Embedded newform 648.2.bb.b.59.13

• Label: 648.2.bb.b.59.13
• Level: $648$
• Weight: $2$
• Character: 648.59
• 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			59.13
Character	\(\chi\)	\(=\)	648.59
Dual form			648.2.bb.b.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32861 - 0.484545i) q^{2} +(-1.07834 - 1.35543i) q^{3} +(1.53043 + 1.28755i) q^{4} +(1.48986 - 0.174139i) q^{5} +(0.775925 + 2.32335i) q^{6} +(-0.383526 + 0.763663i) q^{7} +(-1.40948 - 2.45222i) q^{8} +(-0.674386 + 2.92322i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{41}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32861	−	0.484545i	−0.939472	−	0.342625i
							\(3\)	−1.07834	−	1.35543i	−0.622577	−	0.782558i
\(5\)	1.48986	−	0.174139i	0.666284	−	0.0778774i								0.223774	−	0.974641i	\(-0.428162\pi\)
							0.442510	+	0.896764i	\(0.354088\pi\)
\(7\)	−0.383526	+	0.763663i	−0.144959	+	0.288637i	−0.954382	−	0.298589i	\(-0.903484\pi\)
							0.809423	+	0.587227i	\(0.199780\pi\)
\(11\)	0.247101	−	0.106589i	0.0745037	−	0.0321378i	−0.358502	−	0.933529i	\(-0.616712\pi\)
							0.433006	+	0.901391i	\(0.357453\pi\)
\(13\)	−0.416809	+	1.75866i	−0.115602	+	0.487764i	0.884185	+	0.467137i	\(0.154714\pi\)
							−0.999787	+	0.0206271i	\(0.993434\pi\)
\(17\)	7.49712	+	1.32194i	1.81832	+	0.320619i	0.975902	−	0.218208i	\(-0.0700213\pi\)
							0.842416	+	0.538827i	\(0.181132\pi\)
\(19\)	0.656155	+	3.72124i	0.150532	+	0.853710i	0.962757	+	0.270367i	\(0.0871449\pi\)
							−0.812225	+	0.583344i	\(0.801744\pi\)
\(23\)	−7.57780	+	3.80572i	−1.58008	+	0.793547i	−0.999782	−	0.0208572i	\(-0.993360\pi\)
							−0.580299	+	0.814404i	\(0.697064\pi\)
\(29\)	1.06817	−	3.56792i	0.198353	−	0.662546i	−0.799713	−	0.600383i	\(-0.795015\pi\)
							0.998066	−	0.0621633i	\(-0.0198000\pi\)
\(31\)	−5.55393	+	8.44434i	−0.997515	+	1.51665i	−0.148065	+	0.988978i	\(0.547305\pi\)
							−0.849450	+	0.527670i	\(0.823066\pi\)
\(37\)	1.39309	+	3.82748i	0.229022	+	0.629234i	0.999971	−	0.00766395i	\(-0.00243954\pi\)
							−0.770948	+	0.636898i	\(0.780217\pi\)
\(41\)	4.70314	+	4.43718i	0.734507	+	0.692972i	0.959601	−	0.281366i	\(-0.0907873\pi\)
							−0.225094	+	0.974337i	\(0.572269\pi\)
\(43\)	4.69896	−	6.31180i	0.716585	−	0.962540i	−0.283409	−	0.958999i	\(-0.591465\pi\)
							0.999994	−	0.00354098i	\(-0.00112713\pi\)
\(47\)	10.2207	−	6.72224i	1.49084	−	0.980539i	0.497234	−	0.867616i	\(-0.334349\pi\)
							0.993604	−	0.112923i	\(-0.0360213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.59.13	yes	1872
8.3	odd	2	inner	648.2.bb.b.59.63	yes	1872
81.11	odd	54	inner	648.2.bb.b.11.63	yes	1872
648.11	even	54	inner	648.2.bb.b.11.13	✓	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.11.13	✓	1872	648.11	even	54	inner
648.2.bb.b.11.63	yes	1872	81.11	odd	54	inner
648.2.bb.b.59.13	yes	1872	1.1	even	1	trivial
648.2.bb.b.59.63	yes	1872	8.3	odd	2	inner