Embedded newform 81.2.g.a.13.6

• Label: 81.2.g.a.13.6
• Level: $81$
• Weight: $2$
• Character: 81.13
• Analytic conductor: $0.647$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	81.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.646788256372\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(8\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			13.6
Character	\(\chi\)	\(=\)	81.13
Dual form			81.2.g.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.744407 + 0.373855i) q^{2} +(0.703902 + 1.58257i) q^{3} +(-0.779943 - 1.04765i) q^{4} +(0.345929 + 1.15548i) q^{5} +(-0.0676620 + 1.44123i) q^{6} +(-0.520803 - 1.20736i) q^{7} +(-0.478230 - 2.71217i) q^{8} +(-2.00904 + 2.22795i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.744407	+	0.373855i	0.526375	+	0.264356i	0.692081	−	0.721820i	\(-0.256694\pi\)
							−0.165706	+	0.986175i	\(0.552990\pi\)
\(3\)	0.703902	+	1.58257i	0.406398	+	0.913696i
							\(5\)	0.345929	+	1.15548i	0.154704	+	0.516747i	0.999842	−	0.0178015i	\(-0.00566669\pi\)
−0.845137	+	0.534549i	\(0.820482\pi\)
\(7\)	−0.520803	−	1.20736i	−0.196845	−	0.456338i	0.790993	−	0.611825i	\(-0.209565\pi\)
							−0.987838	+	0.155487i	\(0.950305\pi\)
\(11\)	2.11479	−	0.501215i	0.637634	−	0.151122i	0.100931	−	0.994893i	\(-0.467818\pi\)
							0.536703	+	0.843771i	\(0.319670\pi\)
\(13\)	−3.80561	−	2.50299i	−1.05549	−	0.694204i	−0.101738	−	0.994811i	\(-0.532440\pi\)
							−0.953748	+	0.300607i	\(0.902811\pi\)
\(17\)	3.54217	−	1.28924i	0.859102	−	0.312688i	0.125356	−	0.992112i	\(-0.459993\pi\)
							0.733746	+	0.679424i	\(0.237770\pi\)
\(19\)	−2.50517	−	0.911807i	−0.574725	−	0.209183i	0.0382728	−	0.999267i	\(-0.487814\pi\)
							−0.612998	+	0.790084i	\(0.710037\pi\)
\(23\)	−2.38967	+	5.53988i	−0.498281	+	1.15514i	0.464674	+	0.885482i	\(0.346171\pi\)
							−0.962955	+	0.269663i	\(0.913088\pi\)
\(29\)	0.241756	+	4.15079i	0.0448929	+	0.770782i	0.943366	+	0.331753i	\(0.107640\pi\)
							−0.898473	+	0.439028i	\(0.855323\pi\)
\(31\)	−7.40674	+	0.865723i	−1.33029	+	0.155488i	−0.751362	−	0.659891i	\(-0.770603\pi\)
							−0.578927	+	0.815379i	\(0.696529\pi\)
\(37\)	7.47819	+	6.27494i	1.22941	+	1.03159i	0.998277	+	0.0586711i	\(0.0186863\pi\)
							0.231129	+	0.972923i	\(0.425758\pi\)
\(41\)	5.17242	−	2.59769i	0.807797	−	0.405691i	0.00355548	−	0.999994i	\(-0.498868\pi\)
							0.804241	+	0.594303i	\(0.202572\pi\)
\(43\)	3.46904	+	3.67697i	0.529024	+	0.560732i	0.935680	−	0.352849i	\(-0.114787\pi\)
							−0.406657	+	0.913581i	\(0.633305\pi\)
\(47\)	2.97767	+	0.348040i	0.434338	+	0.0507669i	0.330453	−	0.943822i	\(-0.392798\pi\)
							0.103885	+	0.994589i	\(0.466873\pi\)