Embedded newform 243.2.i.a.103.19

• Label: 243.2.i.a.103.19
• 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			103.19
Character	\(\chi\)	\(=\)	243.103
Dual form			243.2.i.a.151.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.16160 + 0.323353i) q^{2} +(-1.51067 + 0.847281i) q^{3} +(-0.467580 - 0.282186i) q^{4} +(-1.06011 + 0.0411373i) q^{5} +(-2.02876 + 0.495721i) q^{6} +(-5.09437 - 0.796745i) q^{7} +(-2.10679 - 2.23306i) q^{8} +(1.56423 - 2.55992i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.16160	+	0.323353i	0.821374	+	0.228645i	0.653271	−	0.757124i	\(-0.273396\pi\)
							0.168103	+	0.985769i	\(0.446236\pi\)
\(3\)	−1.51067	+	0.847281i	−0.872184	+	0.489178i
							\(5\)	−1.06011	+	0.0411373i	−0.474098	+	0.0183971i	−0.274717	−	0.961525i	\(-0.588584\pi\)
−0.199380	+	0.979922i	\(0.563893\pi\)
\(7\)	−5.09437	−	0.796745i	−1.92549	−	0.301141i	−0.928038	−	0.372485i	\(-0.878506\pi\)
							−0.997452	+	0.0713435i	\(0.977271\pi\)
\(11\)	0.460015	+	3.36790i	0.138700	+	1.01546i	0.920170	+	0.391519i	\(0.128050\pi\)
							−0.781470	+	0.623942i	\(0.785530\pi\)
\(13\)	1.07510	−	1.56753i	0.298178	−	0.434754i	−0.646421	−	0.762981i	\(-0.723735\pi\)
							0.944599	+	0.328227i	\(0.106451\pi\)
\(17\)	−1.45551	+	0.730985i	−0.353013	+	0.177290i	−0.616464	−	0.787383i	\(-0.711436\pi\)
							0.263451	+	0.964673i	\(0.415139\pi\)
\(19\)	0.362880	+	6.23041i	0.0832504	+	1.42935i	0.738659	+	0.674080i	\(0.235460\pi\)
							−0.655408	+	0.755275i	\(0.727503\pi\)
\(23\)	1.67377	−	4.33516i	0.349004	−	0.903944i	−0.641582	−	0.767055i	\(-0.721722\pi\)
							0.990586	−	0.136890i	\(-0.0437106\pi\)
\(29\)	−5.05124	−	3.61041i	−0.937991	−	0.670436i	0.00614886	−	0.999981i	\(-0.498043\pi\)
							−0.944140	+	0.329546i	\(0.893104\pi\)
\(31\)	−3.36645	+	3.85752i	−0.604632	+	0.692831i	−0.971391	−	0.237485i	\(-0.923677\pi\)
							0.366760	+	0.930316i	\(0.380467\pi\)
\(37\)	−3.97044	−	5.33323i	−0.652737	−	0.876778i	0.345358	−	0.938471i	\(-0.387757\pi\)
							−0.998095	+	0.0616933i	\(0.980350\pi\)
\(41\)	1.40191	−	5.44257i	0.218942	−	0.849986i	−0.760678	−	0.649129i	\(-0.775133\pi\)
							0.979621	−	0.200857i	\(-0.0643727\pi\)
\(43\)	3.72490	+	3.38017i	0.568042	+	0.515471i	0.904639	−	0.426178i	\(-0.140140\pi\)
							−0.336597	+	0.941649i	\(0.609276\pi\)
\(47\)	−3.71155	−	4.25297i	−0.541386	−	0.620359i	0.415932	−	0.909396i	\(-0.363455\pi\)
							−0.957318	+	0.289036i	\(0.906665\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.103.19	✓	1404
3.2	odd	2		729.2.i.a.685.8		1404
243.92	odd	162		729.2.i.a.613.8		1404
243.151	even	81	inner	243.2.i.a.151.19	yes	1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.103.19	✓	1404	1.1	even	1	trivial
243.2.i.a.151.19	yes	1404	243.151	even	81	inner
729.2.i.a.613.8		1404	243.92	odd	162
729.2.i.a.685.8		1404	3.2	odd	2