2992i	0.618456	−	0.146577i	0.0905672	−	0.995890i	\(-0.471132\pi\)
							0.527889	+	0.849314i	\(0.322984\pi\)
\(37\)	67.7864	+	384.436i	0.301190	+	1.70813i	0.640919	+	0.767608i	\(0.278553\pi\)
							−0.339730	+	0.940523i	\(0.610336\pi\)
\(41\)	−234.304	+	314.725i	−0.892492	+	1.19882i	0.0867702	+	0.996228i	\(0.472345\pi\)
							−0.979262	+	0.202597i	\(0.935062\pi\)
\(43\)	−3.91473	+	67.2133i	−0.0138835	+	0.238371i	0.984226	+	0.176914i	\(0.0566115\pi\)
							−0.998110	+	0.0614563i	\(0.980426\pi\)
\(47\)	459.602	+	108.928i	1.42638	+	0.338058i	0.870075	−	0.492919i	\(-0.164070\pi\)
							0.556306	+	0.830978i	\(0.312218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.b.7.9	✓	252
81.58	even	27	inner	162.4.g.b.139.9	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.b.7.9	✓	252	1.1	even	1	trivial
162.4.g.b.139.9	yes	252	81.58	even	27	inner