Embedded newform 27.8.e.a.22.17

• Label: 27.8.e.a.22.17
• Level: $27$
• Weight: $8$
• Character: 27.22
• 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			22.17
Character	\(\chi\)	\(=\)	27.22
Dual form			27.8.e.a.16.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(13.0699 - 10.9670i) q^{2} +(-46.6522 + 3.25098i) q^{3} +(28.3216 - 160.620i) q^{4} +(-370.391 + 134.811i) q^{5} +(-574.088 + 554.124i) q^{6} +(150.101 + 851.264i) q^{7} +(-299.413 - 518.598i) q^{8} +(2165.86 - 303.331i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.0699	−	10.9670i	1.15523	−	0.969353i	0.155401	−	0.987851i	\(-0.450333\pi\)
							0.999829	+	0.0184983i	\(0.00588853\pi\)
\(3\)	−46.6522	+	3.25098i	−0.997581	+	0.0695169i
							\(5\)	−370.391	+	134.811i	−1.32515	+	0.482315i	−0.905105	−	0.425188i	\(-0.860208\pi\)
−0.420045	+	0.907503i	\(0.637986\pi\)
\(7\)	150.101	+	851.264i	0.165402	+	0.938039i	0.948649	+	0.316330i	\(0.102451\pi\)
							−0.783248	+	0.621710i	\(0.786438\pi\)
\(11\)	−5773.92	−	2101.54i	−1.30797	−	0.476061i	−0.408384	−	0.912810i	\(-0.633907\pi\)
							−0.899583	+	0.436749i	\(0.856130\pi\)
\(13\)	−8046.32	−	6751.66i	−1.01577	−	0.852332i	−0.0266799	−	0.999644i	\(-0.508493\pi\)
							−0.989090	+	0.147312i	\(0.952938\pi\)
\(17\)	−11834.6	+	20498.1i	−0.584228	+	1.01191i	0.410743	+	0.911751i	\(0.365269\pi\)
							−0.994971	+	0.100162i	\(0.968064\pi\)
\(19\)	−15551.2	−	26935.4i	−0.520146	−	0.900919i	−0.999726	−	0.0234206i	\(-0.992544\pi\)
							0.479580	−	0.877498i	\(-0.340789\pi\)
\(23\)	−8075.27	+	45797.1i	−0.138392	+	0.784858i	0.834046	+	0.551695i	\(0.186019\pi\)
							−0.972438	+	0.233163i	\(0.925092\pi\)
\(29\)	49489.5	−	41526.6i	0.376808	−	0.316180i	−0.434640	−	0.900604i	\(-0.643124\pi\)
							0.811448	+	0.584425i	\(0.198680\pi\)
\(31\)	28692.7	−	162724.i	0.172984	−	0.981040i	−0.767461	−	0.641095i	\(-0.778480\pi\)
							0.940445	−	0.339945i	\(-0.110409\pi\)
\(37\)	−92960.3	+	161012.i	−0.301711	+	0.522579i	−0.976524	−	0.215410i	\(-0.930891\pi\)
							0.674812	+	0.737989i	\(0.264224\pi\)
\(41\)	−101325.	−	85021.8i	−0.229601	−	0.192658i	0.520728	−	0.853722i	\(-0.325660\pi\)
							−0.750329	+	0.661065i	\(0.770105\pi\)
\(43\)	514578.	+	187291.i	0.986987	+	0.359234i	0.784553	−	0.620062i	\(-0.212893\pi\)
							0.202434	+	0.979296i	\(0.435115\pi\)
\(47\)	106272.	+	602700.i	0.149306	+	0.846757i	0.963808	+	0.266596i	\(0.0858990\pi\)
							−0.814502	+	0.580160i	\(0.802990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.22.17	yes	120
3.2	odd	2		81.8.e.a.37.4		120
27.11	odd	18		81.8.e.a.46.4		120
27.16	even	9	inner	27.8.e.a.16.17	✓	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.16.17	✓	120	27.16	even	9	inner
27.8.e.a.22.17	yes	120	1.1	even	1	trivial
81.8.e.a.37.4		120	3.2	odd	2
81.8.e.a.46.4		120	27.11	odd	18