Embedded newform 27.8.e.a.13.18

• Label: 27.8.e.a.13.18
• Level: $27$
• Weight: $8$
• Character: 27.13
• Analytic conductor: $8.434$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43439568807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			13.18
Character	\(\chi\)	\(=\)	27.13
Dual form			27.8.e.a.25.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.12991 + 17.7506i) q^{2} +(14.7882 + 44.3656i) q^{3} +(-185.007 + 67.3369i) q^{4} +(339.472 + 284.851i) q^{5} +(-741.231 + 401.360i) q^{6} +(-71.6195 - 26.0674i) q^{7} +(-620.762 - 1075.19i) q^{8} +(-1749.62 + 1312.18i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.12991	+	17.7506i	0.276648	+	1.56895i	0.733679	+	0.679496i	\(0.237801\pi\)
							−0.457032	+	0.889450i	\(0.651087\pi\)
\(3\)	14.7882	+	44.3656i	0.316222	+	0.948685i
							\(5\)	339.472	+	284.851i	1.21453	+	1.01911i	0.999092	+	0.0425956i	\(0.0135627\pi\)
0.215439	+	0.976517i	\(0.430882\pi\)
\(7\)	−71.6195	−	26.0674i	−0.0789202	−	0.0287246i	0.302258	−	0.953226i	\(-0.402259\pi\)
							−0.381179	+	0.924501i	\(0.624482\pi\)
\(11\)	5972.67	−	5011.67i	1.35299	−	1.13529i	0.374909	−	0.927062i	\(-0.377674\pi\)
							0.978080	−	0.208231i	\(-0.0667706\pi\)
\(13\)	2431.44	−	13789.4i	0.306946	−	1.74078i	−0.307256	−	0.951627i	\(-0.599411\pi\)
							0.614202	−	0.789149i	\(-0.289478\pi\)
\(17\)	3486.14	−	6038.17i	0.172097	−	0.298081i	−0.767056	−	0.641581i	\(-0.778279\pi\)
							0.939153	+	0.343499i	\(0.111612\pi\)
\(19\)	3709.36	+	6424.81i	0.124069	+	0.214893i	0.921368	−	0.388690i	\(-0.127072\pi\)
							−0.797300	+	0.603583i	\(0.793739\pi\)
\(23\)	−39214.2	+	14272.8i	−0.672041	+	0.244603i	−0.655426	−	0.755259i	\(-0.727511\pi\)
							−0.0166147	+	0.999862i	\(0.505289\pi\)
\(29\)	20290.9	+	115075.i	0.154493	+	0.876171i	0.959248	+	0.282565i	\(0.0911852\pi\)
							−0.804756	+	0.593606i	\(0.797704\pi\)
\(31\)	91907.3	−	33451.5i	0.554095	−	0.201674i	−0.0497700	−	0.998761i	\(-0.515849\pi\)
							0.603865	+	0.797087i	\(0.293627\pi\)
\(37\)	52581.2	−	91073.4i	0.170657	−	0.295587i	−0.767993	−	0.640459i	\(-0.778744\pi\)
							0.938650	+	0.344872i	\(0.112078\pi\)
\(41\)	−43379.1	+	246015.i	−0.0982962	+	0.557466i	0.895391	+	0.445281i	\(0.146896\pi\)
							−0.993687	+	0.112185i	\(0.964215\pi\)
\(43\)	−51000.0	+	42794.1i	−0.0978207	+	0.0820813i	−0.690386	−	0.723441i	\(-0.742559\pi\)
							0.592566	+	0.805522i	\(0.298115\pi\)
\(47\)	−239765.	−	87267.2i	−0.336855	−	0.122605i	0.168054	−	0.985778i	\(-0.446252\pi\)
							−0.504909	+	0.863173i	\(0.668474\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.13.18	✓	120
3.2	odd	2		81.8.e.a.10.3		120
27.2	odd	18		81.8.e.a.73.3		120
27.25	even	9	inner	27.8.e.a.25.18	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.13.18	✓	120	1.1	even	1	trivial
27.8.e.a.25.18	yes	120	27.25	even	9	inner
81.8.e.a.10.3		120	3.2	odd	2
81.8.e.a.73.3		120	27.2	odd	18