Embedded newform 27.8.e.a.7.17

• Label: 27.8.e.a.7.17
• 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			7.17
Character	\(\chi\)	\(=\)	27.7
Dual form			27.8.e.a.4.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.3397 - 5.21922i) q^{2} +(-40.6699 + 23.0860i) q^{3} +(80.3327 - 67.4072i) q^{4} +(-74.8052 - 424.241i) q^{5} +(-462.703 + 543.310i) q^{6} +(-616.615 - 517.401i) q^{7} +(-176.505 + 305.716i) q^{8} +(1121.08 - 1877.81i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.3397	−	5.21922i	1.26746	−	0.461318i	0.381195	−	0.924495i	\(-0.375513\pi\)
							0.886266	+	0.463176i	\(0.153290\pi\)
\(3\)	−40.6699	+	23.0860i	−0.869658	+	0.493655i
							\(5\)	−74.8052	−	424.241i	−0.267631	−	1.51781i	−0.761437	−	0.648239i	\(-0.775506\pi\)
0.493805	−	0.869572i	\(-0.335606\pi\)
\(7\)	−616.615	−	517.401i	−0.679471	−	0.570144i	0.236381	−	0.971661i	\(-0.424039\pi\)
							−0.915852	+	0.401517i	\(0.868483\pi\)
\(11\)	808.205	−	4583.56i	0.183083	−	1.03831i	−0.745311	−	0.666717i	\(-0.767699\pi\)
							0.928394	−	0.371597i	\(-0.121190\pi\)
\(13\)	−2908.51	−	1058.61i	−0.367171	−	0.133639i	0.151844	−	0.988405i	\(-0.451479\pi\)
							−0.519015	+	0.854765i	\(0.673701\pi\)
\(17\)	2758.38	+	4777.66i	0.136171	+	0.235854i	0.926044	−	0.377416i	\(-0.123187\pi\)
							−0.789873	+	0.613270i	\(0.789854\pi\)
\(19\)	−15604.9	+	27028.5i	−0.521944	+	0.904033i	0.477730	+	0.878507i	\(0.341460\pi\)
							−0.999674	+	0.0255268i	\(0.991874\pi\)
\(23\)	60272.5	−	50574.6i	1.03293	−	0.866733i	0.0417354	−	0.999129i	\(-0.486711\pi\)
							0.991197	+	0.132396i	\(0.0422669\pi\)
\(29\)	86153.5	−	31357.3i	0.655964	−	0.238751i	0.00747109	−	0.999972i	\(-0.497622\pi\)
							0.648493	+	0.761221i	\(0.275400\pi\)
\(31\)	210318.	−	176478.i	1.26798	−	1.06396i	0.273192	−	0.961959i	\(-0.411921\pi\)
							0.994784	−	0.101999i	\(-0.0325239\pi\)
\(37\)	−114885.	−	198986.i	−0.372868	−	0.645827i	0.617137	−	0.786856i	\(-0.288292\pi\)
							−0.990006	+	0.141029i	\(0.954959\pi\)
\(41\)	−803134.	−	292317.i	−1.81989	−	0.662385i	−0.995321	−	0.0966205i	\(-0.969197\pi\)
							−0.824567	−	0.565765i	\(-0.808581\pi\)
\(43\)	70970.3	−	402493.i	0.136125	−	0.772002i	−0.837945	−	0.545755i	\(-0.816243\pi\)
							0.974070	−	0.226247i	\(-0.0726458\pi\)
\(47\)	603573.	+	506458.i	0.847983	+	0.711542i	0.959344	−	0.282238i	\(-0.0910767\pi\)
							−0.111362	+	0.993780i	\(0.535521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.7.17	yes	120
3.2	odd	2		81.8.e.a.19.4		120
27.4	even	9	inner	27.8.e.a.4.17	✓	120
27.23	odd	18		81.8.e.a.64.4		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.17	✓	120	27.4	even	9	inner
27.8.e.a.7.17	yes	120	1.1	even	1	trivial
81.8.e.a.19.4		120	3.2	odd	2
81.8.e.a.64.4		120	27.23	odd	18