Embedded newform 27.8.e.a.16.9

• Label: 27.8.e.a.16.9
• 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			16.9
Character	\(\chi\)	\(=\)	27.16
Dual form			27.8.e.a.22.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.51200 - 2.94692i) q^{2} +(23.1027 - 40.6604i) q^{3} +(-18.5771 - 105.356i) q^{4} +(493.913 + 179.770i) q^{5} +(-200.959 + 74.7176i) q^{6} +(70.9526 - 402.392i) q^{7} +(-538.647 + 932.964i) q^{8} +(-1119.53 - 1878.73i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.51200	−	2.94692i	−0.310420	−	0.260473i	0.474246	−	0.880393i	\(-0.342721\pi\)
							−0.784666	+	0.619919i	\(0.787165\pi\)
\(3\)	23.1027	−	40.6604i	0.494012	−	0.869455i
							\(5\)	493.913	+	179.770i	1.76708	+	0.643163i	0.999999	+	0.00161009i	\(0.000512509\pi\)
0.767078	+	0.641553i	\(0.221710\pi\)
\(7\)	70.9526	−	402.392i	0.0781853	−	0.443411i	−0.920435	−	0.390896i	\(-0.872165\pi\)
							0.998620	−	0.0525148i	\(-0.0167237\pi\)
\(11\)	2185.90	−	795.602i	0.495172	−	0.180228i	−0.0823495	−	0.996604i	\(-0.526242\pi\)
							0.577521	+	0.816376i	\(0.304020\pi\)
\(13\)	1995.74	−	1674.63i	0.251943	−	0.211405i	−0.508065	−	0.861318i	\(-0.669639\pi\)
							0.760008	+	0.649913i	\(0.225195\pi\)
\(17\)	−19021.0	−	32945.3i	−0.938990	−	1.62638i	−0.767359	−	0.641218i	\(-0.778429\pi\)
							−0.171632	−	0.985161i	\(-0.554904\pi\)
\(19\)	−13174.4	+	22818.8i	−0.440650	+	0.763228i	−0.997738	−	0.0672255i	\(-0.978585\pi\)
							0.557088	+	0.830454i	\(0.311919\pi\)
\(23\)	11610.5	+	65846.5i	0.198977	+	1.12846i	0.906640	+	0.421906i	\(0.138639\pi\)
							−0.707662	+	0.706551i	\(0.750250\pi\)
\(29\)	45531.7	+	38205.7i	0.346674	+	0.290894i	0.799453	−	0.600729i	\(-0.205123\pi\)
							−0.452779	+	0.891623i	\(0.649567\pi\)
\(31\)	−25015.2	−	141868.i	−0.150813	−	0.855301i	−0.962515	−	0.271229i	\(-0.912570\pi\)
							0.811702	−	0.584072i	\(-0.198541\pi\)
\(37\)	198784.	+	344304.i	0.645172	+	1.11747i	0.984262	+	0.176716i	\(0.0565475\pi\)
							−0.339090	+	0.940754i	\(0.610119\pi\)
\(41\)	143096.	−	120071.i	0.324252	−	0.272080i	−0.466101	−	0.884732i	\(-0.654342\pi\)
							0.790353	+	0.612652i	\(0.209897\pi\)
\(43\)	−91637.9	+	33353.5i	−0.175766	+	0.0639737i	−0.428404	−	0.903587i	\(-0.640924\pi\)
							0.252638	+	0.967561i	\(0.418702\pi\)
\(47\)	62036.4	−	351826.i	0.0871574	−	0.494294i	−0.909713	−	0.415238i	\(-0.863698\pi\)
							0.996870	−	0.0790560i	\(-0.0251906\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.16.9	✓	120
3.2	odd	2		81.8.e.a.46.12		120
27.5	odd	18		81.8.e.a.37.12		120
27.22	even	9	inner	27.8.e.a.22.9	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.16.9	✓	120	1.1	even	1	trivial
27.8.e.a.22.9	yes	120	27.22	even	9	inner
81.8.e.a.37.12		120	27.5	odd	18
81.8.e.a.46.12		120	3.2	odd	2