Embedded newform 27.3.f.a.2.2

• Label: 27.3.f.a.2.2
• Level: $27$
• Weight: $3$
• Character: 27.2
• Analytic conductor: $0.736$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.735696713773\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.2
Character	\(\chi\)	\(=\)	27.2
Dual form			27.3.f.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.31604 - 0.408381i) q^{2} +(-1.62484 - 2.52189i) q^{3} +(1.43851 + 0.523575i) q^{4} +(-3.71692 - 4.42965i) q^{5} +(2.73330 + 6.50435i) q^{6} +(4.57693 - 1.66587i) q^{7} +(5.02894 + 2.90346i) q^{8} +(-3.71981 + 8.19530i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 15 q^{5} - 18 q^{6} - 6 q^{7} - 9 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.31604	−	0.408381i	−1.15802	−	0.204190i	−0.438546	−	0.898709i	\(-0.644507\pi\)
							−0.719475	+	0.694518i	\(0.755618\pi\)
\(3\)	−1.62484	−	2.52189i	−0.541612	−	0.840629i
							\(5\)	−3.71692	−	4.42965i	−0.743384	−	0.885930i	0.253293	−	0.967390i	\(-0.418486\pi\)
−0.996677	+	0.0814592i	\(0.974042\pi\)
\(7\)	4.57693	−	1.66587i	0.653847	−	0.237981i	0.00626964	−	0.999980i	\(-0.498004\pi\)
							0.647578	+	0.761999i	\(0.275782\pi\)
\(11\)	1.90678	−	2.27241i	0.173343	−	0.206583i	−0.672377	−	0.740209i	\(-0.734727\pi\)
							0.845720	+	0.533626i	\(0.179171\pi\)
\(13\)	−3.38239	−	19.1825i	−0.260184	−	1.47558i	−0.782408	−	0.622767i	\(-0.786009\pi\)
							0.522224	−	0.852808i	\(-0.325103\pi\)
\(17\)	21.7258	−	12.5434i	1.27799	−	0.737848i	0.301512	−	0.953462i	\(-0.402509\pi\)
							0.976479	+	0.215614i	\(0.0691752\pi\)
\(19\)	−8.92251	+	15.4542i	−0.469606	+	0.813381i	−0.999396	−	0.0347477i	\(-0.988937\pi\)
							0.529790	+	0.848129i	\(0.322271\pi\)
\(23\)	−6.02912	+	16.5649i	−0.262136	+	0.720212i	0.736887	+	0.676016i	\(0.236295\pi\)
							−0.999023	+	0.0441963i	\(0.985927\pi\)
\(29\)	−3.42469	−	0.603866i	−0.118093	−	0.0208230i	0.114289	−	0.993448i	\(-0.463541\pi\)
							−0.232382	+	0.972625i	\(0.574652\pi\)
\(31\)	47.5319	+	17.3002i	1.53329	+	0.558071i	0.964424	−	0.264360i	\(-0.0851606\pi\)
							0.568865	+	0.822431i	\(0.307383\pi\)
\(37\)	−11.1918	−	19.3847i	−0.302480	−	0.523911i	0.674217	−	0.738533i	\(-0.264481\pi\)
							−0.976697	+	0.214622i	\(0.931148\pi\)
\(41\)	−2.61958	+	0.461903i	−0.0638923	+	0.0112659i	−0.205503	−	0.978657i	\(-0.565883\pi\)
							0.141611	+	0.989922i	\(0.454772\pi\)
\(43\)	19.4411	+	16.3130i	0.452118	+	0.379372i	0.840221	−	0.542244i	\(-0.182425\pi\)
							−0.388103	+	0.921616i	\(0.626870\pi\)
\(47\)	−7.07073	−	19.4267i	−0.150441	−	0.413334i	0.841464	−	0.540313i	\(-0.181694\pi\)
							−0.991905	+	0.126979i	\(0.959472\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.3.f.a.2.2	✓	30
3.2	odd	2		81.3.f.a.8.4		30
4.3	odd	2		432.3.bc.a.353.4		30
9.2	odd	6		243.3.f.b.188.2		30
9.4	even	3		243.3.f.d.107.4		30
9.5	odd	6		243.3.f.a.107.2		30
9.7	even	3		243.3.f.c.188.4		30
27.4	even	9		243.3.f.a.134.2		30
27.5	odd	18		243.3.f.c.53.4		30
27.11	odd	18		729.3.b.a.728.23		30
27.13	even	9		81.3.f.a.71.4		30
27.14	odd	18	inner	27.3.f.a.14.2	yes	30
27.16	even	9		729.3.b.a.728.8		30
27.22	even	9		243.3.f.b.53.2		30
27.23	odd	18		243.3.f.d.134.4		30
108.95	even	18		432.3.bc.a.257.4		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.2.2	✓	30	1.1	even	1	trivial
27.3.f.a.14.2	yes	30	27.14	odd	18	inner
81.3.f.a.8.4		30	3.2	odd	2
81.3.f.a.71.4		30	27.13	even	9
243.3.f.a.107.2		30	9.5	odd	6
243.3.f.a.134.2		30	27.4	even	9
243.3.f.b.53.2		30	27.22	even	9
243.3.f.b.188.2		30	9.2	odd	6
243.3.f.c.53.4		30	27.5	odd	18
243.3.f.c.188.4		30	9.7	even	3
243.3.f.d.107.4		30	9.4	even	3
243.3.f.d.134.4		30	27.23	odd	18
432.3.bc.a.257.4		30	108.95	even	18
432.3.bc.a.353.4		30	4.3	odd	2
729.3.b.a.728.8		30	27.16	even	9
729.3.b.a.728.23		30	27.11	odd	18