Embedded newform 27.8.e.a.4.9

• Label: 27.8.e.a.4.9
• Level: $27$
• Weight: $8$
• Character: 27.4
• 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			4.9
Character	\(\chi\)	\(=\)	27.4
Dual form			27.8.e.a.7.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.79327 - 1.01667i) q^{2} +(-34.1687 - 31.9296i) q^{3} +(-91.2849 - 76.5971i) q^{4} +(17.7337 - 100.573i) q^{5} +(62.9806 + 123.926i) q^{6} +(-71.4548 + 59.9577i) q^{7} +(367.352 + 636.273i) q^{8} +(147.997 + 2181.99i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.79327	−	1.01667i	−0.246893	−	0.0898616i	0.215610	−	0.976480i	\(-0.430826\pi\)
							−0.462502	+	0.886618i	\(0.653048\pi\)
\(3\)	−34.1687	−	31.9296i	−0.730641	−	0.682762i
							\(5\)	17.7337	−	100.573i	0.0634461	−	0.359821i	−0.936512	−	0.350636i	\(-0.885965\pi\)
0.999958	−	0.00918458i	\(-0.00292358\pi\)
\(7\)	−71.4548	+	59.9577i	−0.0787387	+	0.0660696i	−0.681308	−	0.731997i	\(-0.738588\pi\)
							0.602569	+	0.798067i	\(0.294144\pi\)
\(11\)	148.107	+	839.955i	0.0335506	+	0.190275i	0.996977	−	0.0776985i	\(-0.0247571\pi\)
							−0.963426	+	0.267973i	\(0.913646\pi\)
\(13\)	−2485.38	+	904.604i	−0.313755	+	0.114198i	−0.494098	−	0.869406i	\(-0.664502\pi\)
							0.180343	+	0.983604i	\(0.442279\pi\)
\(17\)	−3788.34	+	6561.60i	−0.187016	+	0.323921i	−0.944254	−	0.329218i	\(-0.893215\pi\)
							0.757238	+	0.653139i	\(0.226548\pi\)
\(19\)	10954.9	+	18974.4i	0.366411	+	0.634643i	0.989002	−	0.147905i	\(-0.0472529\pi\)
							−0.622590	+	0.782548i	\(0.713920\pi\)
\(23\)	−44040.8	−	36954.6i	−0.754758	−	0.633317i	0.181998	−	0.983299i	\(-0.441743\pi\)
							−0.936757	+	0.349981i	\(0.886188\pi\)
\(29\)	−163583.	−	59539.3i	−1.24550	−	0.453326i	−0.366624	−	0.930369i	\(-0.619486\pi\)
							−0.878880	+	0.477043i	\(0.841709\pi\)
\(31\)	−44764.6	−	37562.0i	−0.269879	−	0.226455i	0.497797	−	0.867294i	\(-0.334142\pi\)
							−0.767676	+	0.640838i	\(0.778587\pi\)
\(37\)	−160525.	+	278037.i	−0.520999	+	0.902396i	0.478703	+	0.877977i	\(0.341107\pi\)
							−0.999702	+	0.0244192i	\(0.992226\pi\)
\(41\)	−178794.	+	65075.5i	−0.405143	+	0.147460i	−0.536550	−	0.843869i	\(-0.680273\pi\)
							0.131407	+	0.991328i	\(0.458051\pi\)
\(43\)	111455.	+	632091.i	0.213776	+	1.21238i	0.883018	+	0.469339i	\(0.155508\pi\)
							−0.669242	+	0.743045i	\(0.733381\pi\)
\(47\)	−705189.	+	591724.i	−0.990747	+	0.831336i	−0.985676	−	0.168651i	\(-0.946059\pi\)
							−0.00507165	+	0.999987i	\(0.501614\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.4.9	✓	120
3.2	odd	2		81.8.e.a.64.12		120
27.7	even	9	inner	27.8.e.a.7.9	yes	120
27.20	odd	18		81.8.e.a.19.12		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.9	✓	120	1.1	even	1	trivial
27.8.e.a.7.9	yes	120	27.7	even	9	inner
81.8.e.a.19.12		120	27.20	odd	18
81.8.e.a.64.12		120	3.2	odd	2