Embedded newform 27.8.e.a.4.12

• Label: 27.8.e.a.4.12
• Level: $27$
• Weight: $8$
• Character: 27.4
• 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			4.12
Character	\(\chi\)	\(=\)	27.4
Dual form			27.8.e.a.7.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.29826 + 1.56444i) q^{2} +(11.1293 + 45.4218i) q^{3} +(-82.0261 - 68.8281i) q^{4} +(-20.3422 + 115.366i) q^{5} +(-23.2230 + 212.646i) q^{6} +(-610.300 + 512.103i) q^{7} +(-537.635 - 931.211i) q^{8} +(-1939.28 + 1011.02i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.29826	+	1.56444i	0.379916	+	0.138278i	0.524917	−	0.851153i	\(-0.324096\pi\)
							−0.145001	+	0.989431i	\(0.546319\pi\)
\(3\)	11.1293	+	45.4218i	0.237981	+	0.971270i
							\(5\)	−20.3422	+	115.366i	−0.0727786	+	0.412748i	0.926552	+	0.376166i	\(0.122758\pi\)
−0.999331	+	0.0365813i	\(0.988353\pi\)
\(7\)	−610.300	+	512.103i	−0.672512	+	0.564305i	−0.913808	−	0.406147i	\(-0.866872\pi\)
							0.241295	+	0.970452i	\(0.422428\pi\)
\(11\)	−574.721	−	3259.40i	−0.130192	−	0.738353i	−0.978088	−	0.208191i	\(-0.933242\pi\)
							0.847897	−	0.530162i	\(-0.177869\pi\)
\(13\)	−6697.04	+	2437.52i	−0.845437	+	0.307714i	−0.728179	−	0.685387i	\(-0.759633\pi\)
							−0.117259	+	0.993101i	\(0.537411\pi\)
\(17\)	−11610.6	+	20110.2i	−0.573172	+	0.992764i	0.423065	+	0.906099i	\(0.360954\pi\)
							−0.996238	+	0.0866645i	\(0.972379\pi\)
\(19\)	−4586.04	−	7943.25i	−0.153391	−	0.265681i	0.779081	−	0.626923i	\(-0.215686\pi\)
							−0.932472	+	0.361242i	\(0.882353\pi\)
\(23\)	33136.2	+	27804.5i	0.567878	+	0.476506i	0.880941	−	0.473226i	\(-0.156911\pi\)
							−0.313063	+	0.949732i	\(0.601355\pi\)
\(29\)	10873.9	+	3957.76i	0.0827925	+	0.0301340i	0.383084	−	0.923713i	\(-0.374862\pi\)
							−0.300292	+	0.953847i	\(0.597084\pi\)
\(31\)	191932.	+	161050.i	1.15713	+	0.970948i	0.999862	−	0.0166083i	\(-0.00528682\pi\)
							0.157268	+	0.987556i	\(0.449731\pi\)
\(37\)	168615.	−	292051.i	0.547257	−	0.947877i	−0.451204	−	0.892421i	\(-0.649005\pi\)
							0.998461	−	0.0554563i	\(-0.0176613\pi\)
\(41\)	−225331.	+	82013.7i	−0.510596	+	0.185842i	−0.584454	−	0.811427i	\(-0.698691\pi\)
							0.0738579	+	0.997269i	\(0.476469\pi\)
\(43\)	−14599.4	−	82797.6i	−0.0280025	−	0.158810i	0.967600	−	0.252488i	\(-0.0812488\pi\)
							−0.995603	+	0.0936778i	\(0.970138\pi\)
\(47\)	−867282.	+	727736.i	−1.21848	+	1.02242i	−0.219575	+	0.975596i	\(0.570467\pi\)
							−0.998903	+	0.0468293i	\(0.985088\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.4.12	✓	120
3.2	odd	2		81.8.e.a.64.9		120
27.7	even	9	inner	27.8.e.a.7.12	yes	120
27.20	odd	18		81.8.e.a.19.9		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.12	✓	120	1.1	even	1	trivial
27.8.e.a.7.12	yes	120	27.7	even	9	inner
81.8.e.a.19.9		120	27.20	odd	18
81.8.e.a.64.9		120	3.2	odd	2