Embedded newform 243.2.i.a.106.4

• Label: 243.2.i.a.106.4
• Level: $243$
• Weight: $2$
• Character: 243.106
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $1404$
• 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			106.4
Character	\(\chi\)	\(=\)	243.106
Dual form			243.2.i.a.94.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.839855 + 2.17528i) q^{2} +(-1.70437 - 0.308407i) q^{3} +(-2.54539 - 2.30982i) q^{4} +(0.847134 + 0.605495i) q^{5} +(2.10229 - 3.44847i) q^{6} +(-3.28884 - 2.54904i) q^{7} +(2.99475 - 1.50402i) q^{8} +(2.80977 + 1.05128i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{50}{81}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.839855	+	2.17528i	−0.593867	+	1.53815i	0.231075	+	0.972936i	\(0.425776\pi\)
							−0.824942	+	0.565218i	\(0.808792\pi\)
\(3\)	−1.70437	−	0.308407i	−0.984020	−	0.178059i
							\(5\)	0.847134	+	0.605495i	0.378850	+	0.270786i	0.754935	−	0.655799i	\(-0.227668\pi\)
−0.376085	+	0.926585i	\(0.622730\pi\)
\(7\)	−3.28884	−	2.54904i	−1.24306	−	0.963447i	−0.243085	−	0.970005i	\(-0.578159\pi\)
							−0.999978	+	0.00655790i	\(0.997913\pi\)
\(11\)	2.67047	−	3.89364i	0.805176	−	1.17398i	−0.176665	−	0.984271i	\(-0.556531\pi\)
							0.981841	−	0.189704i	\(-0.0607530\pi\)
\(13\)	0.212803	−	0.0417931i	0.0590209	−	0.0115913i	−0.163115	−	0.986607i	\(-0.552154\pi\)
							0.222136	+	0.975016i	\(0.428697\pi\)
\(17\)	−0.685232	−	1.58855i	−0.166193	−	0.385279i	0.814684	−	0.579904i	\(-0.196910\pi\)
							−0.980878	+	0.194625i	\(0.937651\pi\)
\(19\)	2.35155	+	3.15868i	0.539483	+	0.724652i	0.985368	−	0.170438i	\(-0.0545184\pi\)
							−0.445885	+	0.895090i	\(0.647111\pi\)
\(23\)	2.08194	+	0.850567i	0.434115	+	0.177355i	0.584714	−	0.811239i	\(-0.301207\pi\)
							−0.150599	+	0.988595i	\(0.548120\pi\)
\(29\)	−0.970192	+	0.535329i	−0.180160	+	0.0994082i	−0.570629	−	0.821208i	\(-0.693301\pi\)
							0.390469	+	0.920616i	\(0.372313\pi\)
\(31\)	−4.69139	−	8.90638i	−0.842598	−	1.59963i	−0.799580	−	0.600559i	\(-0.794945\pi\)
							−0.0430180	−	0.999074i	\(-0.513697\pi\)
\(37\)	1.98010	+	2.09878i	0.325526	+	0.345038i	0.869382	−	0.494141i	\(-0.164517\pi\)
							−0.543855	+	0.839179i	\(0.683036\pi\)
\(41\)	3.47998	−	4.31450i	0.543482	−	0.673812i	−0.429933	−	0.902861i	\(-0.641463\pi\)
							0.973415	+	0.229049i	\(0.0735615\pi\)
\(43\)	−6.55201	−	6.42616i	−0.999172	−	0.979981i	0.000628052	−	1.00000i	\(-0.499800\pi\)
							−0.999800	+	0.0200193i	\(0.993627\pi\)
\(47\)	3.82969	−	7.27049i	0.558618	−	1.06051i	−0.428952	−	0.903327i	\(-0.641117\pi\)
							0.987570	−	0.157183i	\(-0.0502411\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.106.4	yes	1404
3.2	odd	2		729.2.i.a.559.23		1404
243.94	even	81	inner	243.2.i.a.94.4	✓	1404
243.149	odd	162		729.2.i.a.253.23		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.94.4	✓	1404	243.94	even	81	inner
243.2.i.a.106.4	yes	1404	1.1	even	1	trivial
729.2.i.a.253.23		1404	243.149	odd	162
729.2.i.a.559.23		1404	3.2	odd	2