Embedded newform 54.8.e.b.7.3

• Label: 54.8.e.b.7.3
• 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			7.3
Character	\(\chi\)	\(=\)	54.7
Dual form			54.8.e.b.31.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.51754 - 2.73616i) q^{2} +(-31.2009 + 34.8354i) q^{3} +(49.0268 - 41.1384i) q^{4} +(-25.2309 - 143.092i) q^{5} +(-139.239 + 347.247i) q^{6} +(591.389 + 496.234i) q^{7} +(256.000 - 443.405i) q^{8} +(-240.006 - 2173.79i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 213 q^{5} - 1320 q^{6} - 1677 q^{7} + 16896 q^{8} + 6468 q^{9}+O(q^{10}) \)

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
							\(3\)	−31.2009	+	34.8354i	−0.667180	+	0.744897i
\(5\)	−25.2309	−	143.092i	−0.0902689	−	0.511940i								−0.996095	−	0.0882902i	\(-0.971860\pi\)
							0.905826	−	0.423650i	\(-0.139251\pi\)
\(7\)	591.389	+	496.234i	0.651673	+	0.546819i	0.907578	−	0.419883i	\(-0.137929\pi\)
							−0.255905	+	0.966702i	\(0.582373\pi\)
\(11\)	−1082.56	+	6139.52i	−0.245233	+	1.39078i	0.574718	+	0.818351i	\(0.305112\pi\)
							−0.819951	+	0.572433i	\(0.805999\pi\)
\(13\)	822.776	+	299.466i	0.103867	+	0.0378047i	0.393431	−	0.919354i	\(-0.371288\pi\)
							−0.289564	+	0.957159i	\(0.593510\pi\)
\(17\)	13734.8	+	23789.4i	0.678033	+	1.17439i	0.975572	+	0.219678i	\(0.0705006\pi\)
							−0.297540	+	0.954709i	\(0.596166\pi\)
\(19\)	−21741.2	+	37656.9i	−0.727188	+	1.25953i	0.230879	+	0.972982i	\(0.425840\pi\)
							−0.958067	+	0.286544i	\(0.907494\pi\)
\(23\)	−78527.1	+	65892.1i	−1.34577	+	1.12924i	−0.365672	+	0.930744i	\(0.619161\pi\)
							−0.980102	+	0.198495i	\(0.936395\pi\)
\(29\)	184173.	−	67033.7i	1.40228	−	0.510387i	0.473424	−	0.880834i	\(-0.343018\pi\)
							0.928854	+	0.370447i	\(0.120795\pi\)
\(31\)	−106146.	+	89067.3i	−0.639939	+	0.536973i	−0.904000	−	0.427533i	\(-0.859383\pi\)
							0.264060	+	0.964506i	\(0.414938\pi\)
\(37\)	−6338.58	−	10978.7i	−0.0205724	−	0.0356325i	0.855556	−	0.517710i	\(-0.173216\pi\)
							−0.876128	+	0.482078i	\(0.839882\pi\)
\(41\)	381181.	+	138738.i	0.863749	+	0.314379i	0.735633	−	0.677380i	\(-0.236885\pi\)
							0.128116	+	0.991759i	\(0.459107\pi\)
\(43\)	44010.6	−	249596.i	0.0844146	−	0.478739i	−0.913067	−	0.407810i	\(-0.866293\pi\)
							0.997481	−	0.0709288i	\(-0.0225963\pi\)
\(47\)	177652.	+	149068.i	0.249590	+	0.209431i	0.758996	−	0.651095i	\(-0.225690\pi\)
							−0.509406	+	0.860527i	\(0.670135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.8.e.b.7.3	✓	66
27.4	even	9	inner	54.8.e.b.31.3	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.e.b.7.3	✓	66	1.1	even	1	trivial
54.8.e.b.31.3	yes	66	27.4	even	9	inner