Embedded newform 27.5.f.a.14.1

• Label: 27.5.f.a.14.1
• Level: $27$
• Weight: $5$
• Character: 27.14
• 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			14.1
Character	\(\chi\)	\(=\)	27.14
Dual form			27.5.f.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.11993 + 1.25544i) q^{2} +(6.30423 + 6.42314i) q^{3} +(34.0822 - 12.4049i) q^{4} +(-16.3502 + 19.4854i) q^{5} +(-52.9495 - 37.8177i) q^{6} +(-60.7132 - 22.0978i) q^{7} +(-126.911 + 73.2722i) q^{8} +(-1.51338 + 80.9859i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.11993	+	1.25544i	−1.77998	+	0.313859i	−0.964334	−	0.264688i	\(-0.914731\pi\)
							−0.815649	+	0.578547i	\(0.803620\pi\)
\(3\)	6.30423	+	6.42314i	0.700470	+	0.713682i
							\(5\)	−16.3502	+	19.4854i	−0.654007	+	0.779416i	−0.986513	−	0.163686i	\(-0.947662\pi\)
0.332505	+	0.943101i	\(0.392106\pi\)
\(7\)	−60.7132	−	22.0978i	−1.23905	−	0.450976i	−0.362359	−	0.932038i	\(-0.618029\pi\)
							−0.876686	+	0.481063i	\(0.840251\pi\)
\(11\)	−86.4395	−	103.015i	−0.714376	−	0.851360i	0.279695	−	0.960089i	\(-0.409766\pi\)
							−0.994071	+	0.108729i	\(0.965322\pi\)
\(13\)	−27.2976	+	154.812i	−0.161524	+	0.916048i	0.791052	+	0.611748i	\(0.209534\pi\)
							−0.952576	+	0.304300i	\(0.901578\pi\)
\(17\)	−124.536	−	71.9009i	−0.430920	−	0.248792i	0.268818	−	0.963191i	\(-0.413367\pi\)
							−0.699739	+	0.714399i	\(0.746700\pi\)
\(19\)	79.9738	+	138.519i	0.221534	+	0.383708i	0.955274	−	0.295722i	\(-0.0955603\pi\)
							−0.733740	+	0.679430i	\(0.762227\pi\)
\(23\)	193.010	+	530.290i	0.364857	+	1.00244i	0.977288	+	0.211914i	\(0.0679695\pi\)
							−0.612431	+	0.790524i	\(0.709808\pi\)
\(29\)	929.364	−	163.872i	1.10507	−	0.194854i	0.408793	−	0.912627i	\(-0.365950\pi\)
							0.696277	+	0.717773i	\(0.254839\pi\)
\(31\)	298.389	−	108.605i	0.310498	−	0.113012i	−0.182071	−	0.983285i	\(-0.558280\pi\)
							0.492569	+	0.870273i	\(0.336058\pi\)
\(37\)	−1074.59	+	1861.25i	−0.784946	+	1.35957i	0.144085	+	0.989565i	\(0.453976\pi\)
							−0.929031	+	0.370001i	\(0.879357\pi\)
\(41\)	−162.159	−	28.5930i	−0.0964658	−	0.0170095i	0.125207	−	0.992131i	\(-0.460041\pi\)
							−0.221673	+	0.975121i	\(0.571152\pi\)
\(43\)	795.721	−	667.689i	0.430352	−	0.361108i	−0.401732	−	0.915757i	\(-0.631592\pi\)
							0.832085	+	0.554649i	\(0.187147\pi\)
\(47\)	−879.993	+	2417.76i	−0.398367	+	1.09450i	0.564712	+	0.825288i	\(0.308987\pi\)
							−0.963080	+	0.269217i	\(0.913235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.f.a.14.1	yes	66
3.2	odd	2		81.5.f.a.71.11		66
27.2	odd	18	inner	27.5.f.a.2.1	✓	66
27.25	even	9		81.5.f.a.8.11		66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.2.1	✓	66	27.2	odd	18	inner
27.5.f.a.14.1	yes	66	1.1	even	1	trivial
81.5.f.a.8.11		66	27.25	even	9
81.5.f.a.71.11		66	3.2	odd	2