Embedded newform 27.2.e.a.16.2

• Label: 27.2.e.a.16.2
• Level: $27$
• Weight: $2$
• Character: 27.16
• Analytic conductor: $0.216$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.215596085457\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	12.0.1952986685049.1
Defining polynomial:	x^12 - 6*x^11 + 27*x^10 - 80*x^9 + 186*x^8 - 330*x^7 + 463*x^6 - 504*x^5 + 420*x^4 - 258*x^3 + 108*x^2 - 27*x + 3
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			16.2
Root			\(0.500000 - 2.22827i\) of defining polynomial
Character	\(\chi\)	\(=\)	27.16
Dual form			27.2.e.a.22.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.318266 - 0.267057i) q^{2} +(0.159815 + 1.72466i) q^{3} +(-0.317323 - 1.79963i) q^{4} +(-2.08159 - 0.757639i) q^{5} +(0.409719 - 0.591580i) q^{6} +(-0.229151 + 1.29958i) q^{7} +(-0.795075 + 1.37711i) q^{8} +(-2.94892 + 0.551252i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.318266	−	0.267057i	−0.225048	−	0.188837i	0.523291	−	0.852154i	\(-0.324704\pi\)
							−0.748339	+	0.663316i	\(0.769148\pi\)
\(3\)	0.159815	+	1.72466i	0.0922690	+	0.995734i
							\(5\)	−2.08159	−	0.757639i	−0.930917	−	0.338826i	−0.168344	−	0.985728i	\(-0.553842\pi\)
−0.762573	+	0.646902i	\(0.776064\pi\)
\(7\)	−0.229151	+	1.29958i	−0.0866110	+	0.491195i	0.910386	+	0.413759i	\(0.135784\pi\)
							−0.996997	+	0.0774361i	\(0.975327\pi\)
\(11\)	4.90067	−	1.78370i	1.47761	−	0.537805i	0.527454	−	0.849584i	\(-0.323147\pi\)
							0.950155	+	0.311778i	\(0.100925\pi\)
\(13\)	−0.0138336	+	0.0116078i	−0.00383676	+	0.00321942i	−0.644704	−	0.764432i	\(-0.723019\pi\)
							0.640867	+	0.767652i	\(0.278575\pi\)
\(17\)	1.56640	+	2.71308i	0.379907	+	0.658019i	0.991048	−	0.133503i	\(-0.0426226\pi\)
							−0.611141	+	0.791522i	\(0.709289\pi\)
\(19\)	−0.208676	+	0.361438i	−0.0478736	+	0.0829195i	−0.888969	−	0.457967i	\(-0.848578\pi\)
							0.841096	+	0.540886i	\(0.181911\pi\)
\(23\)	−0.179619	−	1.01867i	−0.0374532	−	0.212408i	0.960338	−	0.278839i	\(-0.0899497\pi\)
							−0.997791	+	0.0664316i	\(0.978839\pi\)
\(29\)	−5.98068	−	5.01839i	−1.11058	−	0.931891i	−0.112493	−	0.993652i	\(-0.535884\pi\)
							−0.998091	+	0.0617615i	\(0.980328\pi\)
\(31\)	0.647649	+	3.67300i	0.116321	+	0.659691i	0.986088	+	0.166227i	\(0.0531584\pi\)
							−0.869766	+	0.493464i	\(0.835730\pi\)
\(37\)	−2.21238	−	3.83195i	−0.363713	−	0.629969i	0.624856	−	0.780740i	\(-0.285158\pi\)
							−0.988569	+	0.150771i	\(0.951824\pi\)
\(41\)	−2.81517	+	2.36221i	−0.439655	+	0.368915i	−0.835580	−	0.549368i	\(-0.814868\pi\)
							0.395925	+	0.918283i	\(0.370424\pi\)
\(43\)	7.80685	−	2.84146i	1.19053	−	0.433319i	0.330622	−	0.943763i	\(-0.392741\pi\)
							0.859911	+	0.510445i	\(0.170519\pi\)
\(47\)	−1.23254	+	6.99008i	−0.179784	+	1.01961i	0.752691	+	0.658374i	\(0.228755\pi\)
							−0.932475	+	0.361234i	\(0.882356\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.2.e.a.16.2	✓	12
3.2	odd	2		81.2.e.a.46.1		12
4.3	odd	2		432.2.u.c.97.1		12
5.2	odd	4		675.2.u.b.124.3		24
5.3	odd	4		675.2.u.b.124.2		24
5.4	even	2		675.2.l.c.151.1		12
9.2	odd	6		243.2.e.a.217.1		12
9.4	even	3		243.2.e.c.55.1		12
9.5	odd	6		243.2.e.b.55.2		12
9.7	even	3		243.2.e.d.217.2		12
27.2	odd	18		729.2.c.b.244.4		12
27.4	even	9		243.2.e.d.28.2		12
27.5	odd	18		81.2.e.a.37.1		12
27.7	even	9		729.2.a.a.1.4		6
27.11	odd	18		729.2.c.b.487.4		12
27.13	even	9		243.2.e.c.190.1		12
27.14	odd	18		243.2.e.b.190.2		12
27.16	even	9		729.2.c.e.487.3		12
27.20	odd	18		729.2.a.d.1.3		6
27.22	even	9	inner	27.2.e.a.22.2	yes	12
27.23	odd	18		243.2.e.a.28.1		12
27.25	even	9		729.2.c.e.244.3		12
108.103	odd	18		432.2.u.c.49.1		12
135.22	odd	36		675.2.u.b.49.2		24
135.49	even	18		675.2.l.c.76.1		12
135.103	odd	36		675.2.u.b.49.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.16.2	✓	12	1.1	even	1	trivial
27.2.e.a.22.2	yes	12	27.22	even	9	inner
81.2.e.a.37.1		12	27.5	odd	18
81.2.e.a.46.1		12	3.2	odd	2
243.2.e.a.28.1		12	27.23	odd	18
243.2.e.a.217.1		12	9.2	odd	6
243.2.e.b.55.2		12	9.5	odd	6
243.2.e.b.190.2		12	27.14	odd	18
243.2.e.c.55.1		12	9.4	even	3
243.2.e.c.190.1		12	27.13	even	9
243.2.e.d.28.2		12	27.4	even	9
243.2.e.d.217.2		12	9.7	even	3
432.2.u.c.49.1		12	108.103	odd	18
432.2.u.c.97.1		12	4.3	odd	2
675.2.l.c.76.1		12	135.49	even	18
675.2.l.c.151.1		12	5.4	even	2
675.2.u.b.49.2		24	135.22	odd	36
675.2.u.b.49.3		24	135.103	odd	36
675.2.u.b.124.2		24	5.3	odd	4
675.2.u.b.124.3		24	5.2	odd	4
729.2.a.a.1.4		6	27.7	even	9
729.2.a.d.1.3		6	27.20	odd	18
729.2.c.b.244.4		12	27.2	odd	18
729.2.c.b.487.4		12	27.11	odd	18
729.2.c.e.244.3		12	27.25	even	9
729.2.c.e.487.3		12	27.16	even	9