Embedded newform 162.3.h.a.65.2

• Label: 162.3.h.a.65.2
• Level: $162$
• Weight: $3$
• Character: 162.65
• 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			65.2
Character	\(\chi\)	\(=\)	162.65
Dual form			162.3.h.a.5.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.326140 + 1.37609i) q^{2} +(-2.75282 + 1.19248i) q^{3} +(-1.78727 - 0.897598i) q^{4} +(-1.51997 - 1.13158i) q^{5} +(-0.743160 - 4.17705i) q^{6} +(-2.05983 - 1.35477i) q^{7} +(1.81808 - 2.16670i) q^{8} +(6.15598 - 6.56535i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

Character values

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

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{31}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.326140	+	1.37609i	−0.163070	+	0.688047i
							\(3\)	−2.75282	+	1.19248i	−0.917605	+	0.397493i
\(5\)	−1.51997	−	1.13158i	−0.303994	−	0.226315i								0.434342	−	0.900748i	\(-0.356981\pi\)
							−0.738336	+	0.674433i	\(0.764388\pi\)
\(7\)	−2.05983	−	1.35477i	−0.294262	−	0.193539i	0.393790	−	0.919200i	\(-0.371164\pi\)
							−0.688052	+	0.725661i	\(0.741534\pi\)
\(11\)	0.177149	+	1.51561i	0.0161045	+	0.137783i	0.998899	−	0.0469220i	\(-0.0149412\pi\)
							−0.982794	+	0.184705i	\(0.940867\pi\)
\(13\)	4.60013	−	15.3655i	0.353856	−	1.18196i	−0.577052	−	0.816707i	\(-0.695797\pi\)
							0.930908	−	0.365254i	\(-0.119018\pi\)
\(17\)	13.4044	−	2.36356i	0.788495	−	0.139033i	0.235121	−	0.971966i	\(-0.424451\pi\)
							0.553374	+	0.832933i	\(0.313340\pi\)
\(19\)	1.74697	−	9.90758i	0.0919460	−	0.521451i	−0.903695	−	0.428178i	\(-0.859156\pi\)
							0.995641	−	0.0932737i	\(-0.0297332\pi\)
\(23\)	3.40001	+	5.16947i	0.147827	+	0.224759i	0.901809	−	0.432136i	\(-0.142240\pi\)
							−0.753982	+	0.656895i	\(0.771869\pi\)
\(29\)	0.504434	+	0.475909i	0.0173943	+	0.0164106i	0.694890	−	0.719116i	\(-0.255453\pi\)
							−0.677495	+	0.735527i	\(0.736935\pi\)
\(31\)	−2.90952	−	49.9546i	−0.0938556	−	1.61144i	−0.636364	−	0.771389i	\(-0.719562\pi\)
							0.542509	−	0.840050i	\(-0.317475\pi\)
\(37\)	−42.8060	−	15.5801i	−1.15692	−	0.421084i	−0.308922	−	0.951087i	\(-0.599968\pi\)
							−0.847995	+	0.530004i	\(0.822191\pi\)
\(41\)	−3.48374	−	14.6991i	−0.0849694	−	0.358514i	0.913906	−	0.405926i	\(-0.133051\pi\)
							−0.998875	+	0.0474118i	\(0.984903\pi\)
\(43\)	−9.40174	+	21.7957i	−0.218645	+	0.506876i	−0.991870	−	0.127256i	\(-0.959383\pi\)
							0.773225	+	0.634132i	\(0.218642\pi\)
\(47\)	−36.2386	−	2.11066i	−0.771033	−	0.0449076i	−0.331882	−	0.943321i	\(-0.607684\pi\)
							−0.439151	+	0.898413i	\(0.644721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.65.2	yes	324
81.5	odd	54	inner	162.3.h.a.5.2	✓	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.2	✓	324	81.5	odd	54	inner
162.3.h.a.65.2	yes	324	1.1	even	1	trivial