Embedded newform 648.2.bb.b.11.19

• Label: 648.2.bb.b.11.19
• 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			11.19
Character	\(\chi\)	\(=\)	648.11
Dual form			648.2.bb.b.59.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23855 - 0.682644i) q^{2} +(-0.996621 - 1.41660i) q^{3} +(1.06799 + 1.69097i) q^{4} +(0.173565 + 0.0202869i) q^{5} +(0.267330 + 2.43486i) q^{6} +(1.54089 + 3.06816i) q^{7} +(-0.168429 - 2.82341i) q^{8} +(-1.01349 + 2.82362i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23855	−	0.682644i	−0.875785	−	0.482702i
							\(3\)	−0.996621	−	1.41660i	−0.575399	−	0.817873i
\(5\)	0.173565	+	0.0202869i	0.0776208	+	0.00907257i								0.154815	−	0.987944i	\(-0.450522\pi\)
							−0.0771938	+	0.997016i	\(0.524596\pi\)
\(7\)	1.54089	+	3.06816i	0.582400	+	1.15965i	0.971185	+	0.238325i	\(0.0765985\pi\)
							−0.388785	+	0.921328i	\(0.627105\pi\)
\(11\)	−1.13606	−	0.490047i	−0.342534	−	0.147755i	0.217862	−	0.975980i	\(-0.430092\pi\)
							−0.560396	+	0.828225i	\(0.689351\pi\)
\(13\)	−0.290570	−	1.22601i	−0.0805896	−	0.340034i	0.917782	−	0.397085i	\(-0.129978\pi\)
							−0.998371	+	0.0570512i	\(0.981830\pi\)
\(17\)	−7.14162	+	1.25926i	−1.73210	+	0.305416i	−0.948717	−	0.316126i	\(-0.897618\pi\)
							−0.783381	+	0.621542i	\(0.786507\pi\)
\(19\)	0.248121	−	1.40717i	0.0569230	−	0.322826i	−0.943028	−	0.332713i	\(-0.892036\pi\)
							0.999951	+	0.00988676i	\(0.00314711\pi\)
\(23\)	−2.44153	−	1.22618i	−0.509093	−	0.255676i	0.175660	−	0.984451i	\(-0.443794\pi\)
							−0.684754	+	0.728775i	\(0.740090\pi\)
\(29\)	−2.80610	−	9.37302i	−0.521079	−	1.74053i	−0.661022	−	0.750367i	\(-0.729877\pi\)
							0.139943	−	0.990160i	\(-0.455308\pi\)
\(31\)	−5.37686	−	8.17511i	−0.965712	−	1.46829i	−0.881690	−	0.471830i	\(-0.843594\pi\)
							−0.0840224	−	0.996464i	\(-0.526777\pi\)
\(37\)	−3.47429	+	9.54554i	−0.571170	+	1.56928i	0.231488	+	0.972838i	\(0.425641\pi\)
							−0.802658	+	0.596440i	\(0.796582\pi\)
\(41\)	−4.55764	+	4.29991i	−0.711783	+	0.671533i	−0.954380	−	0.298594i	\(-0.903482\pi\)
							0.242597	+	0.970127i	\(0.422001\pi\)
\(43\)	1.71656	+	2.30574i	0.261773	+	0.351622i	0.913505	−	0.406827i	\(-0.133365\pi\)
							−0.651732	+	0.758449i	\(0.725957\pi\)
\(47\)	8.76466	+	5.76461i	1.27846	+	0.840854i	0.993066	−	0.117557i	\(-0.0375063\pi\)
							0.285391	+	0.958411i	\(0.407877\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.11.19	✓	1872
8.3	odd	2	inner	648.2.bb.b.11.32	yes	1872
81.59	odd	54	inner	648.2.bb.b.59.32	yes	1872
648.59	even	54	inner	648.2.bb.b.59.19	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.11.19	✓	1872	1.1	even	1	trivial
648.2.bb.b.11.32	yes	1872	8.3	odd	2	inner
648.2.bb.b.59.19	yes	1872	648.59	even	54	inner
648.2.bb.b.59.32	yes	1872	81.59	odd	54	inner