Embedded newform 243.2.i.a.103.20

• Label: 243.2.i.a.103.20
• 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			103.20
Character	\(\chi\)	\(=\)	243.103
Dual form			243.2.i.a.151.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.17597 + 0.327353i) q^{2} +(-0.565604 + 1.63710i) q^{3} +(-0.436594 - 0.263486i) q^{4} +(3.30824 - 0.128375i) q^{5} +(-1.20104 + 1.74002i) q^{6} +(3.02899 + 0.473725i) q^{7} +(-2.10253 - 2.22855i) q^{8} +(-2.36018 - 1.85190i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.17597	+	0.327353i	0.831535	+	0.231474i	0.657680	−	0.753298i	\(-0.271538\pi\)
							0.173855	+	0.984771i	\(0.444378\pi\)
\(3\)	−0.565604	+	1.63710i	−0.326552	+	0.945179i
							\(5\)	3.30824	−	0.128375i	1.47949	−	0.0574110i	0.713678	−	0.700474i	\(-0.247028\pi\)
0.765813	+	0.643063i	\(0.222337\pi\)
\(7\)	3.02899	+	0.473725i	1.14485	+	0.179051i	0.698367	−	0.715740i	\(-0.253911\pi\)
							0.446484	+	0.894792i	\(0.352676\pi\)
\(11\)	0.345643	+	2.53055i	0.104215	+	0.762991i	0.966518	+	0.256598i	\(0.0826016\pi\)
							−0.862303	+	0.506393i	\(0.830979\pi\)
\(13\)	−2.19930	+	3.20665i	−0.609975	+	0.889366i	−0.999499	−	0.0316506i	\(-0.989924\pi\)
							0.389524	+	0.921016i	\(0.372640\pi\)
\(17\)	−2.28530	+	1.14772i	−0.554267	+	0.278363i	−0.703806	−	0.710392i	\(-0.748517\pi\)
							0.149539	+	0.988756i	\(0.452221\pi\)
\(19\)	−0.0846874	−	1.45403i	−0.0194286	−	0.333577i	−0.993991	−	0.109461i	\(-0.965088\pi\)
							0.974563	−	0.224116i	\(-0.0719494\pi\)
\(23\)	1.73750	−	4.50023i	0.362293	−	0.938362i	−0.625167	−	0.780491i	\(-0.714969\pi\)
							0.987460	−	0.157871i	\(-0.0504630\pi\)
\(29\)	−4.11031	−	2.93787i	−0.763265	−	0.545549i	0.131911	−	0.991262i	\(-0.457889\pi\)
							−0.895176	+	0.445713i	\(0.852950\pi\)
\(31\)	5.18413	−	5.94035i	0.931097	−	1.06692i	−0.0665100	−	0.997786i	\(-0.521186\pi\)
							0.997607	−	0.0691335i	\(-0.0220234\pi\)
\(37\)	−5.96054	−	8.00640i	−0.979907	−	1.31624i	−0.948514	−	0.316736i	\(-0.897413\pi\)
							−0.0313935	−	0.999507i	\(-0.509994\pi\)
\(41\)	−1.91809	+	7.44647i	−0.299555	+	1.16294i	0.624606	+	0.780940i	\(0.285260\pi\)
							−0.924161	+	0.382003i	\(0.875234\pi\)
\(43\)	−8.37244	−	7.59758i	−1.27679	−	1.15862i	−0.978972	−	0.203997i	\(-0.934607\pi\)
							−0.297814	−	0.954624i	\(-0.596257\pi\)
\(47\)	4.91942	+	5.63703i	0.717572	+	0.822246i	0.990168	−	0.139883i	\(-0.0446725\pi\)
							−0.272596	+	0.962128i	\(0.587882\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.103.20	✓	1404
3.2	odd	2		729.2.i.a.685.7		1404
243.92	odd	162		729.2.i.a.613.7		1404
243.151	even	81	inner	243.2.i.a.151.20	yes	1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.103.20	✓	1404	1.1	even	1	trivial
243.2.i.a.151.20	yes	1404	243.151	even	81	inner
729.2.i.a.613.7		1404	243.92	odd	162
729.2.i.a.685.7		1404	3.2	odd	2