Embedded newform 27.3.f.a.14.1

• Label: 27.3.f.a.14.1
• 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			14.1
Character	\(\chi\)	\(=\)	27.14
Dual form			27.3.f.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.46291 + 0.610605i) q^{2} +(1.47633 - 2.61160i) q^{3} +(7.86014 - 2.86086i) q^{4} +(3.16671 - 3.77394i) q^{5} +(-3.51774 + 9.94519i) q^{6} +(-3.18911 - 1.16074i) q^{7} +(-13.2912 + 7.67367i) q^{8} +(-4.64090 - 7.71116i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 15 q^{5} - 18 q^{6} - 6 q^{7} - 9 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.46291	+	0.610605i	−1.73146	+	0.305302i	−0.948500	−	0.316776i	\(-0.897400\pi\)
							−0.782955	+	0.622078i	\(0.786289\pi\)
\(3\)	1.47633	−	2.61160i	0.492110	−	0.870533i
							\(5\)	3.16671	−	3.77394i	0.633342	−	0.754787i	−0.349961	−	0.936764i	\(-0.613805\pi\)
0.983303	+	0.181977i	\(0.0582496\pi\)
\(7\)	−3.18911	−	1.16074i	−0.455588	−	0.165820i	0.104025	−	0.994575i	\(-0.466828\pi\)
							−0.559613	+	0.828754i	\(0.689050\pi\)
\(11\)	10.2080	+	12.1655i	0.928002	+	1.10595i	0.994136	+	0.108139i	\(0.0344893\pi\)
							−0.0661334	+	0.997811i	\(0.521066\pi\)
\(13\)	−0.621316	+	3.52366i	−0.0477935	+	0.271051i	−0.999335	−	0.0364707i	\(-0.988388\pi\)
							0.951541	+	0.307521i	\(0.0994995\pi\)
\(17\)	−9.99899	−	5.77292i	−0.588176	−	0.339583i	0.176200	−	0.984354i	\(-0.443619\pi\)
							−0.764376	+	0.644771i	\(0.776953\pi\)
\(19\)	9.59953	+	16.6269i	0.505238	+	0.875098i	0.999982	+	0.00605928i	\(0.00192874\pi\)
							−0.494743	+	0.869039i	\(0.664738\pi\)
\(23\)	5.28374	+	14.5170i	0.229728	+	0.631172i	0.999978	−	0.00656926i	\(-0.00209107\pi\)
							−0.770251	+	0.637741i	\(0.779869\pi\)
\(29\)	−11.0239	+	1.94382i	−0.380136	+	0.0670282i	−0.360451	−	0.932778i	\(-0.617377\pi\)
							−0.0196846	+	0.999806i	\(0.506266\pi\)
\(31\)	23.0339	−	8.38364i	0.743027	−	0.270440i	0.0573588	−	0.998354i	\(-0.481732\pi\)
							0.685669	+	0.727914i	\(0.259510\pi\)
\(37\)	21.2827	−	36.8628i	0.575209	−	0.996292i	−0.420810	−	0.907149i	\(-0.638254\pi\)
							0.996019	−	0.0891427i	\(-0.0284127\pi\)
\(41\)	6.57199	+	1.15882i	0.160293	+	0.0282639i	0.253218	−	0.967409i	\(-0.418511\pi\)
							−0.0929259	+	0.995673i	\(0.529622\pi\)
\(43\)	−36.6672	+	30.7674i	−0.852725	+	0.715521i	−0.960388	−	0.278666i	\(-0.910108\pi\)
							0.107663	+	0.994187i	\(0.465663\pi\)
\(47\)	−17.5573	+	48.2383i	−0.373559	+	1.02635i	0.600415	+	0.799688i	\(0.295002\pi\)
							−0.973975	+	0.226657i	\(0.927220\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.3.f.a.14.1	yes	30
3.2	odd	2		81.3.f.a.71.5		30
4.3	odd	2		432.3.bc.a.257.2		30
9.2	odd	6		243.3.f.a.134.1		30
9.4	even	3		243.3.f.c.53.5		30
9.5	odd	6		243.3.f.b.53.1		30
9.7	even	3		243.3.f.d.134.5		30
27.2	odd	18	inner	27.3.f.a.2.1	✓	30
27.5	odd	18		729.3.b.a.728.2		30
27.7	even	9		243.3.f.a.107.1		30
27.11	odd	18		243.3.f.c.188.5		30
27.16	even	9		243.3.f.b.188.1		30
27.20	odd	18		243.3.f.d.107.5		30
27.22	even	9		729.3.b.a.728.29		30
27.25	even	9		81.3.f.a.8.5		30
108.83	even	18		432.3.bc.a.353.2		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.2.1	✓	30	27.2	odd	18	inner
27.3.f.a.14.1	yes	30	1.1	even	1	trivial
81.3.f.a.8.5		30	27.25	even	9
81.3.f.a.71.5		30	3.2	odd	2
243.3.f.a.107.1		30	27.7	even	9
243.3.f.a.134.1		30	9.2	odd	6
243.3.f.b.53.1		30	9.5	odd	6
243.3.f.b.188.1		30	27.16	even	9
243.3.f.c.53.5		30	9.4	even	3
243.3.f.c.188.5		30	27.11	odd	18
243.3.f.d.107.5		30	27.20	odd	18
243.3.f.d.134.5		30	9.7	even	3
432.3.bc.a.257.2		30	4.3	odd	2
432.3.bc.a.353.2		30	108.83	even	18
729.3.b.a.728.2		30	27.5	odd	18
729.3.b.a.728.29		30	27.22	even	9