Embedded newform 243.2.i.a.103.13

• Label: 243.2.i.a.103.13
• 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			103.13
Character	\(\chi\)	\(=\)	243.103
Dual form			243.2.i.a.151.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.129177 - 0.0359588i) q^{2} +(-1.73116 - 0.0555705i) q^{3} +(-1.69694 - 1.02411i) q^{4} +(-2.51593 + 0.0976297i) q^{5} +(0.221627 + 0.0694288i) q^{6} +(4.08453 + 0.638809i) q^{7} +(0.366413 + 0.388376i) q^{8} +(2.99382 + 0.192403i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.129177	−	0.0359588i	−0.0913418	−	0.0254267i	0.222200	−	0.975001i	\(-0.428676\pi\)
							−0.313542	+	0.949574i	\(0.601516\pi\)
\(3\)	−1.73116	−	0.0555705i	−0.999485	−	0.0320837i
							\(5\)	−2.51593	+	0.0976297i	−1.12516	+	0.0436613i	−0.594626	−	0.804002i	\(-0.702700\pi\)
−0.530534	+	0.847664i	\(0.678009\pi\)
\(7\)	4.08453	+	0.638809i	1.54381	+	0.241447i	0.868350	−	0.495951i	\(-0.165181\pi\)
							0.675458	+	0.737399i	\(0.263946\pi\)
\(11\)	0.654096	+	4.78883i	0.197217	+	1.44389i	0.777198	+	0.629256i	\(0.216640\pi\)
							−0.579981	+	0.814630i	\(0.696940\pi\)
\(13\)	−0.189852	+	0.276812i	−0.0526556	+	0.0767737i	−0.850109	−	0.526607i	\(-0.823464\pi\)
							0.797453	+	0.603381i	\(0.206180\pi\)
\(17\)	1.56801	−	0.787482i	0.380297	−	0.190993i	−0.248373	−	0.968664i	\(-0.579896\pi\)
							0.628670	+	0.777672i	\(0.283600\pi\)
\(19\)	0.123676	+	2.12343i	0.0283732	+	0.487149i	0.982324	+	0.187187i	\(0.0599369\pi\)
							−0.953951	+	0.299962i	\(0.903026\pi\)
\(23\)	−0.620409	+	1.60690i	−0.129364	+	0.335062i	−0.982400	−	0.186791i	\(-0.940191\pi\)
							0.853035	+	0.521853i	\(0.174759\pi\)
\(29\)	−0.238461	−	0.170442i	−0.0442811	−	0.0316502i	0.559136	−	0.829076i	\(-0.311133\pi\)
							−0.603417	+	0.797426i	\(0.706195\pi\)
\(31\)	5.27798	−	6.04789i	0.947953	−	1.08623i	−0.0482206	−	0.998837i	\(-0.515355\pi\)
							0.996174	−	0.0873969i	\(-0.0278548\pi\)
\(37\)	6.18920	+	8.31354i	1.01750	+	1.36674i	0.928760	+	0.370682i	\(0.120876\pi\)
							0.0887385	+	0.996055i	\(0.471716\pi\)
\(41\)	−1.90818	+	7.40802i	−0.298008	+	1.15694i	0.627625	+	0.778516i	\(0.284027\pi\)
							−0.925633	+	0.378422i	\(0.876467\pi\)
\(43\)	−0.684120	−	0.620806i	−0.104327	−	0.0946720i	0.618175	−	0.786041i	\(-0.287872\pi\)
							−0.722502	+	0.691369i	\(0.757008\pi\)
\(47\)	4.38427	+	5.02381i	0.639511	+	0.732799i	0.978100	−	0.208136i	\(-0.0667396\pi\)
							−0.338589	+	0.940935i	\(0.609949\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.103.13	✓	1404
3.2	odd	2		729.2.i.a.685.14		1404
243.92	odd	162		729.2.i.a.613.14		1404
243.151	even	81	inner	243.2.i.a.151.13	yes	1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.103.13	✓	1404	1.1	even	1	trivial
243.2.i.a.151.13	yes	1404	243.151	even	81	inner
729.2.i.a.613.14		1404	243.92	odd	162
729.2.i.a.685.14		1404	3.2	odd	2