Embedded newform 27.3.f.a.5.4

• Label: 27.3.f.a.5.4
• Level: $27$
• Weight: $3$
• Character: 27.5
• 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			5.4
Character	\(\chi\)	\(=\)	27.5
Dual form			27.3.f.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.837612 + 0.998227i) q^{2} +(-0.987122 + 2.83295i) q^{3} +(0.399729 - 2.26698i) q^{4} +(0.149473 + 0.410673i) q^{5} +(-3.65475 + 1.38754i) q^{6} +(-1.05651 - 5.99176i) q^{7} +(7.11182 - 4.10601i) q^{8} +(-7.05118 - 5.59293i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 15 q^{5} - 18 q^{6} - 6 q^{7} - 9 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.837612	+	0.998227i	0.418806	+	0.499113i	0.933658	−	0.358166i	\(-0.116598\pi\)
							−0.514852	+	0.857279i	\(0.672153\pi\)
\(3\)	−0.987122	+	2.83295i	−0.329041	+	0.944316i
							\(5\)	0.149473	+	0.410673i	0.0298945	+	0.0821346i	0.953742	−	0.300626i	\(-0.0971956\pi\)
−0.923848	+	0.382761i	\(0.874973\pi\)
\(7\)	−1.05651	−	5.99176i	−0.150930	−	0.855965i	−0.962413	−	0.271590i	\(-0.912451\pi\)
							0.811483	−	0.584376i	\(-0.198660\pi\)
\(11\)	−5.19297	+	14.2676i	−0.472088	+	1.29705i	0.443983	+	0.896035i	\(0.353565\pi\)
							−0.916071	+	0.401016i	\(0.868657\pi\)
\(13\)	−7.31452	−	6.13761i	−0.562655	−	0.472124i	0.316544	−	0.948578i	\(-0.397477\pi\)
							−0.879199	+	0.476454i	\(0.841922\pi\)
\(17\)	4.20474	+	2.42761i	0.247338	+	0.142800i	0.618545	−	0.785750i	\(-0.287723\pi\)
							−0.371207	+	0.928550i	\(0.621056\pi\)
\(19\)	17.7795	+	30.7949i	0.935761	+	1.62079i	0.773271	+	0.634075i	\(0.218619\pi\)
							0.162490	+	0.986710i	\(0.448048\pi\)
\(23\)	−26.7326	−	4.71368i	−1.16229	−	0.204943i	−0.440954	−	0.897530i	\(-0.645360\pi\)
							−0.721334	+	0.692587i	\(0.756471\pi\)
\(29\)	−13.7480	−	16.3843i	−0.474070	−	0.564975i	0.475022	−	0.879974i	\(-0.342440\pi\)
							−0.949092	+	0.314999i	\(0.897996\pi\)
\(31\)	−2.87818	+	16.3229i	−0.0928444	+	0.526547i	0.902542	+	0.430601i	\(0.141698\pi\)
							−0.995387	+	0.0959453i	\(0.969413\pi\)
\(37\)	5.31270	−	9.20187i	0.143587	−	0.248699i	−0.785258	−	0.619168i	\(-0.787470\pi\)
							0.928845	+	0.370469i	\(0.120803\pi\)
\(41\)	−14.1800	+	16.8991i	−0.345854	+	0.412173i	−0.910729	−	0.413003i	\(-0.864480\pi\)
							0.564875	+	0.825176i	\(0.308924\pi\)
\(43\)	5.22687	+	1.90242i	0.121555	+	0.0442424i	0.402082	−	0.915604i	\(-0.368287\pi\)
							−0.280527	+	0.959846i	\(0.590509\pi\)
\(47\)	85.0411	−	14.9950i	1.80938	−	0.319043i	0.836088	−	0.548595i	\(-0.184837\pi\)
							0.973296	+	0.229552i	\(0.0737262\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.3.f.a.5.4	✓	30
3.2	odd	2		81.3.f.a.44.2		30
4.3	odd	2		432.3.bc.a.113.3		30
9.2	odd	6		243.3.f.a.53.2		30
9.4	even	3		243.3.f.c.215.2		30
9.5	odd	6		243.3.f.b.215.4		30
9.7	even	3		243.3.f.d.53.4		30
27.2	odd	18		243.3.f.d.188.4		30
27.4	even	9		729.3.b.a.728.20		30
27.7	even	9		243.3.f.b.26.4		30
27.11	odd	18	inner	27.3.f.a.11.4	yes	30
27.16	even	9		81.3.f.a.35.2		30
27.20	odd	18		243.3.f.c.26.2		30
27.23	odd	18		729.3.b.a.728.11		30
27.25	even	9		243.3.f.a.188.2		30
108.11	even	18		432.3.bc.a.65.3		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.5.4	✓	30	1.1	even	1	trivial
27.3.f.a.11.4	yes	30	27.11	odd	18	inner
81.3.f.a.35.2		30	27.16	even	9
81.3.f.a.44.2		30	3.2	odd	2
243.3.f.a.53.2		30	9.2	odd	6
243.3.f.a.188.2		30	27.25	even	9
243.3.f.b.26.4		30	27.7	even	9
243.3.f.b.215.4		30	9.5	odd	6
243.3.f.c.26.2		30	27.20	odd	18
243.3.f.c.215.2		30	9.4	even	3
243.3.f.d.53.4		30	9.7	even	3
243.3.f.d.188.4		30	27.2	odd	18
432.3.bc.a.65.3		30	108.11	even	18
432.3.bc.a.113.3		30	4.3	odd	2
729.3.b.a.728.11		30	27.23	odd	18
729.3.b.a.728.20		30	27.4	even	9