Embedded newform 54.8.e.b.13.2

• Label: 54.8.e.b.13.2
• Level: $54$
• Weight: $8$
• Character: 54.13
• Analytic conductor: $16.869$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• 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			13.2
Character	\(\chi\)	\(=\)	54.13
Dual form			54.8.e.b.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38919 - 7.87846i) q^{2} +(-44.6459 - 13.9191i) q^{3} +(-60.1403 + 21.8893i) q^{4} +(200.637 + 168.354i) q^{5} +(-47.6395 + 371.077i) q^{6} +(-437.811 - 159.350i) q^{7} +(256.000 + 443.405i) q^{8} +(1799.52 + 1242.86i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 213 q^{5} - 1320 q^{6} - 1677 q^{7} + 16896 q^{8} + 6468 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38919	−	7.87846i	−0.122788	−	0.696364i
							\(3\)	−44.6459	−	13.9191i	−0.954679	−	0.297637i
\(5\)	200.637	+	168.354i	0.717821	+	0.602323i								0.926782	−	0.375601i	\(-0.122564\pi\)
							−0.208961	+	0.977924i	\(0.567008\pi\)
\(7\)	−437.811	−	159.350i	−0.482440	−	0.175594i	0.0893395	−	0.996001i	\(-0.471524\pi\)
							−0.571779	+	0.820408i	\(0.693747\pi\)
\(11\)	4957.59	−	4159.91i	1.12304	−	0.942345i	0.124289	−	0.992246i	\(-0.460335\pi\)
							0.998754	+	0.0499014i	\(0.0158907\pi\)
\(13\)	−2293.84	+	13009.0i	−0.289575	+	1.64226i	0.398895	+	0.916997i	\(0.369394\pi\)
							−0.688470	+	0.725265i	\(0.741717\pi\)
\(17\)	12862.4	−	22278.3i	0.634965	−	1.09979i	−0.351558	−	0.936166i	\(-0.614348\pi\)
							0.986523	−	0.163625i	\(-0.0523188\pi\)
\(19\)	−24342.5	−	42162.4i	−0.814193	−	1.41022i	−0.909906	−	0.414814i	\(-0.863847\pi\)
							0.0957136	−	0.995409i	\(-0.469487\pi\)
\(23\)	−72161.9	+	26264.8i	−1.23669	+	0.450118i	−0.875883	−	0.482523i	\(-0.839721\pi\)
							−0.360806	+	0.932641i	\(0.617498\pi\)
\(29\)	−760.219	−	4311.42i	−0.00578823	−	0.0328267i	0.981777	−	0.190037i	\(-0.0608609\pi\)
							−0.987565	+	0.157211i	\(0.949750\pi\)
\(31\)	258033.	−	93916.2i	1.55564	−	0.566206i	0.585907	−	0.810378i	\(-0.300738\pi\)
							0.969732	+	0.244172i	\(0.0785162\pi\)
\(37\)	79879.0	−	138354.i	0.259255	−	0.449042i	−0.706788	−	0.707426i	\(-0.749856\pi\)
							0.966042	+	0.258383i	\(0.0831898\pi\)
\(41\)	27659.0	−	156862.i	0.0626747	−	0.355446i	−0.937301	−	0.348520i	\(-0.886684\pi\)
							0.999976	−	0.00692588i	\(-0.00220459\pi\)
\(43\)	718387.	−	602798.i	1.37790	−	1.15620i	0.407923	−	0.913017i	\(-0.366253\pi\)
							0.969981	−	0.243182i	\(-0.0781911\pi\)
\(47\)	269870.	+	98224.6i	0.379151	+	0.138000i	0.524563	−	0.851372i	\(-0.324229\pi\)
							−0.145412	+	0.989371i	\(0.546451\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.8.e.b.13.2	✓	66
27.25	even	9	inner	54.8.e.b.25.2	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.e.b.13.2	✓	66	1.1	even	1	trivial
54.8.e.b.25.2	yes	66	27.25	even	9	inner