Embedded newform 27.5.f.a.2.1

• Label: 27.5.f.a.2.1
• Level: $27$
• Weight: $5$
• Character: 27.2
• 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			2.1
Character	\(\chi\)	\(=\)	27.2
Dual form			27.5.f.a.14.1

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.2.1	✓	66	1.1	even	1	trivial
27.5.f.a.14.1	yes	66	27.14	odd	18	inner
81.5.f.a.8.11		66	3.2	odd	2
81.5.f.a.71.11		66	27.13	even	9