Embedded newform 54.5.f.a.41.6

• Label: 54.5.f.a.41.6
• Level: $54$
• Weight: $5$
• Character: 54.41
• Analytic conductor: $5.582$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	54.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58197800653\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			41.6
Character	\(\chi\)	\(=\)	54.41
Dual form			54.5.f.a.29.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.78546 + 0.491151i) q^{2} +(8.96144 - 0.832223i) q^{3} +(7.51754 - 2.73616i) q^{4} +(16.6260 - 19.8141i) q^{5} +(-24.5530 + 6.71954i) q^{6} +(-37.8941 - 13.7923i) q^{7} +(-19.5959 + 11.3137i) q^{8} +(79.6148 - 14.9158i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 18 q^{5} - 96 q^{6} + 204 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.78546	+	0.491151i	−0.696364	+	0.122788i
							\(3\)	8.96144	−	0.832223i	0.995716	−	0.0924692i
\(5\)	16.6260	−	19.8141i	0.665041	−	0.792565i								−0.323059	−	0.946379i	\(-0.604711\pi\)
							0.988100	+	0.153814i	\(0.0491557\pi\)
\(7\)	−37.8941	−	13.7923i	−0.773348	−	0.281476i	−0.0749521	−	0.997187i	\(-0.523880\pi\)
							−0.698396	+	0.715711i	\(0.746103\pi\)
\(11\)	34.3945	+	40.9897i	0.284252	+	0.338758i	0.889210	−	0.457499i	\(-0.151255\pi\)
							−0.604958	+	0.796257i	\(0.706810\pi\)
\(13\)	55.9149	−	317.109i	0.330857	−	1.87639i	−0.133971	−	0.990985i	\(-0.542773\pi\)
							0.464829	−	0.885401i	\(-0.346116\pi\)
\(17\)	393.006	+	226.902i	1.35988	+	0.785129i	0.989608	−	0.143794i	\(-0.0459302\pi\)
							0.370275	+	0.928922i	\(0.379264\pi\)
\(19\)	−58.2492	−	100.891i	−0.161355	−	0.279475i	0.774000	−	0.633186i	\(-0.218253\pi\)
							−0.935355	+	0.353710i	\(0.884920\pi\)
\(23\)	48.9201	+	134.407i	0.0924766	+	0.254077i	0.977304	−	0.211842i	\(-0.0679461\pi\)
							−0.884827	+	0.465919i	\(0.845724\pi\)
\(29\)	−748.634	+	132.004i	−0.890172	+	0.156961i	−0.599987	−	0.800009i	\(-0.704828\pi\)
							−0.290184	+	0.956971i	\(0.593717\pi\)
\(31\)	−1046.37	+	380.848i	−1.08884	+	0.396304i	−0.823190	−	0.567767i	\(-0.807808\pi\)
							−0.265646	+	0.964071i	\(0.585585\pi\)
\(37\)	−1043.50	+	1807.39i	−0.762233	+	1.32023i	0.179463	+	0.983765i	\(0.442564\pi\)
							−0.941697	+	0.336462i	\(0.890769\pi\)
\(41\)	1604.28	+	282.878i	0.954360	+	0.168279i	0.629082	−	0.777339i	\(-0.283431\pi\)
							0.325278	+	0.945618i	\(0.394542\pi\)
\(43\)	920.778	−	772.625i	0.497987	−	0.417861i	−0.358891	−	0.933379i	\(-0.616845\pi\)
							0.856878	+	0.515519i	\(0.172401\pi\)
\(47\)	−375.095	+	1030.56i	−0.169803	+	0.466530i	−0.995181	−	0.0980500i	\(-0.968740\pi\)
							0.825379	+	0.564580i	\(0.190962\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.5.f.a.41.6	yes	72
3.2	odd	2		162.5.f.a.71.12		72
27.2	odd	18	inner	54.5.f.a.29.6	✓	72
27.25	even	9		162.5.f.a.89.12		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.5.f.a.29.6	✓	72	27.2	odd	18	inner
54.5.f.a.41.6	yes	72	1.1	even	1	trivial
162.5.f.a.71.12		72	3.2	odd	2
162.5.f.a.89.12		72	27.25	even	9