Embedded newform 243.2.i.a.103.2

• Label: 243.2.i.a.103.2
• 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			103.2
Character	\(\chi\)	\(=\)	243.103
Dual form			243.2.i.a.151.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.50078 - 0.696141i) q^{2} +(1.62962 - 0.586805i) q^{3} +(4.05698 + 2.44840i) q^{4} +(0.716991 - 0.0278225i) q^{5} +(-4.48383 + 0.333029i) q^{6} +(2.72859 + 0.426743i) q^{7} +(-4.87841 - 5.17082i) q^{8} +(2.31132 - 1.91254i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.50078	−	0.696141i	−1.76832	−	0.492246i	−0.777671	−	0.628671i	\(-0.783599\pi\)
							−0.990650	+	0.136425i	\(0.956439\pi\)
\(3\)	1.62962	−	0.586805i	0.940861	−	0.338792i
							\(5\)	0.716991	−	0.0278225i	0.320648	−	0.0124426i	0.122051	−	0.992524i	\(-0.461053\pi\)
0.198597	+	0.980081i	\(0.436362\pi\)
\(7\)	2.72859	+	0.426743i	1.03131	+	0.161294i	0.647458	−	0.762101i	\(-0.275832\pi\)
							0.383851	+	0.923395i	\(0.374598\pi\)
\(11\)	0.742549	+	5.43642i	0.223887	+	1.63914i	0.668710	+	0.743523i	\(0.266847\pi\)
							−0.444823	+	0.895618i	\(0.646733\pi\)
\(13\)	−1.59335	+	2.32316i	−0.441915	+	0.644328i	−0.980101	−	0.198498i	\(-0.936394\pi\)
							0.538186	+	0.842826i	\(0.319110\pi\)
\(17\)	−1.84965	+	0.928931i	−0.448607	+	0.225299i	−0.658735	−	0.752375i	\(-0.728908\pi\)
							0.210128	+	0.977674i	\(0.432612\pi\)
\(19\)	0.137773	+	2.36548i	0.0316074	+	0.542678i	0.976530	+	0.215383i	\(0.0690999\pi\)
							−0.944922	+	0.327295i	\(0.893863\pi\)
\(23\)	2.77393	−	7.18467i	0.578405	−	1.49811i	−0.266846	−	0.963739i	\(-0.585982\pi\)
							0.845252	−	0.534368i	\(-0.179450\pi\)
\(29\)	2.18645	+	1.56278i	0.406014	+	0.290202i	0.766105	−	0.642716i	\(-0.222192\pi\)
							−0.360090	+	0.932917i	\(0.617254\pi\)
\(31\)	4.19476	−	4.80666i	0.753401	−	0.863301i	−0.240830	−	0.970567i	\(-0.577420\pi\)
							0.994230	+	0.107266i	\(0.0342096\pi\)
\(37\)	−3.11994	−	4.19081i	−0.512916	−	0.688965i	0.467927	−	0.883767i	\(-0.345001\pi\)
							−0.980843	+	0.194802i	\(0.937594\pi\)
\(41\)	1.92690	−	7.48070i	0.300932	−	1.16829i	−0.621903	−	0.783094i	\(-0.713640\pi\)
							0.922835	−	0.385195i	\(-0.125866\pi\)
\(43\)	−5.74986	−	5.21772i	−0.876846	−	0.795695i	0.103160	−	0.994665i	\(-0.467105\pi\)
							−0.980006	+	0.198970i	\(0.936240\pi\)
\(47\)	1.85151	+	2.12160i	0.270071	+	0.309467i	0.872411	−	0.488773i	\(-0.162555\pi\)
							−0.602341	+	0.798239i	\(0.705765\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.103.2	✓	1404
3.2	odd	2		729.2.i.a.685.25		1404
243.92	odd	162		729.2.i.a.613.25		1404
243.151	even	81	inner	243.2.i.a.151.2	yes	1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.103.2	✓	1404	1.1	even	1	trivial
243.2.i.a.151.2	yes	1404	243.151	even	81	inner
729.2.i.a.613.25		1404	243.92	odd	162
729.2.i.a.685.25		1404	3.2	odd	2