Embedded newform 648.2.bb.b.11.18

• Label: 648.2.bb.b.11.18
• Level: $648$
• Weight: $2$
• Character: 648.11
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1872$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.bb (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(1872\)
Relative dimension:	\(104\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			11.18
Character	\(\chi\)	\(=\)	648.11
Dual form			648.2.bb.b.59.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24556 - 0.669769i) q^{2} +(-1.70483 - 0.305889i) q^{3} +(1.10282 + 1.66847i) q^{4} +(2.63396 + 0.307866i) q^{5} +(1.91858 + 1.52284i) q^{6} +(-0.382909 - 0.762434i) q^{7} +(-0.256135 - 2.81681i) q^{8} +(2.81286 + 1.04297i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{13}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24556	−	0.669769i	−0.880741	−	0.473598i
							\(3\)	−1.70483	−	0.305889i	−0.984282	−	0.176605i
\(5\)	2.63396	+	0.307866i	1.17794	+	0.137682i								0.682451	−	0.730931i	\(-0.260914\pi\)
							0.495491	+	0.868613i	\(0.334988\pi\)
\(7\)	−0.382909	−	0.762434i	−0.144726	−	0.288173i	0.809577	−	0.587013i	\(-0.199696\pi\)
							−0.954303	+	0.298841i	\(0.903400\pi\)
\(11\)	0.252798	+	0.109047i	0.0762215	+	0.0328788i	0.433849	−	0.900985i	\(-0.357155\pi\)
							−0.357628	+	0.933864i	\(0.616414\pi\)
\(13\)	−0.477261	−	2.01372i	−0.132368	−	0.558506i	−0.998297	−	0.0583382i	\(-0.981420\pi\)
							0.865929	−	0.500168i	\(-0.166728\pi\)
\(17\)	3.06877	−	0.541106i	0.744285	−	0.131238i	0.211370	−	0.977406i	\(-0.432208\pi\)
							0.532915	+	0.846169i	\(0.321096\pi\)
\(19\)	−0.419935	+	2.38157i	−0.0963398	+	0.546370i	0.897989	+	0.440018i	\(0.145028\pi\)
							−0.994329	+	0.106352i	\(0.966083\pi\)
\(23\)	2.10446	+	1.05690i	0.438811	+	0.220379i	0.654468	−	0.756090i	\(-0.272893\pi\)
							−0.215656	+	0.976469i	\(0.569189\pi\)
\(29\)	2.63911	+	8.81523i	0.490070	+	1.63695i	0.742876	+	0.669429i	\(0.233461\pi\)
							−0.252806	+	0.967517i	\(0.581353\pi\)
\(31\)	−0.0573756	−	0.0872354i	−0.0103050	−	0.0156679i	0.830299	−	0.557319i	\(-0.188170\pi\)
							−0.840604	+	0.541651i	\(0.817799\pi\)
\(37\)	1.49973	−	4.12046i	0.246553	−	0.677400i	−0.753253	−	0.657731i	\(-0.771516\pi\)
							0.999807	−	0.0196692i	\(-0.00626130\pi\)
\(41\)	7.38303	−	6.96553i	1.15304	−	1.08783i	0.158089	−	0.987425i	\(-0.449467\pi\)
							0.994946	−	0.100408i	\(-0.0320147\pi\)
\(43\)	−3.48580	−	4.68225i	−0.531580	−	0.714036i	0.452500	−	0.891765i	\(-0.350532\pi\)
							−0.984079	+	0.177729i	\(0.943125\pi\)
\(47\)	6.40021	+	4.20948i	0.933567	+	0.614016i	0.922531	−	0.385923i	\(-0.126117\pi\)
							0.0110354	+	0.999939i	\(0.496487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.11.18	✓	1872
8.3	odd	2	inner	648.2.bb.b.11.34	yes	1872
81.59	odd	54	inner	648.2.bb.b.59.34	yes	1872
648.59	even	54	inner	648.2.bb.b.59.18	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.11.18	✓	1872	1.1	even	1	trivial
648.2.bb.b.11.34	yes	1872	8.3	odd	2	inner
648.2.bb.b.59.18	yes	1872	648.59	even	54	inner
648.2.bb.b.59.34	yes	1872	81.59	odd	54	inner