Embedded newform 27.10.e.a.4.12

• Label: 27.10.e.a.4.12
• Level: $27$
• Weight: $10$
• Character: 27.4
• Analytic conductor: $13.906$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.9059675764\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(26\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.12
Character	\(\chi\)	\(=\)	27.4
Dual form			27.10.e.a.7.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.86289 - 1.76995i) q^{2} +(133.111 + 44.3232i) q^{3} +(-371.700 - 311.893i) q^{4} +(375.168 - 2127.68i) q^{5} +(-568.853 - 451.138i) q^{6} +(-8670.18 + 7275.14i) q^{7} +(2580.30 + 4469.20i) q^{8} +(15753.9 + 11799.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.86289	−	1.76995i	−0.214911	−	0.0782214i	0.232321	−	0.972639i	\(-0.425368\pi\)
							−0.447232	+	0.894418i	\(0.647590\pi\)
\(3\)	133.111	+	44.3232i	0.948784	+	0.315926i
							\(5\)	375.168	−	2127.68i	0.268449	−	1.52245i	−0.490584	−	0.871394i	\(-0.663216\pi\)
0.759032	−	0.651053i	\(-0.225672\pi\)
\(7\)	−8670.18	+	7275.14i	−1.36486	+	1.14525i	−0.390406	+	0.920643i	\(0.627665\pi\)
							−0.974449	+	0.224607i	\(0.927890\pi\)
\(11\)	−4593.95	−	26053.6i	−0.0946062	−	0.536538i	−0.994867	−	0.101188i	\(-0.967736\pi\)
							0.900261	−	0.435350i	\(-0.143375\pi\)
\(13\)	−123720.	+	45030.4i	−1.20142	+	0.437281i	−0.863719	−	0.503973i	\(-0.831871\pi\)
							−0.337700	+	0.941254i	\(0.609649\pi\)
\(17\)	−161974.	+	280547.i	−0.470354	+	0.814677i	−0.999425	−	0.0339004i	\(-0.989207\pi\)
							0.529071	+	0.848577i	\(0.322540\pi\)
\(19\)	−66172.8	−	114615.i	−0.116490	−	0.201766i	0.801884	−	0.597479i	\(-0.203831\pi\)
							−0.918374	+	0.395713i	\(0.870498\pi\)
\(23\)	−1.63207e6	−	1.36947e6i	−1.21608	−	1.02041i	−0.999020	−	0.0442542i	\(-0.985909\pi\)
							−0.217060	−	0.976158i	\(-0.569647\pi\)
\(29\)	996089.	+	362547.i	0.261521	+	0.0951860i	0.469453	−	0.882957i	\(-0.344451\pi\)
							−0.207932	+	0.978143i	\(0.566673\pi\)
\(31\)	−5.87685e6	−	4.93127e6i	−1.14292	−	0.959027i	−0.143393	−	0.989666i	\(-0.545801\pi\)
							−0.999531	+	0.0306387i	\(0.990246\pi\)
\(37\)	2.26181e6	−	3.91757e6i	0.198403	−	0.343644i	−0.749608	−	0.661882i	\(-0.769758\pi\)
							0.948011	+	0.318238i	\(0.103091\pi\)
\(41\)	1.31544e6	−	478781.i	0.0727016	−	0.0264612i	−0.305413	−	0.952220i	\(-0.598795\pi\)
							0.378115	+	0.925759i	\(0.376572\pi\)
\(43\)	3.51066e6	+	1.99099e7i	0.156596	+	0.888099i	0.957312	+	0.289056i	\(0.0933413\pi\)
							−0.800716	+	0.599044i	\(0.795548\pi\)
\(47\)	2.25607e7	−	1.89307e7i	0.674391	−	0.565881i	−0.239971	−	0.970780i	\(-0.577138\pi\)
							0.914361	+	0.404899i	\(0.132693\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.e.a.4.12	✓	156
3.2	odd	2		81.10.e.a.64.15		156
27.7	even	9	inner	27.10.e.a.7.12	yes	156
27.20	odd	18		81.10.e.a.19.15		156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.e.a.4.12	✓	156	1.1	even	1	trivial
27.10.e.a.7.12	yes	156	27.7	even	9	inner
81.10.e.a.19.15		156	27.20	odd	18
81.10.e.a.64.15		156	3.2	odd	2