Embedded newform 54.8.e.b.7.11

• Label: 54.8.e.b.7.11
• Level: $54$
• Weight: $8$
• Character: 54.7
• 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			7.11
Character	\(\chi\)	\(=\)	54.7
Dual form			54.8.e.b.31.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.51754 - 2.73616i) q^{2} +(43.9672 + 15.9339i) q^{3} +(49.0268 - 41.1384i) q^{4} +(72.2267 + 409.618i) q^{5} +(374.123 - 0.517653i) q^{6} +(-152.227 - 127.734i) q^{7} +(256.000 - 443.405i) q^{8} +(1679.22 + 1401.13i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 213 q^{5} - 1320 q^{6} - 1677 q^{7} + 16896 q^{8} + 6468 q^{9}+O(q^{10}) \)

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
							\(3\)	43.9672	+	15.9339i	0.940165	+	0.340720i
\(5\)	72.2267	+	409.618i	0.258406	+	1.46549i								0.787176	+	0.616729i	\(0.211542\pi\)
							−0.528770	+	0.848765i	\(0.677346\pi\)
\(7\)	−152.227	−	127.734i	−0.167745	−	0.140755i	0.555051	−	0.831816i	\(-0.312699\pi\)
							−0.722796	+	0.691062i	\(0.757143\pi\)
\(11\)	−938.612	+	5323.14i	−0.212624	+	1.20585i	0.672359	+	0.740225i	\(0.265281\pi\)
							−0.884983	+	0.465624i	\(0.845830\pi\)
\(13\)	−7254.96	−	2640.59i	−0.915869	−	0.333349i	−0.159275	−	0.987234i	\(-0.550916\pi\)
							−0.756594	+	0.653885i	\(0.773138\pi\)
\(17\)	8063.48	+	13966.3i	0.398062	+	0.689464i	0.993487	−	0.113947i	\(-0.0363494\pi\)
							−0.595425	+	0.803411i	\(0.703016\pi\)
\(19\)	25405.5	−	44003.7i	0.849749	−	1.47181i	−0.0316837	−	0.999498i	\(-0.510087\pi\)
							0.881432	−	0.472310i	\(-0.156580\pi\)
\(23\)	−34815.3	+	29213.5i	−0.596653	+	0.500652i	−0.890368	−	0.455241i	\(-0.849553\pi\)
							0.293715	+	0.955893i	\(0.405108\pi\)
\(29\)	177240.	−	64510.0i	1.34949	−	0.491172i	0.436698	−	0.899608i	\(-0.356148\pi\)
							0.912787	+	0.408436i	\(0.133926\pi\)
\(31\)	142294.	−	119398.i	0.857865	−	0.719835i	−0.103642	−	0.994615i	\(-0.533050\pi\)
							0.961507	+	0.274780i	\(0.0886051\pi\)
\(37\)	286706.	+	496590.i	0.930532	+	1.61173i	0.782413	+	0.622760i	\(0.213989\pi\)
							0.148119	+	0.988970i	\(0.452678\pi\)
\(41\)	−192028.	−	69892.5i	−0.435132	−	0.158375i	0.115161	−	0.993347i	\(-0.463262\pi\)
							−0.550293	+	0.834972i	\(0.685484\pi\)
\(43\)	105674.	−	599308.i	0.202689	−	1.14950i	−0.698347	−	0.715759i	\(-0.746081\pi\)
							0.901036	−	0.433745i	\(-0.142808\pi\)
\(47\)	−487813.	−	409324.i	−0.685348	−	0.575075i	0.232216	−	0.972664i	\(-0.425402\pi\)
							−0.917564	+	0.397589i	\(0.869847\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.8.e.b.7.11	✓	66
27.4	even	9	inner	54.8.e.b.31.11	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.e.b.7.11	✓	66	1.1	even	1	trivial
54.8.e.b.31.11	yes	66	27.4	even	9	inner