Embedded newform 243.2.i.a.103.10

• Label: 243.2.i.a.103.10
• 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			103.10
Character	\(\chi\)	\(=\)	243.103
Dual form			243.2.i.a.151.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04528 - 0.290973i) q^{2} +(1.59268 + 0.680716i) q^{3} +(-0.704391 - 0.425102i) q^{4} +(-2.06977 + 0.0803163i) q^{5} +(-1.46672 - 1.17497i) q^{6} +(-4.16283 - 0.651054i) q^{7} +(2.10177 + 2.22774i) q^{8} +(2.07325 + 2.16832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04528	−	0.290973i	−0.739124	−	0.205749i	−0.121906	−	0.992542i	\(-0.538901\pi\)
							−0.617218	+	0.786792i	\(0.711740\pi\)
\(3\)	1.59268	+	0.680716i	0.919534	+	0.393011i
							\(5\)	−2.06977	+	0.0803163i	−0.925627	+	0.0359186i	−0.497179	−	0.867648i	\(-0.665631\pi\)
−0.428448	+	0.903566i	\(0.640940\pi\)
\(7\)	−4.16283	−	0.651054i	−1.57340	−	0.246075i	−0.693224	−	0.720722i	\(-0.743810\pi\)
							−0.880177	+	0.474646i	\(0.842576\pi\)
\(11\)	0.767098	+	5.61615i	0.231289	+	1.69333i	0.630272	+	0.776374i	\(0.282943\pi\)
							−0.398983	+	0.916958i	\(0.630637\pi\)
\(13\)	−3.39021	+	4.94305i	−0.940275	+	1.37095i	−0.0116516	+	0.999932i	\(0.503709\pi\)
							−0.928624	+	0.371023i	\(0.879007\pi\)
\(17\)	1.18240	−	0.593821i	0.286773	−	0.144023i	−0.299598	−	0.954066i	\(-0.596852\pi\)
							0.586371	+	0.810043i	\(0.300556\pi\)
\(19\)	−0.383327	−	6.58146i	−0.0879412	−	1.50989i	−0.697175	−	0.716901i	\(-0.745560\pi\)
							0.609234	−	0.792991i	\(-0.291477\pi\)
\(23\)	−0.756746	+	1.96002i	−0.157792	+	0.408692i	−0.989124	−	0.147084i	\(-0.953011\pi\)
							0.831332	+	0.555777i	\(0.187579\pi\)
\(29\)	−3.30770	−	2.36420i	−0.614224	−	0.439021i	0.231356	−	0.972869i	\(-0.425684\pi\)
							−0.845580	+	0.533848i	\(0.820745\pi\)
\(31\)	0.0367254	−	0.0420827i	0.00659609	−	0.00755828i	−0.750128	−	0.661292i	\(-0.770008\pi\)
							0.756724	+	0.653734i	\(0.226798\pi\)
\(37\)	−2.13975	−	2.87418i	−0.351772	−	0.472512i	0.590655	−	0.806924i	\(-0.298870\pi\)
							−0.942427	+	0.334413i	\(0.891462\pi\)
\(41\)	−0.587714	+	2.28165i	−0.0917856	+	0.356333i	−0.997752	−	0.0670163i	\(-0.978652\pi\)
							0.905966	+	0.423350i	\(0.139146\pi\)
\(43\)	2.40574	+	2.18309i	0.366872	+	0.332919i	0.834501	−	0.551006i	\(-0.185756\pi\)
							−0.467629	+	0.883925i	\(0.654892\pi\)
\(47\)	0.143743	+	0.164711i	0.0209670	+	0.0240255i	0.763824	−	0.645425i	\(-0.223320\pi\)
							−0.742857	+	0.669451i	\(0.766530\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.103.10	✓	1404
3.2	odd	2		729.2.i.a.685.17		1404
243.92	odd	162		729.2.i.a.613.17		1404
243.151	even	81	inner	243.2.i.a.151.10	yes	1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.103.10	✓	1404	1.1	even	1	trivial
243.2.i.a.151.10	yes	1404	243.151	even	81	inner
729.2.i.a.613.17		1404	243.92	odd	162
729.2.i.a.685.17		1404	3.2	odd	2