Embedded newform 27.2.e.a.22.2

• Label: 27.2.e.a.22.2
• Level: $27$
• Weight: $2$
• Character: 27.22
• 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			22.2
Root			\(0.500000 + 2.22827i\) of defining polynomial
Character	\(\chi\)	\(=\)	27.22
Dual form			27.2.e.a.16.2

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.16.2	✓	12	27.16	even	9	inner
27.2.e.a.22.2	yes	12	1.1	even	1	trivial
81.2.e.a.37.1		12	3.2	odd	2
81.2.e.a.46.1		12	27.11	odd	18
243.2.e.a.28.1		12	9.5	odd	6
243.2.e.a.217.1		12	27.20	odd	18
243.2.e.b.55.2		12	27.2	odd	18
243.2.e.b.190.2		12	9.2	odd	6
243.2.e.c.55.1		12	27.25	even	9
243.2.e.c.190.1		12	9.7	even	3
243.2.e.d.28.2		12	9.4	even	3
243.2.e.d.217.2		12	27.7	even	9
432.2.u.c.49.1		12	4.3	odd	2
432.2.u.c.97.1		12	108.43	odd	18
675.2.l.c.76.1		12	5.4	even	2
675.2.l.c.151.1		12	135.124	even	18
675.2.u.b.49.2		24	5.2	odd	4
675.2.u.b.49.3		24	5.3	odd	4
675.2.u.b.124.2		24	135.43	odd	36
675.2.u.b.124.3		24	135.97	odd	36
729.2.a.a.1.4		6	27.4	even	9
729.2.a.d.1.3		6	27.23	odd	18
729.2.c.b.244.4		12	27.5	odd	18
729.2.c.b.487.4		12	27.14	odd	18
729.2.c.e.244.3		12	27.22	even	9
729.2.c.e.487.3		12	27.13	even	9