Embedded newform 162.3.h.a.5.14

• Label: 162.3.h.a.5.14
• 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			5.14
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.326140 + 1.37609i) q^{2} +(-0.0281193 - 2.99987i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(1.73126 - 1.28887i) q^{5} +(4.11893 - 1.01707i) q^{6} +(3.67681 - 2.41828i) q^{7} +(-1.81808 - 2.16670i) q^{8} +(-8.99842 + 0.168708i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	−0.0281193	−	2.99987i	−0.00937309	−	0.999956i
\(5\)	1.73126	−	1.28887i	0.346251	−	0.257774i								−0.409958	−	0.912104i	\(-0.634457\pi\)
							0.756209	+	0.654330i	\(0.227049\pi\)
\(7\)	3.67681	−	2.41828i	0.525259	−	0.345468i	−0.259025	−	0.965871i	\(-0.583401\pi\)
							0.784284	+	0.620402i	\(0.213031\pi\)
\(11\)	1.53316	−	13.1170i	0.139378	−	1.19246i	−0.724687	−	0.689078i	\(-0.758016\pi\)
							0.864066	−	0.503379i	\(-0.167910\pi\)
\(13\)	−2.24917	−	7.51277i	−0.173013	−	0.577905i	−0.999880	−	0.0155066i	\(-0.995064\pi\)
							0.826866	−	0.562399i	\(-0.190121\pi\)
\(17\)	32.0200	+	5.64599i	1.88353	+	0.332117i	0.992541	−	0.121908i	\(-0.0389013\pi\)
							0.890989	+	0.454025i	\(0.150012\pi\)
\(19\)	−4.44548	−	25.2116i	−0.233973	−	1.32693i	−0.844766	−	0.535136i	\(-0.820260\pi\)
							0.610793	−	0.791790i	\(-0.290851\pi\)
\(23\)	−2.91523	+	4.43240i	−0.126749	+	0.192713i	−0.893287	−	0.449486i	\(-0.851607\pi\)
							0.766538	+	0.642199i	\(0.221978\pi\)
\(29\)	7.70155	−	7.26604i	0.265571	−	0.250553i	−0.541925	−	0.840427i	\(-0.682304\pi\)
							0.807495	+	0.589874i	\(0.200823\pi\)
\(31\)	−1.26519	+	21.7226i	−0.0408127	+	0.700728i	0.914401	+	0.404810i	\(0.132662\pi\)
							−0.955214	+	0.295917i	\(0.904375\pi\)
\(37\)	−43.5271	+	15.8426i	−1.17641	+	0.428178i	−0.854932	−	0.518739i	\(-0.826402\pi\)
							−0.321477	+	0.946917i	\(0.604179\pi\)
\(41\)	−3.35858	+	14.1709i	−0.0819165	+	0.345633i	−0.998534	−	0.0541331i	\(-0.982760\pi\)
							0.916617	+	0.399766i	\(0.130909\pi\)
\(43\)	5.82117	+	13.4950i	0.135376	+	0.313837i	0.972396	−	0.233335i	\(-0.0749638\pi\)
							−0.837020	+	0.547172i	\(0.815704\pi\)
\(47\)	91.8452	−	5.34938i	1.95415	−	0.113817i	0.965149	−	0.261700i	\(-0.0842832\pi\)
							0.989005	+	0.147884i	\(0.0472462\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.14	✓	324
81.65	odd	54	inner	162.3.h.a.65.14	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.14	✓	324	1.1	even	1	trivial
162.3.h.a.65.14	yes	324	81.65	odd	54	inner