Embedded newform 27.5.f.a.5.6

• Label: 27.5.f.a.5.6
• Level: $27$
• Weight: $5$
• Character: 27.5
• Analytic conductor: $2.791$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79098900326\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	27.5
Dual form			27.5.f.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.712681 + 0.849340i) q^{2} +(5.75088 + 6.92296i) q^{3} +(2.56491 - 14.5463i) q^{4} +(9.60722 + 26.3956i) q^{5} +(-1.78140 + 9.81832i) q^{6} +(2.66962 + 15.1401i) q^{7} +(29.5458 - 17.0583i) q^{8} +(-14.8546 + 79.6262i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.712681	+	0.849340i	0.178170	+	0.212335i	0.847737	−	0.530417i	\(-0.177965\pi\)
							−0.669567	+	0.742752i	\(0.733520\pi\)
\(3\)	5.75088	+	6.92296i	0.638987	+	0.769217i
							\(5\)	9.60722	+	26.3956i	0.384289	+	1.05582i	0.969532	+	0.244966i	\(0.0787769\pi\)
−0.585243	+	0.810858i	\(0.699001\pi\)
\(7\)	2.66962	+	15.1401i	0.0544820	+	0.308983i	0.999855	−	0.0170069i	\(-0.00541372\pi\)
							−0.945373	+	0.325989i	\(0.894303\pi\)
\(11\)	47.3205	−	130.012i	0.391078	−	1.07448i	−0.575431	−	0.817850i	\(-0.695166\pi\)
							0.966510	−	0.256629i	\(-0.0826119\pi\)
\(13\)	−143.299	−	120.242i	−0.847922	−	0.711491i	0.111409	−	0.993775i	\(-0.464464\pi\)
							−0.959331	+	0.282283i	\(0.908908\pi\)
\(17\)	−115.641	−	66.7656i	−0.400143	−	0.231023i	0.286403	−	0.958109i	\(-0.407541\pi\)
							−0.686546	+	0.727087i	\(0.740874\pi\)
\(19\)	−188.124	−	325.840i	−0.521119	−	0.902605i	−0.999698	−	0.0245603i	\(-0.992181\pi\)
							0.478579	−	0.878044i	\(-0.341152\pi\)
\(23\)	428.991	+	75.6427i	0.810947	+	0.142992i	0.563720	−	0.825966i	\(-0.309370\pi\)
							0.247227	+	0.968958i	\(0.420481\pi\)
\(29\)	442.518	+	527.372i	0.526181	+	0.627078i	0.962031	−	0.272942i	\(-0.0879966\pi\)
							−0.435850	+	0.900019i	\(0.643552\pi\)
\(31\)	−298.888	+	1695.08i	−0.311017	+	1.76387i	0.282714	+	0.959204i	\(0.408765\pi\)
							−0.593731	+	0.804663i	\(0.702346\pi\)
\(37\)	1033.37	−	1789.85i	0.754837	−	1.30742i	−0.190619	−	0.981664i	\(-0.561049\pi\)
							0.945455	−	0.325752i	\(-0.105617\pi\)
\(41\)	−2103.88	+	2507.30i	−1.25156	+	1.49156i	−0.450739	+	0.892656i	\(0.648839\pi\)
							−0.800824	+	0.598899i	\(0.795605\pi\)
\(43\)	−1201.06	−	437.152i	−0.649575	−	0.236426i	−0.00384622	−	0.999993i	\(-0.501224\pi\)
							−0.645729	+	0.763566i	\(0.723447\pi\)
\(47\)	21.5344	−	3.79709i	0.00974846	−	0.00171892i	−0.168772	−	0.985655i	\(-0.553980\pi\)
							0.178520	+	0.983936i	\(0.442869\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.f.a.5.6	✓	66
3.2	odd	2		81.5.f.a.44.6		66
27.11	odd	18	inner	27.5.f.a.11.6	yes	66
27.16	even	9		81.5.f.a.35.6		66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.5.6	✓	66	1.1	even	1	trivial
27.5.f.a.11.6	yes	66	27.11	odd	18	inner
81.5.f.a.35.6		66	27.16	even	9
81.5.f.a.44.6		66	3.2	odd	2