Embedded newform 27.8.e.a.4.14

• Label: 27.8.e.a.4.14
• 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			4.14
Character	\(\chi\)	\(=\)	27.4
Dual form			27.8.e.a.7.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.46251 + 2.71613i) q^{2} +(42.7022 + 19.0664i) q^{3} +(-49.7420 - 41.7385i) q^{4} +(71.4552 - 405.243i) q^{5} +(266.878 + 258.268i) q^{6} +(898.235 - 753.709i) q^{7} +(-766.085 - 1326.90i) q^{8} +(1459.95 + 1628.35i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.46251	+	2.71613i	0.659599	+	0.240074i	0.650063	−	0.759880i	\(-0.274742\pi\)
							0.00953567	+	0.999955i	\(0.496965\pi\)
\(3\)	42.7022	+	19.0664i	0.913115	+	0.407703i
							\(5\)	71.4552	−	405.243i	0.255646	−	1.44984i	−0.538763	−	0.842457i	\(-0.681108\pi\)
0.794409	−	0.607383i	\(-0.207781\pi\)
\(7\)	898.235	−	753.709i	0.989799	−	0.830540i	0.00426019	−	0.999991i	\(-0.498644\pi\)
							0.985539	+	0.169451i	\(0.0541995\pi\)
\(11\)	1289.55	+	7313.38i	0.292121	+	1.65670i	0.678679	+	0.734435i	\(0.262553\pi\)
							−0.386558	+	0.922265i	\(0.626336\pi\)
\(13\)	−3791.99	+	1380.17i	−0.478702	+	0.174233i	−0.570090	−	0.821582i	\(-0.693092\pi\)
							0.0913885	+	0.995815i	\(0.470869\pi\)
\(17\)	7380.03	−	12782.6i	0.364323	−	0.631026i	−0.624344	−	0.781149i	\(-0.714634\pi\)
							0.988667	+	0.150123i	\(0.0479670\pi\)
\(19\)	−1194.93	−	2069.69i	−0.0399674	−	0.0692256i	0.845350	−	0.534213i	\(-0.179392\pi\)
							−0.885317	+	0.464988i	\(0.846059\pi\)
\(23\)	−782.904	−	656.935i	−0.0134172	−	0.0112583i	0.636054	−	0.771644i	\(-0.280565\pi\)
							−0.649472	+	0.760386i	\(0.725010\pi\)
\(29\)	−127204.	−	46298.5i	−0.968519	−	0.352512i	−0.191153	−	0.981560i	\(-0.561223\pi\)
							−0.777366	+	0.629048i	\(0.783445\pi\)
\(31\)	123510.	+	103637.i	0.744623	+	0.624813i	0.934075	−	0.357077i	\(-0.116227\pi\)
							−0.189452	+	0.981890i	\(0.560671\pi\)
\(37\)	−120547.	+	208793.i	−0.391246	+	0.677658i	−0.992614	−	0.121314i	\(-0.961289\pi\)
							0.601368	+	0.798972i	\(0.294622\pi\)
\(41\)	154404.	−	56198.5i	0.349877	−	0.127345i	−0.161103	−	0.986938i	\(-0.551505\pi\)
							0.510980	+	0.859593i	\(0.329283\pi\)
\(43\)	177068.	+	1.00420e6i	0.339625	+	1.92611i	0.375655	+	0.926759i	\(0.377418\pi\)
							−0.0360301	+	0.999351i	\(0.511471\pi\)
\(47\)	797496.	−	669179.i	1.12043	−	0.940155i	0.121807	−	0.992554i	\(-0.461131\pi\)
							0.998626	+	0.0523986i	\(0.0166866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.4.14	✓	120
3.2	odd	2		81.8.e.a.64.7		120
27.7	even	9	inner	27.8.e.a.7.14	yes	120
27.20	odd	18		81.8.e.a.19.7		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.14	✓	120	1.1	even	1	trivial
27.8.e.a.7.14	yes	120	27.7	even	9	inner
81.8.e.a.19.7		120	27.20	odd	18
81.8.e.a.64.7		120	3.2	odd	2