Embedded newform 648.2.bb.b.59.77

• Label: 648.2.bb.b.59.77
• 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			59.77
Character	\(\chi\)	\(=\)	648.59
Dual form			648.2.bb.b.11.77

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.949273 + 1.04827i) q^{2} +(1.35074 + 1.08421i) q^{3} +(-0.197760 + 1.99020i) q^{4} +(-2.04070 + 0.238523i) q^{5} +(0.145675 + 2.44515i) q^{6} +(-0.409521 + 0.815423i) q^{7} +(-2.27400 + 1.68194i) q^{8} +(0.648994 + 2.92896i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.949273	+	1.04827i	0.671238	+	0.741242i
							\(3\)	1.35074	+	1.08421i	0.779850	+	0.625967i
\(5\)	−2.04070	+	0.238523i	−0.912628	+	0.106671i								−0.559430	−	0.828878i	\(-0.688980\pi\)
							−0.353198	+	0.935549i	\(0.614906\pi\)
\(7\)	−0.409521	+	0.815423i	−0.154784	+	0.308201i	−0.957638	−	0.287976i	\(-0.907018\pi\)
							0.802853	+	0.596176i	\(0.203314\pi\)
\(11\)	2.73975	−	1.18181i	0.826067	−	0.356331i	0.0592874	−	0.998241i	\(-0.481117\pi\)
							0.766780	+	0.641910i	\(0.221858\pi\)
\(13\)	−0.613865	+	2.59010i	−0.170256	+	0.718365i	0.819198	+	0.573511i	\(0.194419\pi\)
							−0.989453	+	0.144853i	\(0.953729\pi\)
\(17\)	1.01238	+	0.178510i	0.245539	+	0.0432951i	0.295063	−	0.955478i	\(-0.404659\pi\)
							−0.0495242	+	0.998773i	\(0.515770\pi\)
\(19\)	−0.421435	−	2.39007i	−0.0966837	−	0.548321i	−0.994218	−	0.107377i	\(-0.965755\pi\)
							0.897535	−	0.440944i	\(-0.145356\pi\)
\(23\)	0.183620	−	0.0922173i	0.0382874	−	0.0192286i	−0.429553	−	0.903042i	\(-0.641329\pi\)
							0.467840	+	0.883813i	\(0.345032\pi\)
\(29\)	1.31748	−	4.40070i	0.244651	−	0.817190i	−0.744158	−	0.668003i	\(-0.767149\pi\)
							0.988809	−	0.149187i	\(-0.0476656\pi\)
\(31\)	−2.30574	+	3.50571i	−0.414123	+	0.629644i	−0.980965	−	0.194183i	\(-0.937795\pi\)
							0.566842	+	0.823826i	\(0.308165\pi\)
\(37\)	0.675658	+	1.85636i	0.111078	+	0.305183i	0.982759	−	0.184889i	\(-0.0591926\pi\)
							−0.871682	+	0.490072i	\(0.836970\pi\)
\(41\)	3.98948	+	3.76388i	0.623051	+	0.587819i	0.931643	−	0.363376i	\(-0.118376\pi\)
							−0.308591	+	0.951195i	\(0.599857\pi\)
\(43\)	−0.203126	+	0.272846i	−0.0309764	+	0.0416086i	−0.817342	−	0.576153i	\(-0.804553\pi\)
							0.786365	+	0.617762i	\(0.211960\pi\)
\(47\)	2.49712	−	1.64238i	0.364242	−	0.239566i	−0.354174	−	0.935180i	\(-0.615238\pi\)
							0.718416	+	0.695614i	\(0.244868\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.59.77	yes	1872
8.3	odd	2	inner	648.2.bb.b.59.27	yes	1872
81.11	odd	54	inner	648.2.bb.b.11.27	✓	1872
648.11	even	54	inner	648.2.bb.b.11.77	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.11.27	✓	1872	81.11	odd	54	inner
648.2.bb.b.11.77	yes	1872	648.11	even	54	inner
648.2.bb.b.59.27	yes	1872	8.3	odd	2	inner
648.2.bb.b.59.77	yes	1872	1.1	even	1	trivial