Embedded newform 54.8.e.b.7.2

• Label: 54.8.e.b.7.2
• Level: $54$
• Weight: $8$
• Character: 54.7
• Analytic conductor: $16.869$
• Analytic rank: $0$
• Dimension: $66$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	54.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.8687913761\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.2
Character	\(\chi\)	\(=\)	54.7
Dual form			54.8.e.b.31.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.51754 - 2.73616i) q^{2} +(-45.6144 - 10.3117i) q^{3} +(49.0268 - 41.1384i) q^{4} +(-8.01649 - 45.4638i) q^{5} +(-371.122 + 47.2898i) q^{6} +(1037.92 + 870.920i) q^{7} +(256.000 - 443.405i) q^{8} +(1974.34 + 940.721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 213 q^{5} - 1320 q^{6} - 1677 q^{7} + 16896 q^{8} + 6468 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.51754	−	2.73616i	0.664463	−	0.241845i
							\(3\)	−45.6144	−	10.3117i	−0.975387	−	0.220498i
\(5\)	−8.01649	−	45.4638i	−0.0286807	−	0.162656i								0.967104	−	0.254383i	\(-0.0818724\pi\)
							−0.995784	+	0.0917269i	\(0.970761\pi\)
\(7\)	1037.92	+	870.920i	1.14373	+	0.959700i	0.999554	−	0.0298530i	\(-0.00950392\pi\)
							0.144171	+	0.989553i	\(0.453948\pi\)
\(11\)	1188.38	−	6739.64i	0.269204	−	1.52673i	−0.487587	−	0.873074i	\(-0.662123\pi\)
							0.756791	−	0.653657i	\(-0.226766\pi\)
\(13\)	−12837.4	−	4672.43i	−1.62060	−	0.589849i	−0.637101	−	0.770780i	\(-0.719867\pi\)
							−0.983496	+	0.180931i	\(0.942089\pi\)
\(17\)	−11457.2	−	19844.4i	−0.565596	−	0.979641i	−0.996994	−	0.0774793i	\(-0.975313\pi\)
							0.431398	−	0.902162i	\(-0.358021\pi\)
\(19\)	8851.46	−	15331.2i	0.296058	−	0.512788i	−0.679172	−	0.733979i	\(-0.737661\pi\)
							0.975231	+	0.221191i	\(0.0709943\pi\)
\(23\)	12871.5	−	10800.5i	0.220588	−	0.185096i	−0.525796	−	0.850611i	\(-0.676233\pi\)
							0.746384	+	0.665515i	\(0.231788\pi\)
\(29\)	124020.	−	45139.6i	0.944276	−	0.343689i	0.176423	−	0.984314i	\(-0.443547\pi\)
							0.767853	+	0.640626i	\(0.221325\pi\)
\(31\)	−97585.0	+	81883.5i	−0.588325	+	0.493663i	−0.887669	−	0.460482i	\(-0.847677\pi\)
							0.299344	+	0.954145i	\(0.403232\pi\)
\(37\)	68300.9	+	118301.i	0.221677	+	0.383956i	0.955317	−	0.295582i	\(-0.0955137\pi\)
							−0.733640	+	0.679538i	\(0.762180\pi\)
\(41\)	−308334.	−	112224.i	−0.698679	−	0.254298i	−0.0318320	−	0.999493i	\(-0.510134\pi\)
							−0.666847	+	0.745195i	\(0.732356\pi\)
\(43\)	166949.	−	946812.i	0.320216	−	1.81604i	−0.221140	−	0.975242i	\(-0.570978\pi\)
							0.541356	−	0.840794i	\(-0.317911\pi\)
\(47\)	−513226.	−	430647.i	−0.721051	−	0.605033i	0.206625	−	0.978420i	\(-0.433752\pi\)
							−0.927676	+	0.373387i	\(0.878196\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.8.e.b.7.2	✓	66
27.4	even	9	inner	54.8.e.b.31.2	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.e.b.7.2	✓	66	1.1	even	1	trivial
54.8.e.b.31.2	yes	66	27.4	even	9	inner