Embedded newform 27.8.e.a.16.19

• Label: 27.8.e.a.16.19
• Level: $27$
• Weight: $8$
• Character: 27.16
• 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			16.19
Character	\(\chi\)	\(=\)	27.16
Dual form			27.8.e.a.22.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.6306 + 12.2765i) q^{2} +(-26.1976 - 38.7387i) q^{3} +(41.1141 + 233.170i) q^{4} +(344.707 + 125.463i) q^{5} +(92.2895 - 888.385i) q^{6} +(-186.148 + 1055.70i) q^{7} +(-1038.66 + 1799.02i) q^{8} +(-814.369 + 2029.72i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.6306	+	12.2765i	1.29317	+	1.08510i	0.991282	+	0.131759i	\(0.0420624\pi\)
							0.301891	+	0.953342i	\(0.402382\pi\)
\(3\)	−26.1976	−	38.7387i	−0.560193	−	0.828362i
							\(5\)	344.707	+	125.463i	1.23326	+	0.448870i	0.874713	−	0.484641i	\(-0.161050\pi\)
0.358548	+	0.933511i	\(0.383272\pi\)
\(7\)	−186.148	+	1055.70i	−0.205123	+	1.16331i	0.692122	+	0.721780i	\(0.256676\pi\)
							−0.897246	+	0.441532i	\(0.854435\pi\)
\(11\)	5688.07	−	2070.29i	1.28852	−	0.468983i	0.395278	−	0.918561i	\(-0.370648\pi\)
							0.893241	+	0.449579i	\(0.148426\pi\)
\(13\)	−6914.76	+	5802.17i	−0.872922	+	0.732468i	−0.964711	−	0.263310i	\(-0.915186\pi\)
							0.0917894	+	0.995778i	\(0.470741\pi\)
\(17\)	−9766.87	−	16916.7i	−0.482152	−	0.835112i	0.517638	−	0.855600i	\(-0.326811\pi\)
							−0.999790	+	0.0204878i	\(0.993478\pi\)
\(19\)	15962.9	−	27648.6i	0.533918	−	0.924773i	−0.465297	−	0.885155i	\(-0.654052\pi\)
							0.999215	−	0.0396187i	\(-0.0126143\pi\)
\(23\)	−6446.96	−	36562.5i	−0.110486	−	0.626598i	−0.988887	−	0.148672i	\(-0.952500\pi\)
							0.878400	−	0.477925i	\(-0.158611\pi\)
\(29\)	7822.01	+	6563.45i	0.0595560	+	0.0499734i	0.672080	−	0.740479i	\(-0.265401\pi\)
							−0.612524	+	0.790452i	\(0.709846\pi\)
\(31\)	8199.29	+	46500.5i	0.0494323	+	0.280344i	0.999497	−	0.0317081i	\(-0.0100947\pi\)
							−0.950065	+	0.312052i	\(0.898984\pi\)
\(37\)	−234297.	−	405814.i	−0.760432	−	1.31711i	−0.942628	−	0.333845i	\(-0.891654\pi\)
							0.182196	−	0.983262i	\(-0.441680\pi\)
\(41\)	564113.	−	473347.i	1.27827	−	1.07260i	0.284790	−	0.958590i	\(-0.408076\pi\)
							0.993480	−	0.114006i	\(-0.0363684\pi\)
\(43\)	−802607.	+	292125.i	−1.53944	+	0.560311i	−0.965912	−	0.258872i	\(-0.916649\pi\)
							−0.573531	+	0.819184i	\(0.694427\pi\)
\(47\)	−45071.4	+	255613.i	−0.0633226	+	0.359120i	0.936638	+	0.350298i	\(0.113920\pi\)
							−0.999961	+	0.00882282i	\(0.997192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.16.19	✓	120
3.2	odd	2		81.8.e.a.46.2		120
27.5	odd	18		81.8.e.a.37.2		120
27.22	even	9	inner	27.8.e.a.22.19	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.16.19	✓	120	1.1	even	1	trivial
27.8.e.a.22.19	yes	120	27.22	even	9	inner
81.8.e.a.37.2		120	27.5	odd	18
81.8.e.a.46.2		120	3.2	odd	2