Embedded newform 162.3.h.a.5.3

• Label: 162.3.h.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.326140 - 1.37609i) q^{2} +(-2.45330 + 1.72665i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(-4.47588 + 3.33217i) q^{5} +(3.17615 + 2.81284i) q^{6} +(7.95611 - 5.23282i) q^{7} +(1.81808 + 2.16670i) q^{8} +(3.03734 - 8.47199i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	−2.45330	+	1.72665i	−0.817766	+	0.575551i
\(5\)	−4.47588	+	3.33217i	−0.895177	+	0.666434i								−0.943133	−	0.332415i	\(-0.892137\pi\)
							0.0479565	+	0.998849i	\(0.484729\pi\)
\(7\)	7.95611	−	5.23282i	1.13659	−	0.747545i	0.165486	−	0.986212i	\(-0.447081\pi\)
							0.971101	+	0.238667i	\(0.0767104\pi\)
\(11\)	0.259654	−	2.22149i	0.0236049	−	0.201953i	−0.976290	−	0.216468i	\(-0.930546\pi\)
							0.999895	+	0.0145148i	\(0.00462037\pi\)
\(13\)	−6.44003	−	21.5112i	−0.495387	−	1.65471i	−0.730632	−	0.682771i	\(-0.760775\pi\)
							0.235245	−	0.971936i	\(-0.424411\pi\)
\(17\)	23.0288	+	4.06060i	1.35464	+	0.238859i	0.803375	−	0.595474i	\(-0.203036\pi\)
							0.551261	+	0.834333i	\(0.314147\pi\)
\(19\)	−2.79692	−	15.8621i	−0.147206	−	0.834848i	−0.965569	−	0.260146i	\(-0.916229\pi\)
							0.818363	−	0.574702i	\(-0.194882\pi\)
\(23\)	−1.70366	+	2.59028i	−0.0740720	+	0.112621i	−0.870624	−	0.491949i	\(-0.836285\pi\)
							0.796552	+	0.604570i	\(0.206655\pi\)
\(29\)	7.80788	−	7.36635i	0.269237	−	0.254012i	−0.539768	−	0.841814i	\(-0.681488\pi\)
							0.809005	+	0.587802i	\(0.200006\pi\)
\(31\)	2.06123	−	35.3900i	0.0664914	−	1.14161i	−0.785777	−	0.618509i	\(-0.787737\pi\)
							0.852269	−	0.523104i	\(-0.175226\pi\)
\(37\)	25.6684	−	9.34254i	0.693741	−	0.252501i	0.0290047	−	0.999579i	\(-0.490766\pi\)
							0.664736	+	0.747078i	\(0.268544\pi\)
\(41\)	11.9342	−	50.3543i	0.291078	−	1.22815i	−0.609263	−	0.792969i	\(-0.708534\pi\)
							0.900341	−	0.435186i	\(-0.143317\pi\)
\(43\)	5.27347	+	12.2253i	0.122639	+	0.284309i	0.968465	−	0.249150i	\(-0.0801512\pi\)
							−0.845826	+	0.533459i	\(0.820892\pi\)
\(47\)	−52.0649	+	3.03243i	−1.10776	+	0.0645198i	−0.602256	−	0.798303i	\(-0.705731\pi\)
							−0.505507	+	0.862823i	\(0.668694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.3	✓	324
81.65	odd	54	inner	162.3.h.a.65.3	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.3	✓	324	1.1	even	1	trivial
162.3.h.a.65.3	yes	324	81.65	odd	54	inner