Embedded newform 27.5.f.a.11.6

• Label: 27.5.f.a.11.6
• Level: $27$
• Weight: $5$
• Character: 27.11
• 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			11.6
Character	\(\chi\)	\(=\)	27.11
Dual form			27.5.f.a.5.6

$q$-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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.5.6	✓	66	27.5	odd	18	inner
27.5.f.a.11.6	yes	66	1.1	even	1	trivial
81.5.f.a.35.6		66	3.2	odd	2
81.5.f.a.44.6		66	27.22	even	9