Embedded newform 27.8.e.a.16.13

• Label: 27.8.e.a.16.13
• 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			16.13
Character	\(\chi\)	\(=\)	27.16
Dual form			27.8.e.a.22.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.72049 + 3.96096i) q^{2} +(-42.6841 + 19.1067i) q^{3} +(-15.6331 - 88.6600i) q^{4} +(139.230 + 50.6754i) q^{5} +(-277.171 - 78.8772i) q^{6} +(26.9258 - 152.704i) q^{7} +(671.762 - 1163.53i) q^{8} +(1456.87 - 1631.11i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.72049	+	3.96096i	0.417237	+	0.350103i	0.827111	−	0.562039i	\(-0.189983\pi\)
							−0.409874	+	0.912142i	\(0.634427\pi\)
\(3\)	−42.6841	+	19.1067i	−0.912729	+	0.408565i
							\(5\)	139.230	+	50.6754i	0.498123	+	0.181302i	0.578849	−	0.815434i	\(-0.303502\pi\)
−0.0807267	+	0.996736i	\(0.525724\pi\)
\(7\)	26.9258	−	152.704i	0.0296705	−	0.168270i	−0.966372	−	0.257148i	\(-0.917217\pi\)
							0.996042	+	0.0888783i	\(0.0283282\pi\)
\(11\)	5841.36	−	2126.08i	1.32324	−	0.481621i	0.418747	−	0.908103i	\(-0.362469\pi\)
							0.904496	+	0.426482i	\(0.140247\pi\)
\(13\)	2572.64	−	2158.70i	0.324771	−	0.272515i	−0.465794	−	0.884893i	\(-0.654231\pi\)
							0.790565	+	0.612378i	\(0.209787\pi\)
\(17\)	−1683.26	−	2915.50i	−0.0830962	−	0.143927i	0.821482	−	0.570234i	\(-0.193148\pi\)
							−0.904578	+	0.426307i	\(0.859814\pi\)
\(19\)	9886.31	−	17123.6i	0.330672	−	0.572740i	−0.651972	−	0.758243i	\(-0.726058\pi\)
							0.982644	+	0.185503i	\(0.0593915\pi\)
\(23\)	−5800.79	−	32897.9i	−0.0994122	−	0.563795i	−0.993306	−	0.115515i	\(-0.963148\pi\)
							0.893894	−	0.448279i	\(-0.147963\pi\)
\(29\)	−61227.3	−	51375.8i	−0.466178	−	0.391170i	0.379220	−	0.925307i	\(-0.376192\pi\)
							−0.845398	+	0.534136i	\(0.820637\pi\)
\(31\)	6384.07	+	36205.9i	0.0384886	+	0.218280i	0.997986	−	0.0634387i	\(-0.0202067\pi\)
							−0.959497	+	0.281718i	\(0.909096\pi\)
\(37\)	283033.	+	490228.i	0.918611	+	1.59108i	0.801527	+	0.597959i	\(0.204021\pi\)
							0.117084	+	0.993122i	\(0.462645\pi\)
\(41\)	−54162.7	+	45447.9i	−0.122732	+	0.102984i	−0.702088	−	0.712090i	\(-0.747749\pi\)
							0.579356	+	0.815075i	\(0.303304\pi\)
\(43\)	743122.	−	270474.i	1.42535	−	0.518784i	0.489752	−	0.871862i	\(-0.337087\pi\)
							0.935594	+	0.353078i	\(0.114865\pi\)
\(47\)	101965.	−	578271.i	0.143254	−	0.812435i	−0.825498	−	0.564405i	\(-0.809106\pi\)
							0.968752	−	0.248030i	\(-0.0797833\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.16.13	✓	120
3.2	odd	2		81.8.e.a.46.8		120
27.5	odd	18		81.8.e.a.37.8		120
27.22	even	9	inner	27.8.e.a.22.13	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.16.13	✓	120	1.1	even	1	trivial
27.8.e.a.22.13	yes	120	27.22	even	9	inner
81.8.e.a.37.8		120	27.5	odd	18
81.8.e.a.46.8		120	3.2	odd	2