Embedded newform 27.3.f.a.14.2

• Label: 27.3.f.a.14.2
• Level: $27$
• Weight: $3$
• Character: 27.14
• 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			14.2
Character	\(\chi\)	\(=\)	27.14
Dual form			27.3.f.a.2.2

$q$-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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.2.2	✓	30	27.2	odd	18	inner
27.3.f.a.14.2	yes	30	1.1	even	1	trivial
81.3.f.a.8.4		30	27.25	even	9
81.3.f.a.71.4		30	3.2	odd	2
243.3.f.a.107.2		30	27.7	even	9
243.3.f.a.134.2		30	9.2	odd	6
243.3.f.b.53.2		30	9.5	odd	6
243.3.f.b.188.2		30	27.16	even	9
243.3.f.c.53.4		30	9.4	even	3
243.3.f.c.188.4		30	27.11	odd	18
243.3.f.d.107.4		30	27.20	odd	18
243.3.f.d.134.4		30	9.7	even	3
432.3.bc.a.257.4		30	4.3	odd	2
432.3.bc.a.353.4		30	108.83	even	18
729.3.b.a.728.8		30	27.5	odd	18
729.3.b.a.728.23		30	27.22	even	9