Embedded newform 27.8.e.a.7.3

• Label: 27.8.e.a.7.3
• Level: $27$
• Weight: $8$
• Character: 27.7
• 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			7.3
Character	\(\chi\)	\(=\)	27.7
Dual form			27.8.e.a.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.4594 + 5.99073i) q^{2} +(7.02376 - 46.2349i) q^{3} +(136.969 - 114.931i) q^{4} +(-40.0449 - 227.106i) q^{5} +(161.374 + 803.077i) q^{6} +(882.649 + 740.631i) q^{7} +(-444.907 + 770.602i) q^{8} +(-2088.33 - 649.486i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.4594	+	5.99073i	−1.45482	+	0.529511i	−0.943933	−	0.330137i	\(-0.892905\pi\)
							−0.510887	+	0.859648i	\(0.670683\pi\)
\(3\)	7.02376	−	46.2349i	0.150191	−	0.988657i
							\(5\)	−40.0449	−	227.106i	−0.143269	−	0.812520i	−0.968741	−	0.248076i	\(-0.920202\pi\)
0.825472	−	0.564444i	\(-0.190909\pi\)
\(7\)	882.649	+	740.631i	0.972624	+	0.816129i	0.982960	−	0.183818i	\(-0.0588457\pi\)
							−0.0103360	+	0.999947i	\(0.503290\pi\)
\(11\)	647.465	−	3671.95i	0.146670	−	0.831808i	−0.819341	−	0.573307i	\(-0.805660\pi\)
							0.966011	−	0.258501i	\(-0.0832285\pi\)
\(13\)	−7567.07	−	2754.19i	−0.955270	−	0.347690i	−0.183092	−	0.983096i	\(-0.558610\pi\)
							−0.772178	+	0.635406i	\(0.780833\pi\)
\(17\)	−10746.9	−	18614.2i	−0.530532	−	0.918909i	−0.999365	−	0.0356221i	\(-0.988659\pi\)
							0.468833	−	0.883287i	\(-0.344675\pi\)
\(19\)	−5562.25	+	9634.10i	−0.186043	+	0.322236i	−0.943927	−	0.330153i	\(-0.892900\pi\)
							0.757885	+	0.652389i	\(0.226233\pi\)
\(23\)	−79672.0	+	66852.8i	−1.36540	+	1.14570i	−0.391121	+	0.920339i	\(0.627913\pi\)
							−0.974275	+	0.225364i	\(0.927643\pi\)
\(29\)	−132448.	+	48207.1i	−1.00844	+	0.367044i	−0.792835	−	0.609436i	\(-0.791396\pi\)
							−0.215609	+	0.976480i	\(0.569174\pi\)
\(31\)	−10109.4	+	8482.81i	−0.0609481	+	0.0511415i	−0.672753	−	0.739867i	\(-0.734888\pi\)
							0.611805	+	0.791009i	\(0.290444\pi\)
\(37\)	−203213.	−	351975.i	−0.659547	−	1.14237i	−0.980733	−	0.195353i	\(-0.937415\pi\)
							0.321186	−	0.947016i	\(-0.395919\pi\)
\(41\)	−121630.	−	44269.8i	−0.275612	−	0.100315i	0.200516	−	0.979690i	\(-0.435738\pi\)
							−0.476129	+	0.879376i	\(0.657960\pi\)
\(43\)	6197.05	−	35145.2i	0.0118863	−	0.0674104i	−0.978288	−	0.207252i	\(-0.933548\pi\)
							0.990174	+	0.139841i	\(0.0446592\pi\)
\(47\)	814928.	+	683806.i	1.14492	+	0.960705i	0.999589	−	0.0286771i	\(-0.00912944\pi\)
							0.145335	+	0.989382i	\(0.453574\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.7.3	yes	120
3.2	odd	2		81.8.e.a.19.18		120
27.4	even	9	inner	27.8.e.a.4.3	✓	120
27.23	odd	18		81.8.e.a.64.18		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.3	✓	120	27.4	even	9	inner
27.8.e.a.7.3	yes	120	1.1	even	1	trivial
81.8.e.a.19.18		120	3.2	odd	2
81.8.e.a.64.18		120	27.23	odd	18