Embedded newform 648.2.bb.b.131.86

• Label: 648.2.bb.b.131.86
• Level: $648$
• Weight: $2$
• Character: 648.131
• 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			131.86
Character	\(\chi\)	\(=\)	648.131
Dual form			648.2.bb.b.371.86

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.18306 - 0.774838i) q^{2} +(1.10899 + 1.33047i) q^{3} +(0.799253 - 1.83336i) q^{4} +(-0.124111 + 2.13091i) q^{5} +(2.34289 + 0.714734i) q^{6} +(0.538907 + 2.27383i) q^{7} +(-0.474991 - 2.78826i) q^{8} +(-0.540289 + 2.95095i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{47}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.18306	−	0.774838i	0.836548	−	0.547893i
							\(3\)	1.10899	+	1.33047i	0.640275	+	0.768146i
\(5\)	−0.124111	+	2.13091i	−0.0555042	+	0.952971i								0.849506	+	0.527580i	\(0.176900\pi\)
							−0.905010	+	0.425391i	\(0.860137\pi\)
\(7\)	0.538907	+	2.27383i	0.203688	+	0.859426i	0.974863	+	0.222803i	\(0.0715208\pi\)
							−0.771176	+	0.636622i	\(0.780331\pi\)
\(11\)	0.345772	+	0.525721i	0.104254	+	0.158511i	0.883847	−	0.467776i	\(-0.154944\pi\)
							−0.779593	+	0.626286i	\(0.784574\pi\)
\(13\)	−0.0566596	−	0.484754i	−0.0157146	−	0.134447i	0.983104	−	0.183050i	\(-0.0585970\pi\)
							−0.998818	+	0.0486034i	\(0.984523\pi\)
\(17\)	−0.616334	+	0.734519i	−0.149483	+	0.178147i	−0.835590	−	0.549354i	\(-0.814874\pi\)
							0.686107	+	0.727501i	\(0.259318\pi\)
\(19\)	0.827944	−	0.694728i	0.189943	−	0.159381i	−0.542857	−	0.839825i	\(-0.682657\pi\)
							0.732800	+	0.680444i	\(0.238213\pi\)
\(23\)	−2.91721	−	0.691392i	−0.608281	−	0.144165i	−0.0850792	−	0.996374i	\(-0.527114\pi\)
							−0.523202	+	0.852209i	\(0.675262\pi\)
\(29\)	2.05891	−	2.76560i	0.382331	−	0.513559i	−0.568746	−	0.822513i	\(-0.692571\pi\)
							0.951077	+	0.308954i	\(0.0999788\pi\)
\(31\)	9.21908	−	2.76001i	1.65579	−	0.495712i	0.683594	−	0.729862i	\(-0.260416\pi\)
							0.972201	+	0.234150i	\(0.0752306\pi\)
\(37\)	−3.18294	−	0.561239i	−0.523273	−	0.0922671i	−0.0942285	−	0.995551i	\(-0.530038\pi\)
							−0.429044	+	0.903283i	\(0.641150\pi\)
\(41\)	2.02322	+	0.872731i	0.315973	+	0.136298i	0.548160	−	0.836374i	\(-0.315329\pi\)
							−0.232186	+	0.972671i	\(0.574588\pi\)
\(43\)	−3.48884	−	1.75216i	−0.532043	−	0.267202i	0.162431	−	0.986720i	\(-0.448066\pi\)
							−0.694474	+	0.719518i	\(0.744363\pi\)
\(47\)	2.40577	−	8.03583i	0.350917	−	1.17215i	−0.582342	−	0.812944i	\(-0.697864\pi\)
							0.933260	−	0.359202i	\(-0.116951\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.131.86	✓	1872
8.3	odd	2	inner	648.2.bb.b.131.97	yes	1872
81.47	odd	54	inner	648.2.bb.b.371.97	yes	1872
648.371	even	54	inner	648.2.bb.b.371.86	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.131.86	✓	1872	1.1	even	1	trivial
648.2.bb.b.131.97	yes	1872	8.3	odd	2	inner
648.2.bb.b.371.86	yes	1872	648.371	even	54	inner
648.2.bb.b.371.97	yes	1872	81.47	odd	54	inner