Embedded newform 81.2.g.a.13.5

• Label: 81.2.g.a.13.5
• 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			13.5
Character	\(\chi\)	\(=\)	81.13
Dual form			81.2.g.a.25.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.507233 - 0.254742i) q^{2} +(1.73070 - 0.0683184i) q^{3} +(-1.00193 - 1.34582i) q^{4} +(-0.832798 - 2.78174i) q^{5} +(-0.895274 - 0.406230i) q^{6} +(1.23966 + 2.87385i) q^{7} +(0.362501 + 2.05585i) q^{8} +(2.99067 - 0.236478i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.507233	−	0.254742i	−0.358668	−	0.180130i	0.260335	−	0.965518i	\(-0.416167\pi\)
							−0.619003	+	0.785388i	\(0.712463\pi\)
\(3\)	1.73070	−	0.0683184i	0.999222	−	0.0394437i
							\(5\)	−0.832798	−	2.78174i	−0.372439	−	1.24403i	−0.914807	−	0.403891i	\(-0.867657\pi\)
0.542368	−	0.840141i	\(-0.317528\pi\)
\(7\)	1.23966	+	2.87385i	0.468546	+	1.08621i	0.974649	+	0.223739i	\(0.0718263\pi\)
							−0.506103	+	0.862473i	\(0.668914\pi\)
\(11\)	−3.33963	+	0.791507i	−1.00694	+	0.238648i	−0.700814	−	0.713344i	\(-0.747180\pi\)
							−0.306122	+	0.951992i	\(0.599032\pi\)
\(13\)	3.33176	+	2.19133i	0.924064	+	0.607766i	0.919869	−	0.392225i	\(-0.128294\pi\)
							0.00419461	+	0.999991i	\(0.498665\pi\)
\(17\)	0.878443	−	0.319727i	0.213054	−	0.0775452i	−0.233289	−	0.972408i	\(-0.574949\pi\)
							0.446342	+	0.894862i	\(0.352726\pi\)
\(19\)	−4.55845	−	1.65914i	−1.04578	−	0.380633i	−0.238711	−	0.971091i	\(-0.576725\pi\)
							−0.807068	+	0.590458i	\(0.798947\pi\)
\(23\)	−2.43678	+	5.64910i	−0.508104	+	1.17792i	0.450397	+	0.892828i	\(0.351283\pi\)
							−0.958501	+	0.285089i	\(0.907977\pi\)
\(29\)	−0.104644	−	1.79666i	−0.0194318	−	0.333631i	−0.993988	−	0.109488i	\(-0.965079\pi\)
							0.974556	−	0.224143i	\(-0.0719583\pi\)
\(31\)	−0.671302	+	0.0784640i	−0.120569	+	0.0140925i	−0.176164	−	0.984361i	\(-0.556369\pi\)
							0.0555942	+	0.998453i	\(0.482295\pi\)
\(37\)	−8.73079	−	7.32600i	−1.43533	−	1.20439i	−0.942475	−	0.334277i	\(-0.891508\pi\)
							−0.492858	−	0.870110i	\(-0.664048\pi\)
\(41\)	5.94501	−	2.98570i	0.928455	−	0.466288i	0.0807376	−	0.996735i	\(-0.474272\pi\)
							0.847718	+	0.530448i	\(0.177976\pi\)
\(43\)	1.53472	+	1.62671i	0.234042	+	0.248071i	0.833741	−	0.552155i	\(-0.186194\pi\)
							−0.599699	+	0.800226i	\(0.704713\pi\)
\(47\)	3.14822	+	0.367974i	0.459214	+	0.0536745i	0.342556	−	0.939498i	\(-0.388707\pi\)
							0.116659	+	0.993172i	\(0.462782\pi\)