Embedded newform 162.3.h.a.65.5

• Label: 162.3.h.a.65.5
• 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			65.5
Character	\(\chi\)	\(=\)	162.65
Dual form			162.3.h.a.5.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.326140 + 1.37609i) q^{2} +(-0.663139 - 2.92579i) q^{3} +(-1.78727 - 0.897598i) q^{4} +(0.605521 + 0.450794i) q^{5} +(4.24244 + 0.0416761i) q^{6} +(1.59852 + 1.05136i) q^{7} +(1.81808 - 2.16670i) q^{8} +(-8.12049 + 3.88041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	−0.663139	−	2.92579i	−0.221046	−	0.975263i
\(5\)	0.605521	+	0.450794i	0.121104	+	0.0901587i								0.656007	−	0.754755i	\(-0.272244\pi\)
							−0.534903	+	0.844914i	\(0.679652\pi\)
\(7\)	1.59852	+	1.05136i	0.228360	+	0.150195i	0.658533	−	0.752552i	\(-0.271177\pi\)
							−0.430173	+	0.902746i	\(0.641548\pi\)
\(11\)	−1.86600	−	15.9646i	−0.169636	−	1.45133i	−0.763001	−	0.646397i	\(-0.776275\pi\)
							0.593365	−	0.804934i	\(-0.297799\pi\)
\(13\)	5.81481	−	19.4228i	0.447293	−	1.49406i	−0.376094	−	0.926581i	\(-0.622733\pi\)
							0.823387	−	0.567480i	\(-0.192082\pi\)
\(17\)	−24.4120	+	4.30450i	−1.43600	+	0.253206i	−0.836851	−	0.547430i	\(-0.815606\pi\)
							−0.599151	+	0.800636i	\(0.704495\pi\)
\(19\)	5.59322	−	31.7207i	0.294380	−	1.66951i	−0.375332	−	0.926891i	\(-0.622471\pi\)
							0.669712	−	0.742621i	\(-0.266418\pi\)
\(23\)	4.73528	+	7.19964i	0.205882	+	0.313028i	0.923638	−	0.383267i	\(-0.125201\pi\)
							−0.717756	+	0.696295i	\(0.754831\pi\)
\(29\)	14.0691	+	13.2736i	0.485143	+	0.457709i	0.889528	−	0.456880i	\(-0.151033\pi\)
							−0.404385	+	0.914589i	\(0.632515\pi\)
\(31\)	3.26584	+	56.0722i	0.105350	+	1.80878i	0.475400	+	0.879770i	\(0.342304\pi\)
							−0.370050	+	0.929012i	\(0.620659\pi\)
\(37\)	22.6310	+	8.23702i	0.611649	+	0.222622i	0.629225	−	0.777223i	\(-0.283372\pi\)
							−0.0175754	+	0.999846i	\(0.505595\pi\)
\(41\)	2.75433	+	11.6214i	0.0671787	+	0.283449i	0.996258	−	0.0864320i	\(-0.0275465\pi\)
							−0.929079	+	0.369881i	\(0.879398\pi\)
\(43\)	24.1722	−	56.0374i	0.562144	−	1.30320i	−0.365876	−	0.930664i	\(-0.619231\pi\)
							0.928019	−	0.372532i	\(-0.121510\pi\)
\(47\)	21.9201	+	1.27670i	0.466385	+	0.0271638i	0.289729	−	0.957109i	\(-0.406435\pi\)
							0.176656	+	0.984273i	\(0.443472\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.65.5	yes	324
81.5	odd	54	inner	162.3.h.a.5.5	✓	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.5	✓	324	81.5	odd	54	inner
162.3.h.a.65.5	yes	324	1.1	even	1	trivial