Embedded newform 243.2.i.a.103.8

• Label: 243.2.i.a.103.8
• Level: $243$
• Weight: $2$
• Character: 243.103
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $1404$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.i (of order \(81\), degree \(54\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94036476912\)
Analytic rank:	\(0\)
Dimension:	\(1404\)
Relative dimension:	\(26\) over \(\Q(\zeta_{81})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{81}]$

Embedding invariants

Embedding label			103.8
Character	\(\chi\)	\(=\)	243.103
Dual form			243.2.i.a.151.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36720 - 0.380585i) q^{2} +(1.70855 - 0.284335i) q^{3} +(0.0120479 + 0.00727097i) q^{4} +(-2.11465 + 0.0820581i) q^{5} +(-2.44414 - 0.261509i) q^{6} +(1.64831 + 0.257791i) q^{7} +(1.93410 + 2.05002i) q^{8} +(2.83831 - 0.971602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{81}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36720	−	0.380585i	−0.966754	−	0.269114i	−0.251405	−	0.967882i	\(-0.580893\pi\)
							−0.715349	+	0.698768i	\(0.753732\pi\)
\(3\)	1.70855	−	0.284335i	0.986434	−	0.164161i
							\(5\)	−2.11465	+	0.0820581i	−0.945701	+	0.0366975i	−0.507010	−	0.861940i	\(-0.669249\pi\)
−0.438691	+	0.898638i	\(0.644558\pi\)
\(7\)	1.64831	+	0.257791i	0.623002	+	0.0974358i	0.458139	−	0.888881i	\(-0.348516\pi\)
							0.164863	+	0.986316i	\(0.447282\pi\)
\(11\)	−0.699652	−	5.12236i	−0.210953	−	1.54445i	−0.726246	−	0.687435i	\(-0.758736\pi\)
							0.515292	−	0.857014i	\(-0.327683\pi\)
\(13\)	3.44321	−	5.02032i	0.954974	−	1.39239i	0.0354424	−	0.999372i	\(-0.488716\pi\)
							0.919532	−	0.393015i	\(-0.128568\pi\)
\(17\)	−1.16391	+	0.584539i	−0.282290	+	0.141771i	−0.584311	−	0.811530i	\(-0.698635\pi\)
							0.302021	+	0.953301i	\(0.402339\pi\)
\(19\)	0.0591164	+	1.01499i	0.0135622	+	0.232855i	0.998277	+	0.0586793i	\(0.0186889\pi\)
							−0.984715	+	0.174175i	\(0.944274\pi\)
\(23\)	−0.0433815	+	0.112361i	−0.00904566	+	0.0234289i	−0.937333	−	0.348435i	\(-0.886713\pi\)
							0.928287	+	0.371864i	\(0.121281\pi\)
\(29\)	−2.48271	−	1.77454i	−0.461028	−	0.329523i	0.327148	−	0.944973i	\(-0.393912\pi\)
							−0.788176	+	0.615450i	\(0.788974\pi\)
\(31\)	2.39439	−	2.74366i	0.430045	−	0.492777i	−0.496988	−	0.867758i	\(-0.665561\pi\)
							0.927033	+	0.374981i	\(0.122351\pi\)
\(37\)	4.61774	+	6.20270i	0.759151	+	1.01972i	0.998806	+	0.0488587i	\(0.0155584\pi\)
							−0.239655	+	0.970858i	\(0.577034\pi\)
\(41\)	−0.596270	+	2.31486i	−0.0931217	+	0.361521i	−0.997931	−	0.0642967i	\(-0.979520\pi\)
							0.904809	+	0.425817i	\(0.140013\pi\)
\(43\)	3.38429	+	3.07108i	0.516100	+	0.468335i	0.887939	−	0.459962i	\(-0.152137\pi\)
							−0.371839	+	0.928297i	\(0.621273\pi\)
\(47\)	7.20444	+	8.25537i	1.05087	+	1.20417i	0.978383	+	0.206800i	\(0.0663050\pi\)
							0.0724917	+	0.997369i	\(0.476905\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.103.8	✓	1404
3.2	odd	2		729.2.i.a.685.19		1404
243.92	odd	162		729.2.i.a.613.19		1404
243.151	even	81	inner	243.2.i.a.151.8	yes	1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.103.8	✓	1404	1.1	even	1	trivial
243.2.i.a.151.8	yes	1404	243.151	even	81	inner
729.2.i.a.613.19		1404	243.92	odd	162
729.2.i.a.685.19		1404	3.2	odd	2