Embedded newform 27.3.f.a.14.4

• Label: 27.3.f.a.14.4
• 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			14.4
Character	\(\chi\)	\(=\)	27.14
Dual form			27.3.f.a.2.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14332 - 0.201599i) q^{2} +(1.10490 - 2.78912i) q^{3} +(-2.49222 + 0.907094i) q^{4} +(-3.46013 + 4.12362i) q^{5} +(0.700975 - 3.41162i) q^{6} +(9.89907 + 3.60297i) q^{7} +(-6.68824 + 3.86146i) q^{8} +(-6.55839 - 6.16340i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 15 q^{5} - 18 q^{6} - 6 q^{7} - 9 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14332	−	0.201599i	0.571662	−	0.100799i	0.119658	−	0.992815i	\(-0.461820\pi\)
							0.452004	+	0.892016i	\(0.350709\pi\)
\(3\)	1.10490	−	2.78912i	0.368300	−	0.929707i
							\(5\)	−3.46013	+	4.12362i	−0.692026	+	0.824724i	−0.991599	−	0.129349i	\(-0.958711\pi\)
0.299574	+	0.954073i	\(0.403156\pi\)
\(7\)	9.89907	+	3.60297i	1.41415	+	0.514709i	0.932346	−	0.361568i	\(-0.117759\pi\)
							0.481807	+	0.876278i	\(0.339981\pi\)
\(11\)	−7.54888	−	8.99640i	−0.686262	−	0.817855i	0.304636	−	0.952469i	\(-0.401465\pi\)
							−0.990898	+	0.134614i	\(0.957021\pi\)
\(13\)	1.95057	−	11.0622i	0.150044	−	0.850940i	−0.813134	−	0.582076i	\(-0.802241\pi\)
							0.963178	−	0.268864i	\(-0.0866482\pi\)
\(17\)	2.73630	+	1.57980i	0.160959	+	0.0929296i	0.578316	−	0.815813i	\(-0.303710\pi\)
							−0.417357	+	0.908743i	\(0.637044\pi\)
\(19\)	2.26051	+	3.91532i	0.118974	+	0.206069i	0.919361	−	0.393414i	\(-0.128706\pi\)
							−0.800387	+	0.599483i	\(0.795373\pi\)
\(23\)	6.80880	+	18.7070i	0.296035	+	0.813349i	0.995153	+	0.0983427i	\(0.0313541\pi\)
							−0.699118	+	0.715007i	\(0.746424\pi\)
\(29\)	24.9744	−	4.40366i	0.861187	−	0.151851i	0.274424	−	0.961609i	\(-0.411513\pi\)
							0.586763	+	0.809758i	\(0.300402\pi\)
\(31\)	−20.7874	+	7.56598i	−0.670560	+	0.244064i	−0.654789	−	0.755812i	\(-0.727243\pi\)
							−0.0157712	+	0.999876i	\(0.505020\pi\)
\(37\)	−8.82807	+	15.2907i	−0.238596	+	0.413261i	−0.960312	−	0.278929i	\(-0.910021\pi\)
							0.721715	+	0.692190i	\(0.243354\pi\)
\(41\)	−12.2557	−	2.16101i	−0.298920	−	0.0527076i	0.0221770	−	0.999754i	\(-0.492940\pi\)
							−0.321096	+	0.947046i	\(0.604051\pi\)
\(43\)	27.9211	−	23.4286i	0.649329	−	0.544851i	−0.257539	−	0.966268i	\(-0.582911\pi\)
							0.906867	+	0.421417i	\(0.138467\pi\)
\(47\)	−4.24091	+	11.6518i	−0.0902320	+	0.247910i	−0.976597	−	0.215076i	\(-0.931000\pi\)
							0.886365	+	0.462987i	\(0.153222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.3.f.a.14.4	yes	30
3.2	odd	2		81.3.f.a.71.2		30
4.3	odd	2		432.3.bc.a.257.3		30
9.2	odd	6		243.3.f.a.134.4		30
9.4	even	3		243.3.f.c.53.2		30
9.5	odd	6		243.3.f.b.53.4		30
9.7	even	3		243.3.f.d.134.2		30
27.2	odd	18	inner	27.3.f.a.2.4	✓	30
27.5	odd	18		729.3.b.a.728.19		30
27.7	even	9		243.3.f.a.107.4		30
27.11	odd	18		243.3.f.c.188.2		30
27.16	even	9		243.3.f.b.188.4		30
27.20	odd	18		243.3.f.d.107.2		30
27.22	even	9		729.3.b.a.728.12		30
27.25	even	9		81.3.f.a.8.2		30
108.83	even	18		432.3.bc.a.353.3		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.2.4	✓	30	27.2	odd	18	inner
27.3.f.a.14.4	yes	30	1.1	even	1	trivial
81.3.f.a.8.2		30	27.25	even	9
81.3.f.a.71.2		30	3.2	odd	2
243.3.f.a.107.4		30	27.7	even	9
243.3.f.a.134.4		30	9.2	odd	6
243.3.f.b.53.4		30	9.5	odd	6
243.3.f.b.188.4		30	27.16	even	9
243.3.f.c.53.2		30	9.4	even	3
243.3.f.c.188.2		30	27.11	odd	18
243.3.f.d.107.2		30	27.20	odd	18
243.3.f.d.134.2		30	9.7	even	3
432.3.bc.a.257.3		30	4.3	odd	2
432.3.bc.a.353.3		30	108.83	even	18
729.3.b.a.728.12		30	27.22	even	9
729.3.b.a.728.19		30	27.5	odd	18