Embedded newform 243.2.i.a.103.6

• Label: 243.2.i.a.103.6
• 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			103.6
Character	\(\chi\)	\(=\)	243.103
Dual form			243.2.i.a.151.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.83047 - 0.509546i) q^{2} +(1.22146 + 1.22802i) q^{3} +(1.37865 + 0.832017i) q^{4} +(3.83386 - 0.148771i) q^{5} +(-1.61012 - 2.87024i) q^{6} +(0.365157 + 0.0571095i) q^{7} +(0.508193 + 0.538653i) q^{8} +(-0.0160588 + 2.99996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.83047	−	0.509546i	−1.29434	−	0.360303i	−0.448518	−	0.893774i	\(-0.648048\pi\)
							−0.845819	+	0.533470i	\(0.820888\pi\)
\(3\)	1.22146	+	1.22802i	0.705212	+	0.708997i
							\(5\)	3.83386	−	0.148771i	1.71456	−	0.0665325i	0.837356	−	0.546659i	\(-0.184100\pi\)
0.877199	+	0.480126i	\(0.159409\pi\)
\(7\)	0.365157	+	0.0571095i	0.138016	+	0.0215853i	0.223150	−	0.974784i	\(-0.428366\pi\)
							−0.0851333	+	0.996370i	\(0.527132\pi\)
\(11\)	−0.814808	−	5.96545i	−0.245674	−	1.79865i	−0.538648	−	0.842531i	\(-0.681065\pi\)
							0.292974	−	0.956120i	\(-0.405355\pi\)
\(13\)	−3.43162	+	5.00342i	−0.951759	+	1.38770i	−0.0301651	+	0.999545i	\(0.509603\pi\)
							−0.921594	+	0.388154i	\(0.873113\pi\)
\(17\)	3.18666	−	1.60040i	0.772878	−	0.388154i	−0.0182273	−	0.999834i	\(-0.505802\pi\)
							0.791105	+	0.611680i	\(0.209506\pi\)
\(19\)	0.159287	+	2.73485i	0.0365429	+	0.627417i	0.966009	+	0.258509i	\(0.0832312\pi\)
							−0.929466	+	0.368908i	\(0.879732\pi\)
\(23\)	−0.230565	+	0.597178i	−0.0480761	+	0.124520i	−0.954842	−	0.297115i	\(-0.903976\pi\)
							0.906766	+	0.421635i	\(0.138544\pi\)
\(29\)	0.792323	+	0.566318i	0.147131	+	0.105163i	0.652675	−	0.757638i	\(-0.273647\pi\)
							−0.505545	+	0.862801i	\(0.668708\pi\)
\(31\)	3.51822	−	4.03144i	0.631891	−	0.724067i	−0.344809	−	0.938673i	\(-0.612056\pi\)
							0.976701	+	0.214605i	\(0.0688466\pi\)
\(37\)	−3.14819	−	4.22875i	−0.517559	−	0.695202i	0.464114	−	0.885775i	\(-0.346373\pi\)
							−0.981673	+	0.190574i	\(0.938965\pi\)
\(41\)	−0.857298	+	3.32824i	−0.133887	+	0.519783i	0.865920	+	0.500182i	\(0.166733\pi\)
							−0.999808	+	0.0196012i	\(0.993760\pi\)
\(43\)	3.41495	+	3.09891i	0.520776	+	0.472579i	0.889484	−	0.456966i	\(-0.151064\pi\)
							−0.368708	+	0.929545i	\(0.620200\pi\)
\(47\)	−2.84239	−	3.25702i	−0.414605	−	0.475085i	0.507707	−	0.861530i	\(-0.330493\pi\)
							−0.922313	+	0.386445i	\(0.873703\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.103.6	✓	1404
3.2	odd	2		729.2.i.a.685.21		1404
243.92	odd	162		729.2.i.a.613.21		1404
243.151	even	81	inner	243.2.i.a.151.6	yes	1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.103.6	✓	1404	1.1	even	1	trivial
243.2.i.a.151.6	yes	1404	243.151	even	81	inner
729.2.i.a.613.21		1404	243.92	odd	162
729.2.i.a.685.21		1404	3.2	odd	2