Embedded newform 27.2.e.a.22.1

• Label: 27.2.e.a.22.1
• 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			22.1
Root			\(0.500000 - 0.258654i\) of defining polynomial
Character	\(\chi\)	\(=\)	27.22
Dual form			27.2.e.a.16.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62143 + 1.36054i) q^{2} +(-0.986166 + 1.42389i) q^{3} +(0.430663 - 2.44241i) q^{4} +(2.52129 - 0.917674i) q^{5} +(-0.338267 - 3.65046i) q^{6} +(0.168844 + 0.957561i) q^{7} +(0.508086 + 0.880031i) q^{8} +(-1.05495 - 2.80839i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62143	+	1.36054i	−1.14652	+	0.962046i	−0.999633	−	0.0271067i	\(-0.991371\pi\)
							−0.146889	+	0.989153i	\(0.546926\pi\)
\(3\)	−0.986166	+	1.42389i	−0.569363	+	0.822086i
							\(5\)	2.52129	−	0.917674i	1.12755	−	0.410396i	0.290150	−	0.956981i	\(-0.406295\pi\)
0.837404	+	0.546585i	\(0.184072\pi\)
\(7\)	0.168844	+	0.957561i	0.0638170	+	0.361924i	0.999947	+	0.0102706i	\(0.00326930\pi\)
							−0.936130	+	0.351653i	\(0.885620\pi\)
\(11\)	0.297791	+	0.108387i	0.0897872	+	0.0326799i	0.386523	−	0.922280i	\(-0.373676\pi\)
							−0.296736	+	0.954960i	\(0.595898\pi\)
\(13\)	−1.15981	−	0.973200i	−0.321675	−	0.269917i	0.467623	−	0.883928i	\(-0.345111\pi\)
							−0.789297	+	0.614011i	\(0.789555\pi\)
\(17\)	−0.587342	+	1.01731i	−0.142451	+	0.246733i	−0.928419	−	0.371534i	\(-0.878832\pi\)
							0.785968	+	0.618267i	\(0.212165\pi\)
\(19\)	−3.11040	−	5.38737i	−0.713575	−	1.23595i	−0.963507	−	0.267685i	\(-0.913741\pi\)
							0.249931	−	0.968264i	\(-0.419592\pi\)
\(23\)	0.375556	−	2.12988i	0.0783089	−	0.444112i	−0.920292	−	0.391232i	\(-0.872049\pi\)
							0.998601	−	0.0528796i	\(-0.0168399\pi\)
\(29\)	−3.37436	+	2.83142i	−0.626602	+	0.525782i	−0.899871	−	0.436156i	\(-0.856340\pi\)
							0.273269	+	0.961938i	\(0.411895\pi\)
\(31\)	−1.50609	+	8.54146i	−0.270502	+	1.53409i	0.482395	+	0.875954i	\(0.339767\pi\)
							−0.752897	+	0.658138i	\(0.771344\pi\)
\(37\)	2.23332	−	3.86823i	0.367156	−	0.635933i	−0.621964	−	0.783046i	\(-0.713665\pi\)
							0.989120	+	0.147113i	\(0.0469982\pi\)
\(41\)	4.47767	+	3.75721i	0.699295	+	0.586778i	0.921573	−	0.388205i	\(-0.126905\pi\)
							−0.222278	+	0.974983i	\(0.571349\pi\)
\(43\)	−5.25381	−	1.91223i	−0.801199	−	0.291613i	−0.0912158	−	0.995831i	\(-0.529075\pi\)
							−0.709983	+	0.704219i	\(0.751298\pi\)
\(47\)	−0.429965	−	2.43845i	−0.0627168	−	0.355685i	−0.999975	−	0.00704911i	\(-0.997756\pi\)
							0.937258	−	0.348636i	\(-0.113355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.2.e.a.22.1	yes	12
3.2	odd	2		81.2.e.a.37.2		12
4.3	odd	2		432.2.u.c.49.2		12
5.2	odd	4		675.2.u.b.49.1		24
5.3	odd	4		675.2.u.b.49.4		24
5.4	even	2		675.2.l.c.76.2		12
9.2	odd	6		243.2.e.b.190.1		12
9.4	even	3		243.2.e.d.28.1		12
9.5	odd	6		243.2.e.a.28.2		12
9.7	even	3		243.2.e.c.190.2		12
27.2	odd	18		243.2.e.b.55.1		12
27.4	even	9		729.2.a.a.1.2		6
27.5	odd	18		729.2.c.b.244.2		12
27.7	even	9		243.2.e.d.217.1		12
27.11	odd	18		81.2.e.a.46.2		12
27.13	even	9		729.2.c.e.487.5		12
27.14	odd	18		729.2.c.b.487.2		12
27.16	even	9	inner	27.2.e.a.16.1	✓	12
27.20	odd	18		243.2.e.a.217.2		12
27.22	even	9		729.2.c.e.244.5		12
27.23	odd	18		729.2.a.d.1.5		6
27.25	even	9		243.2.e.c.55.2		12
108.43	odd	18		432.2.u.c.97.2		12
135.43	odd	36		675.2.u.b.124.1		24
135.97	odd	36		675.2.u.b.124.4		24
135.124	even	18		675.2.l.c.151.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.16.1	✓	12	27.16	even	9	inner
27.2.e.a.22.1	yes	12	1.1	even	1	trivial
81.2.e.a.37.2		12	3.2	odd	2
81.2.e.a.46.2		12	27.11	odd	18
243.2.e.a.28.2		12	9.5	odd	6
243.2.e.a.217.2		12	27.20	odd	18
243.2.e.b.55.1		12	27.2	odd	18
243.2.e.b.190.1		12	9.2	odd	6
243.2.e.c.55.2		12	27.25	even	9
243.2.e.c.190.2		12	9.7	even	3
243.2.e.d.28.1		12	9.4	even	3
243.2.e.d.217.1		12	27.7	even	9
432.2.u.c.49.2		12	4.3	odd	2
432.2.u.c.97.2		12	108.43	odd	18
675.2.l.c.76.2		12	5.4	even	2
675.2.l.c.151.2		12	135.124	even	18
675.2.u.b.49.1		24	5.2	odd	4
675.2.u.b.49.4		24	5.3	odd	4
675.2.u.b.124.1		24	135.43	odd	36
675.2.u.b.124.4		24	135.97	odd	36
729.2.a.a.1.2		6	27.4	even	9
729.2.a.d.1.5		6	27.23	odd	18
729.2.c.b.244.2		12	27.5	odd	18
729.2.c.b.487.2		12	27.14	odd	18
729.2.c.e.244.5		12	27.22	even	9
729.2.c.e.487.5		12	27.13	even	9