Embedded newform 27.8.e.a.16.6

• Label: 27.8.e.a.16.6
• 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			16.6
Character	\(\chi\)	\(=\)	27.16
Dual form			27.8.e.a.22.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.13331 - 7.66376i) q^{2} +(41.8883 + 20.7935i) q^{3} +(2.45720 + 13.9355i) q^{4} +(132.857 + 48.3560i) q^{5} +(-223.222 - 510.935i) q^{6} +(-128.828 + 730.621i) q^{7} +(-678.696 + 1175.54i) q^{8} +(1322.26 + 1742.01i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.13331	−	7.66376i	−0.807278	−	0.677387i	0.142678	−	0.989769i	\(-0.454429\pi\)
							−0.949956	+	0.312382i	\(0.898873\pi\)
\(3\)	41.8883	+	20.7935i	0.895712	+	0.444635i
							\(5\)	132.857	+	48.3560i	0.475324	+	0.173004i	0.568563	−	0.822640i	\(-0.307500\pi\)
−0.0932386	+	0.995644i	\(0.529722\pi\)
\(7\)	−128.828	+	730.621i	−0.141961	+	0.805099i	0.827797	+	0.561028i	\(0.189594\pi\)
							−0.969757	+	0.244071i	\(0.921517\pi\)
\(11\)	3663.78	−	1333.51i	0.829955	−	0.302079i	0.108115	−	0.994138i	\(-0.465519\pi\)
							0.721840	+	0.692059i	\(0.243296\pi\)
\(13\)	6398.55	−	5369.02i	0.807755	−	0.677787i	−0.142316	−	0.989821i	\(-0.545455\pi\)
							0.950071	+	0.312034i	\(0.101010\pi\)
\(17\)	15420.3	+	26708.8i	0.761241	+	1.31851i	0.942211	+	0.335019i	\(0.108743\pi\)
							−0.180971	+	0.983489i	\(0.557924\pi\)
\(19\)	17641.7	−	30556.3i	0.590068	−	1.02203i	−0.404154	−	0.914691i	\(-0.632434\pi\)
							0.994223	−	0.107337i	\(-0.0342325\pi\)
\(23\)	11961.8	+	67838.7i	0.204998	+	1.16260i	0.897443	+	0.441130i	\(0.145422\pi\)
							−0.692445	+	0.721470i	\(0.743467\pi\)
\(29\)	−30681.6	−	25744.9i	−0.233606	−	0.196019i	0.518468	−	0.855097i	\(-0.326502\pi\)
							−0.752075	+	0.659078i	\(0.770947\pi\)
\(31\)	−8278.69	−	46950.8i	−0.0499109	−	0.283059i	0.949629	−	0.313375i	\(-0.101460\pi\)
							−0.999540	+	0.0303162i	\(0.990349\pi\)
\(37\)	40589.0	+	70302.1i	0.131735	+	0.228172i	0.924346	−	0.381556i	\(-0.124612\pi\)
							−0.792610	+	0.609729i	\(0.791278\pi\)
\(41\)	−186343.	+	156361.i	−0.422251	+	0.354310i	−0.829018	−	0.559221i	\(-0.811100\pi\)
							0.406768	+	0.913532i	\(0.366656\pi\)
\(43\)	−520138.	+	189315.i	−0.997652	+	0.363116i	−0.788679	−	0.614806i	\(-0.789234\pi\)
							−0.208974	+	0.977921i	\(0.567012\pi\)
\(47\)	110727.	−	627964.i	0.155565	−	0.882251i	−0.802703	−	0.596379i	\(-0.796606\pi\)
							0.958268	−	0.285872i	\(-0.0922834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.16.6	✓	120
3.2	odd	2		81.8.e.a.46.15		120
27.5	odd	18		81.8.e.a.37.15		120
27.22	even	9	inner	27.8.e.a.22.6	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.16.6	✓	120	1.1	even	1	trivial
27.8.e.a.22.6	yes	120	27.22	even	9	inner
81.8.e.a.37.15		120	27.5	odd	18
81.8.e.a.46.15		120	3.2	odd	2