Embedded newform 27.3.f.a.2.1

• Label: 27.3.f.a.2.1
• Level: $27$
• Weight: $3$
• Character: 27.2
• 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			2.1
Character	\(\chi\)	\(=\)	27.2
Dual form			27.3.f.a.14.1

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.2.1	✓	30	1.1	even	1	trivial
27.3.f.a.14.1	yes	30	27.14	odd	18	inner
81.3.f.a.8.5		30	3.2	odd	2
81.3.f.a.71.5		30	27.13	even	9
243.3.f.a.107.1		30	9.5	odd	6
243.3.f.a.134.1		30	27.4	even	9
243.3.f.b.53.1		30	27.22	even	9
243.3.f.b.188.1		30	9.2	odd	6
243.3.f.c.53.5		30	27.5	odd	18
243.3.f.c.188.5		30	9.7	even	3
243.3.f.d.107.5		30	9.4	even	3
243.3.f.d.134.5		30	27.23	odd	18
432.3.bc.a.257.2		30	108.95	even	18
432.3.bc.a.353.2		30	4.3	odd	2
729.3.b.a.728.2		30	27.16	even	9
729.3.b.a.728.29		30	27.11	odd	18