Embedded newform 648.2.bb.b.131.8

• Label: 648.2.bb.b.131.8
• 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			131.8
Character	\(\chi\)	\(=\)	648.131
Dual form			648.2.bb.b.371.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39009 + 0.260084i) q^{2} +(1.73188 + 0.0244920i) q^{3} +(1.86471 - 0.723080i) q^{4} +(-0.0414453 + 0.711588i) q^{5} +(-2.41384 + 0.416387i) q^{6} +(-0.543909 - 2.29493i) q^{7} +(-2.40406 + 1.49013i) q^{8} +(2.99880 + 0.0848343i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39009	+	0.260084i	−0.982944	+	0.183907i
							\(3\)	1.73188	+	0.0244920i	0.999900	+	0.0141405i
\(5\)	−0.0414453	+	0.711588i	−0.0185349	+	0.318232i								0.976279	+	0.216517i	\(0.0694698\pi\)
							−0.994814	+	0.101714i	\(0.967567\pi\)
\(7\)	−0.543909	−	2.29493i	−0.205578	−	0.867403i	−0.973834	−	0.227260i	\(-0.927023\pi\)
							0.768256	−	0.640143i	\(-0.221125\pi\)
\(11\)	0.518686	+	0.788623i	0.156390	+	0.237779i	0.905180	−	0.425028i	\(-0.139736\pi\)
							−0.748791	+	0.662806i	\(0.769365\pi\)
\(13\)	−0.389588	−	3.33313i	−0.108052	−	0.924445i	−0.933158	−	0.359467i	\(-0.882959\pi\)
							0.825106	−	0.564978i	\(-0.191116\pi\)
\(17\)	0.293271	−	0.349507i	0.0711288	−	0.0847679i	−0.729307	−	0.684187i	\(-0.760157\pi\)
							0.800435	+	0.599419i	\(0.204602\pi\)
\(19\)	1.44381	−	1.21150i	0.331232	−	0.277937i	−0.461970	−	0.886896i	\(-0.652857\pi\)
							0.793202	+	0.608959i	\(0.208413\pi\)
\(23\)	5.49458	+	1.30224i	1.14570	+	0.271536i	0.759265	−	0.650781i	\(-0.225559\pi\)
							0.386434	+	0.922317i	\(0.373707\pi\)
\(29\)	4.29178	−	5.76486i	0.796963	−	1.07051i	−0.198934	−	0.980013i	\(-0.563748\pi\)
							0.995897	−	0.0904946i	\(-0.0288448\pi\)
\(31\)	−4.14611	+	1.24126i	−0.744663	+	0.222937i	−0.636587	−	0.771205i	\(-0.719654\pi\)
							−0.108076	+	0.994143i	\(0.534469\pi\)
\(37\)	−6.12087	−	1.07927i	−1.00627	−	0.177432i	−0.353857	−	0.935300i	\(-0.615130\pi\)
							−0.652408	+	0.757868i	\(0.726241\pi\)
\(41\)	−1.75237	−	0.755897i	−0.273674	−	0.118051i	0.254820	−	0.966988i	\(-0.417984\pi\)
							−0.528494	+	0.848937i	\(0.677243\pi\)
\(43\)	2.02746	+	1.01823i	0.309184	+	0.155278i	0.596625	−	0.802520i	\(-0.296508\pi\)
							−0.287441	+	0.957798i	\(0.592804\pi\)
\(47\)	−1.07637	+	3.59531i	−0.157004	+	0.524430i	−0.999907	−	0.0136356i	\(-0.995660\pi\)
							0.842903	+	0.538066i	\(0.180845\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.131.8	✓	1872
8.3	odd	2	inner	648.2.bb.b.131.20	yes	1872
81.47	odd	54	inner	648.2.bb.b.371.20	yes	1872
648.371	even	54	inner	648.2.bb.b.371.8	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.131.8	✓	1872	1.1	even	1	trivial
648.2.bb.b.131.20	yes	1872	8.3	odd	2	inner
648.2.bb.b.371.8	yes	1872	648.371	even	54	inner
648.2.bb.b.371.20	yes	1872	81.47	odd	54	inner