Embedded newform 162.3.h.a.5.4

• Label: 162.3.h.a.5.4
• Level: $162$
• Weight: $3$
• Character: 162.5
• 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			5.4
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.326140 - 1.37609i) q^{2} +(-1.15412 - 2.76912i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(5.90063 - 4.39286i) q^{5} +(-3.43416 + 2.49129i) q^{6} +(11.1239 - 7.31631i) q^{7} +(1.81808 + 2.16670i) q^{8} +(-6.33603 + 6.39177i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

Character values

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

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{23}{54}\right)\)

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
							\(3\)	−1.15412	−	2.76912i	−0.384706	−	0.923039i
\(5\)	5.90063	−	4.39286i	1.18013	−	0.878572i								0.185556	−	0.982634i	\(-0.440591\pi\)
							0.994571	+	0.104062i	\(0.0331840\pi\)
\(7\)	11.1239	−	7.31631i	1.58913	−	1.04519i	0.624693	−	0.780871i	\(-0.285224\pi\)
							0.964436	−	0.264316i	\(-0.0851461\pi\)
\(11\)	−1.19877	+	10.2561i	−0.108979	+	0.932373i	0.822547	+	0.568697i	\(0.192552\pi\)
							−0.931526	+	0.363676i	\(0.881522\pi\)
\(13\)	−2.19072	−	7.31750i	−0.168517	−	0.562885i	−0.999973	−	0.00729523i	\(-0.997678\pi\)
							0.831457	−	0.555589i	\(-0.187507\pi\)
\(17\)	−10.5153	−	1.85413i	−0.618547	−	0.109067i	−0.144411	−	0.989518i	\(-0.546129\pi\)
							−0.474137	+	0.880451i	\(0.657240\pi\)
\(19\)	−0.318206	−	1.80464i	−0.0167477	−	0.0949808i	0.975288	−	0.220937i	\(-0.0709114\pi\)
							−0.992036	+	0.125956i	\(0.959800\pi\)
\(23\)	−2.41065	+	3.66521i	−0.104811	+	0.159357i	−0.884085	−	0.467326i	\(-0.845217\pi\)
							0.779274	+	0.626683i	\(0.215588\pi\)
\(29\)	−30.9822	+	29.2302i	−1.06835	+	1.00794i	−0.0684049	+	0.997658i	\(0.521791\pi\)
							−0.999947	+	0.0102806i	\(0.996728\pi\)
\(31\)	−3.32477	+	57.0841i	−0.107251	+	1.84142i	0.332263	+	0.943187i	\(0.392188\pi\)
							−0.439513	+	0.898236i	\(0.644849\pi\)
\(37\)	66.4373	−	24.1812i	1.79560	−	0.653546i	0.796820	−	0.604217i	\(-0.206514\pi\)
							0.998783	−	0.0493292i	\(-0.0157084\pi\)
\(41\)	6.62289	−	27.9442i	0.161534	−	0.681565i	−0.830651	−	0.556793i	\(-0.812032\pi\)
							0.992185	−	0.124773i	\(-0.0398201\pi\)
\(43\)	9.40424	+	21.8015i	0.218703	+	0.507011i	0.991880	−	0.127180i	\(-0.0405925\pi\)
							−0.773176	+	0.634191i	\(0.781333\pi\)
\(47\)	3.55206	−	0.206884i	0.0755757	−	0.00440179i	−0.0203153	−	0.999794i	\(-0.506467\pi\)
							0.0958910	+	0.995392i	\(0.469430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.4	✓	324
81.65	odd	54	inner	162.3.h.a.65.4	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.4	✓	324	1.1	even	1	trivial
162.3.h.a.65.4	yes	324	81.65	odd	54	inner