Embedded newform 27.8.e.a.16.2

• Label: 27.8.e.a.16.2
• Level: $27$
• Weight: $8$
• Character: 27.16
• Analytic conductor: $8.434$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43439568807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			16.2
Character	\(\chi\)	\(=\)	27.16
Dual form			27.8.e.a.22.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.1817 - 11.8998i) q^{2} +(-3.59005 - 46.6274i) q^{3} +(37.2867 + 211.463i) q^{4} +(-47.2604 - 17.2014i) q^{5} +(-503.945 + 703.974i) q^{6} +(-206.697 + 1172.23i) q^{7} +(802.766 - 1390.43i) q^{8} +(-2161.22 + 334.789i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.1817	−	11.8998i	−1.25349	−	1.05181i	−0.996344	−	0.0854343i	\(-0.972772\pi\)
							−0.257150	−	0.966372i	\(-0.582783\pi\)
\(3\)	−3.59005	−	46.6274i	−0.0767672	−	0.997049i
							\(5\)	−47.2604	−	17.2014i	−0.169084	−	0.0615415i	0.256091	−	0.966653i	\(-0.417565\pi\)
−0.425175	+	0.905111i	\(0.639787\pi\)
\(7\)	−206.697	+	1172.23i	−0.227767	+	1.29173i	0.629558	+	0.776953i	\(0.283236\pi\)
							−0.857325	+	0.514776i	\(0.827875\pi\)
\(11\)	3140.49	−	1143.04i	0.711414	−	0.258934i	0.0391378	−	0.999234i	\(-0.487539\pi\)
							0.672276	+	0.740300i	\(0.265317\pi\)
\(13\)	1674.59	−	1405.15i	0.211401	−	0.177387i	−0.530939	−	0.847410i	\(-0.678161\pi\)
							0.742340	+	0.670024i	\(0.233716\pi\)
\(17\)	1028.25	+	1780.98i	0.0507607	+	0.0879201i	0.890289	−	0.455395i	\(-0.150502\pi\)
							−0.839529	+	0.543315i	\(0.817169\pi\)
\(19\)	−20475.8	+	35465.2i	−0.684864	+	1.18622i	0.288616	+	0.957445i	\(0.406805\pi\)
							−0.973480	+	0.228774i	\(0.926528\pi\)
\(23\)	−7712.48	−	43739.6i	−0.132174	−	0.749597i	−0.976786	−	0.214218i	\(-0.931280\pi\)
							0.844612	−	0.535379i	\(-0.179831\pi\)
\(29\)	187740.	+	157532.i	1.42943	+	1.19944i	0.946048	+	0.324026i	\(0.105037\pi\)
							0.483383	+	0.875409i	\(0.339408\pi\)
\(31\)	45015.1	+	255293.i	0.271389	+	1.53912i	0.750203	+	0.661207i	\(0.229956\pi\)
							−0.478815	+	0.877916i	\(0.658933\pi\)
\(37\)	112089.	+	194144.i	0.363796	+	0.630113i	0.988582	−	0.150683i	\(-0.0481472\pi\)
							−0.624786	+	0.780796i	\(0.714814\pi\)
\(41\)	−199081.	+	167049.i	−0.451114	+	0.378530i	−0.839849	−	0.542820i	\(-0.817357\pi\)
							0.388735	+	0.921350i	\(0.372912\pi\)
\(43\)	884528.	−	321942.i	1.69657	−	0.617501i	0.701144	−	0.713020i	\(-0.252673\pi\)
							0.995428	+	0.0955188i	\(0.0304510\pi\)
\(47\)	−147817.	+	838311.i	−0.207674	+	1.17778i	0.685503	+	0.728070i	\(0.259582\pi\)
							−0.893176	+	0.449706i	\(0.851529\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.16.2	✓	120
3.2	odd	2		81.8.e.a.46.19		120
27.5	odd	18		81.8.e.a.37.19		120
27.22	even	9	inner	27.8.e.a.22.2	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.16.2	✓	120	1.1	even	1	trivial
27.8.e.a.22.2	yes	120	27.22	even	9	inner
81.8.e.a.37.19		120	27.5	odd	18
81.8.e.a.46.19		120	3.2	odd	2