Embedded newform 162.3.h.a.5.10

• Label: 162.3.h.a.5.10
• 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			5.10
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.326140 + 1.37609i) q^{2} +(-2.96504 - 0.456664i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(-1.81547 + 1.35157i) q^{5} +(-0.338606 - 4.22911i) q^{6} +(-1.49360 + 0.982359i) q^{7} +(-1.81808 - 2.16670i) q^{8} +(8.58292 + 2.70805i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	−2.96504	−	0.456664i	−0.988346	−	0.152221i
\(5\)	−1.81547	+	1.35157i	−0.363095	+	0.270314i								−0.763210	−	0.646151i	\(-0.776378\pi\)
							0.400115	+	0.916465i	\(0.368970\pi\)
\(7\)	−1.49360	+	0.982359i	−0.213372	+	0.140337i	−0.651697	−	0.758480i	\(-0.725942\pi\)
							0.438325	+	0.898817i	\(0.355572\pi\)
\(11\)	2.37524	−	20.3215i	0.215931	−	1.84741i	−0.257129	−	0.966377i	\(-0.582776\pi\)
							0.473060	−	0.881030i	\(-0.343149\pi\)
\(13\)	−4.99401	−	16.6812i	−0.384155	−	1.28317i	−0.903495	−	0.428599i	\(-0.859007\pi\)
							0.519340	−	0.854568i	\(-0.326178\pi\)
\(17\)	−26.1697	−	4.61443i	−1.53940	−	0.271437i	−0.661373	−	0.750057i	\(-0.730026\pi\)
							−0.878022	+	0.478620i	\(0.841137\pi\)
\(19\)	1.74443	+	9.89314i	0.0918120	+	0.520692i	0.995678	+	0.0928755i	\(0.0296059\pi\)
							−0.903866	+	0.427816i	\(0.859283\pi\)
\(23\)	23.2916	−	35.4131i	1.01268	−	1.53970i	0.181729	−	0.983349i	\(-0.441831\pi\)
							0.830947	−	0.556351i	\(-0.187799\pi\)
\(29\)	−34.8032	+	32.8351i	−1.20011	+	1.13225i	−0.212199	+	0.977226i	\(0.568063\pi\)
							−0.987911	+	0.155020i	\(0.950456\pi\)
\(31\)	−0.786383	+	13.5017i	−0.0253672	+	0.435538i	0.961583	+	0.274513i	\(0.0885168\pi\)
							−0.986950	+	0.161024i	\(0.948520\pi\)
\(37\)	34.9752	−	12.7299i	0.945275	−	0.344052i	0.177028	−	0.984206i	\(-0.443352\pi\)
							0.768247	+	0.640154i	\(0.221129\pi\)
\(41\)	8.80684	−	37.1590i	0.214801	−	0.906317i	−0.753682	−	0.657239i	\(-0.771724\pi\)
							0.968483	−	0.249078i	\(-0.0801276\pi\)
\(43\)	−5.48771	−	12.7219i	−0.127621	−	0.295859i	0.842411	−	0.538835i	\(-0.181136\pi\)
							−0.970032	+	0.242976i	\(0.921876\pi\)
\(47\)	−51.4277	+	2.99533i	−1.09421	+	0.0637303i	−0.595739	−	0.803178i	\(-0.703141\pi\)
							−0.498468	+	0.866908i	\(0.666104\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.10	✓	324
81.65	odd	54	inner	162.3.h.a.65.10	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.10	✓	324	1.1	even	1	trivial
162.3.h.a.65.10	yes	324	81.65	odd	54	inner